A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depression to be 30⁰. If after 100 km, the target has an angle of depression of 60⁰, how far is the target from the fighter jet at that instant?

A man starts his morning walk at a point A reaches two points B and C and finally back to A such that ㄥA = 60⁰ and ㄥB = 45⁰, AC = 4 km in ΔABC. Find the total distance he covered during his morning walk.

If \(\frac { cos^{ 4 }\alpha }{ { cos }^{ 2 }\beta } +\frac { { sin }^{ 4 }\alpha }{ { sin }^{ 2 }\beta } =1\) prove that \(\frac { { cos }^{ 4 }\beta }{ { cos }^{ 2 }\alpha } +\frac { { sin }^{ 4 }\beta }{ { sin }^{ 2 }\alpha } =1\)

If sec \(\theta\) + tan \(\theta\) = p, obtain the values of sec \(\theta\), tan \(\theta\) and sin \(\theta\) in terms of p

Eliminate \(\theta\) from the equation a sec \(\theta\) - c tan \(\theta\) = b and b sec \(\theta\)  + d tan \(\theta\) = C

For what value of a and b is the function \(f\left( x \right) =\begin{cases} { x }^{ 2 },\quad \quad x\le c \\ ax+b,\quad x>c \end{cases}\) is differentiable at x = c.

Differentiate \({ tan }^{ -1 }(secx+tanx),\) \(-\frac{\pi}{ 2 }\) with respect to 'x'.

Differentiate \({ \left( \sin { x } \right) }^{ { \cos { ^{ -1x } } } }\) with respect to 'x'.

If \(x=\tan { \left( \frac { 1 }{ a } \log { y } \right) } \) then show that \(\left( 1+{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +(2x-a)\frac { dy }{ dx } =0.\)

Discuss the differentiability of the functions:
(i) \(f(x)=\{ \begin{matrix} 1,0\le x\le 1 \\ x,x>1 \end{matrix}at=1\)
(ii) \(f(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}=\lim _{h \rightarrow \infty} \frac{1+h-1}{h}=1\)

If sec \(\theta\) + tan \(\theta\) = p, obtain the values of sec \(\theta\), tan \(\theta\) and sin \(\theta\) in terms of p

Eliminate \(\theta\) from the equation a sec \(\theta\) - c tan \(\theta\) = b and b sec \(\theta\)  + d tan \(\theta\) = C

Show that \(cot(A+{ 15 }^{ 0 })-tan(A-{ 15 }^{ 0 })=\frac { 4cos2A }{ 1+2sin2A } \)

If A + B + C = 180⁰, prove that \(tan\frac { A }{ 2 } tan\frac { B }{ 2 } +tan\frac { B }{ 2 } tan\frac { C }{ 2 } +tan\frac { C }{ 2 } tan\frac { A }{ 2 } =1\)

Solve the following equations sin \(\theta\) + sin 3\(\theta\) + sin5\(\theta\) = 0

\(\int { { 3 }^{ x+2 } } \) dx = __________+c.

\(\int { \frac { sin\sqrt { x } }{ x } } \) dx = ________ +c.

\(\int { \frac { 1 }{ 9x^{ 2 }-4 } } \) dx = ________+c.

\(\int { \frac { x }{ 4+{ x }^{ 4 } } } \) dx is equal to________+c.

\(\int { { e }^{ x }\left[ f\left( x \right) +f'\left( x \right) \right] } \) dx = ___________+c.

\(\int { sin } \)e^x . d (e^x) = ________+c.

\(\int { \frac { { e }^{ x } }{ \left( 1+{ e }^{ x } \right) ^{ 2 } } } \) dx =_______+c

\(\int { { tan }^{ 3 } } 2sec2x\) dx = ___________+c.

\(\int { \frac { { 4x }^{ 3 }+1 }{ { x }^{ 4 }+x } } \) dx = _______ + c.

\(\int { \left| x \right| ^{ 3 } } \) dx is equal to ________+c.

Evaluate : \(\int { { 2 }^{ x } } \) e^x dx

If f'(x) = \(\frac { x }{ 2 } +\frac { 2 }{ x } \) and f(1) = \(\frac { 5 }{ 4 } \), find f (x)

Solve : cosec(3 - 2x) cot(3 - 2x)

Integrate the function with respect to x : 5x⁴ + 3(2x + 3)⁴ - 6(4 - 3x)⁵

Integrate the function with respect to x : cos³ 2x - sin 6x

Evaluate : \(\int { \sqrt { x } -{ cos }^{ 2 } } \frac { x }{ 2 } \)

Evaluate : \(\int { \left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 }+1 } \right) } \) dx

Solve : e^{3x+ 2}

Solve : (lx + m)^1I2

Integrate the function with respect to x : \(\cfrac { { e }^{ 2x }+{ e }^{ -2x }+2 }{ { e }^{ x } } \)

Evaluate \(\int { \frac { { sin }^{ 6 }x+cos^{ 6 }x }{ sin^{ 2 }xcos^{ 2 }x } } \)

Evaluate if f'(x) = 3x² - \(\frac { 2 }{ { x }^{ 3 } } \) and f (1) = 0, find f (x)

Evaluate the integrate \(\cfrac { 1 }{ 7-(4x+1)^{ 2 } } \)

Integrate the function with respect to x : \(I=\int { \cfrac { 1 }{ { x }^{ 2 }-3x-3 } dx } \)

Integrate the function with respect to x : \(\sqrt { 1-3x-{ x }^{ 2 } } \)

Evaluate \(\int { \frac { \left( { a }^{ x }+{ b }^{ x } \right) ^{ 2 } }{ { a }^{ x }{ b }^{ x } } } \)dx

Evaluate \(\int { \sqrt { 1+sinx } } \) dx, 0 < x < \(\frac { \pi }{ 2 } \)

Evaluate \(\int { cot^{ 3 } } \) x dx

Evaluate the integrate ;  \(\cfrac { 1 }{ 5-6x-{ 9x }^{ 2 } } \)

Integrate the function with respect to x : \(\sqrt { (2-x)(3+x) } \)

If f'(x) = a sin x + b cos x and f ' (0) = 4, (0) = 3, f \(\left( \frac { \pi }{ 2 } \right) \) = 5, find f (x)

Evaluate \(\int { \frac { { x }^{ 7 } }{ x+1 } } \)dx

Evaluate the integrate
\(\cfrac { 1 }{ { 3x }^{ 2 }-13-10 } \)

Evaluate the integral
\(\cfrac { 4x+1 }{ { x }^{ 2 }+3x+1 } \)

Evaluate the integral
\(\cfrac { 6x+7 }{ \sqrt { (x-4)(x-5) } } \)

Evaluate x cos 5x cos 2x

Integrate the function with respect to x
e^2x sin 3x dx

Integrate the function with respect to x
e^3x sin 2x

Evaluate the integral
\(\cfrac { 2x-1 }{ { 2x }^{ 2 }+x+3 } \)

Evaluate the integral
\(\cfrac { 2x-3 }{ \sqrt { 10-7x-{ x }^{ 2 } } } \)

The probabilities of a student getting I. II and III class in an examination are \(\frac { 1 }{ 10 } ,\frac { 3 }{ 5 } \) and \(\frac { 1 }{ 4 } \) respectively. The probability that the student fails in the examination is

Three integers are chosen at random from the first 20 integers. The probability that their product is even is

The probability that in a year of 22^nd century, chosen at random there will be 53 Sundays is

If A and B are two events such that \(P(A\cap B)=\frac { 7 }{ 10 } \) and P(B) = \(\frac { 17 }{ 20 } \) , then P(A/B) =

Let A and B be two events such that P(A) = 0.6, P(B) = 0.2 and P(A/B) = 0.5, Then P(\(\bar { A } /\bar { B } \)) is

If P(A\(\cup \)B) = 0.8 and P(A\(\cap \)B) = 0.3 then \(P(\bar { A } )+P(\bar { B } )\) =

If A and B are two events such that P(A) = \(\frac { 4 }{ 5 } \) and \(P(A\cap B)=\frac { 7 }{ 10 } \) then P(B/A) = 

If P(B)=\(\frac { 3 }{ 5 } \)P(A/B) = \(\frac { 1 }{ 2 } \) and \(P(A\cup B)=\frac { 4 }{ 5 } \), then P(A) is

If A and B are two independent events with P(A) = \(\frac { 3 }{ 5 } \) and P(B)=\(\frac { 4 }{ 9 } \) then \(P(\bar { A } \cap \bar { B } )\) = equals

Choose the incorrect pair:

The probability that student selected at random from a class will pass in Mathematics is \(\frac { 2 }{ 3 } \) and the probability that he passes in Mathematics and English is \(\frac { 1 }{ 3 } \). What is the probability that he will pass in English if it is known that he has passed in Mathematics?

Events A and B are such that P(A) = \(\frac { 1 }{ 2 } \) , P(B) = \(\frac { 7 }{ 12 } \) and P(not A or not B) = \(\frac { 1 }{ 4 } \). State whether A and B are independent? 

Given that the events A and B are such that P(A) = \(\frac { 1 }{ 2 } \), P(AUB) = \(\frac { 3 }{ 5 } \) and P(B) = p. find P if they are mutually exclusive events. 

An experiment has the four possible mutually exclusive outcomes A, B, C and D, Check whether the following assignments of probability are permissible.
p(A) = 0.32, P(B) = 0.28, P(C) = - 0.06, P(D) = 0.46

If A and B are two events such that \(P(A\cup B)=0.7 ,\) \(P(A\cap B)=0.2\) ,\(P(\bar { B } )=0.5,\) show that A and B are independent.

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

A die is tossed thrice. find the probability of getting an odd number atleast once?

The ratio of the number of boys to the number of girls in a class is 1:2. It is known that the probability of a girl and a boy getting a first class are 0.25 and 0.28 respectively. Find the probability that a student chosen  at random will get first class?

An integers is chosen at random from the first fifty positive integers. What is probability that the integer chosen is a prime or multiple of 4.

Two cards are drawn one by one at random from a deck of 52 playing cards. What is the probability of getting two jacks if
(i) the first card is replaced before the second card is drawn
(ii) the first card is not replaced before the second card is draw?

Two unbiased die are thrown. Find the probability that the sum is 8 or greater if 3 appears on the first die.

A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective. If a person takes out 2 at random what is the probability that either both are apples or both are good?

The probability that a person will get an electric contract  \(\frac { 2 }{ 3 } \) and the probability that he will not get plumbing contract is \(\frac { 4 }{ 7 } \). If the probability of getting atleast one contract is \(\frac { 2 }{ 3 } \). What is the probability tht he will get both? 

A and B are two events such that P(A) \(\neq \) 0. Find P(B/A) if (i) A is a subset of B (ii) A\(\cap \)B = \(\phi \)

In a box containing 10 bulbs, 2 ae defective. What is the probability that among 5 bulbs chosen at random, none is defective?

One card is drawn from a well shuffled pack of 52 cards. If E is the event, "the card drawn is a king or queen" and F is the event "the card drawn is a queen or an ace", then find P(E/F).

Three events A, B and C have probalilities \(\frac { 2 }{ 5 } ,\frac { 1 }{ 3 } \)and \(\frac { 1 }{ 2 } \)  respectively. Given that P(A\(\cap \)C) = \(\frac { 1 }{ 5 } \), \(P(B\cap C)=\frac { 1 }{ 4 } \)find P(C/B) and P(\(\bar { A } \cap \bar { C } \))?

A fair dice is rolled. Consider the following events A = {1, 3, 5}, B = {2, 3} and C ={2, 3, 4, 5} Find (i) P(A/B) and P(B/A) (ii) P(A\(\cap \)B/C)

A die is thrown 3 times. Events A and B are defined as follows.
A: getting 4 on third die
B: getting 6 on the first and 5 on the second throw. Find the probability of A given that B has already occurred.

Given p(A) = 0.5, P(B) = 0.6 and \(P(A\cap B)=0.24\) .Find
(i) \(P(A\cup B)\) 
(ii) \(P(\vec { A } \cap B)\) 
(iii) \(P\left( A\cap \bar { B } \right) \) 
(iv) \(P\left( \bar { A } \cup \bar { B } \right) \) 
(v) \(P(\bar { A } \cap \bar { B } )\)

A couple has two children. Find the probability that
(i) both the children are boys, if it is known that the older child is a boy.
(ii) both the children are girls, if it is known that the older child is a girl.

Evaluate P(AUB) if 2P(A) = P(B) = \(\frac { 5 }{ 13 } \)and P(A/B) = \(\frac { 2 }{ 5 } \).

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let \(\frac { 3 }{ 4 } \) be the probability that he knows the answer and \(\frac { 1 }{ 4 } \) be the probability that he guesses. Assuming that a student who guesse at the answer will be correct with probability \(\frac { 1 }{ 4 } \). What is the probability that the student knows the answer given that he answered it correctly?

A and B are two candidates seeking admission in a college. The probability that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the probability that B is selected.

p(A) = 0.3, P(B) = 0.6 and \(P(A\cap B)=0.25\) .Find
(i) \(P(A\cup B)\) 
(ii) P(A/B)
(iii) \(P(B/\bar { A } )\) 
(iv) \(P(\bar { A } /B)\) 
(v) \(P(\bar { A } /\bar { B } )\)

Two integers are selected at random from integers 1 to 11. If the sum is even, find the probability that both the numbers are odd.

A purse contains 3 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled out at random from one of the two purses, what is the probability that it is a silver coin?

for a loaded die, the probabilities of outcomes are given as under
P(1) = P(2) = \(\frac { 2 }{ 10 } \), P(3) = P(5) = P(6) = \(\frac { 1 }{ 10 } \) and P(4) = \(\frac { 3 }{ 10 } \)
The die is thrown 2 times. Let A and B be the events as defined below
A: Getting same number each time
B: Getting a total score of 10 or more
Discuss the independency of the events A and B

Out of 10 outstanding students in a school there are 6 girls and 4 boys. A team of 4 students is selected at random for a quiz programme. Find the probability that there are atleast two girls.

In a factory, Machine-I produces 45% of the output and Machine-II produces 55% of the output. On the average 10% items produced by I and 5% of the items produced by II are defective. An item is drawn at random from a day's output. (i) Find the probability that it is a defective item (ii) If it is defective, what is the probability that it was produced by Machine-II?

The shaded region in the adjoining diagram represents.

For real numbers x and y, define xRy if x - y + √2 is an irrational number. Then the relation R is __________

The number of reflective relations one set containing n elements is __________

Which one of the following statements is false? The graph of the function \(f(x)={1\over x}\)

Which one of the following is not a singleton set?

Given A = {5,6,7,8}. Which one of the following is incorrect?

The shaded region in the adjoining diagram represents.

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

Let R be the relation over the set of all straight lines in a plane such that l₁Rl₂ ⇔ l₁丄l₂ . Then  R is __________

Which of the following functions from z to itself are bijections (one-one and onto)?

In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

The number of 5 digit numbers all digits of which are odd is

If ^a2-a \(C_2 = ^{a^2-a}\) C₄ then the value of 'a' is

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

Write the following in roster form.
The set of all positive roots of the equation (x-1)(x+1)(x²-1) = 0.

Write the following in roster form.
\(\left\{ x:\frac { x-4 }{ x+2 } =3,x\in R-\{ -2\} \right\} \)

State whether the following sets are finite or infinite.
{x \(\in \) N:x is a rational number}

By taking suitable sets A, B, C, verify the following results:
A \(\times\) (B\(\cup \)C) = (A\(\times\)B) \(\cup \) (A\(\times\)C)

Discuss the following relations for reflexivity, symmetricity and transitivity :
Let A be the set consisting of all the female members of a family. The relation R defined by "aRb if a is not a sister of b".

The sum of the digits at the 10^th place of all numbers formed with the help of 2, 4, 5, 7 taken all at a time is

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

^(n-1)C_r + ^(n-1)C_(r-1) is

The number of ways of choosing 5 cards out of a deck of 52 cards which include at least one king is

The number of rectangles that a chessboard has

Check whether the following sets are disjoint where p = {x : x is a prime < 15} and Q = {x : x is a multiple of 2 and x < 16}

If A⊂B then find A⋂B and A\B (using venn diagram)

Show that the relation R on the set A = {1, 2, 3} given by R = {(1, 1) (2, 2) (3, 3) (1, 2) (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the function f : N➝N given by f(x) = 2x is one-one but not onto.

Let S = {1, 2, 3,....,10}. Define 'm is related to n' if m divides n.

Find the total number of outcomes when 5 coins are tossed once.

Find the value of 5!

Find the value of \(\frac { 8! }{ 5!\times 2! } \).

Evaluate \(\frac { n! }{ r!(n-r)! } \) when n = 7, r = 5.

If \(\frac { 6! }{ n! } \) = 6, then find the value of n.

Find the quotient of the identity function by the modulus function

Show that the function f : R ⟶ R given by f(x) = cos x for all x ∈ R is neither one-one nor onto.

Which of the following sets are finite and which are infinite?
Set of concentric circles in a plane.

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩, ' and \ as necessary.

Draw venn diagram of three sets A, B and C which illustrates the following:
A ∩ B ∩ C

A person went to a restaurant for dinner. In the menu card, the person saw 10 Indian and 7 Chinese food items. In how many ways the person can select either an Indian or a Chinese food?

Three persons enter in to a conference hall in which there are 10 seats. In how many ways they can take their seats?

count the total number of ways of answering 6 objective type questions, each question having 4 choices

Find the value of \(\frac { 12! }{ 9!\times 3! } \)

If ^(n-1)P₃ :ⁿ P₄ = 1 : 10, find n

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩,  and as necessary.

Draw venn diagram of three sets A, B and C which illustrates the following:
A and B disjoint but both are subsets of C.

