List of top Mathematics Questions asked in CBSE CLASS XII

Evaluate : \(\displaystyle\int_{0}^{1} x\tan^{-1}x\,dx\).

Find the particular solution of the differential equation

\(xy\dfrac{dy}{dx} = (x+2)(y+2)\), given that \(y(1) = -1\).

The length of perpendicular drawn from point \( (2, 5, 7) \) on line \( \dfrac{x}{1} = \dfrac{y}{0} = \dfrac{z}{0} \) is

The order and degree of the differential equation \( \dfrac{d}{dx}\left(e^{y}\right) = 0 \) respectively are

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of \(1\ \text{mm}^3/\text{min}\). Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is \(10\ \text{mm}\), if the semi-vertical angle of conical bottle is \(\dfrac{\pi}{6}\).

Assertion (A) : Lines given by \( x = py + q \), \( z = ry + s \) and \( x = p^{\prime}y + q^{\prime} \), \( z = r^{\prime}y + s^{\prime} \) are perpendicular to each other when \( pp^{\prime} + rr^{\prime} = 1 \).

Reason (R) : Two lines \( \vec{r} = \vec{a}_1 + \lambda\vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu\vec{b}_2 \) are perpendicular to each other if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).

The least value of \( f(x) = x^3 - 12x,\ x \in [0, 3] \) is

Find the vector of magnitude \(14\) in the direction of \(\overrightarrow{QP}\), where P and Q are the points \((1,\ 3,\ 2)\) and \((-1,\ 0,\ 8)\) respectively.

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?

If \( 2\cos^{-1}x = y \), then

Which of the following cannot be the order of a row-matrix ?

Find: \[ \int \frac{x^2}{(x^2+9)(x^2+16)}\,dx \]

In a school, the probability of holding a debate competition is \( \frac{1}{3} \) and that of a quiz competition is \( \frac{2}{3} \). In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.

For a square matrix \(A\), \[ (3A)^{-1}= \]

The least value of \[ f(x)=e^{-x} \] in the interval \[ [0,3] \] is:

Assertion (A) : In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is \( \frac{2}{3} \).
Reason (R) : For any two events \( A \) and \( B \), \( P(A|B) = \frac{P(A \cup B)}{P(B)} \).

Check whether the function \[ f(x)= \begin{cases} \dfrac{|x-3|}{2(x-3)}, & x<3,\\[6pt] \dfrac{x-6}{6}, & x\geq 3 \end{cases} \] is continuous at \(x=3\) or not.

Evaluate: \[ \cos\left[\sin^{-1}(-1)-\tan^{-1}(-\sqrt{3})\right]. \]

A function \[ f:\mathbb{R}-\left\{\frac{3}{5}\right\} \to \mathbb{R}-\left\{\frac{3}{5}\right\} \] is defined as \[ f(x)=\frac{3x+2}{5x-3}. \] Show that \(f\) is one-one and onto.

For \[ f(x)=x+\frac{1}{x}, \quad x\neq 0. \]

For two vectors \(\vec{a}\) and \(\vec{b}\):

Assertion (A): \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Reason (R): \[ |\vec{a}\times\vec{b}| = (\vec{a}\cdot\vec{b})\tan\theta, \quad \theta\neq\frac{\pi}{2}. \]

Check whether \[ f:\mathbb{R}-\{3\}\rightarrow\mathbb{R} \] defined as \[ f(x)=\frac{x-2}{x-3} \] is onto or not.

If \[ \frac{d}{dx}(F(x))=\frac{1}{e^x+1}, \] then find \(F(x)\), given that \[ F(0)=\log\left(\frac{1}{2}\right). \]

Find the domain of \[ q(x)=\cos^{-1}(4x^2-3). \] Hence, find the value of \(x\) for which \[ q(x)=0. \] Also, write the range of \[ 3q(x)-\pi. \]

Sports car racing is a form of motorsport which uses sports car prototypes.The competition is held on special tracks designed in various shapes.

The equation of a sports car racing track is given as: \[ f(x)= \begin{cases} x^4-4x^2+4, & 0\leq x<3,\\ x^2+40, & x\geq 3 \end{cases} \] Based on this information: