List of top Questions asked in CBSE CLASS XII

Let two rods placed on the ground be represented by vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $-2\hat{i} + \hat{j} - 2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

Evaluate : $\tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\left(\frac{2\pi}{3}\right)\right)$}

A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with the $x$-axis, $\frac{\pi}{3}$ with the $y$-axis and an acute angle $\theta$ with the $z$-axis. Find $\theta$ and the components of $\vec{a}$.

Direction: Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.

If $(2\hat{i} + 3\hat{j} + \hat{k})$ and $(2\hat{i} + \hat{j} - \hat{k})$ represent the sides $\vec{AB}$ and $\vec{AC}$ respectively of $\Delta ABC$, find the vector representing the median through A.

Direction: Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : A function f : N $\to$ N given by f(x) = $x^3$ + 2, $\forall$ x $\in$ N is one-one but not onto.
Reason (R) : Since $\forall$ y $\in$ N (Codomain), there does not exist x = $(y - 2)^{1/3}$ in N (Domain) such that f(x) = $x^3$ + 2 = y.

The region represented by the system of linear inequalities \( 3x + y \ge 3 \), \( 2x - y \ge -5 \), and \( x, y \ge 0 \) is:

The probability that it will rain tomorrow in cities A, B and C is \(60\%\), \(70\%\) and \(80\%\) respectively. Assuming these events are independent, the probability that it will rain tomorrow in at least one of the cities is:

The sum of the order and the degree of the differential equation \(\left[\frac{d^2y}{dx^2}\right]^2 + \left[\frac{dy}{dx}\right]^4 + 3\left(\frac{dy}{dx}\right) + 1 - y^2 = 0\) is:

Three points \( A(0, 1, 1) \), \( B(2, 0, -1) \) and \( C(1, 0, 3) \) form \(\Delta ABC\). The area of \(\Delta ABC\) is:

The area of the region bounded by the curve \(y = x\) and the x-axis, between the boundaries \(x = 0\) and \(x = 2\) is:

The greatest integer function, \(f(x) = [x]\), evaluated across the open domain boundary \(0 < x < 3\), is not differentiable at how many points?

In a linear programming problem, the feasible region representing the maximization of the objective function \( Z = px + qy \), where \( p, q > 0 \), is a shaded polygonal region. If all points on the line segment joining the corner points \( A(0, 3) \) and \( B(3, 2) \) give the maximum value of \( Z \), then which of the following is true?

If the distance travelled by a particle in \(t\) seconds is given by \(S = 72t + 3t^2 - t^3\), then the time taken by the particle to come to rest is:

If \(\int_{0}^{1} \frac{dx}{e^x + e^{-x}} = \tan^{-1}e + k\), then the constant value of \(k\) is:

The differential of \(e^{e^x}\) with respect to \(x\) is:

The indefinite integral \(\int \frac{dx}{\sqrt{25 - 16x^2}}\) is equal to:

The graph of an inverse trigonometric function passes through the point $(0, \frac{\pi}{2})$ with asymptotes at $y = 0$ and $y = \pi$. This graph represents the function:

If the points \((2, 3)\), \((0, 4)\) and \((p, 2)\) are collinear, then the value of \(p\) is:

Name the place where a major sea port is located in Gujarat.

Name the place where an international airport is located in Assam.

Name the coal mining area in Tamil Nadu.

Mention the easternmost iron-ore mining area in Odisha.

Name the state leading in wheat production.

Name the state leading in the production of rice.