If R is the set of all real numbers, what do the cartesian products R \(\times\) Rand R \(\times\)R \(\times\)R represent?

Solve the inequation x \(\ge\) 2 graphically.

Solve the quadratic equation 5^2x- 5^{x + 3}+ 125 = 5^x.

Four children are running a race.
(i) In how many ways can the first two places be filled?
(ii) In how many different ways could they finish the race?

Count the number of three-digit numbers which can be formed from the digits 2, 4, 6, 8 if
(i) repetitions of digits is allowed.
(ii) repetitions of digits is not allowed

How many three-digit numbers are there with 3 in the unit place?
(i) with repetition
(ii) without repetition.

If ¹⁰P_r-1 = 2 \(\times\) ⁶P_r, find r.

A test consists of 10 multiple choice questions. In how many ways can the test be answered if
(i) Each question has four choices?
(ii) The first four questions have three choices and the remaining have five choices?
(iii) Question number n has n + 1 choices?

Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
Verify using Venn diagram.

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

If a \(\in\) {-1, 2, 3, 4, 5} and b \(\in\) {0,3, 6}. Write the set of all ordered pairs (a, b) such that a + b = 5.

Find the sum and difference of the identity function and the modulus function?

Find the range of the function.
f = {1, x), (1, y), (2, x), (2, y), (3, z)}

There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees
(i) a particular teacher is included?
(ii) a particular student is excluded?

If ⁽ⁿ⁺²⁾P₄ = 42 \(\times\) ⁿP₂, find n.

How many 'letter strings' together can be formed with the letters of the word "VOWELS" so that
(i) the strings begin with E
(ii) the strings begin with E and end with W.

There are 15 candidates for an examination. 7 candidates are appearing for mathematics examination while the remaining 8 are appearing for different subjects. In how many ways can  they be seated in a row so that no two mathematics candidates are together?

How many numbers can be formed using the digits 1, 2, 3, 4, 2, 1 such that, even digits occupies even place?

For A = {0,1,2,3, 4}, B = {1, -2, 3, 4, 5, 6} and C = {2, 4, 6, 7} verify A\(B ∩ C) = (A\B) U(A\C) Using venn diagram.

The cartesian product A \(\times\) A has 9 elements among which are found (-1, 0) and (0, 1).Find the set A and the remaining elements of A \(\times\)A.

Show that the relation R on the set R of all real numbers defined as R = {(a, b): a < b²} is neither reflexive, nor symmetric nor transitive.

Consider the function \(f:[0,{\pi\over 2}]⟶R\) given by f(x) = sin x and \(g:[0,{\pi\over 2}]⟶R\)given by g(x) = cos x. Show that f and g are one-one but (f + g) is not one-one.

A relation R is defined on the set z of integers as follows:
(x, Y) ∈ R ⇔ x² + y² = 25. Express R and R^-1 as the set of ordered pairs and hence find their respective domains.

How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5 ? if
(i) repetition of digits allowed
(ii) the repetition of digits is not allowed.

How many three-digit odd numbers can be formed using the digits 0, 1, 2, 3, 4, 5? if
The Repetition of digits is not allowed

How many three-digit odd numbers can be formed using the digits 0, 1, 2, 3, 4, 5? if 
The repetition of digits is allowed

Count the numbers between 999 and 10000 subject to the condition that there are
(i) no restriction.
(ii) no digit is repeated.
(iii) at least one of the digits is repeated.

To travel from a place A to place B, there are two different bus routes B₁, B₂, two different train routes T₁, T₂ and one air route A₁. From place B to place C there is one bus route say B'₁, two different train routes say T'₁, T'₂ and one air route A'₁. Find the number of routes of commuting from place A to place C via place B without using similar mode of transportation.

Find the number of strings that can be made using all letters of the word THING. If these words are written as in a dictionary, what will be the 85^th string?

Find the sum of all 4-digit numbers that can be formed using digits 1, 2, 3, 4 and 5 repetitions not allowed?

Find the number of strings of 4 letters that can be formed with the letters of the word EXAMINATION?

There are 11 points in a plane. No three of these lies in the same straight line except 4 points, which are collinear. Find,
(i) the number of straight lines that can be obtained from the pairs of these points?
(ii) the number of triangles that can be formed for which the points are their vertices?

A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of
(i) exactly 3 women?
(ii) at least 3 women?
(iii) at most 3 women?

If a, 8, b are in AP, a, 4, b are in GP, and if a, x, b are in HP then x is

The sequence \(\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } +\sqrt { 2 } }, \frac { 1 }{ \sqrt { 3 } +2\sqrt { 2 } },...... \)form an 

If S_n denotes the sum of n terms of an AP whose common difference is d, the value of S_n - 2S_n-1 + S_n-2 is

The remainder when 38¹⁵ is divided by 13 is

The n^th term of the sequence 1, 2, 4, 7, 11,... is

With usual notation C₀ + C₂ + C₄ + ... is ______________

In the expansion of (2x + 3)⁵ the coefficient of x² is ______________

In the expansion of (1 +x )²² which term is the middle term ______________

AM, GM, HM denote the Arithmetic mean, Geometric mean and Harmonic mean respectively the relationship between this is ______________

In the series \(\frac{1}{1+\sqrt 2}+\frac{1}{\sqrt 2+\sqrt 3}+\frac{1}{\sqrt 3+\sqrt 4}+...\) some of first 24 number is ______________

Show that the sum of (m + n)^th and (m - n)^th term of an A.P is equal to twice the m^th term.

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetic-geometric progression, harmonic progression and none of them. \(\frac { 1 }{ 2^{ n+1 } } \)

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetic geometric progression, harmonic progression and none of them \(\frac { \left( n+1 \right) \left( n+2 \right) }{ \left( n+3 \right) (n+4) } \)

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression,arithmetic-geometric progression, harmonic progression and none of them 4\(\left( \frac { 1 }{ 2 } \right) ^{ n }\)

Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetic -geometric progression, harmonic progression and none of them \(\frac { (-1)^{ n } }{ n } \)

Write the n^th term of the following sequences
2,2,4,4,6,6

Write the n^th term of the following sequences
\(\frac { 1 }{ 2 } ,\frac { 2 }{ 3 } ,\frac { 3 }{ 4 } ,\frac { 4 }{ 5 } ,\frac { 5 }{ 6 } \)

Write the n^th term of the following sequences
\(\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 5 }{ 6 } ,\frac { 7 }{ 8 } ,\frac { 9 }{ 10 } \)

Write the n^th term of the following sequences
6,10, 4, 12, 2, 14, 0, 16, -2...

Find the expansion of (2x + 3)⁵.

Expand \(\left( { 2x }^{ 2 }-\frac { 3 }{ x } \right) ^{ 3 }\)

Find the general terms and sum to n terms of the sequence 1, \(\frac{4}{3},\frac{7}{9},\frac{10}{27},....\)

A man repays an amount of Rs. 3250 by paying Rs. 20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

In a race, 20 balls are placed in a line at intervals of 4 meters, with the first ball 24 meters away from the starting point. A contestant is required to bring the balls back to the starting place one at a time. How far would the contestant run to bring back all balls?

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\({ \left( x+2 \right) }^{ -\frac { 2 }{ 3 } }\)

Write nth term of the Sequence \(\frac { 3 }{ { 1 }^{ 2 }{ 2 }^{ 2 } } ,\frac { 5 }{ { 2 }^{ 2 }{ 3 }^{ 2 } } ,\frac { 7 }{ { 3 }^{ 2 }{ 4 }^{ 2 } } \) as a difference of two terms 

Find the coefficient of x⁶ in the expansion of (3 + 2x)¹⁰.

Expand \({\left( 2x-{1\over 2x} \right)}^{4}.\)

If n is an odd positive integer, prove that the coefficients of the middle terms in the expansion of (x+ y)ⁿ are equal.

If the 5^th and 9^th terms of a harmonic progression are \({1\over 19}\) and \({1 \over 35},\) find the 12^th term of the sequence.

If the roots of the equation (q - r) x² + (r - p)x + p - q = 0 are equal, then show that p, q and r are in A.P.

If a, b, c are respectively the p^th q^th and r^th terms of a GP. show that (q - r) log a + (r - p) log b + (p - q) log c = 0.

Expand \(\left( { 2x }^{ 2 }-3\sqrt { 1-{ x }^{ 2 } } \right) ^{ 4 }+({ 2x }^{ 2 }+3\sqrt { 1-{ x }^{ 2 }) } ^{ 4 }\)

Find the sum up to the 17^th term of the series \(\frac { { 1 }^{ 3 } }{ 1 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +...+\frac { { 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 } }{ 1+3+5 } +......\)

Compute the sum of first n terms of the following series 8 + 88 + 888 + .......

if the binomial co-efficients of three consecutive terms in the expansion of ( a + xⁿ) are in the radio 1:7:42 then find n

In the binomial coefficient of (1+x)ⁿ  the Coefficients of the 5^th, 6^th and 7^th terms are in A.P find all values of n

If p - q is small compared to either p or q, then show that \(n\sqrt { \frac { p }{ q } } =\frac { \left( n+1 \right) p+\left( n-1 \right) q }{ \left( n-1 \right) p+\left( n+1 \right) q } \)
Hence find \(8\sqrt { \frac { 15 }{ 16 } } \)

Find the coefficient of x⁴ in the expansion of \(\frac { 3-4x+{ x }^{ 2 } }{ { e }^{ 2x } } \)

The 2^nd, 3^rd and 4^th terms in the binomial expansion of (x + a)ⁿ are 240, 720 and 1080 for a suitable value of x. Find x, a and n.

The equation of the locus of the point whose distance from y-axis is half the distance from origin is

Which of the following equation is the locus of (at², 2at)

Which of the following point lie on the locus of 3x²+ 3y²- 8x - 12y + 17 = 0

If the point (8, -5) lies on the locus \(\frac{x^2}{16}-\frac{y^2}{25}=k\), then the value of k is

The slope of the line which makes an angle 45^o with the line 3x- y = -5 are:

If (1 + x²)² (1 + x)ⁿ = a₀+ a₁x + a₂x² + ... + xⁿ⁺⁴ and if a₀, a₁, a₂ are in AP, then n is

With usual notation C₀ + C₂ + C₄ + ... is ______________

In the expansion of (2x + 3)⁵ the coefficient of x² is ______________

In the expansion of (1 +x )²² which term is the middle term ______________

AM, GM, HM denote the Arithmetic mean, Geometric mean and Harmonic mean respectively the relationship between this is ______________

Find the locus of P, if for all values of \(\alpha\) the co-ordinates of a moving point P is  (9 cos \(\alpha\) 9 sin \(\alpha\))

Find the equation of the lines passing through the point (1, 1) 
(i) with y-intercept (-4)
(ii) with slope 3
(iii) and (-2, 3)
(iv) and the perpendicular from the origin makes an angle 60° with x- axis.

Show that the lines are 3x + 2y + 9 = 0 and 12x + 8y - 15 = 0 are paralle llines.

Find the equation of the straight line parallel to 5x - 4y + 3 = 0 and having x-intercept 3.

Determine the equation of line through the point (-4, -3) and perpendicular to y-axis.

Find the distance between the parallel lines
12x + 5y = 7 and 12x + 5y + 7 = 0.

Find the acute angle between the pair of lines given by 2x²- 5xy - 7y² = 0.

Find the path traced out by the point \((ct,\frac{c}{t})\) , here t ≠ 0 is the parameter and c is a constant.

Find the equation of the line through the point of intersection of the line 5x - 6y = 1 and 3x + 2y + 5 = 0 and cutting off equal intercepts on the coordinate axis.

Find the equation of the line through (1, 2) and which is perpendicular to the line joining (2, -3) (-1, 5)..

If (-4, 7) is one vertex of a rhombus and if the equation of one diagonal is 5x - y + 7 = 0, then find the equation of another diagonal.

Find the locus of a point P that moves at a constant distant of 
(i) two units from the X-axis
(ii) three units from the Y-axis

Find the value of k and b, if the points P(-3, 1) and Q(2, b) lie on the locus of x² - 5x + ky = 0.

If O is origin and R is a variable point on y² = 4x, then find the equation of the locus of the mid-point of the line segment OR.

Find the equation of the lines passing through the point of intersection lines 4x - y + 3 = 0 and 5x + 2y + 7 = 0
(i) through the point (-1, 2)
(ii) Parallel to x - y + 5 = 0
(iii) Perpendicular to x - 2y + 1 = 0.

Show the points \((0,-\frac{3}{2}),(1,-1)\) and \((2,-\frac{1}{2})\) are collinear.

Find the equations of the straight lines, making the y-intercept of 7 and angle between the line and the y-axis is 30°.

The length of the perpendicular drawn from the origin to a line is 12 and makes an angle 150° with positive direction of the x-axis. Find the equation of the line.

Area of the triangle formed by a line with the coordinate axes, is 36 square units. Find the equation of the line if the perpendicular drawn from the origin to the line makes an angle of 45° with positive the x-axis.

Separate the equations 5x² + 6xy + y² = 0.

If θ is a parameter, find the equation of the locus of a moving point, whose coordinates are x = a cos³ θ, y = a sin³ θ.

A straight rod of length 8 units slides with its ends A and B always on the x and y axes respectively. Find the locus of the mid point of the line segment AB.

Find the equation of the locus of a point such that the sum of the squares of the distance from the points (3, 5), (1, -1) is equal to 20.

Find the equation of the locus of the point P such that the line segment AB, joining the points A(1, -6) and B(4,-2), subtends a right angle at P.

lf P(2,-7) is a given point and Q is a point on (2x² + 9y² = 18), then find the equations of the locus of the mid-point of PQ.

A ray of light coming from the point (1, 2)is reflected at a point A on the x-axis and it passes through the point (5, 3). Find the co-ordinates of the point A.

A line is drawn perpendicular to 5x = y + 7. Find the equation of the line if the area of the triangle formed by this line with co-ordinate axes is 10 sq. units.

Show that the points (1, 3), (2, 1) and \((\frac{1}{2},4)\) are collinear, by using
(i) concept of slope
(ii) a straight line
(iii) any other method.

In a shopping mall there is a hall of cuboid shape with dimension 800 \(\times\)800 \(\times\)720 units, which needs to be added the facility of an escalator in the path as shown by the dotted line in the figure. Find 
(i) the minimum total length of the escalator.
(ii) the heights at which the escalator changes its direction.
(iii) the slopes of the escalator at the turning points.

Find the equation of the perpendicular bisector of the line segment joining the points (1, 1) and (2, 3).

If a_ij = \({1\over2}(3i-2j)\) and A = [a_ij]_2x2 is

What must be the matrix X, if 2x +\(\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?\)

If A is a square matrix, then which of the following is not symmetric?

If A = \(\begin{bmatrix}a & x \\ y& a \end{bmatrix}\) and if xy = 1, then det (A A^T ) is equal to

The value of x, for which the matrix A = \(\begin{bmatrix} e^{x-2}& e^{7+x} \\ e^{2+x} & e^{2x+3} \end{bmatrix}\) is singular

A root of the equation \(\begin{vmatrix} 3-x&-6 &3 \\ -6 & 3-x & 3 \\ 3 &3 &-6-x \end{vmatrix}=0 \ is\)

The value of the determinant of A = \(\begin{bmatrix} 0&a &-b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}is\)

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁ ), (x₂, y₂), (x₃, y₃ ) are

If a \(\neq\) b, b, c satisfy \(\begin{vmatrix} a&2b &2c \\3 & b & c \\ 4 & a & b \end{vmatrix}=0,\) then abc =

If A = \(\begin{vmatrix}-1 & 2 &4 \\ 3 &1 &0 \\ -2& 4 &2 \end{vmatrix}\) and B = \(\begin{vmatrix}-2 & 4 &2 \\ 6 &2 &0 \\ -2& 4 &8 \end{vmatrix}\), then B is given by

Suppose that a matrix has 12 elements. What are the possible orders it can have? What if it has 7 elements?

Find x, y, a, and b if \(\begin{bmatrix} 3x+4y & 6 & x-2y \\ a+b & 2a-b & -3 \end{bmatrix}\)=\(\begin{bmatrix} 2 & 6 & 4 \\ 5 & -5 & -3 \end{bmatrix}\)

Compute A + B and A - B if A =\(\begin{bmatrix} 4 & \sqrt { 5 } & 7 \\ -1 & 0 & 0.5 \end{bmatrix}\) and B = \(\begin{bmatrix} \sqrt { 3 } & \sqrt { 5 } & 7.3 \\ 1 & {1\over3} &{1\over4} \end{bmatrix}\) .

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\) verify (A - B)^T = A^T - B^T

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
verify (3A)^T = 3A^T

If A⊆B, then A\B is  ________

The shaded region in the adjoining diagram represents.

Which of the following is not an equivalence relation on z?

Let f : Z➝Z be given by f(x) = \(\begin{cases} \frac { x }{ 2 } \quad if\quad x\quad is\quad even \\ 0\quad if\quad x\quad is\quad odd \end{cases}\). Then f is __________

Domain of the function \(y={x-1\over x+1}\) is __________

Find the value of x if \(\begin{vmatrix} x-1 & x & x-2 \\ 0 &x-2 & x-3 \\ 0 & 0 & x-3 \end{vmatrix}=0\)

Prove that \(\begin{bmatrix} sec^2 \theta & tan ^2 \theta & 1 \\ tan^2 \theta & sec^2 \theta & -1 \\ 38 & 36 & 2 \end{bmatrix}=0\)

Show that \(\begin{vmatrix} x+2a & y+2b & z+2c \\ x & y & z \\ a & b & c \end{vmatrix}=0\) .

Without expanding, evaluate the following determinants:
\(\begin{vmatrix} 2& 3 &4 \\ 5 & 6 & 8 \\ 6x & 9x &12x \end{vmatrix}\) 

If A is a square matrix and | A | = 2, find the value of | AA^T | .

