For any two sets A and B, (A - B) \(\cup\) (B - A)=_____.

If R is a relation from a finite set A having m elements to a finite set B having n elemen.ts, then the number of relations from A to B is ______.

If sin \(\theta\) + sin \(\phi\)= \(\alpha\) and cos\(\theta\) - cos\(\phi\)= b then tan\({\theta -\phi\over 2}\) is equal to ______.

Prove that

\( \frac {1} {9!}+\frac{1} {10!}+ \frac {1}{11!}=\frac {122} {11!}\)

The coefficient of x⁵y⁸ in the expansion of (x + y)¹³ is _____.

If first and last terms ofanAP. are 3 and 18 and the sum of its terms is 84, then number of terms will be ______.

If p be the length of the perpendicular from the origin to the line \({x\over a}+{y\over b}=1\) then ______.

The four distinct points (0,0), (2,0), (0,−2) and (k,−2) are concyclic if k is equal to ______.

In the three dimensional space the equation x² - 7x + 12 = 0 represents ______.

\(\overset{lim}{x\rightarrow \frac{\pi}{2}} \) (sec x-tan x) is equal to ______.

Standard deviation of a data is given by ______.

Two die are thrown simulatneously. The probability of getting a total of 5 is _______.

Taking the set of natural numbers as the universal set, write down the complement of the following set: {x : x is a prime number}

If A = {1, 2, 3}, B = {3, 4} and C = {1, 3, 5}, find A x (B \(\cup\) C)

A circular wire of radius 7.5 cm is cut and bent so as to lie along the circumfeence of a hoop whose radius is 120 cm. Find in degrees the angle which is subtended at the centre of the hoop.

Prove that \(\sum _{ t=1 }^{ n-1 }{ t(t+1) } =\frac { n(n-1)(n+1) }{ 3 } \) , for all natural numbers \(n\ge 2\)

Find the argument of the complex number\(\frac { 1+3i }{ 1-2i } \)

Anu needs a minimum of 450 marks in five tests in statistics course obtain B grade. In her first four tests, she scored 90,94,96 and 88 marks. What should her score be in the 5th test, so that she can make B grade

If ¹⁵C_r : ¹⁵C_r-1=11:5 Find r

Find the value of (1.02)⁶corrrect to five decimal places using binomial theorem.

Find the sum to infinity of the G.P.\(\frac { -5 }{ 4 } ,\frac { 5 }{ 16 } ,\frac { -5 }{ 64 } \),...

the length of the perpendicular from the origin to a line is 7 and the line makes an angle of 120^{\(^{o}\)}   with the positive direction of Y - axis. Find the equation of the line.

Find the equation of the parabola whose, vertex is at (0, 0) and focus is at (5, 0).

Find the distance between the following pairs of points: (5, 6, 7) and (2, 3, 4)

Find the derivative of the following functions.
\(​​​​\frac { { x }^{ 5 }-cosx }{ sinx } \)

Write negation
Mridul is cruel or he is strict.

The analysis of daily wages paid to workers in two firms A and B belonging to the same industry are:

(i) Which firm A or B pays more in total daily wages?
(ii) Which A or B shows greater variability in wages?

If A and B are mutually exclusive exhaustive events such that P(B) = 3P(A); Find P(A).

In a group of 65 people, 40 like nutrition Indian food, 10 like both Indian nutrition food and fast food.How many like only fast food?What is your opinion about this set of people?

Find the domain of \(f\left( x \right) =\frac { 1 }{ \sqrt { x+\left| x \right| } } \)

Find the general solution for each of the following equations:cosec x = - 2

If p(n): "49ⁿ +16ⁿ +k is divisible by 64 for n\(\epsilon \)N" is true, then find the least negative integral value of K.

Find the value of \(\sqrt { -25 } +3\sqrt { -4 } +2\sqrt { -9 } \)

Solve \(|x-2|\ge 6\).

If \(\frac { 1 }{ 6! } +\frac { 1 }{ 7! } =\frac { x }{ 8! } \), find x

Prove that there is no term involving y⁶ in the expansion of \(\left( 2y^{ 2 }-\frac { 3 }{ y } \right) ^{ 11 }\), where \(y\neq 0\).

If a,b and c be positive numbers, then prove that a² + b² + c² is greater than ab + bc + ca.

Find the distance of the point (2, 3) from the line 2x-3y+9=0 measure along a line x-y+1=0.

Find the equation of the circle which passes through the point of intersection of the lines 3x - 2y - 1 = 0 and 4x + y - 27 = 0 and whose centre is (2, -3).

Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by YZ-plane.

