List of top Mathematics Questions asked in CBSE CLASS XII

Let \(A=\begin{bmatrix}1&-2&1\\ -2&3&1\\ 1&1&5\end{bmatrix}\)verify that -
\((i)[adjA]^{-1}=adj(A^{-1})\)
\((ii)(A^{-1})^{-1}=A\)

It is given that at x = 1, the function x⁴−62x²+ax+9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
A. R is reflexive and symmetric but not transitive.
B. R is reflexive and transitive but not symmetric.
C. R is symmetric and transitive but not reflexive.
D. R is an equivalence relation.

Find the maximum profit that a company can make, if the profit function is given by p(x) = 41−24x−18x²

The two equal sides of an isosceles triangle with fixed base \(b\) are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

Show that the given differential equation is homogeneous and solve:\((x^2+xy)dy=(x^2+y^2)dx\)

Find \(|\vec{a}\times\vec{b}|\) if \(\vec{a}=\hat{i}-7\hat{j}+7\hat{k}\space and\space \vec{b}=3\hat{i}-2\hat{j}+2\hat{k}.\)

\(\text {Find} \ \frac {dy}{dx}:\) \(y=sin^{-1}(\frac {2x}{1+x^2})\)

Without expanding the determinant, prove that
\(\begin{vmatrix} a &a^2 &bc \\ b&b^2 &ca \\ c&c^2 &ab \end{vmatrix}=\begin{vmatrix} 1 &a^2 &a^3 \\ 1&b^2 &b^3 \\ 1&c^2 &c^3 \end{vmatrix}\)

A company manufactures two types of novelty souvenirs made of polywood.Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and building.3 hours 20 minutes are available for cutting and 4 hours of assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many types of souvenirs of each type should the company manufacture in order to maximize the profit?

Using the property of determinants and without expanding, prove that: \(\begin{vmatrix}a-b&b-c&c-a\\b-c&c-a&a-b\\c-a&a-b&b-c\end{vmatrix}\)=0

\(∫\frac{dx}{e^x+e^{-x}}\) is equal to

An edge of a variable cube is increasing at the rate of \(3 cm/s\). How fast is the volume of the cube increasing when the edge is \(10 cm\) long?

Find the projection of the vector \(\hat{i}+3\hat{j}+7\hat{k}\) on the vector \(7\hat{i}-\hat{j}+8\hat{k}.\)

Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

f : R→R defined by f(x)=3−4x
f : R→R defined by f(x)=1+x2

If A and B are square matrices of the same order such that \(AB=BA\),then prove by induction that \(AB^n=B^nA\).Further, prove that \((AB)^n=A^nB^n\) for all \(n∈N\)

Find x, if \(\begin{bmatrix} x &-5&-1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 0 & 2\\ 0 & 2 & 1 \\2&1&3 \end{bmatrix}\)\(\begin{bmatrix} x \\ 4\\1 \end{bmatrix}=O\)

A coin is biased so that the head is 3 times as likely to occur as tail.If the coin is tossed twice,find the probability distribution of number of tails.

Evaluate the definite integral as limit of sums: \(∫^3_2 x^2dx\)

Find the intervals in which the function f given by f(x) = 2x³ − 3x² − 36x + 7 is (a) strictly increasing (b) strictly decreasing

Sand is pouring from a pipe at the rate of \(12 cm^3 /s\). The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is \(4 cm\)?

Find \(|\vec{a}|\) and \(|\vec{b}|\) ,if \((\vec{a}+\vec{b}).(\vec{a}-\vec{b})=8\) and \(|\vec{a}|=8|\vec{b}|.\)

State whether the following statements are true or false. Justify.
(i) For an arbitrary binary operation * on a set N, a * a=a ∀ a * N.
(ii) If * is a commutative binary operation on N, then a * (b * c)= (a * b)* a

Integrate the function: \(\frac{sin^8x-cos^8x}{1-2sin^2xcos^2x}\)