Given A = {5,6,7,8}. Which one of the following is incorrect?

Let R be the relation over the set of all straight lines in a plane such that l₁Rl₂ ⇔ l₁丄l₂ . Then  R is __________

Which of the following functions from z to itself are bijections (one-one and onto)?

If \(f:R\rightarrow R\) is defined by \(f(x)=2x-3\) __________

If \(f(x)={1-x\over 1+x},(x\neq0)\) then f^-1(x) =

If A = {x : x is a multiple of 5, x ≤ 30 and x ∈ N}
B = {1, 3, 7, 10, 12, 15, 18, 25} then find A⋂B

If U = {x : 1 ≤ x ≤ 10, x ∈ N}, A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 9, 10} then find A'UB'.

Using Venn diagram verify (AUB)'=A'⋂B' 

Let C be the set of all circles in a plane and define a circle C is related to a circle C', if the radius of C is equal to the radius of C'

Let A be the set consisting of children and elders of a family. Let R be the relation defined by aRb if a is a sister of b.

Construct a 2 \(\times\) 3 matrix whose (i, j)^th element is given by \(a_ij={\sqrt{3}\over 2}|2i-3j|(1\le i\le2,1\le j\le3)\) .

Solve for x if \(\left[\begin{array}{lll} x & 2 & -1 \end{array}\right]\)\(\begin{bmatrix} 1&1 &2 \\ -1 & -4 &1 \\ -1 &-1 &-2 \end{bmatrix}\)\(\begin{bmatrix} x \\ 2 \\ 1 \end{bmatrix}\)=0

Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)
Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

If A =\(\begin{bmatrix} 1 &0 &0 \\0 & 1 & 0 \\a &b &-1 \end{bmatrix}\) , show that A² is a unit matrix.

For n, m\(\in \)N, n/m means that n is a factor of n & m. Then find whether the given relation is an equivalence relation.

Find the domain of each of the following functions given by:
f(x) = \(\frac { { x }^{ 3 }-x+3 }{ { x }^{ 2 }-1 } \).

Find the domain and range of the function f(x) = \(\frac { { x }^{ 2 }-9 }{ x-3 } \).

If A = { 0, 1, 2, 3, 4, 5, 6, 7 } is a set. Then,

Draw the curves of
(i) y = x² + 1
(ii) Y = (x + 1)² by using the graph of curve y = x.

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
\(\begin{bmatrix} 4 & -2 \\ 3& -5 \end{bmatrix}\)

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
\(\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)

Construct the matrix \(A=[a_{ij}]_{3\times 3}\), where \(a_{ij}=i-j.\) State whether A is symmetric or skew-symmetric.

A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins, and almonds.
Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds.
Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds.
Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds.
The cost of 50 gm of cashew nuts is Rs.50, 50 gm of raisins is Rs.10, and 50 gm of almonds is Rs.60. What is the cost of each gift pack?

If a, b, c and x are positive real numbers, then show that \(\begin{vmatrix} (a^x+a^{-x})^2 &(a^x-a^{-x})^2 &1 \\ (b^x+b^{-x})^2 & (b^x-b^{-x})^2 & 1 \\ (c^x+c^{-x})^2 & (c^x-c^{-x})^2 & 1 \end{vmatrix}\) is zero.

Find the quotient of the identity function by the modulus function

Show that the function f : R ⟶ R given by f(x) = cos x for all x ∈ R is neither one-one nor onto.

Which of the following sets are finite and which are infinite?
Set of concentric circles in a plane.

If A \(\times\) B = {(a, 1) (b, 3) (a, 3) (b, 1) (a, 2) (b, 2)}, then find A and B

Show that the relation is congruent to on the set of all triangles in a plane is an equivalence relation.

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩, ' and \ as necessary.

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩, ' and \ as necessary.

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩,  and as necessary.

If R is the set of all real numbers, what do the cartesian products R \(\times\) Rand R \(\times\)R \(\times\)R represent?

See the figure below, here letters of the English alphabets are mapped onto.

Express the matrix A =\(\begin{bmatrix} 1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5 \end{bmatrix}\)as the sum of a symmetric and a skew-symmetric matrices.

If A =\(\begin{bmatrix} 1 &0 &2 \\0 & 2 & 1 \\2 &0 &3 \end{bmatrix}\) and A³ - 6A² + 7A + KI = O, find the value of k.

Show that f(x) f(y) = f(x + y), where f(x) =\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

Compute all minors, cofactors of A and hence compute |A| if A =\(\begin{bmatrix} 1& 3 &-2 \\4 & -5 &6 \\ -3 & 5 & 2 \end{bmatrix}\) .
Also check that | A | remains unaltered by expanding along any row or any column.

Without expanding the determinants, show that | B | = 2| A |.
Where B =\(\begin{bmatrix} b+c & c+a & a+b \\ c+a & a+b &b+c \\a+b & b+c & c+a \end{bmatrix}\)and A =\(\begin{bmatrix} a& b & c \\ b & c & a \\ c & a & b \end{bmatrix}\)

Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
Verify using Venn diagram.

Show that the relation R defined on the set A of all polygons as R = {(P₁ P₂) : P₁ and P₂ have same number of sides} is an equivalence relation.

Let f and g be real functions defined by \(f(x)=\sqrt{x+2}\)and \(g(x)=\sqrt{4-x^2}\). Find f + g

For the curve \(y={ -x }^{ \left( \frac { 1 }{ 3 } \right) }\) given in figure, draw.

Let A = {2, 3, 5} and relation R = {(2, 5)} write down the minimum number of ordered pairs to be included to R to make it an equivalence relation.

Prove that \(\begin{vmatrix} 1 &x^2 &x^3 \\ 1 & y^2 &y^3 \\1 &z^2 &z^3 \end{vmatrix}\) = (x - y)(y - z)(z - x)(xy + yz + zx).

In a triangle ABC, if \(\begin{vmatrix} 1& 1 &1 \\1+sin A &1+sin B &1+sin C \\ sinA(1+sin A) &sin B(1+sin B) &sin C(1+sin C) \end{vmatrix}=0,\)
prove that \(\triangle\)ABC is an isosceles triangle.

Solve the following problems by using Factor Theorem :
Show that \(\begin{vmatrix}x & a & a \\ a & x & a \\ a &a & x \end{vmatrix}=(x-a)^2(x+2a)\)

Solve the following problems by using Factor Theorem :
Show that \(\begin{vmatrix} b+c & a-c & a-b \\ b-c & c+a & b-a \\ c-b & c-a & a+b \end{vmatrix}=8abc\)

Solve the following problems by using Factor Theorem :
Solve \(\begin{vmatrix} x+a &b &c \\ a & x+b & c \\ a & b &x+c \end{vmatrix}=0\)

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

If a \(\in\) {-1, 2, 3, 4, 5} and b \(\in\) {0,3, 6}. Write the set of all ordered pairs (a, b) such that a + b = 5.

Find the sum and difference of the identity function and the modulus function?

Let f and g be real functions defined by \(f(x)=\sqrt{x+2}\)and \(g(x)=\sqrt{4-x^2}\). Find f-g 

Let A = R - [2] and B = R - [1]. If f : A ⟶ B is a mapping defined by \(f(x)={x-1\over x-2}\) Show that f is one-one and onto.

If x is a real number and |x| < 5 then ___________

The logarithmic form of 5² = 25 is ___________

The value of log₁₀⁸ + log₁₀⁵- log₁₀⁴ = ___________

Solve \(\sqrt{7+6x-x^2}=x+1\)

If P(x) = x³ + 3x² + 2x + 1, then the remainder on dividing p(x) by (x - 1) is ___________

If - 3x + 17 < -13 then ___________

If |x + 3| ≥ 10 then ___________

The Value of \({ log }_{ 3/4 }^{ (4/3) }\) is ___________

(x²-2x+2)(x²+2x+2) are the factors of the polynomial ___________

If \(\alpha\) and \(\beta\) are the roots of 2x² + 4x + 5 = 0 the equation where roots are 2\(\alpha\) and 2\(\beta\) is ___________

If k > 0, then solve the inequation |x| ≤ K

Solve the equation \(\frac { x+2 }{ x+3 } =\frac { x+4 }{ 2x+3 } \)

If a³+ b³= ab(8 - 3a - 3b), show that log \(\left( \frac { a+b }{ 2 } \right) =\frac { 1 }{ 3 } \)  (log a + log b)

Solve \(\sqrt{x+5}\)

Resolve into partial function \(\frac{2}{x^2-1}\).

A solution is to be kept between 68^oF and 77^oF. What is the range in temperature in degree Celsius (c) or Fahrenheit (F), conversion formula is given by \(F=\frac { 9 }{ 5 } \)C + 32?

Solve :x²+ 2|x| - 8 = 0

Given log₂¹⁶ = 4. Find log₁₆²

Find the value of \(\frac { 2-\sqrt { 3 } }{ \sqrt { 3 } } \) when \(\sqrt { 3 } \)  = 1.732

Find the square root of 9-4\(\sqrt{5}\)

A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the length is to be twice as long as the shortest. What are the possible lengths for the shortest board if the third piece is to be at least 5 cm longer than the second?

Solve the equation x^2/3 + x^1/3 - 2 = 0.

Solve \(\sqrt [ 8 ]{{{x}\over{x+3}} } -\sqrt{{{x+3}\over{x}}}=2.\)

Find the value of log₂ \(\left({{\sqrt [ 3 ]{4 } }\over{4^2\sqrt{8}}} \right).\)

Simplify: \(\sqrt { 98 } +\sqrt { 50 } -\sqrt { 18 } +\sqrt { 75 } -\sqrt { 27 } \)

Solve the inequation x \(\ge\) 2 graphically.

Solve the quadratic equation 5^2x- 5^{x + 3}+ 125 = 5^x.

Solve \(\frac { x+1 }{ x-1 } >0\)

Simplify : \(\frac { 1 }{ 2+\sqrt { 3 } } +\frac { 3 }{ 4-\sqrt { 5 } } +\frac { 6 }{ 7-\sqrt { 8 } } \)

Solve: \(\frac { |x|-1 }{ |x|-3 } \ge 0,x\epsilon R,\quad x\neq \pm 3\)

The largest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is atleast 61 cm, find the minimum length of the shortest side?

Solve the equation \(8+9\sqrt{(3x-1)(x-2)}=3x^2-7x.\)

If \({{{log}_{e}^{x}}\over{b-c}}={{{log}_{e}^{y}}\over{c-a}}={{{log}_{e}^{z}}\over{a-b}},\) show that x^ay^bz^c = 1

If x = 2 is one root x³+ 2x²- 5x - 6 = 0 then find the other roots of the equation

Solve the equation x³+ 5x²-16x-14 = 0. Given x + 7 is a root

Solve: \(\sqrt{x+5}+\sqrt{x+21}=\sqrt{6x+40}\)

Solve for x⁴-7x³+ 8x²+ 8x- 8 = 0. Given 3 -\(\sqrt { 5 } \) is a root

Solve \(\frac { x-2 }{ x+4 } \ge \frac { 5 }{ x+3 } \)

Solve \((x+1)^{ \frac { 1 }{ 3 } }=\sqrt { x-3 } \)

Solve \((x+1)^{ \frac { 1 }{ 3 } }=\sqrt { x-3 } \)

The angle between the minute and hour hands of a clock at 8.30 is ___________

If cos x = \(\frac { -1 }{ 2 } \) \(0 < x < 2\pi\) and, then the solutions are _______________

(sec A + tan A-1) (sec A - tan A+1)-2 tan A = _______________

If sin θ = sin \(\alpha\), then the angles θ and \(\alpha\) are related by _______________

If cos θ + \(\sqrt{3}\) sin θ = 2 and θ∈[0, 2π] then θ is _______________

If \(\overrightarrow{a}+2\overrightarrow{b}\) and \(3\overrightarrow{a}+m\overrightarrow{b}\) are parallel, then the value of m is

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}-2\hat{j}+\hat{k}\) is

A vector \(\overrightarrow{OP}\) makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between \(\overrightarrow{OP}\)and the z-axis is

If \(\overrightarrow{BA}=3\hat{i}+2\hat{j}+\hat{k}\) and the position vector of B is \(\hat{i}+3\hat{j}-\hat{k}\) ,then the position vector of A is

A vector makes equal angle with the positive direction of the coordinate axes. Then each angle is equal to

Vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are inclined at an angle \(\theta =120^o\). If \(|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,\) then \([(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2\) is equal to

If the points whose position vectors \(10\hat{i}+3\hat{j},12\hat{i}-5\hat{j}\) and \(a\hat{i}+11\hat{j}\) are collinear then a is equal to

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

The value of m for which the vectors \(3\hat { i } -6\hat { j } +\hat { k } \) and \(2\hat { i } -4\hat { j } +\lambda \hat { k } \) are parallel is __________ .

Represent graphically the displacement of (i) 30 km 60° west of north (ii) 60 km 50° south of east.

Represent graphically the displacement of 45cm 30°north of east.

Prove that the relation R defined on the set V of all vectors by ‘ \(\overrightarrow{a}\ R\ \overrightarrow{b} \ if \ \overrightarrow{a}=\overrightarrow{b}\) is an equivalence relation on V.

If G is the centroid of a triangle ABC, prove that \(\overrightarrow{GA}\) + \(\overrightarrow{GB}\)  + \(\overrightarrow{GC}\) = \(\overrightarrow{0}\).

Find the direction cosines and direction ratios for the following vectors. 5\(\hat{i}\) - 3\(\hat{j}\) - 48\(\hat{k}\)

Find the direction cosines and direction ratios for the following vectors. \(\hat{i}\) - \(\hat{k}\)

Find \(\overrightarrow{a}.\overrightarrow{b}\) when \(\overrightarrow{a}\)= \(\hat{i}-\hat{j}+5\hat{k}\)  and \(\overrightarrow{b}=3\hat{i}-2\hat{k}\)

If \(\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},\) \(\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}\) and \(\overrightarrow{c}=3\hat{i}+\hat{j}\) be such that \(\overrightarrow{a}+\lambda \overrightarrow{b}\) is perpendicular to \(\overrightarrow{c}\) then find \(\lambda\).

If \(|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\) prove that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular.

Let A and B be two points with position vectors 2\(\overrightarrow{a}\)+ 4\(\overrightarrow{b}\) and 2\(\overrightarrow{a}\) − 8\(\overrightarrow{b}\). Find the position vectors of the points which divide the line segment joining A and B in the ratio 1:3 internally and externally.

If D and E are the midpoints of the sides AB and AC of a triangle ABC, prove that \(\overrightarrow{BE}+\overrightarrow{DC}={3\over2}\overrightarrow{BC}\) 

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\) represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal.

Let A, B and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that \(\overrightarrow{AD}\) + \(\overrightarrow{BE}\) +\(\overrightarrow{CF}\) = \(\overrightarrow{0}\).

Show that the points whose position vectors are 2\(\hat{i}\) + 3\(\hat{j}\) − 5\(\hat{k}\), 3\(\hat{i}\) + \(\hat{j}\) − 2\(\hat{k}\) and, 6\(\hat{i}\) − 5\(\hat{j}\) + 7\(\hat{k}\) are collinear

If \(\overrightarrow{a},\overrightarrow{b}\) are unit vectors and \(\theta\) is the angle between them, show that \(cos {\theta \over 2}={1\over2}|\overrightarrow{a}+\overrightarrow{b}|\)

If \(\overrightarrow{a}=-3\hat{i}+4\hat{j}-7\hat{k}\) and \(\overrightarrow{b}=6\hat{i}+2\hat{j}-3\hat{k},\) verify \(\overrightarrow{a}\) are \(\overrightarrow{a}\times \overrightarrow{b}\) perpendicular to each other.

Find the unit vectors perpendicular to each of the vectors \(\overrightarrow{a}+\overrightarrow{b}\) and \(\overrightarrow{a}-\overrightarrow{b}\), where \(\overrightarrow{a}=\hat{i}+\hat{j} +\hat{k} \) and \(\overrightarrow{b} =\hat{i}+2\hat{j} +3\hat{k} \).

If \({1\over2},{1\over \sqrt{2}}\), a are the direction cosines of some vector, then find a.

Find the unit vector in the direction of the vector \(\overrightarrow { a } -2\overrightarrow { b } +3\overrightarrow { c } \) if \(\overrightarrow { a } =\hat { i } +\hat { j } ,\overrightarrow { b } =\hat { j } +\hat { k } \) and \(\overrightarrow { c } =\hat { i } +\hat { k } \) .

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

Prove that the points whose position vectors \(2\hat{i}+4\hat{j}+3\hat{k},4\hat{i}+\hat{j}+9\hat{k}\) and \(10\hat{i}-\hat{j}+6\hat{k}\) form a right angled triangle.

Show that the vectors \(5\hat{i}+6\hat{j}+7\hat{k},7\hat{i}-8\hat{j}+9\hat{k},3\hat{i}+20\hat{j}+5\hat{k}\) are coplanar.

Show that the following vectors are coplanar \(\hat{i}\) − 2\(\hat{j}\) + 3\(\hat{k}\), - 2\(\hat{i}\) + 3\(\hat{j}\) - 4\(\hat{k}\) ,-\(\hat{j}\) + 2\(\hat{k}\) .

Show that the following vectors are coplanar 5\(\hat{i}\) +6\(\hat{j}\) +7\(\hat{k}\) ,7 \(\hat{i}\) -8\(\hat{j}\) +9 \(\hat{k}\),3\(\hat{i}\)+20\(\hat{j}\) +5\(\hat{k}\) .