Evaluate the following limits \(\lim _{x \rightarrow 1} \frac{x-1}{\log _{e} x}\)

Consider the statement, S : 80 is a multiple of 5 and 4. Check its validity.

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results. Number of observation =25, mean=18.2s , standard deviation=3.25 s. Further, another set of 15 observation=3.25 s .Further, another set of 15 Observations,\({ x }_{ 1 },{ x }_{ 2 },.....{ x }_{ 15 }\) also in seconds is now available and we have\(\sum _{ i=1 }^{ 15 }{ { x }_{ i }=279 } \) and \(\sum _{ i=1 }^{ 15 }{ { x }_{ 1 }^{ 2 } } =5524\) Calculate the standard derivation based on all 40 observation

If E and F are events such that P(E)=\(\frac { 1 }{ 4 } \) , P(F)=\(\frac { 1 }{ 2 } \) and P(E and F)=\(\frac { 1 }{ 8 } \)find.
i) P(E and F) ii) P(not E and not F)

Is g = {(2, 3), (4, 5), (6, 7)} a function? If this is described by the formula \(g(x)=\alpha x+2\beta \) then what values would be assigned to a \(\alpha \quad and\quad \beta \) ?

Prove that \(cos\frac { \pi }{ 7 } cos\frac { 2\pi }{ 7 } cos\frac { 4\pi }{ 7 } =\frac { -1 }{ 8 } \)

Prove that 2n<(n+2)! for all natural numbers n.

Solve \(5{ x }^{ 2 }-4ix+9=0\)

Solve graphically \(4x+3y\ge 12\quad and\quad 4x-5y\ge -20\)

The straight lines l₁, l₂ and l₃ are parallel and lie in the same plane. A total numbers of m points are taken on l₁ : n points on l₂ : k points on l₃ . Find the maximum number of triangles formed with vertices at these points.

If there is a term independent of x, in the expansion of\({ \left( { x }^{ 2 }+\frac { 1 }{ x } \right) }^{ n }\), then show that the term is\(\frac { n! }{ \left( \frac { n }{ 3 } \right) !\left( \frac { 2n }{ 3 } \right) ! } \).

If in an AP, S_n=qn² and S_m=qm², where Sr denotes the sum of r terms of the AP, then find S_q 

Show that the points A(7,10),B(-2,5) and C(3,-4) are the vertices of an isosceles right angled triangle.

Find  the equation of the hyperbola whose one directrix is x + =9, the corresponding focus is (2,2) and eccentricity is 2. 

Three points A(1,2,3), B(0,4,1) and C(-1,-1,-3) are the vertices of \(\Delta ABC\). Find the point in which the bisector of \(\angle BAC\) meets BC.

Evaluate e^2x

Write the following statement in five different ways, conveying the same meaning.
p :If a triangle is equiangular, then it is an obtuse angled triangle.

The frequency distribution

Where, A is a positive integer, has a variance of 160. Determine the value of A. 

One card is drawn from a well-shuffled deck of 52 cards.Calculate the probability that the card will be
not an ace

Show that the following four conditions are equivalent:
(i) A ⊂ B
(ii) A - B = ф
(iii) A \(\cup\) B =B
(iv) A \(\cap\) B =A

Find the domain and range of the following real functions 
(i) \(f(x)=-|x|\)
(ii) \(f(x)=\sqrt{9-x^2}\)

Find the general solutions of the following equations:
sec x-1 = (\(\sqrt{2}\) - 1) tan x where x ± (2n + 1)\(\pi\over2\), n \(\in\) Z.

Prove by the principle of mathematical induction that for all n∊N \(\frac { 1 }{ 1.2 } +\frac { 1 }{ 2.3 } +\frac { 1 }{ 3.4 } +...+\frac { 1 }{ n(n+1) } =\frac { n }{ (n+1) } \)

Convert the following in the polar form \(\frac { 1+3i }{ 1-2i } \)

Solve the inequalities graphically x +y \(\le\) 6, x + y \(\ge\) 4

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

Find the middle terms in the expansions of \(\left( 3-\frac { { x }^{ 3 } }{ 6 } \right) ^{ 7 }\)

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2,3).

Find a point in XY plane which is equidistant from three points (2, 0, 3), (0, 3, 2) and (0, 0, 1).

Evaluate: \(\overset{Lt}{x\rightarrow 0} [tan(\frac{\pi}{4}+x)]^{\frac{1}{cotx}}\)

Give three examples of sentences which are not statements. Give reasons for the answers.

Find the C.V of the following data:

A die is thrown, find the probability of following events:
(i) A prime number will appear.
(ii) A number greater than or equal to 3 will appear.
(iii) A number less than or equal to one will appear.
(iv) A number more than 6 will appear.
(v) A number less than 6 will appear.