If \(A\cap B=B\) then _____.

If the set A has m elements, B has n elements then the number of elements in A x B is ______.

The solution of the equation cos² \(\theta\) -+sin\(\theta\)+ 1=0lies in the interval ______.

In how many ways can a student choose programme of 5 courses, if 9 ourses are available and 2 specific courses are compulsory for every student?

The middle term in the expansion of \(\left( \frac { { 2x }^{ 2 } }{ 3 } +\frac { 3 }{ { 2x }^{ 2 } } \right) ^{ 10 }\)is _____.

If the first term of an AP. is 5 and common difference is - 3 then sum of its 60 terms is equal to ______.

The area of a triangle whose vertices are (3, -2), (5, 6) and (-2, -5) is ______.

The eccentricity of the hyperbola whose latus rectum is half of its transverse axis is _______.

The ratio in which the line joining (4, -3, 2) and (6, -5, -1) is divided by YZ-plane is _______.

\(\overset{lim}{x\rightarrow 0} \frac{x}{tan x}\) is _______.

Six boys and six girls sit in a row. The probability that all girls sit together is _______.

Write all the proper subsets of the set A = {1, 2, 3}.

Which of the following are functions, if X = {a,b,c,d} and Y = {1, 2, 3, 4, 5}?
f₁ = {(a,1),(b,1),(c,3),(d,4)}

Find the radian measures corresponding to following degree measures.
5^o37' 30''

Prove that the sum of first n even numbers is n(n+1).

Express the complex number in the form a+ib: (1-i)⁴

Solve the inequalities 3(1- x)< 2 (x +4)

Find r if ⁵⁴P_r+3 : ⁵⁴P_r+3 = 30800 : 1

Using binomial theorem, expand the following expressions.
\(\left( 2x-3y \right) ^{ 4 }\)

Three numbers are in AP. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.

Find the equation of the line which is parallel to 2x - 3y + 1 = 0 and passes through the point (6, 2).

Find the equation of the ellipse that satisfies the given conditions:
Foci (0, ±3), a = 5

If (2,6, -4) and (4, -2,3) are two vertices of a triangle whose centroid is (7, -2,5). Find the coordinates of the third vertex.

If \(f(x)=\left\{\begin{array}{l} x+2, x \leq-1 \\ c x^{2}, x>-1 \end{array}\right.\) then find c when \(\lim _{ x\rightarrow -1 }{ f(x) } \) exists.

Find the component statement of the following compound statements and check whether they are true or false.
Number 3 is prime or it is odd. 

Find the mean deviation about the mean for the following data

A die is rolled. If the outcome is an odd number, then what is the probability that it is a prime number?

Find the symmetric difference of sets A={1,3,5,6,7} and B ={3,7,8,9}

If \(y=f(x)=\frac { 1-x }{ 1+x } ,\) then show that x = f(y).

If a \(\tan \alpha =\frac { 1 }{ \sqrt { x\left( { x }^{ 2 }+x+1 \right) } } \tan\beta =\frac { \sqrt { x } }{ \sqrt { { x }^{ 2 }+x+1 } } and \tan\gamma =\sqrt { { x }^{ -3 }+{ x }^{ -2 }+{ x }^{ -1 } } ,\) then prove that α + β = γ

Prove by the principle of mathematical induction that \(1\times 1!+2\times 2!+3\times 3!+....+n\times n!=(n+1)!-1\)for all natural numbers n.

Express the following in the form of a + ib, where a, b \(\in \) R
\(({ i }^{ 29 }+\frac { 1 }{ { i }^{ 29 } } )\)

Solve the inequalities graphically 2x - 3y > 6

Evaluate  \(^{13}{C}{_{6}}\)+  \(^{13}{C}{_{5}}\)

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

Prove that \(\left( { 9 }^{ \frac { 1 }{ 3 } }\times { 9 }^{ \frac { 1 }{ 9 } }\times { 9 }^{ \frac { 1 }{ 27 } }\times ...\infty \right) =3\)

Find the equation of line passing through the points(-1,1) and (2,-4)

Find the centre and radius of each of the following circles.
\((x-\frac { 1 }{ 2 } { ) }^{ 2 }+(y+\frac { 1 }{ 3 } { ) }^{ 2 }=\frac { 1 }{ 4 } \)

Three students are standing in a park with signboards "SAVE ENVIRONMENT". "DONT'T LITTER", "KEEP YOUR PLACE CLEAN". Their positions are marked by the points A(0, 7, 10), B(-1, 6, 6) and C(-4, 9, 6). The three students are holding GREEN coloured ribbon together. Does the ribbons form sides of a right angled triangle? Do you feel the need to promote? What message is given from this question to the society?