Three vectors \(\overrightarrow{a},\overrightarrow{b}\)and \(\overrightarrow{c}\) are such that \(|\overrightarrow{a}|=2,|\overrightarrow{b}|=3,|\overrightarrow{c}|=4,\) and \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\) .Find \(4\overrightarrow{a}.\overrightarrow{b}+​​3\overrightarrow{b}.\overrightarrow{c}+3\overrightarrow{c}.\overrightarrow{a}.\)

Prove that the smallar angle between any two diagonals of a cube is cos^-1 \(({1\over3})\).

Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a² = b² + c² - 2bc cos A (Law of cosines)

Let \(\overrightarrow { a } =\hat { i } +\hat { j } +2\hat { k } \) and \(\overrightarrow { b } =\hat { i } +2\hat { j } +\hat { k } \) and \(\overrightarrow { c } \)  be a unit vectorin the plane determined by \(\overrightarrow { a } \) and \(\overrightarrow { b } \). If \(\overrightarrow { c } \) is perpendicular to the vector \(\hat { i } +\hat { j } +\hat { k } \) and makes an obtuse angle with \(\overrightarrow { a } \), then prove that \(\overrightarrow { c } =\frac { \hat { j } -\hat { k } }{ \sqrt { 2 } } \)

Let A, Band C represent the angles of a \(\triangle\)ABC and a, b, c represent the lengths of the sides opposite to them, then prove that a = b cos C + c cos B (Projection formula)

\(lim_{x\rightarrow {\pi/2}}{2x-\pi\over cosx} \)

\(lim_{\theta\rightarrow0}{Sin\sqrt{\theta}\over \sqrt{sin \theta}} \)

\(\underset { x\rightarrow \infty }{ lim } \left( \cfrac { { x }^{ 2 }+5x+3 }{ { x }^{ 2 }+x+3 } \right) ^{ x }\)is

\(lim_{x \rightarrow \infty}{\sqrt{x^2-1}\over 2x+1}=\)

\(lim_{x \rightarrow 0}{a^x-b^x\over x}=\)

\(lim_{\alpha \rightarrow {\pi/4}}{sin \alpha -cos \alpha \over \alpha -{\pi\over 4}}\) is

\(lim_{n \rightarrow \infty}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})\) is

\(lim_{x \rightarrow 0}{e^{sin \ x}-1\over x}=\)

\(lim_{x \rightarrow 0}{e^{tan \ x}-e^x\over tan x-x}=\)

The value of \(lim_{x\rightarrow k^-}x-\left\lfloor x \right\rfloor \) , where k is an integer is

Calculate \(\lim _{ x\rightarrow0}{|x| } \).

Consider the function f(x) = \(\sqrt{x},x\ge0.\) Does\(lim_{x\rightarrow0}f(x)\) exist?

Evaluate \(lim_{x\rightarrow 2^-}\left\lfloor x \right\rfloor \) and \(lim_{x\rightarrow 2^+}\left\lfloor x \right\rfloor \) .

In problems 1-6, using the table estimate the value of the limit.
\(lim_{x\rightarrow 2}{x-2\over x^2-x-2}\)

In problem, using the table estimate the value of the limit
\(lim_{x\rightarrow 0}{sin x\over x}\)

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
\(lim_{x\rightarrow3}{1\over x-3}\)

If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?

Compute\(lim_{x\rightarrow-2}(-{3\over 2}x)\)

Compute \(lim_{x\rightarrow1}{\sqrt{x}-1\over x-1}\) .

Evaluate the following limits :
\(lim_{x\rightarrow1}{x^m-1\over x^n-1}\) ,m and n are integers.

Evaluate:\(lim_{x\rightarrow{3}}{x^2-9\over x-3}\) if it exists by finding f(3^-)and f(3⁺).

Verify the existence of \(lim_{x\rightarrow1}f(x),\) where \(f(x)= \begin{cases}\frac{|x-1|}{x-1}, & \text { for } x \neq 1 \\ 0, & \text { for } x=1\end{cases}\)

Calculate \(lim_{x\rightarrow3}{(x^2-6x+5)\over x^3-8x+7}\)

Find \(lim_{t\rightarrow0}{\sqrt{t^2+9}-3\over t^2}.\)

Find the relation between a and b if \(lim_{x\rightarrow3}f(x)\) exists where \(f(x)= \begin{cases}a x+b & \text { if } x>3 \\ 3 a x-4 b+1 & \text { if } x<3\end{cases}\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{sin \alpha x\over sin \beta x}\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{2^x-3^x\over x}\)

Evaluate the following limits :\(\)\(lim_{x \rightarrow \infty}\{ x[log(x+a)-log(x)]\}\)

Evaluate the following limits :\(lim_{x\rightarrow {\pi\over 2}}(1+sin x)^{2cosec \ x}\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{\sqrt{2}-\sqrt{1+cos x}\over sin^2x}\)

The velocity in ft/sec of a falling object is modeled by \(r(t)=-\sqrt{32\over k}{1-e^{2t\sqrt{32k}}\over1+e^{-2r\sqrt{32k}}}\), where k is a constant that depends upon the size and shape of the object and the density of the air. Find the  limiting velocity of the object, that is, find \(lim_{t\rightarrow \infty}r(t).\)

Show that \(lim_{x\rightarrow\infty}{1+2+3+...+n\over 3n^2+7n+2}={1\over6}\)

Show that \(lim_{x\rightarrow\infty} {1\over 1.2}+{1\over 2.3}+{1\over 3.4}+...+{1\over n(n+1)}=1\)

Show that \(lim_{x\rightarrow 0^+}x[\left\lfloor {1\over x} \right\rfloor+\left\lfloor {2\over x} \right\rfloor +....+\left\lfloor {15\over x}\right\rfloor ]=120\)

Do the limits of following functions exist as x\(\rightarrow 0?\) State reasons for your answer.\(sin(x -\left\lfloor x \right\rfloor) \over x- \left\lfloor x \right\rfloor\)

Show that \(lim_{x\rightarrow\infty}{1^2+2^2+....+(3n)^2\over (1+2+...+5n)(2n+3)}={9\over25}\)

Evaluate : \(lim_{x \rightarrow 0}{3^x-1\over \sqrt{1+x}-1}.\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{\sqrt{x^2+a^2}-a\over \sqrt{x^2+b^2}-b}\)

Evaluate the following limits :\(limx_{x\rightarrow \infty}x[{3^{1\over x}+1-cos({1\over x}) -e^{1\over x}}]\)

Describe the interval(s) on which each function is continuous.
\(h(x)= \begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{cases}\)

\(\frac{d}{d x}\left(\frac{2}{\pi} \sin x^{\circ}\right)\) is

If y = f(x²+2) and f '(3) = 5, then \({dy\over dx}\) at x = 1 is

If y = mx + c and f(0) =\(f '(0)=1\), then f(2) is

If f(x) = x tan^-1 x, then f '(1) is

\({d\over dx}(e^{x+5log \ x})\) is

The differential coefficient of log₁₀ x with respect to log_x10 is

If f(x) = x + 2, then f '(f(x)) at x = 4 is

It is given that f '(a) exists, then \(lim_{x\rightarrow a}{xf(a)-af(x)\over x-a}\) is

If \(f(x)=\left\{\begin{array}{l} x+1, \quad \text { when } x<2 \\ 2 x-1 \text { when } x \geq 2 \end{array}\right.\), then f'(2) is

If g(x) = (x² + 2x + 3) f(x) and f(0) = 5 and \(lim_{x \rightarrow 0}{f(x)-5\over x}=4\), then g'(0) is

Show that the greatest integer function \(f(x)=\left\lfloor x \right\rfloor \) is not differentiable at any integer?

Differentiate the following with respect to x : y = x³ + 5x² + 3x + 7

Differentiate the following with respect to x : y = e^x + sin x + 2

Differentiate the following with respect to x : \(y=(x-{1\over x})^2\)

Differentiate the following with respect to x : y = xe^x log x

Show that the greatest integer function \(f(x)=\left\lfloor x \right\rfloor \) is not differentiable at any integer?

Differentiate the following with respect to x : y = x³ + 5x² + 3x + 7

Find the derivatives of the following functions with respect to corresponding independent variables: f(x) = x sin x

Find the derivatives of the following functions with respect to corresponding independent variables: g(t) = 4 sec t + tan t

Find the derivatives of the following functions with respect to corresponding independent variables : y = e^x sin x

Find the slope of tangent line to the graph of f(x) = - 5x² + 7x at (5, f(5)).

Find the derivatives of the following functions using first principle. f(x) = 6

Find the derivatives of the following functions using first principle. f(x) = - 4x + 7

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
\(f(x)=|x-1|\)

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
\(f(x)=\sqrt{1-x^2}\)

Differentiate the following: \(f(x)=\frac{x}{\sqrt{7-3 x}}\)

Differentiate the following: \(y=5^{\frac{-1}{x}}\)

Differentiate the following: y = sin³ x + cos³ x

Find f'(x) if f(x) = cos^-1(4x³ - 3x).

Find \({dy\over dx}\) if x = at² ; y = 2at, t\(\neq 0.\)

Show that the following functions are not differentiable at the indicated value of x.

Find the derivative of the function g(t) = \(({t-2\over 2t+1})^9\) .

Differentiate the following: \(y=\left(x^2+1\right) \sqrt[3]{x^2+2}\)

Differentiate: y = sin (tan(\(\sqrt{sin x}\)))

Find \({dy\over dx}\) if sin y = ycos 2x

Find \({d^2y\over dx^2}\) if x² + y² = 4.

Find the derivatives of the following : y = x^cosx

Find the derivatives of the following :  \(\sqrt{xy}=e^{(x-y)}\)

Find the derivative of the tan (x + y) + tan (x - y) = x

Find the derivatives of the following : \(tan^{-1}\sqrt{1-cos \ x \over 1+ cos \ x}\)

If \(\int f(x) d x=g(x)+c\), then \(\int f(x) g^{\prime}(x) d x\)

If \(\int {3^{1\over x}\over x^2}dx=k(3^{1\over x})+c\), then the value of k is

If \(\int f '(x)e^{x^2}dx=(x-1)e^{x^2}+c\), then f(x) is

The gradient (slope) of a curve at any point (x, y) is\({x^2-4\over x^2}\). If the curve passes through the point (2, 7), then the equation of the curve is

\(\int \sin ^3 x d x\) is

\(\int \sqrt{\frac{1-x}{1+x}} d x\) is

\(\int \frac{d x}{e^x-1}\) is

\(\int e^{-4 x} \cos x d x\) is

\(\int \frac{\sec ^2 x}{\tan ^2 x-1} d x\) is

\(\int e^{-7 x} \sin 5 x d x\) is

Integrate the following with respect to x : x¹⁰

Integrate the following with respect to x : \(\sqrt{x}\)

IIntegrate the following with respect to x : \({cot \ x \over sin \ x}\)

Integrate the following with respect to x : \({sin \ x \over cos ^2 \ x}\)

Integrate the following with respect to x : \({1\over \sqrt{1-x^2}}\)

Integrate the following with respect to x : sec²(3 + 4x)

Integrate the following with respect to x : cosec(ax + b)cot (ax + b)

Integrate the following functions with respect to x : \({1\over (2-3x)^4}\)

Integrate the following functions with respect to x : \({1\over \sqrt{1-(4x)^2}}\)

Integrate the following functions with respect to x : \({1\over \sqrt{1-81x^2}}\)

Integrate the following with respect to x : 2cos x - 4sin x + 5 sec² x + cosec² x

Evaluate the following integrals : \({12\over (4x-5)^3}+{6\over 3x+2}+16e^{4x+3}\)

Integrate the following with respect to x : \((x+4)^5+{5\over (2-5x)^4}-cosec^2(3x-1)\)

Integrate the following with respect to x : \(4cos(5-2x)+9e^{3x-6}+{24\over 6-4x}\)

Integrate the following with respect to x : \(sec^2{x\over 5}+18cos 2x+10sec(5x+3)tan(5x+3)\)

Integrate the following functions with respect to x : \({cos 2x\over sin^2x cos^2x}\)

Evaluate the following integrals : \(\int{sin x\over 1+cos \ x}dx\)

Evaluate the following integrals : \(\int{1\over 1+x^2}dx\)

Evaluate the following integrals : \(\int {x(a-x)^8}dx\)

Integrate the following with respect to x : \(tan \ x\sqrt{sec \ x}\)

If f'(x) = 3x² - 4x + 5 and f(1) = 3, then find f(x).

A train started from Madurai Junction towards Coimbatore at 3 pm (time t = 0) with velocity v(t) = 20t + 50 kilometre per hour, where t is measured in hours. Find the distance covered by the train at 5 pm.

A tree is growing so that, after t - years its height is increasing at a rate of \({18\over \sqrt{t}}\) cm per year Assume that when t = 0, the height is 5 cm.
(i) Find the height of the tree after 4 years.
(ii) After how many years will the height be 149 cm?

At a particular moment, a student needs to stop his speedy bike to avoid a collision with the barrier ahead at a distance 40 metres away from him. Immediately he slows (retardation) the bike under braking at a rate of 8 metre/second². If the bike is moving at a speed of 24m/s, when the brakes are applied, would it stop before collision?

The rate of change of weight of person w in kg with respect to their height h in centimetres is given approximately by \({dw\over dh}=4.364 \times 10^{-5}h^2\). Find weight as a function of height. Also find the weight of a person whose height is 150 cm.

A tree is growing so that, after t - years its height is increasing at a rate of \({18\over \sqrt{t}}\) cm per year Assume that when t = 0, the height is 5 cm.
(i) Find the height of the tree after 4 years.
(ii) After how many years will the height be 149 cm?

Integrate the following functions with respect to x : \((3x+4)\sqrt{3x+7}\)

Integrate the following functions with respect to x :\({1\over (x-1)(x+2)^2}\)

Integrate the following with respect to x : \(x^3 e^{-x}\)

A number is selected from the set {1,2,3,...,20}. The probability that the selected number is divisible by 3 or 4 is

A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are\({3\over4},{1\over2},{5\over 8}\). The probability that the target is hit by A or B but not by C is

If A and B are any two events, then the probability that exactly one of them occur is

Let A and B be two events such that \(P(\overline{A\cup B})={1\over6}, P(A\cap B)={1\over4}\) and \({P(\overline{A})}={1\over4}\). Then the events A and B are

Two items are chosen from a lot containing twelve items of which four are defective, then the probability that at least one of the item is defective

A number x is chosen at random from the first 100 natural numbers. Let A be the event of numbers which satisfies\({(x-10)(x-50)\over x-30}\ge0\), then P(A) is

If two events A and B are independent such that P(A) = 0.35 and \(P(A\cup B)=0.6\), then P(B) is

If A and B are two events such that P(A) = 0.4, P(B) = 0.8 and P(B/A) = 0.6, then \(P(\overline{A}\cap B )\) is

There are three events A, B, and C of which one and only one can happen. If the odds are 7 to 4 against A and 5 to 3 against B, then odds against C is

If a and b are chosen randomly from the set {1,2,3,4} with replacement, then the probability of the real roots of the equation \(x^2+ax+b=0\) is

Can two events be mutually exclusive and independent simultaneously?

If an experiment has exactly the three possible mutually exclusive outcomes A, B, and C, check in each case whether the assignment of probability is permissible.
\(P(A)=\frac { 4 }{ 7 } ,P(B)=\frac { 1 }{ 7 } ,P(C)=\frac { 2 }{ 7 } \)

If an experiment has exactly the three possible mutually exclusive outcomes A, B, and C, check in each case whether the assignment of probability is permissible
\(P(A)=\frac { 2 }{ 5 } ,\quad P(B)=\frac { 1 }{ 5 } ,\quad P(C)=\frac { 3 }{ 5 } \)

If an experiment has exactly the three possible mutually exclusive outcomes A, B, and C, check in each case whether the assignment of probability is permissible.
\(P(A)=0.3,P(B)=0.9,P(C)=-0.2\)

If an experiment has exactly the three possible mutually exclusive outcomes A, B, and C, check in each case whether the assignment of probability is permissible.
\(P(A)=\frac { 1 }{ \sqrt { 3 } } ,\quad P(B)-1-\frac { 1 }{ \sqrt { 3 } } ,\quad P(C)-0\)

If two coins are tossed simultaneously, then find the probability of getting (i) one head and one tail (ii) at most two tails

A single card is drawn from a pack of 52 cards. What is the probability that 
The card is an ace or a king?

A single card is drawn from a pack of 52 cards. What is the probability that
The card is either a queen or 9?

If \(P(A)=0.6, P(B)=0.5\), and \(P(A \cap B)=0.2\) Find \( P(\bar{A} / B)\)

Find the probability of getting the number 7, when a usual die is rolled.

If P(A) = 0.5, P(B) = 0.8 and P(B/A) = 0.8, find P(A/B) and P(A\(\cup \)B)

If A and B are two independent events such that P(A\(\cup \)B) = 0.6, P(A) = 0.2,  find P(B).

If for two events A and B, P(A) = \(\frac{3}{4}\), P(B) = \(\frac{2}{5}\) and A\(\cup \)B = S (sample space), find the conditional probability P(A/B).

The probability that a car being filled with petrol will also need an oil change is 0.30; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and filter need changing is 0.15.
(i) If the oil had to be changed, what is the probability that a new oil filter is needed?
(ii) If a new oil filter is needed, what is the probability that the oil has to be changed?

Two thirds of students in a class are boys and rest girls. It is known that the probability of a girl getting a first grade is 0.85 and that of boys is 0.70. Find the probability that a student chosen at random will get first grade marks.

If A and B are two independent events such that, P(A) = 0.4 and P\((A\cup B)\) = 0.9. Find P(B).

Suppose a fair die is rolled. Find the probability of getting (i) an even number (ii) multiple of three.

