List of top Questions asked in CBSE CLASS XII

A line passing through the points \(A(1,2,3)\) and \(B(6,8,11)\) intersects the line \[ \vec r = 4\hat i + \hat j + \lambda(6\hat i + 2\hat j + \hat k) \] Find the coordinates of the point of intersection. Hence write the equation of a line passing through the point of intersection and perpendicular to both the lines.

Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).

Evaluate : \[ \int_{\frac{1}{12}}^{\frac{5}{12}} \frac{dx}{1+\sqrt{\cot x}} \]

Find : \[ \int \frac{2x+1}{\sqrt{x^2+6x}}\,dx \]

Mother, Father and Son line up at random for a family picture. Let events \(E\): Son on one end and \(F\): Father in the middle. Find \(P(E/F)\).

Solve the following linear programming problem graphically:
Maximize \[ Z = 8000x + 12000y \] Subject to the constraints
\[ 3x + 4y \le 60 \] \[ x + 3y \le 30 \] \[ x \ge 0,\; y \ge 0 \]

Find the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]

Solve the differential equation \[ x\frac{dy}{dx}=y-x\sin^2\left(\frac{y}{x}\right), \quad \text{given that } y(1)=\frac{\pi}{6}. \]

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

If $x=\sin t-\cos t$ and $y=\sin t\cos t$, find $\dfrac{dy}{dx}$ at $t=\dfrac{\pi}{4}$.

Check whether $f:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}\times\mathbb{Z}$ defined by $f(x,y)=(2y,3x)$ is injective or not.

Check whether $f:\mathbb{R}-\{3\}\rightarrow\mathbb{R}$ defined by $f(x)=\dfrac{x-2}{x-3}$ is onto or not.

Find the coordinates of the foot of perpendicular drawn from $(0,0,0)$ to the line \[ \frac{x}{1}=\frac{y+1}{-1}=\frac{z-3}{-2} \]

If $\overrightarrow{AB}=\hat{j}+\hat{k}$ and $\overrightarrow{AC}=3\hat{i}-\hat{j}+4\hat{k}$ represent the two vectors along the sides $AB$ and $AC$ of $\triangle ABC$, prove that the median $\overrightarrow{AD}=\dfrac{\overrightarrow{AB}+\overrightarrow{AC}}{2}$ where $D$ is the midpoint of $BC$. Hence, find the length of the median $AD$.

If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.

Find the absolute maximum value of $f(x)=\cos x+\sin^2 x$, where $x\in[0,\pi]$.

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): $|\vec a\times \vec b|=|\vec a||\vec b|\sin\theta$, where $\theta$ is the angle between them.

Assertion (A): A line can have direction cosines $\langle 1,1,1\rangle$.
Reason (R): $\cos\theta=1$ is possible for $\theta=0^\circ$.

If $e^{x+y}+y=3x$, then $\dfrac{dy}{dx}$ is

If \[ \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & 4 \\ 0 & 4 & 2a \end{vmatrix} = -86 \] then the sum of all possible values of \( a \) is

For a square matrix $A$, $(3A)^{-1}$ is

If \[ A = \begin{bmatrix} \frac{1}{2}\cos x & -\sin x \\ \sin x & \frac{1}{2}\cos x \end{bmatrix} \] and \(A + A^T = I\), then the value of \(x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) is

If \[ A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix} \] is a symmetric matrix, then the value of \(3a + b + c\) is

If $A$ and $B$ are square matrices of same order, then which of the following statements is/are always true?
(i) $(A+B)(A-B)=A^2-B^2$
(ii) $AB=BA$
(iii) $(A+B)^2=A^2+AB+BA+B^2$
(iv) $AB=0 \Rightarrow A=0$ or $B=0$

A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is