Find the derivative of \(\frac{2x+3}{3x+2}\) from first principle.

Write the negation of the following statements and check whether the resulting statements are true.
(i) The sum of 2 and 3 is 4.
(ii) \(\sqrt { 3 } \) is rational
(iii) New Delhi is a city.
(iv) Every natural number is greater than 0

Two plants A and B of a factory show following results about the number of workers and the wages paid to them.

Which plant, A or B shows greater variability in individual wages?

A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.

If X={1,2,3} and n represents any member of X, write the following sets containing all numbers represented by n-1

If f(x) = x², find \(\frac { f(1.1)-f(1) }{ 1.1-1 } \)

Prove that \(sin4A=4sinAcos^{ 3 }A-4cosAsin^{ 3 }A.\)

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

Find real values of x and y for which the complex numbers - 3 + ix² y and x² + y + 4i are conjugate of each other.

Find the linear inequalities for which the shaded region in given figure is a solution set.

How many 3 digit numbers can be formed usingthe digits 0, 2, 3, 6, 8, when the digits may be repeated any number of times ? 

Prove that 11ⁿ-10n, when divided by 100, always leave a remainder 1 where n\(\in\)+N

If the first and the n^th term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that p² = (ab)ⁿ.

Transform the equation of the line 2x + 6y - 8 = 0 to intercept form and also find the intercepts on the coordinate axes

Draw the shape of ellipse \(\frac { { x }^{ 2 } }{ 49 } +\frac { { y }^{ 2 } }{ 16 } =1\) and find the eccentricity.

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 log(x+k)

Show that the following statement is true. p:For any real numbers x,y if x = y, then 2x + a = 2y + a when a \(\in\) Z.

Find the mean and standard deviation of the following frequency distribution

A card is selected from a pack of 52 cards .How many points are there in the sample space.

Draw the Venn diagrams to illustrate the following relationship among sets, E, M and U, where E is the set of students studying English in a school, M is the set of students studying Mathematics in the same school, U is the set of all students in that school.
There is no student who studies both Mathematics and English.
 

Let A = {1, 2, 3, 4} and B = {5, 6, 7}. If R = {(a, b) ; a \(\in\) A, b \(\in\) B} and a - b is even} then find R.

Solve the equation: cos \(\theta\) + cos 3\(\theta\) - 2 cos 2\(\theta\) = 0.

Prove the following by using the principle of mathematical induction for all n∊ N (2n+7) < (n+3)²

If a+i=\(\frac { ({ x }^{ 2 }+i)^{ 2 } }{ { 2x }^{ 2 }+1 } \) ,prove that a²+b² =\(\frac { ({ x }^{ 2 }+i)^{ 2 } }{ ({ 2x }^{ 2 }+1)^{ 2 } } \)

Solve the inequalities graphically 4x+3y\(\le\)60, y\(\ge\)2x, x\(\ge\)3, x,y\(\ge\)0

Find if  \(^{ 5 }P_{ r }=^{ 6 }P_{ r-1 }\)

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

The ratio of the sum of m and n terms of an A.p. is m²:n². Show that the ratio of m^th and n^th term is (2m - 1): (2n - 1).

The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle which are parallel to the axes.

Find the equation of the hyperbola satisfying the given conditions.
Foci (0, ± \(\sqrt10\) ),passing through (2, 3)

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

Suppose \( f(x)=\left\{\begin{array}{ll} a+b x, & x<1 \\ 4, & x=1 \text { and if } \lim _{x \rightarrow 1} f(x)=f(1) \\ h-a x & x>1 \end{array}\right.\)what are possible values of a and b?

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

Find the mean deviation about the median for the data
13,17,16,14,11,13,10,16,11,18,12,17

Refer to question 6 above, state true or false (give reason for your answer)
(i) A and B are mutually exclusive
(ii) A and B are mutually exclusive and exhaustive
(iii) A = B'
(iv) A and C are mutually exclusive
(v) A and B' are mutually exclusive
(vi) A', B', C are mutually exclusive and exhaustive.