A main road in a City has 4 crossroads with traffic lights. Each traffic light opens or closes the traffic with the probability of 0.4 and 0.6 respectively. Determine the probability of
(i) a car crossing the first crossroad without stopping
(ii) a car crossing first two crossroads without stopping
(iii) a car crossing all the crossroads, stopping at third cross.
(iv) a car crossing all the crossroads, stopping at exactly one cross.

Three letters are written to three different persons and addresses on three envelopes are also written. Without looking at the addresses, what is the probability that (i) exactly one letter goes to the right envelopes (ii) none of the letters go into the right envelopes?

Let the matrix M = \(\left[ \begin{matrix} x & y \\ z & 1 \end{matrix} \right] \), If x,y and z are chosen at random from the set {1, 2,3, } and repetition is allowed (i.e., x = y = z ), what is the probability that the given matrix M is a singular matrix?

A problem in Mathematics is given to three students whose chances of solving  \(\frac { 1 }{ 3 } ,\frac { 1 }{ 4 } \) and \(\frac { 1 }{ 5 } \) (i) What is the probability that the problem is solved? (ii) What is the probability that exactly one of them will solve it?

One bag contains 5 white and 3 black balls. Another bag contains 4 white and 6 black balls. If one ball is drawn from each bag, find the probability that (i) both are white (ii) both are black (iii) one white and one black.

A year is selected at random. What is the probability that
(i) it contains 53 Sundays (ii) it is a leap year which contains 53 Sundays.

A coin is tossed twice. Events E and F are defined as follows E= Head on first toss, F = Head on second toss. Find.
(i) \(P(E \cup F)\)
(ii) \(P(E / F)\)
(iii) \(P(\bar{E} / F)\)
(iv) Are the events E and F independent

A factory has two Machines-I and II. Machine-I produces 60% of items and Machine-II produces 40% of the items of the total output. Further 2% of the items produced by Machine-I are defective whereas 4% produced by Machine-II are defective. If an item is drawn at random what is the probability that it is defective?

X speaks truth in 70 percent of cases, and Y in 90 percent of cases. What is the probability that they likely to contradict each other in stating the same fact?

A factory has two machines I and II. Machine-I produces 40% of items of the output and Machine-II produces 60% of the items. Further 4% of items produced by Machine-I are defective and 5% produced by Machine-II are defective. If an item is drawn at random, find the probability that it is a defective item.

A factory has two machines I and II. Machine I produces 40% of items of the output and Machine II produces 60% of the items. Further 4% of items produced by Machine I are defective and 5% produced by Machine II are defective. An item is drawn at random. If the drawn item is defective, find the probability that it was produced by Machine II. (See the previous example, compare the questions).

Three candidates X, Y, and Z are going to play in a chess competition to win FIDE (World chess Federation) cup this year. X is thrice as likely to win as Y and Y is twice as likely as to win Z. Find the respective probability of X,Y and Z to win the cup.

A construction company employs 2 executive engineers. Engineer-1 does the work for 60% of jobs of the company. Engineer-2 does the work for 40% of jobs of the company. It is known from the past experience that the probability of an error when engineer-1 does the work is 0.03, whereas the probability of an error in the work of engineer-2 is 0.04. Suppose a serious error occurs in the work, which engineer would you guess did the work?

The function f:[0,2π]➝[-1,1] defined by f(x) = sin x is

If the function f:[-3,3]➝S defined by f(x) = x² is onto, then S is

The inverse of f(x) = \(\begin{cases} x\quad if\quad x<1 \\ { x }^{ 2 }\quad if\quad 1\le x\le 4 \\ 8\sqrt { x } \quad if\quad x>4 \end{cases}\) is

Let f:R➝R be defined by f(x) = 1 - |x|. Then the range of f is

If |x+2| \(\le\) 9, then x belongs to

If f(x) = |x - 2| + |x + 2|, x ∈ R, then

Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

The solution set of the following inequality |x-1| \(\ge\) |x-3| is

The value of \({ log }_{ \sqrt { 2 } }512\) is

If \(\pi <2\theta <\frac { 3\pi }{ 2 } \), then \(\sqrt { 2+\sqrt { 2+2cos4\theta } } \) equals to

Write the following in roster form {x\(\in \)N : 4x + 9 < 52}

Write the following in roster form.
\(\left\{ x:\frac { x-4 }{ x+2 } =3,x\in R-\{ -2\} \right\} \)

Discuss the following relations for reflexivity, symmetricity and transitivity :
On the set of natural numbers, the relation R is defined by "xRy if x + 2y = 1".

Let X = {a, b, c, d}, and R = {(a, a) (b, b) (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it 
(i) reflexive
(ii) symmetric
(iii) transitive
(iv) equivalence

Solve for x \(\left| x \right| -10<-3\)

Write the following in roster form.
The set of all positive roots of the equation (x-1)(x+1)(x²-1) = 0.

State whether the following sets are finite or infinite.
{x \(\in \) N : x is an even prime number}

Justify the trueness of the statement "An element of a set can never be a subset of itself".

Discuss the following relations for reflexivity, symmetricity and transitivity :
The relation R defined on the set of all positive integers by "mRn if m divided n".

Discuss the following relations for reflexivity, symmetricity and transitivity:
Let P denote the set of all straight lines in a plane. The relation R defined by "lRm if l is perpendicular to m".

The owner of a small restaurant can prepare a particular meal at a cost of Rupee 100. He estimate that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200 - x. Express his day revenue total cost and profit on this meal as a function of x.

Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A\(\rightarrow\)B for each of the following:
one-to-one but not onto.

Consider the functions: 
i) \(f(x)=x^2,\)
ii) \(f(x)={1\over 2}x^2,\)
iii) \(f(x)=2x^2\)

Solve \(-{ x }^{ 2 }+3x-2\ge 0\)

If x = -2 is one root of x³ - x²- 17x = 22, then find the other roots of the equation.

Find the largest possible domain for the real valued function f defined by \(f(x)=\sqrt{x^2-5x+6}.\)

By using the same concept applied in previous example, graphs of y = sin x and y = sin 2x, and also their combined graphs are given figures (a), (b) and (c). The minimum and maximum values of sin x and sin 2x are the same. But they have different x-intercepts. The x-intercepts for y = sin x are \(\pm n\pi\) and for y = sin 2x are \(\pm{1\over 2}n\pi,\ n\in Z.\) ​​​

A model rocket is launched from the ground. The height 'h' reached by the rocket after t seconds from lift off is given by h(t) = -5t² + 100t , \(0\le t\le 20\).  At what time the rocket is 495 feet above the ground?

Solve the following system of linear inequalities 3x - 9 ≥ 0, 4x -10 ≤ 6;

Solve x = \(\sqrt{x+20}\) for x ∈ R

A simple cipher takes a number and codes it, using the function f(x) = 3x - 4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x(by drawing the lines)

Graph the function f(x) = x³ and \(g(x)=\sqrt[3]x\) on the same co-ordinate plane. Find f o g and graph it on the plane as well. Explain your results.

From the curve y = sin x, graph the functions.
(i) y = sin(-x)
(ii) y = -sin(-x)
(iii) ​\(y=sin\left( {\pi\over 2}+x\right)\) which is cos x​
(iv) ​\(y=sin\left({\pi\over 2}-x \right)\)​ which is also cos x (refer trigonometry)

From the curve y = sin x, draw y = sin |x|. (Hint: sin (-x) = -sin x)

Write the values of f at -4, 1, -2, 7, 0 if
\(f(x)=\left\{ \begin{matrix} -x+4& if -\infty <x\leq -3\\ x+4& if -3<x<-2\\ x^{2}-x& if -2\leq x < 1 \\ x-x^{2}& if 1\leq x<7\\ 0& otherwise\\ \end{matrix}\right.\)

Show that f(x) f(y) = f(x + y), where f(x) =\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

Prove that |A| = \(\begin{vmatrix} (q+r)^2& p^2 &p^2 \\ q^2 & (r+p)^2 & q^2 \\ r^2 &r^2 & (p+q)^2 \end{vmatrix}\) = 2pqr(p + q + r)³.

Show that \(\begin{vmatrix} 1 &1 &1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix}\) = (x - y)( y - z)(z - x).

Prove that the smallar angle between any two diagonals of a cube is cos^-1 \(({1\over3})\).

Evaluate the following limits : \(lim_{x\rightarrow2}{2-\sqrt{x+2}\over 3\sqrt{2}-3\sqrt{4-x}}\)

The angle between the minute and hour hands of a clock at 8.30 is ___________

Which of the following is incorrect?

\(\frac { cos3x }{ 2cos2x-1 } \) is _______________

If cos x = \(\frac { -1 }{ 2 } \) \(0 < x < 2\pi\) and, then the solutions are _______________

The maximum value of 3 sin θ+4 cos θ is _______________

Find the radian measures 40⁰ 20'

Find the values of cos x and tan x if \(\sin x=-\frac{3}{5}\) and \(\pi < x < \frac{3\pi}{2}\)

Prove that \(\frac { 1+sinx-cosx }{ 1+sinx+cosx } =tan\frac { x }{ 2 } \) .

In a ΔABC if a = 3, b = 5 and c = 7, find cos A and cos B.

Prove that \({ tan }^{ -1 }\left( \frac { 1 }{ 7 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 13 } \right) ={ tan }^{ -1 }\frac { 2 }{ 9 } \)

Prove that sin⁶x + cos⁶x = 1 - 3 sin²x cos²x

Evaluate tan 480⁰

Prove that \(\frac { cos(2\pi +x)cosec(2\pi +x)tan\left( \frac { \pi }{ 2 } +x \right) }{ sec\left( \frac { \pi }{ 2 } +x \right) cos.cot(\pi +x) } \)= 1

Find the value of tan\(\frac { \pi }{ 2 } \).

Prove that \(\frac { cos9x-cos5x }{ sin17x-sin3x } =-\frac { sin2x }{ cos10x } \)

Two slopes leave a port at the same time one goes 24 km/hr in the direction N 45^o E and other travels 32 km/hr in the direction S 75^o E. Find the distance between the ships at the end of 3 hours.

Prove that cos^-1 x = \(2\sin ^{ -1 }{ \sqrt { \frac { 1-x }{ 2 } } } =2\cos ^{ -1 }{ \sqrt { \frac { 1+x }{ 2 } } } \)

Prove \(\frac { cosA }{ a } +\frac { cosB }{ b } +\frac { cosC }{ c } =\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ 2abc } \)

Prove that \(cos\frac { B-C }{ 2 }= \frac { b+c }{ a } sin\frac { A }{ 2 } \)

In ∆ABC, if tan \(\frac{A}{2}=\frac{5}{6}\) and tan \(\frac{C}{2}=\frac{2}{5}\), then show that a, b, c, are in A.P.

Solve: sin 2x + cos x = 0

Prove that \(\tan ^{ -1 }{ \left( \frac { x }{ \sqrt { { { a }^{ 2 }-{ x }^{ 2 } } } } \right) } =\sin ^{ -1 }{ \left( \frac { x }{ a } \right) } \)

Prove that 4 cos 12° cos 48° cos 72° = cos 36°

Prove that cos 20° cos 40° cos 60° cos 80°

Prove that \(\frac{sin11AsinA+sin7Asin3A}{cos11AsinA+cos7Asin3A}=tan8A\)

If 3 tan A tan B = 1, prove that 2 cos(A+B) = cos(A-B) 

If tan \(\frac { \theta }{ 2 } =\sqrt { \frac { a-b }{ a+b } } tan\frac { \emptyset }{ 2 } ,prove\quad that\quad cos\theta =\frac { acos\emptyset +b }{ a+bcos\emptyset } .\)

A + B + C =\(\pi\), prove that sin 2A - sin 2B + sin 2C = 4 cos A sin B cos C

Solve: sin²θ - 2cos θ +\(\frac{1}{4}=0\)

Prove that cos²x + cos² \(\\ \left( x+\frac { \pi }{ 3 } \right) +{ cos }^{ 2 }\left( x-\frac { \pi }{ 3 } \right) =\frac { 3 }{ 2 } \)

Prove that \(\cos x\cos \left( \frac { \pi }{ 3 } -x \right) cos\left( \frac { \pi }{ 3 } +x \right)=\frac14 cos3x\)

The minute hand of a watch is 1.5 cm long. How far does its top move in 40 minutes?

Prove that \(\cos { 5x } =16\cos ^{ 5 }{ x } -20\cos ^{ 3 }{ x } +5\cos { x } \)

If the sides of a \(\triangle\)ABC are a = 4, b = 6, and c = 8, show that \(4\cos { B } +3\cos { C } =2\)

Solve: tan^-1 (x + 1) + tan^-1 (x - 1) = tan^-1\(\frac{4}{7}\).0

The number of different signals which can be give from 6 flags of different colours taking one or more at a time is _________

Number of all four digit numbers having different digits formed of the digits 1, 2, 3, 4 and 5 and divisible by 4 is _________

If 10ⁿ + 3 \(\times\) 4ⁿ⁺²+\(\lambda \) is divisible by 9 for all n \(\in \)N, then the least positive integral value of \(\lambda \) is _________

If ⁿP_r=k x ^n-1P_r-1 what is k:

The number of rectangles than can be formed on a chess board is  _________

The number of ways to average the letters of the word CHEESE are _________

If p(n):49ⁿ + 16ⁿ +\(\lambda \) is divisible by 64 for n \(\in \) N is true, then the least negative integral value of \(\lambda \) is _________

The number of ways of selecting of 3 poets and 4 scientists such that poets are in even places _________

The number of ways of disturbing 7 identical balls in 3 distinct boxes, so that no box is empty is  _________

The number of ways in which we can post 5 letters in 10 letter boxes is  _________

A room has 6 doors. In how many ways can a man enter the room through one door and come out through a different door?

If nP₄ = 20 \(\times\) 3 nP₂, then find n.

If the ratio 2nC₃: nC₃ = 11 : 1, find n.

How many chord can be drawn through 21 points on a circle?

How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7 if no digit is repeated?

Write down all the permutations of the vowels A, E, I, O, U in English alphabets taking there at a time starting with A.

In how many ways can the letters of the word PENCIL be arranged so that N is always next to E.

There are six periods in each working day of a school. In how many ways can one arrange 5 subjects such that each subject is allowed atleast one period?

In how many ways a cricket team of eleven be chosen out of a batch of 15 players if there is no restriction on the selection?

How many words can be formed by using the letters of the word ORIENTAL so that A and E always occupy the odd places?

Prove that n!(n + 2) = n! + (n + 1)!

Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the points.

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of exactly 3 girls

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: almost 3 girls?

3²ⁿ - 1 is divisible by 8

If (n+2)! = 60(n-1)! find n.

In how many ways can 9 examination papers be arranged so that the best and the worst papers are never together?

A question paper has two parts A and B, each containing 10 questions. If a student has to choose 8 from part A, 5 from Part B, in how many ways can he choose the questions?

How many words (with or without dictionary meaning) can be made from the letters of the word MONDAY, assuming that no letter is repeated, if

Find n if ^{n - 1}P₃ : ⁿP₄ = 1 : 9

How many numbers are there between 100 and 1000 such that atleast one of the their digits in 7?

In how many ways can the letters of the word PERMUTATIONS be arranged if the words start with P and end with S.

Determine n if ²ⁿC₃ : ⁿC₂ = 12 : 1

2n < (n + 2)! for all natural number n.

If the letters of the word APPLE are permuted in all possible ways and the strings then formed are arranged in the dictionary order show that the rank of the word APPLE is 12.

If 2n+1 P_n-1 : 2n-1 P_n= 3 : 5, find n.

In how many ways can the letters of the word PERMUTATIONS be arranged if vowels are all together.

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if atleast 5 women have to be included in a committee? In how many of these committees the women are in majority?

Prove by the principle of mathematical induction that for every natural number n, 3^{2n + 2} - 8ⁿ - 9 is divisible by 8.

1 + 5 + 9 + ... + (4n - 3) = n(2n -1), \(\forall\)n \(\in\)N.

The number of constant functions from a set containing m elements to a set containing n elements is

The function f:[0,2π]➝[-1,1] defined by f(x) = sin x is

If the function f:[-3,3]➝S defined by f(x) = x² is onto, then S is

Let X = {1, 2, 3, 4}, Y = {a, b, c, d} and f = {(1, a), (4, b), (2, c), (3, d), (2, d)}. Then f is

The inverse of f(x) = \(\begin{cases} x\quad if\quad x<1 \\ { x }^{ 2 }\quad if\quad 1\le x\le 4 \\ 8\sqrt { x } \quad if\quad x>4 \end{cases}\) is

The Co-efficient of x^-17 in \({ \left( { x }^{ 4 }-\frac { 1 }{ { x }^{ 3 } } \right) }^{ 15 }\)is _____________ 

The term without x in \({ \left( 2x-\frac { 1 }{ 2{ x }^{ 2 } } \right) }^{ 12 }\) is ______________

Sum of n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } ..is\) ______________

The series for log \(\left( \frac { 1+x }{ 1-x } \right) is\) ______________

\(\left(1+\frac{1}{\lfloor2}+\frac{1}{\lfloor4}+\frac{1}{\lfloor6}+...\right)^2-\left(1+\frac{1}{\lfloor3}+\frac{1}{\lfloor5}+\frac{1}{\lfloor7}+...\right)^2=\)______________

If the sum of n terms of an A. P. be 3n² - n and its common difference is 6, then its first term is ______________

The series 1+4x+8x² + \(\frac { 32 }{ 3 } { x }^{ 3 }+.....+\infty \ is\) ______________

The Co-efficient of x³ in \(\sqrt { \frac { 1-x }{ 1+x } } ,\left| x \right| <1\ is\ \)______________

\(\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+. . .\infty =\) ______________

2^1/4 4^1/8 8^1/16 16^1/32 . . . = ______________

Find a negative value of m if the Co-efficient of x² in the expansion of (1+x)m, |x|<1 is 6

If a, b, c are in A.P., show that (a-c)² = 4(b² - ac).

If H be the H. M. between a and b, then show that (H - 2a) (H - 2b) = H²

Find a positive value of m for which the coefficient of x² in the expansion of (1 + x)^m is 6.

Find the \(\sqrt [ 3 ]{ 126 } \) approximately to two decimal places.

Show that \(n!>\left( \frac { n }{ e } \right) ^{ 2 }\) for n ∈ N

Find the middle term in \({ \left( x-\frac { 1 }{ 2y } \right) }^{ 10 }\)

Find the 5^th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

Find the n^th term of the series 3 - 6 + 9 -12 + ...

In the binomial expansion of (1+a)^m+n, Prove  that the coefficients of a^m and aⁿ are equal.

The first three terms in the expansion of (1 + ax)ⁿ are 1 + 12x + 64x². Find n and a

Show that the sequence where log a,\(log\frac { { a }^{ 2 } }{ b^{ 1 } } log\frac { { a }^{ 2 } }{ { b }^{ 2 } } \)  ..is an A.P

If the p^th, q^th and r^th terms of an A.P. are a, b, c respectively, prove that a (q - r) + b (r - p) + c (p - q) = 0.

The sum of first three terms of a G.P. is to the sum of the first six terms as 125: 152. Find the common ratio of the G.P.

If x so large prove that \(\sqrt { { x }^{ 2 }+25 } -\sqrt { { x }^{ 2 }+9 } =\frac { 8 }{ x } \) nearly.

Prove that in the expansion of (1+x)ⁿ, the Co-efficient of terms equidistant from the beginning and from the end are equal

For what value of n, the nth term of the series "3 + 10 + 17 +..+ and 63 + 65 + 67 +... are equal

Find the co-efficient of xⁿ  in the series 1 + (a+bx) + \(\frac { (a+bx)^2}{ 2! } +\frac { (a+bx)^{ 3 } }{ 3! } \)

The sum of two members is\(\frac { 13 }{ 6 } \). An even number A.M.S are being inserted between them and their sum exceeds their number by 1. Find the number of A.M.S inserted.

Write the first six terms of the sequences given by a₁ = 4, a_n+1 = 2na_n.

If the Co-efficients of three successive terms in the expansion of (1 +x)ⁿ are in the ratio 1 : 3 : 5, then find the value of n

If (p+1) ^th  term of an A.P is twice the (q+1)^th terms prove that the (3p+1)^th term is twice the  (p+q+1)^th term

If S _n denotes that Sum of n terms of a G. P., prove that (s₁₀-s₂₀ )² = s₁₀ (s₃₀ - s₂₀)

Show that the coefficient of the middle term in the expansion of (1+x)²ⁿ is equal to the sum of the coefficients of the two middle terms in the expansion of (1+x)^2n-1.

If S₁, S₂, S₃ be respectively the sums of n, 2n, 3n, terms of a G.P. , then prove that S₁ (S₃ - S₂) = (S₂ - S₁)².

If n((A \(\times\) B) ∩(A \(\times\) C)) = 8 and n(B ∩ C) = 2, then n(A) is

If n(A) = 2 and n(B ∪ C) = 3, then n[(A \(\times\) B) ∪ (A \(\times\) C)] is

If two sets A and B have 17 elements in common, then the number of elements common to the set A \(\times\)B and B \(\times\)A is

For non-empty sets A and B, if A ⊂ B then (A \(\times\)B) ⋂ (B \(\times\)A) is equal to

The number of relations on a set containing 3 elements is

If x = 0.001, prove that \(\frac { { \left( 1-2x \right) }^{ \frac { 2 }{ 3 } }{ \left( 4+5x \right) }^{ \frac { 3 }{ 2 } } }{ \sqrt { 1-x } } \) = 8.01 up to two places of decimals 

If A and G be respectively the A. M and G. M between two positive numbers, find the numbers

If \(\alpha ,\beta \)are the roots of the equation x²-px + q = 0, then prove that \(\log { (1+px+q{ x }^{ 2 }) } =(\alpha +\beta )x=\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } { x }^{ 2 }+\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 3 } { x }^{ 3 }-....\infty \)

Find the value of \((a^{2}+\sqrt{a^{2}-1})^{4}+(a^{2}-\sqrt{a^{2}-1})^{4}\)

If sum of the n terms of a G.P be S, their product P and the sum of their reciprocals R, then prove that \(P^{2}=(\frac{S}{R})^{n}\)

Write the following in roster form. 
{x\(\in \)N : x²<121 and x is a prime}

Write the following in roster form.
The set of all positive roots of the equation (x-1)(x+1)(x²-1) = 0.

Write the following in roster form {x\(\in \)N : 4x + 9 < 52}

Write the following in roster form.
\(\left\{ x:\frac { x-4 }{ x+2 } =3,x\in R-\{ -2\} \right\} \)

Write the set {-1, 1} in set builder form.

The inclination to the x-axis and intercept on y-axis of the line \(\sqrt {2y}=x+2\sqrt 2\) ______________

The co-ordinates of the foot of the perpendicular drawn from the point (2, 3) to the line 3x - y + 4 = 0 is ______________

The image of the point (1, 2) with respect to the line y = x is ______________

The equation of the bisectors of the angle between the lines represented by 3x²- 5xy + 4y² = 0 is ______________

The gradient of one of the lines of ax²+ 2hxy + by² = 0 is twice that of the other, then ______________

The equation of the bisectors of the angle between the co-ordinate axes are ______________

The lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0(a ≠ b ≠ c ≠ 1) are concurrent, then the value of \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\) ______________

Which one of the following statements is false?

If h² = ab, then the lines represented by ax²+ 2hx + by² = 0 are ______________

The equation x²+ kxy + y²- 5x - 7y + 6 = 0 represents a pair of straight lines then k = ______________

By taking suitable sets A, B, C, verify the following results:
A \(\times\) (B\(\cap \)C) = (A\(\times\)B) \(\cap \) (A\(\times\)C)

By taking suitable sets A, B, C, verify the following results:
A \(\times\) (B\(\cup \)C) = (A\(\times\)B) \(\cup \) (A\(\times\)C)

Let A = {a, b, c}, and R = {(a, a) (b, b) (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it 
(i) reflexive
(ii) symmetric
(iii) transitive
(iv) equivalence

Prove that the relation "friendship" is not an equivalence relation on the set of all people in Chennai.

If the equation 12x² - 10xy + 2y² + 14x - 5y + k = 0 represents a pair of straight lines, find k, find separate equation and also angle between them.

Find the combined equation of the straight lines whose separate equations are x - 2y - 3 = 0 and x + y + 5 = 0.

A line passing through the points (a, 2a) and (-2, 3) is perpendicular to the line 4x+3y+ 5 = 0, find the value of a.

Find the angle between the pair of straight lines given by
(a² - 3b²)x² + 8ab xy+(b² -3a²)y² =0.

Transform the equation 3x + 4y + 12 = 0 in to normal form.

The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day function for exchanging Singapore dollar to Indian Rupee is g(y) = 50.50y, Where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee

The owner of a small restaurant can prepare a particular meal at a cost of Rupee 100. He estimate that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200 - x. Express his day revenue total cost and profit on this meal as a function of x.

The formula for converting from Fahrenheit to Celsius temperatures is \(y={5x\over 9}-{160\over 9}\). Find the inverse of this function and determine whether the inverse is also a function.

If n(A\(\cap\)B) = 3 and n(A\(\cup\)B) = 10 then find n(P(A \(\Delta \) B))

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is reflexive

Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, -1).

Find the equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of 120^o with the positive direction of x-axis.

Find the equation of the straight line passing through intersection of the straight lines 5x - 6y = 1 and 3x + 2y + 5 = 0 and perpendicular to the straight line 3x - 5y + 11=0.

Show that 9x² + 24xy +16y² +21x +28y +6 = 0 represents a pair of parallel straight lines and find the distance between them.

Show that 3x²+10xy+8y²+14x+22y+15=0 represents a pair of straight lines and the angle between them is tan^-1\(\left( \frac { 2 }{ 11 } \right) \).

If the sum of the distance of a moving point in a plane from the axis is 1, then find the locus of the point.

Find the equation of the straight line which passes through the intersection of the straight lines 2x + Y= 8 and 3x - 2y + 7 = 0 and is parallel to the straight line 4x+ y-11 =0.

Find the equation of the line which passes through the point (- 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5: 3 by this point.

Find the equation of the straight line which passes through the point (1, -2) and cuts off equal intercepts from axes.

The line 2x - y = 5 turns about the point on it, whose ordinate and abscissae are equal, through an angle of 45° in the anti-clockwise direction. find the equation of the line in the new position.

If \(f:R-\{ -1,1\}\rightarrow R\) is defined by \(f(x)={x \over x^2-1},\) verify whether f is one-to-one or not.

Let f and g be the two functions from R to R defined by f(x) = 3x - 4 and g(x) = x²+ 3. Find g o f and f o g.

Consider the positive branches y² = x and y² = -x.

A point moves so that square of its distance from the point (3, -2) is numerically equal to its distance from the line 5x -12y = 3. The equation of its locus is .................

Locus of the mid points of the portion of the line \(x\sin\theta+y\cos\theta=p\) intercepted between the axis is ............

Show that the locus of the mid-point of the segment intercepted between the axes of the variable line x cos \(\alpha\) + y sin \(\alpha\) = p is \(\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}\) where p is a constant.

The line \(\frac{x}{a}+\frac{x}{b}=1\) moves in such a way that \(\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2},\) where c is a constant. Find the locus of the foot of the perpendicular from the origin on the given line.

If the equation 12x²-10xy+2y²+14x-5y+c=0 represents a pair of straight lines, find the value of c. Find the separate equations of the straight lines and also the angle between them.

Find the points on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10.

Find the equation of the line passing through the point of intersection 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x +4y = 7.

If the intercept of a line between the coordinate axes is divided by the point (-5, 4) in the ratio 1: 2, then find the equation of the line.

If the equation 12x²-10xy+2y²+14x-5y+c=0 represents a pair of straight lines, find the value of c. Find the separate equations of the straight lines and also the angle between them.

For what value of k does 12x²+7xy+ky²+13x-y+3=0 represents a pair of straight lines? Also write the separate equations

A salesperson whose annual earnings can be represented by the function A(x) = 30,000 + 0.04x, where x is the rupee value of the merchandise he sells. His son also in sales and his earnings are represented by the function S(x) = 25,000 + 0.05x. Find (A+S) (x) and determine the total family income if they each sell Rs. 1,50,00,000 worth of merchandise.

A simple cipher takes a number and codes it, using the function f(x) = 3x - 4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x(by drawing the lines)

From the curve y = sin x, graph the functions.
(i) y = sin(-x)
(ii) y = -sin(-x)
(iii) ​\(y=sin\left( {\pi\over 2}+x\right)\) which is cos x​
(iv) ​\(y=sin\left({\pi\over 2}-x \right)\)​ which is also cos x (refer trigonometry)

From the curve y = x, draw
(i) y = - x
(ii) y = 2x
(iii) y = x + 1
(iv) ​\(y={1\over 2}x+1\)​
(v) 2x + y + 3 = 0

Write the values of f at -3, 5, 2, -1, 0 if
\(f(x)=\begin{cases} x^2+x-5\quad if\ x \in(-\infty, 0) \\x^2+3x-2\quad if\ x\in(3,\infty) \\x^2\quad \quad \quad \quad \quad if\ x\ \in(0,2) \\x^2-3 \quad \quad \quad otherwise \end{cases}\)

If A(B + C) = AB + AC where A, B, C are matrices of the same order than the property applied is matrix multipication is _______ .

The product of any matrix by the scalar_________is the null matrix.

If A is a matrix 3 \(\times\) 3, then \({ { (A }^{ 2 }) }^{ -1 }\) =____________

If \(\left( \begin{matrix} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{matrix} \right) \) is a singular matrix, then \(\lambda \) is_____________

If \(\begin{bmatrix} 4 & 3 \\ -2 & x \end{bmatrix}\) is singular then the value of x is _____________

A matrix which is not a square matrix is called a_________matrix.

The value of \(\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right| \) =_____________ 0, where a, b, c are in AP is 

The value of\(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1+sin\theta & 1 \\ 1 & 1 & 1+cos\theta \end{matrix} \right| \) is _____________

If f(x) = \(\left| \begin{matrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{matrix} \right| \)  then _____________

Choose the correct statement

Solve\(\left[ \begin{matrix} { x }^{ 2 } \\ { y }^{ 2 } \end{matrix} \right] -3\left[ \begin{matrix} x \\ 2y \end{matrix} \right] =\left[ \begin{matrix} -2 \\ 9 \end{matrix} \right] \)

Find the value of x such that [1 \(\times\) 1]\(\left[ \begin{matrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 2 \\ x \end{matrix} \right] =0\)

Using properties of determinant, show that \(\triangle =\left| \begin{matrix} { cosec }^{ 2 }\theta & -{ cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & -cose{ c }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

Prove that \(\left| \begin{matrix} 1 & 1+p & 1+p+q \\ 2 & 3+2p & 4+4p+2q \\ 3 & 6+3p & 10+6p+3q \end{matrix} \right| =1\)

Prove that \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{matrix} \right| =xy\)

For the given curve y = x³ given in figure draw, try to draw with the same scale
(i) y = -x³ 
(ii) y = x³+1
(iii) y = x³-1
(iv) y = (x + 1)³

For the given curve, \(y=x^{1\over 3}\)given in  figure draw
(i) \(y=-x^{ \left( \frac { 1 }{ 3 } \right) }\)
(ii) ​\(y=x^{ \left( \frac { 1 }{ 3 } \right) }+1\)
(iii)​ \(y=x^{ \left( \frac { 1 }{ 3 } \right) }-1\)
(iii) ​\(y=(x+1)^{1\over 3}\)​

From the curve y = x, draw
(i) y = - x
(ii) y = 2x
(iii) y = x + 1
(iv) ​\(y={1\over 2}x+1\)​
(v) 2x + y + 3 = 0

Write the values of f at -3, 5, 2, -1, 0 if
\(f(x)=\begin{cases} x^2+x-5\quad if\ x \in(-\infty, 0) \\x^2+3x-2\quad if\ x\in(3,\infty) \\x^2\quad \quad \quad \quad \quad if\ x\ \in(0,2) \\x^2-3 \quad \quad \quad otherwise \end{cases}\)

Find the range of the function \(\frac { 1 }{ 2cosx-1 } \)

In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section of Express the given information as a column matrix. Using sclar multiplication find the total number of each kind in all the colleges.

Show that all positive integral powers of a symmetric are symmetric.

Without expanding evaluate the determinant \(\left| \begin{matrix} 41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{matrix} \right| \)

If A is a skew-symmetric matrix of odd order n, then |A| = 0.

Prove that \(\left[ \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right] =\left( a-b \right) \left( b-c \right) \left( c-a \right) \)

Find non-Zero values of x satisfying the matrix equation, \(x\left[ \begin{matrix} 2x & 2 \\ 3 & x \end{matrix} \right] +2\left[ \begin{matrix} 8 & 5x \\ 4 & 4x \end{matrix} \right] =\left[ \begin{matrix} { x }^{ 2 }+8 & 24 \\ 10 & 6x \end{matrix} \right] \)

Under what condition is the matrix equation A² - B² = (A - B)(A + B) is true?

Prove that the determinant\(\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right| \) is independent of \(\theta\) ?

Prove that \(\left| \begin{matrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{matrix} \right| =\left( a+b+c \right) ^{ 3 }\) 

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

Find non-Zero values of x satisfying the matrix equation, \(x\left[ \begin{matrix} 2x & 2 \\ 3 & x \end{matrix} \right] +2\left[ \begin{matrix} 8 & 5x \\ 4 & 4x \end{matrix} \right] =\left[ \begin{matrix} { x }^{ 2 }+8 & 24 \\ 10 & 6x \end{matrix} \right] \)

If AB = A and BA = B, then show that A² = A and B² = B.

Prove that \(\left| \begin{matrix} 1 & a & { a }^{ 3 } \\ 1 & b & { b }^{ 3 } \\ 1 & c & { c }^{ 3 } \end{matrix} \right| =\left( a-b \right) \left( b-c \right) \left( c-a \right) \left( a+b+c \right) \) 

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

Show that \(\left| \begin{matrix} 1 & a & { a } \\ a & 1 & a \\ a & a & 1 \end{matrix} \right| ^{ 2 }=\left| \begin{matrix} 1-2{ a }^{ 2 } & -{ a }^{ 2 } & -{ a }^{ 2 } \\ -{ a }^{ 2 } & -1 & { a }^{ 2 }-2a \\ -{ a }^{ 2 } & { a }^{ 2 }-2a & -1 \end{matrix} \right| \)

If A is a square matrix such that A² = I, then find the simplified value of (A-I)³+(A+I)³-7A.

Without expanding evaluate the determinant\(\left| \begin{matrix} sin\alpha & cos\alpha & sin(\alpha +\delta ) \\ sin\beta & cos\beta & sin(\beta +\delta ) \\ sin\gamma & cos\gamma & sin(\gamma +\delta ) \end{matrix} \right| \)

Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k,0) is a point such that area of \(\triangle\)ABC is 3 sq. units.

If \(A=\left[ \begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix} \right] \) find A² - 7A - 21.

If \(A=\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \), Show that k so that A² -4A- 51 = 0

Prove that a matrix which is both symmetric as well as skew-symmetric is a null matrix.

If \(A=\left[ \begin{matrix} 3 & -2 \\ 4 & -2 \end{matrix} \right] \), find k so that A² = kA - 2I.

 If \(A=\left[ \begin{matrix} 1 & 2 \\ 2 & 0 \end{matrix} \right] ,B=\left[ \begin{matrix} 3 & -1 \\ 1 & 0 \end{matrix} \right] \) verify the following:

Prove that \(\left| \begin{matrix} { a }^{ 2 }+\lambda & ab & ac \\ ab & { b }^{ 2 }+\lambda & bc \\ ac & bc & { c }^{ 2 }+\lambda \end{matrix} \right| ={ \lambda }^{ 2 }\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+\lambda \right) \)  

Factorise \(\left| \begin{matrix} a & b & c \\ { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \\ bc & ca & ab \end{matrix} \right| \) .

If m \(\left( \overset { \rightarrow }{ 2 } +\overset { \rightarrow }{ j } +\overset { \rightarrow }{ k } \right) \) is a unit vector then the value of m is ___________ .

Assertion (A): If ABCD is a parallelogram, \(\overset { \rightarrow }{ AB } +\overset { \rightarrow }{ AD } +\overset { \rightarrow }{ CB } +\overset { \rightarrow }{ CD } \) then is equal zero.
Reason (R): \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ CD } \) are equal in magnitude and opposite in direction. Also\( \overset { \rightarrow }{ AD } \) and \( \overset { \rightarrow }{ CB } \) are equal in magnitude and opposite in direction

Find the odd one out of the following

Assertion (A) : \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are the position vector three collinear points then 2 \(\overset { \rightarrow }{ a }=\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)
Reason (R): Collinear points, have same direction

Find the odd one out of the following

Find λ so that the vectors λ so that the vectors \(2\hat { i } +\lambda \hat { j } +\hat { k } \) and \(\hat { i } -2\hat { j } +\hat { k } \) are perpendicular to each other.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three mutually perpendicular unit vectors, then prove that \(|\vec { a } +\vec { b } +\vec { c } |=\sqrt { 3 } \) 

If \(\vec { a } ,\vec { b } \) are any two vectors, then prove that \(\left| \vec { a } \times \vec { b } \right| ^{ 2 }+\left( \vec { a } .\vec { b } \right) ^{ 2 }=\left| \vec { a } \right| ^{ 2 }\left| \vec { b } \right| ^{ 2 }\) 

Find the vectors of magnitude 6 which are perpendicular to both the vectors \(4\vec { i } -\vec { j } +3\vec { k } \) and \(-2\vec { i } +\vec { j } -2\vec { k } \)

Find the angle between two vectors \(\vec { a } \) and \(\vec { b } \) if \(\left| \vec { a } \times \vec { b } \right| =\vec { a } .\vec { b } \) 

If |x+2| \(\le\) 9, then x belongs to

Given that x, y and b are real numbers x < y, b > 0, then

If \(\frac { |x-2| }{ x-2 } \ge 0\), then x belongs to

The solution 5x-1<24 and 5x+1 > -24 is

The solution set of the following inequality |x-1| \(\ge\) |x-3| is

Show that the vectors \(3\hat { i } -2\hat { j } +\hat { k } \) ,\(\hat { i } -3\hat { j } +5\hat { k } \) and \(2\hat { i } +\hat { j } -4\hat { k } \) form a right angled triangle.

If \(\vec { a } ,\vec { b } \) are any two vectors, then prove that \(\left| \vec { a } \times \vec { b } \right| ^{ 2 }+\left( \vec { a } .\vec { b } \right) ^{ 2 }=\left| \vec { a } \right| ^{ 2 }\left| \vec { b } \right| ^{ 2 }\) 

Find the angle between the vectors \(2\vec { i } +\vec { j } -\vec { k } \) and \(\vec { i } +2\vec { j } +\vec { k } \) by using cross product.

If \(\vec { a } \times \vec { b } =\vec { c } \times \vec { d } \) and \(\vec { a } \times \vec { c } =\vec { b } \times \vec { d } \) show that \(\vec { a } -\vec { d } \) and \(\vec { b } -\vec { d } \)are parallel.

If \(\left| \vec { a } \right| =2,\) ,\(\left| \vec { b } \right| =7\) and \(\vec { a } \times \vec { b } =3\hat { i } -2\hat { j } +6\hat { k } \) find the angle between \(\vec { a } \) and \(\vec { b } \)

Find a so that the sum and product of the roots of the equation 2x²+ (a - 3) x + 3a - 5 = 0 are equal is

If a and b are the roots of the equation x²- kx + 16 = 0 and a²+ b² = 32, then the value of k is

The number of solution of x² + |x - 1| = 1 is

The equation whose roots are numerically equal but opposite in sign to the roots 3x²- 5x -7 = 0 is

If 8 and 2 are the roots of x²+ ax + c = 0 and 3, 3 are the roots of x² + dx + b = 0; then the roots of the equation x²+ ax + b = 0 are

Prove using vectors the mid-points of two opposite sides of a quadrilateral and the mid-points of the diagonals are the vertices of a parallelogram.

Find the unit vectors parallel to the sum of \(3\vec { i } -5\vec { j } +8\vec { k } \) and \(-2\vec { i } -2\vec { k } \) 

Prove that the points \(2\hat { i } +3\hat { j } +4\hat { k } ,3\hat { i } +4\hat { j } +2\hat { k } ,4\hat { i } +2\hat { j } +3\hat { k } \) form an equilateral triangle.

If \(\left| \vec { a } +\vec { b } \right| =60\),\(\left| \vec { a } -\vec { b } \right| =40;\) and \(\left| \vec { b } \right| =46\) find \(\left| \vec { a } \right| \)

Show that the vector \(\hat { i } +\hat { j } +\hat { k } \) is equally inclined with the coordinate axes.

Simplify \(\left( 125 \right) ^{ \frac { 2 }{ 3 } }\)

Evaluate \(\left( \left[ (256)^{ \frac { -1 }{ 2 } } \right] ^{ \frac { -1 }{ 4 } } \right) ^{ 3 }\)

Simplify and hence find the value of n: \(3^{2 n} 9^{2} 3^{-n} / 3^{3 n}=27\)

Find the radius of the spherical tank whose volume is  \(\frac { 32\pi }{ 3 } \) units

Solve for x  \(\left| 3-x \right| <7\)

Show that the points whose position vectors given by 
(i) \(-2\hat { i } +3\hat { j } +5,\hat { i } +2\hat { j } +3\hat { k } ,7\hat { i } -\hat { k } \) 
(ii) \(\hat { i } -2\hat { j } +3\hat { k } ,2\hat { i } +3\hat { j } -4\hat { k } \) and\(-7\vec { j } +10\vec { k } \)  are collinear.

The vertices of a triangle have position vectors \(4\hat { i } +5\hat { j } +6\hat { k } ,5\hat { i } +6\hat { j } +4\hat { k } ,6\hat { i } +4\hat { j } +5\hat { k } \) Prove that the triangle is equilateral.

Examine whether the vectors \(\hat { i } +3\hat { j } +\hat { k } ,2\hat { i } -\hat { j } -\hat { k } \) and  \(7\hat { j } +5\hat { k } \) are coplanar 

Show that the points whose positions vectors \(4\hat { i } -3\hat { j } +\hat { k } \) ,\(2\hat { i } -4\hat { j } +5\hat { k } \) ,\(\hat { i } -\hat { j } \) from a right angled triangle.

Find the vectors whose length 5 and which are perpendicular to the vectors \(\vec { a } =3\vec { i } +\vec { j } -4\vec { k } \) and \(\vec { b } =6\vec { i } +5\vec { j } -2\vec { k } \)

Discuss the nature of roots of -x² + 3x + 1 = 0

Without sketching the graphs, find whether the graphs of the following functions will intersect the x-axis and if so in how many points. y = x² + 6x + 9

Solve |2x- 17| = 3 for x.

Solve 3|x - 2| + 7 = 19 for x.

Solve 3x - 5 ≤ x + 1 for x.

\(\lim _{ x\rightarrow 1 }{ \frac { { x }^{ m }-1 }{ { x }^{ n }-1 } } is\)

The points of discontinuity of the function \(\frac { { x }^{ 2 }+6x+8\quad }{ { x }^{ 2 }-5x+6\quad } is\)

Find the odd one of the following

Find the odd one out of the following

Find the odd one of out of the following

If \(\left( { x }^{ \frac { 1 }{ 2 } }+{ x }^{ -\frac { 1 }{ 2 } } \right) ^{ 2 }=\frac { 9 }{ 2 } \), then find the value of \(\left( { x }^{ \frac { 1 }{ 2 } }-{ x }^{ -\frac { 1 }{ 2 } } \right) \)for x > 1

Classify each element of \(\left\{ \sqrt { 7 } ,\frac { -1 }{ 4 } ,0,3.14,4,\frac { 22 }{ 7 } \right\} \) as a member of N, Q, R, -Q or Z.

Prove that \(\sqrt { 3 } \) is an irrational number. (Hint: Follow the method that we have used to prove \(\sqrt { 2 } \notin Q\))

Are there two distinct irrational numbers such that their difference is a rational number? Justify.

Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number

\(\lim _{ x\rightarrow \infty }{ \left( \frac { 1 }{ x } +2 \right) } \)is equal to

If y= 6x -x³ and x increases at the ratio of 5 units per second, the rate of change of slope when x = 3 is ______ units/sec.

The slope of the graph of \(f\left( x \right) =\frac { \left| x \right| }{ x } ,x>0\quad is\)

Choose the incorrect pair

Choose the incorrect statement

If \(\left( { x }^{ \frac { 1 }{ 2 } }+{ x }^{ -\frac { 1 }{ 2 } } \right) ^{ 2 }=\frac { 9 }{ 2 } \), then find the value of \(\left( { x }^{ \frac { 1 }{ 2 } }-{ x }^{ -\frac { 1 }{ 2 } } \right) \)for x > 1

Prove that \(\sqrt { 3 } \) is an irrational number. (Hint: Follow the method that we have used to prove \(\sqrt { 2 } \notin Q\))

Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number

Solve for x  \(\left| 3-\frac { 3 }{ 4 } x \right| \le \frac { 1 }{ 4 } \)

Solve log₈ x + log₄ x + log₂ x = 11

Evaluate \(\lim _{ x\rightarrow a }{ \frac { \sqrt { x } +\sqrt { a } }{ x+a } } \)

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ \frac { 2 }{ 3 } }-9 }{ x-27 } } \)

\(If\lim _{ x\rightarrow 2 }{ \frac { { x }^{ n }-{ 2 }^{ n } }{ x-2 } } =80\quad and\quad n\in N,\quad find\quad n.\)

Show that the function is \(f\left( x \right) =\begin{cases} \frac { \sin { x } }{ x } +\cos { x,\quad x\neq 0 } \\ 2,\quad \quad \quad x=0 \end{cases}\) continuous at x =0.

Evaluate \(\underset { n\rightarrow \infty }{ lim } \cfrac { 1+2+3+...+n }{ { n }^{ 2 } } \)

If x=\(\sqrt { 2 } +\sqrt { 3 } \)  find \(\frac { { x }^{ 2 }+1 }{ { x }^{ 2 }-2 } \)

Prove that \(log_{10}2+16log_{10}\frac { 16 }{ 15 } +12log_{10}\frac { 25 }{ 24 } +7log_{10}\frac { 81 }{ 80 } =1\)

Find the condition that one of the roots of ax²+ bx + c may be negative of the other.

A manufacturer has 600 litres of a 12 percent solution of acid. How many litres of a 30 percent acid solution must be added to it so that the acid content in the resulting mixture will be more than 15 percent but less than 18 percent?

Find all values of x for which \({{x^3(x-1)}\over{x-2}}>0.\)

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1+x } +\sqrt { 1-x } }{ 1+x } } \)

Evaluate \(\lim _{ x\rightarrow 2 }{ \frac { { x }^{ 2 }-3x+2 }{ { x }^{ 2 }-x-2 } } \)

For what value of k is the function \(f\left( x \right) =\begin{cases} \frac { \sin { 5x } }{ 3x } \quad if\quad x\neq 0 \\ k,\quad \quad \quad if\quad x=0 \end{cases}\) is continuous at x = 0.

Find \(\underset { x\rightarrow 1 }{ lim } \) fix), if  \(f(x)=\{ \begin{matrix} { x }^{ 2 }-1 & x\le 1 \\ -{ x }^{ 2 }-1 & x>1 \end{matrix}\)

Evaluate: \(\underset { x\rightarrow \infty }{ lim } \cfrac { \left( x+1 \right) ^{ 10 }+\left( x+2 \right) ^{ 10 }+...+\left( x+100 \right) ^{ 10 } }{ { x }^{ 10 }+{ x }^{ 10 } } \)

Evaluate \(\lim _{ x\rightarrow 1 }{ \frac { \sqrt { { x }^{ 2 }-1 } +\sqrt { x-1 } }{ \sqrt { { x }^{ 2 }-1 } } } if\quad x>1\)

Evaluate \(\lim _{ x\rightarrow 1 }{ \frac { (2x-3)\sqrt { x } -1 }{ { 2x }^{ 2 }+x-3 } } \)

Examine the continuity of \(f\left( x \right) =\begin{cases} \frac { \sin { 2x } }{ \sin { 3x } } \quad if\quad x\neq 0 \\ 2\quad \quad \quad if\quad x=0 \end{cases}at\quad x=0\)

If \(f\left( x \right) =\frac { 2x+3\sin { x } }{ 3x+2\sin { x } } ,\quad x\neq 0\) is continuous at x = 0, then find f(0).

Evaluate \(\underset { x\rightarrow 0 }{ lim } f(x)\) ,where \(f(x)=\{ \begin{matrix} \frac { \left| x \right| }{ 0 } & x\neq 0 \\ 0 & x=0 \end{matrix}\)

Resolve the following rational expressions into partial fractions.
\({{x^2+x+1}\over{x^2-5x+6}}\)

Determine the region in the plane determined by the inequalities.
\(2x+3y\le 6,\ x+4y\le 4,\ x\ge 0,\ y\ge 0.\)

Determine the region in the plane determined by the inequalities.
\(x-2y\ge 0,\ 2x-y\le -2,\ x\ge 0,\ y\ge 0.\)

Resolve the following rational expressions into partial fractions.
\({{x+12}\over{(x+1)^{2}(x-2)}}\)

Resolve the following rational expressions into partial fractions.
\({{7+x}\over{(1+x)(1+x^2)}}\)

Evaluate \(\lim _{ x\rightarrow 1 }{ \frac { \sqrt { { x }^{ 2 }-1 } +\sqrt { x-1 } }{ \sqrt { { x }^{ 2 }-1 } } } if\quad x>1\)

\(It\quad \lim _{ x\rightarrow a }{ \frac { { x }^{ 9 }-{ a }^{ 9 } }{ x-a } } =9,\)find all possible values of a.

Evaluate \(\lim _{ x\rightarrow \pi }{ \frac { \sin { x } }{ x-\pi } } \)

Suppose \(f(x)=\{ \begin{matrix} a+bx, & x<1 \\ 4, & x=1 \\ b-ax & x>1 \end{matrix}\) and,if \(\underset { x\rightarrow 1 }{ lim } f(x)=f(1)\) .What are possible values of a and b?

Evaluate \(\underset { x\rightarrow \infty }{ lim } \left( \sqrt { { x }^{ 2 }-x+1 } +x \right) \)

Evaluate \(\lim _{ x\rightarrow \sqrt { 2 } }{ \frac { { x }^{ 9 }-3{ x }^{ 8 }+{ x }^{ 6 }-9{ x }^{ 4 }-4{ x }^{ 2 }-16x+84 }{ { x }^{ 5 }-3{ x }^{ 4 }-4x+12 } } \)

Find k if \(\lim _{ x\rightarrow 1 }{ \frac { { x }^{ 4 }-1 }{ x-1 } } =\lim _{ x\rightarrow k }{ \left( \frac { { x }^{ 3 }-{ k }^{ 3 } }{ { x }^{ 2 }-{ k }^{ 2 } } \right) } \)

Evaluate \(\lim _{ x\rightarrow \frac { \pi }{ 2 } }{ \left( \frac { \pi }{ 2 } -x \right) \tan { x } } \)

Examine the continuity of \(f\left( x \right) \quad at\quad x=\frac { 1 }{ 2 } where\quad f\left( x \right) =\begin{cases} \frac { 1 }{ 2 } -x,\quad 0\le x\le \frac { 1 }{ 2 } \quad \\ 1\quad ,\quad \quad x=\frac { 1 }{ 2 } \\ \frac { 3 }{ 2 } -x,\quad \frac { 1 }{ 2 }

If \(f\left( x \right) =\begin{cases} 1,\quad \quad x\le 3 \\ ax+b,\quad 3 is continuous, prove that a = 3 and b = - 8.

Evaluate \(\lim _{ x\rightarrow 0 }{ \left[ \frac { 1 }{ x } -\log { \frac { (1+x) }{ { x }^{ 2 } } } \right] } \)

Evaluate \(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1-{ x }^{ 2 } } -\sqrt { 1+{ x }^{ 2 } } }{ 2{ x }^{ 2 } } } \)

Evaluate \(\lim _{ x\rightarrow 0 }{ \frac { { 4 }^{ x }-1 }{ \sqrt { 1+x } -1 } } \)

Evaluate \(\lim _{ x\rightarrow \frac { 1 }{ \sqrt { 2 } } }{ \frac { x-\cos { (\sin ^{ -1 }{ (x) } ) } }{ 1-\tan { (\sin ^{ -1 }{ x } ) } } } \)

Determine k, so that  \(f\left( x \right) =\begin{cases} k{ x }^{ 2 },\quad x\le 2 \\ 3,\quad x>2 \end{cases}\) is continuous.

\(\frac { 1 }{ cos{ 80 }^{ 0 } } -\frac { \sqrt { 3 } }{ sin{ 80 }^{ 0 } } \)=

The maximum value of 4sin²x + 3 cos²x + \(sin\frac { x }{ 2 } +cos\frac { x }{ 2 } \) is

Let f_k(x) = \(\frac { 1 }{ k } \)[sin^kx + cos^kx] where x\(\in \)R and k ≥ 1. Then f₄(x) - f₆(x) =

Which of the following is not true?

cos 2ፀ cos 2ф + sin²(ፀ - ф) - sin²(ፀ + ф) is equal to

 Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sin ^{ -1 }{ x } +\cos ^{ -1 }{ x } \) then \(\frac { dy }{ dx } \) is _____

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sqrt { \sin { x+y } } \quad then\quad \frac { dy }{ dx }\) is _________

Choose the correct or the most suitable answer from the given four alternatives.
If f(x) is an even functions, thenf^'(x) is an ______function

Choose the correct or the most suitable answer from the given four alternatives.
If \(x=a(\theta+\sin \theta), y=a(1+\cos \theta)\) then \(\frac{dy}{dx}\) is ______

Assertion (A) : f (x) =\(\begin{cases} \begin{matrix} x+1, & x<2 \end{matrix} \\ \begin{matrix} 2x-1, & x\ge 2 \end{matrix} \end{cases}\) then f'(2) does not exist.
Reason (R) : f(x) is not continuous at 2.

If sin \(\theta\) + cos \(\theta\) = 1 then sin⁶ \(\theta\) + cos⁶ \(\theta\) is _______________

If the arcs of same lengths in two circles sustend central angles 30° and 40° find the ratio of their radii _______________

If sin(45 ° + 10°) - sin(45° -10°) = \(\sqrt{2}\)sin x then x is ___________ 

The quadratic equation whose roots are tan 75° and cot 75° is _______________

sin² \((22{1\over 2}^o)\) is ____________ 

 Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\log _{ a }{ x } \) then \(\frac { dy }{ dx } \) is ______

Choose the correct or the most suitable answer from the given four alternatives.
The derivative of \(\cos ^{ -1 }{ (2{ x }^{ 2 } } -1)\) with respect to \(\cos ^{ -1 }{ x } \) is _____

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\log \left(\frac{1-x^2}{1+x^2}\right)\) then \(\frac{dy}{dx}\) is ______

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) then \(\frac{dy}{dx}\) is ______

Choose the correct or the most suitable answer from the given four alternatives.
If, \(y=a+b{ x }^{ 2 }\) where a, b are arbitrary constants, then ____

Identify the quadrant in which an angle of each given measure lies; 25⁰

Find the value of sin 105^o

Prove that \(\cos { \left( \pi +\theta \right) } =-\cos { \theta } \)

Find the principal value of \(sin^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \).

Find the principal value of cosec^-1(-1)

Differentiate \(\sin { (\sqrt { 3 } \sin { x } +\cos { x } ) } \) with respect to x.

Find \(\frac { dy }{ dx } if\quad { x }^{ 4 }+{ x }^{ 2 }{ y }^{ 2 }+{ y }^{ 4 }=50.\)

Find the derivation : 3 sin x + 4 cos x - e^x

Find the derivation (x⁴ - 6x³ + 7x² + 4x + 2) (x³ -1)

Find the derivation  : x² e^x sin x

If \(\triangle ABC\) is a right triangle and if \(\angle A=\frac{\pi}{2}\), then prove that \(\cos^2B+\cos^2C=1\)

Find the values of other five trigonometric functions for the following
sin \(\theta\) = -\(\frac { 2 }{ 3 },\) \(\theta\) = lies in the IV quadrant

Prove that \(\sin { 4\alpha } =4\tan { \alpha } \frac { 1-\tan ^{ 2 }{ \alpha } }{ { \left( 1+\tan ^{ 2 }{ \alpha } \right) }^{ 2 } } \)

Express each of the following as a sum or difference. sin 4x cos 2x

Express each of the following as a product.
cos 65^o + cos 15^o

Differentiate \(\frac { \sin { (ax+b) } }{ \cos { (cx+d) } } .\)

Differentiate \(\mathrm{y}=\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right), \quad \frac{-1}{\sqrt{2}}

Differentiate x² (x + 1)³ (x + 2)⁴ with respect to 'x'.

Find the derivation : sin 5 + log₁₀ x + 2 sec x

Find the derivation \(y=\cfrac { cosx+logx }{ { x }^{ 2 }+{ e }^{ x } } \)

Show that \(\frac { (cos\theta -cos3\theta )(sin8\theta +sin2\theta ) }{ (sin5\theta -sin\theta )(cos4\theta -cos6\theta ) } =1\)

If \(\sin { x } =\frac { 15 }{ 17 } \) and \(\cos {y } =\frac { 12 }{ 13 } \), 0 < x < \(\frac{\pi}{2}\), 0 < y < \(\frac{\pi}{2}\), find the value of sin (x + y)

Find cos(x - y), given that cos x = \(-\frac{4}{5}\) with \(\pi<x<{{3\pi}\over{2}}\) and \(sin \ y = -{{24}\over{25}}\) with \(\pi<x<{{3\pi}\over{2}}\). 

For each given Angle, find a coterminal angle with a measure of \(\theta\) such that \(0^o\le \theta \le 360°\) 
395⁰ 

Prove that \(sinx+sin2x+sin3x=sin2x(1+2cosx)\)

Show  that\(f\left( x \right) ={ x }^{ 2 }\) is differentiable at x = 1 and find \(f^{ ' }\left( 1 \right) \)

If xy = 4, Prove that \(x\left( \frac { dy }{ dx } +{ y }^{ 2 } \right) =3y.\)

If \({ x }^{ 2 }+2xy+{ y }^{ 3 }=42,\) find \(\frac { dy }{ dx } \)

Differentiate \(\log { (1+{ x }^{ 2 } } )\) with respect to \(\tan ^{ -1 }{ x } \)

Find the derivation : 6 sin x log₁₀ x + e

In a circular of diameter 40 cm, a chord is of length 20 cm. FInd the length of the minor arc of the chord?

A train is moving on a circular track of 1500 m radius at the rate of 66 Km/hr. What angle will it turn in 20 seconds?

Prove that cos (A + B) cos C - cos (B + c) cos A = sin B sin (C - A)

Find the degree measure of the angle subtended at the center of circle of radius 100 cm by an arc of length 22 cm.

Prove that sin² (A + B) - sin² (A - B) = sin 2A sin 2B

Differentiate \(f\left( x \right) ={ e }^{ 2x }\)from first principles.

If \(y=\sqrt { x+1 } +\sqrt { x-1 } \) prove that\(\sqrt { { x }^{ 2 }+1 } \frac { dy }{ dx } =\frac { 1 }{ 2 } y.\)

Differentiate \((\sec ^{-1}\left(\frac{1}{2 x^{2}-1}\right), \quad 0)\)

If x = \(a\sec ^{ 3 }{ \theta }\) and \(y=a\tan ^{ 3 }{ \theta }\) find \(\frac { dy }{ dx }\) at \(\theta =\frac { \pi }{ 3 }\)

If f(x) = 2x² + 3x - 5, then prove that f' (0) + 3 f' (-1) = 0

If \(f\left( 2 \right) =4\ and f^{ ' }\left( 2 \right) =1, \) then find \(\lim _{ x\rightarrow 2 }{ \frac { xf\left( 2 \right) -(2)f\left( x \right) }{ x } }\)

If \(y=\sqrt { \frac { 1+{ e }^{ x } }{ 1-{ e }^{ x } } } \) , show that \(\frac { dy }{ dx } =\frac { { e }^{ x } }{ (1-{ e }^{ x })\sqrt { 1-{ e }^{ 2x } } } \)

If \(\log { ({ x }^{ 2 }+{ y }^{ 2 }) } =2\tan ^{ -1 }{ \frac { y }{ x } , } \) Show that \(\frac { dy }{ dx } =\frac { x+y }{ x-y } .\)

Differentiate x^x with respect to x log x

\(If\quad x=4{ z }^{ 2 }+5,y=6{ z }^{ 2 }+7z+3,\quad find\quad \frac { d^{ 2 }y }{ dx^{ 2 } } \)

For A = {0,1,2,3, 4}, B = {1, -2, 3, 4, 5, 6} and C = {2, 4, 6, 7} verify A\(B ∩ C) = (A\B) U(A\C) Using venn diagram.

The cartesian product A \(\times\) A has 9 elements among which are found (-1, 0) and (0, 1).Find the set A and the remaining elements of A \(\times\)A.

Show that the relation R on the set R of all real numbers defined as R = {(a, b): a < b²} is neither reflexive, nor symmetric nor transitive.

Consider the function \(f:[0,{\pi\over 2}]⟶R\) given by f(x) = sin x and \(g:[0,{\pi\over 2}]⟶R\)given by g(x) = cos x. Show that f and g are one-one but (f + g) is not one-one.

A relation R is defined on the set z of integers as follows:
(x, Y) ∈ R ⇔ x² + y² = 25. Express R and R^-1 as the set of ordered pairs and hence find their respective domains.

Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
Verify using Venn diagram.

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

If a \(\in\) {-1, 2, 3, 4, 5} and b \(\in\) {0,3, 6}. Write the set of all ordered pairs (a, b) such that a + b = 5.

Find the sum and difference of the identity function and the modulus function?

Find the range of the function.
f = {1, x), (1, y), (2, x), (2, y), (3, z)}

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩,  and as necessary.

Draw venn diagram of three sets A, B and C which illustrates the following:
A and B disjoint but both are subsets of C.

If R is the set of all real numbers, what do the cartesian products R \(\times\) Rand R \(\times\)R \(\times\)R represent?

Solve the inequation x \(\ge\) 2 graphically.

Solve the quadratic equation 5^2x- 5^{x + 3}+ 125 = 5^x.

Find the quotient of the identity function by the modulus function

Show that the function f : R ⟶ R given by f(x) = cos x for all x ∈ R is neither one-one nor onto.

Which of the following sets are finite and which are infinite?
Set of concentric circles in a plane.

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩, ' and \ as necessary.

Draw venn diagram of three sets A, B and C which illustrates the following:
A ∩ B ∩ C

Check whether the following sets are disjoint where p = {x : x is a prime < 15} and Q = {x : x is a multiple of 2 and x < 16}

If A⊂B then find A⋂B and A\B (using venn diagram)

Show that the relation R on the set A = {1, 2, 3} given by R = {(1, 1) (2, 2) (3, 3) (1, 2) (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the function f : N➝N given by f(x) = 2x is one-one but not onto.

Let S = {1, 2, 3,....,10}. Define 'm is related to n' if m divides n.

Write the following in roster form.
The set of all positive roots of the equation (x-1)(x+1)(x²-1) = 0.

Write the following in roster form.
\(\left\{ x:\frac { x-4 }{ x+2 } =3,x\in R-\{ -2\} \right\} \)

State whether the following sets are finite or infinite.
{x \(\in \) N:x is a rational number}

By taking suitable sets A, B, C, verify the following results:
A \(\times\) (B\(\cup \)C) = (A\(\times\)B) \(\cup \) (A\(\times\)C)

Discuss the following relations for reflexivity, symmetricity and transitivity :
Let A be the set consisting of all the female members of a family. The relation R defined by "aRb if a is not a sister of b".

Given A = {5,6,7,8}. Which one of the following is incorrect?

The shaded region in the adjoining diagram represents.

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

Let R be the relation over the set of all straight lines in a plane such that l₁Rl₂ ⇔ l₁丄l₂ . Then  R is __________

Which of the following functions from z to itself are bijections (one-one and onto)?

The shaded region in the adjoining diagram represents.

For real numbers x and y, define xRy if x - y + √2 is an irrational number. Then the relation R is __________

The number of reflective relations one set containing n elements is __________

Which one of the following statements is false? The graph of the function \(f(x)={1\over x}\)

Which one of the following is not a singleton set?

Two integers are selected at random from integers 1 to 11. If the sum is even, find the probability that both the numbers are odd.

A purse contains 3 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled out at random from one of the two purses, what is the probability that it is a silver coin?

for a loaded die, the probabilities of outcomes are given as under
P(1) = P(2) = \(\frac { 2 }{ 10 } \), P(3) = P(5) = P(6) = \(\frac { 1 }{ 10 } \) and P(4) = \(\frac { 3 }{ 10 } \)
The die is thrown 2 times. Let A and B be the events as defined below
A: Getting same number each time
B: Getting a total score of 10 or more
Discuss the independency of the events A and B

Out of 10 outstanding students in a school there are 6 girls and 4 boys. A team of 4 students is selected at random for a quiz programme. Find the probability that there are atleast two girls.

In a factory, Machine-I produces 45% of the output and Machine-II produces 55% of the output. On the average 10% items produced by I and 5% of the items produced by II are defective. An item is drawn at random from a day's output. (i) Find the probability that it is a defective item (ii) If it is defective, what is the probability that it was produced by Machine-II?

A couple has two children. Find the probability that
(i) both the children are boys, if it is known that the older child is a boy.
(ii) both the children are girls, if it is known that the older child is a girl.

Evaluate P(AUB) if 2P(A) = P(B) = \(\frac { 5 }{ 13 } \)and P(A/B) = \(\frac { 2 }{ 5 } \).

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let \(\frac { 3 }{ 4 } \) be the probability that he knows the answer and \(\frac { 1 }{ 4 } \) be the probability that he guesses. Assuming that a student who guesse at the answer will be correct with probability \(\frac { 1 }{ 4 } \). What is the probability that the student knows the answer given that he answered it correctly?

A and B are two candidates seeking admission in a college. The probability that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the probability that B is selected.

p(A) = 0.3, P(B) = 0.6 and \(P(A\cap B)=0.25\) .Find
(i) \(P(A\cup B)\) 
(ii) P(A/B)
(iii) \(P(B/\bar { A } )\) 
(iv) \(P(\bar { A } /B)\) 
(v) \(P(\bar { A } /\bar { B } )\)

One card is drawn from a well shuffled pack of 52 cards. If E is the event, "the card drawn is a king or queen" and F is the event "the card drawn is a queen or an ace", then find P(E/F).

Three events A, B and C have probalilities \(\frac { 2 }{ 5 } ,\frac { 1 }{ 3 } \)and \(\frac { 1 }{ 2 } \)  respectively. Given that P(A\(\cap \)C) = \(\frac { 1 }{ 5 } \), \(P(B\cap C)=\frac { 1 }{ 4 } \)find P(C/B) and P(\(\bar { A } \cap \bar { C } \))?

A fair dice is rolled. Consider the following events A = {1, 3, 5}, B = {2, 3} and C ={2, 3, 4, 5} Find (i) P(A/B) and P(B/A) (ii) P(A\(\cap \)B/C)

A die is thrown 3 times. Events A and B are defined as follows.
A: getting 4 on third die
B: getting 6 on the first and 5 on the second throw. Find the probability of A given that B has already occurred.

Given p(A) = 0.5, P(B) = 0.6 and \(P(A\cap B)=0.24\) .Find
(i) \(P(A\cup B)\) 
(ii) \(P(\vec { A } \cap B)\) 
(iii) \(P\left( A\cap \bar { B } \right) \) 
(iv) \(P\left( \bar { A } \cup \bar { B } \right) \) 
(v) \(P(\bar { A } \cap \bar { B } )\)

Two unbiased die are thrown. Find the probability that the sum is 8 or greater if 3 appears on the first die.

A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective. If a person takes out 2 at random what is the probability that either both are apples or both are good?

The probability that a person will get an electric contract  \(\frac { 2 }{ 3 } \) and the probability that he will not get plumbing contract is \(\frac { 4 }{ 7 } \). If the probability of getting atleast one contract is \(\frac { 2 }{ 3 } \). What is the probability tht he will get both? 

A and B are two events such that P(A) \(\neq \) 0. Find P(B/A) if (i) A is a subset of B (ii) A\(\cap \)B = \(\phi \)

In a box containing 10 bulbs, 2 ae defective. What is the probability that among 5 bulbs chosen at random, none is defective?

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

A die is tossed thrice. find the probability of getting an odd number atleast once?

The ratio of the number of boys to the number of girls in a class is 1:2. It is known that the probability of a girl and a boy getting a first class are 0.25 and 0.28 respectively. Find the probability that a student chosen  at random will get first class?

An integers is chosen at random from the first fifty positive integers. What is probability that the integer chosen is a prime or multiple of 4.

Two cards are drawn one by one at random from a deck of 52 playing cards. What is the probability of getting two jacks if
(i) the first card is replaced before the second card is drawn
(ii) the first card is not replaced before the second card is draw?

The probability that student selected at random from a class will pass in Mathematics is \(\frac { 2 }{ 3 } \) and the probability that he passes in Mathematics and English is \(\frac { 1 }{ 3 } \). What is the probability that he will pass in English if it is known that he has passed in Mathematics?

Events A and B are such that P(A) = \(\frac { 1 }{ 2 } \) , P(B) = \(\frac { 7 }{ 12 } \) and P(not A or not B) = \(\frac { 1 }{ 4 } \). State whether A and B are independent? 

Given that the events A and B are such that P(A) = \(\frac { 1 }{ 2 } \), P(AUB) = \(\frac { 3 }{ 5 } \) and P(B) = p. find P if they are mutually exclusive events. 

An experiment has the four possible mutually exclusive outcomes A, B, C and D, Check whether the following assignments of probability are permissible.
p(A) = 0.32, P(B) = 0.28, P(C) = - 0.06, P(D) = 0.46

If A and B are two events such that \(P(A\cup B)=0.7 ,\) \(P(A\cap B)=0.2\) ,\(P(\bar { B } )=0.5,\) show that A and B are independent.