List of practice Questions

Let $\mathbb{N}, \mathbb{Z}$ and $\mathbb{R}$ be the set of natural numbers, integers and real numbers respectively, $[\cdot]$ denotes the greatest integer function. Match List-I with List-II:}

The determinant \( \begin{vmatrix} \lambda & \sin\theta & \cos\theta \\ -\sin\theta & -\lambda & 1 \cos\theta & 1 & \lambda \end{vmatrix} \) is equal to:

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}+\vec{b}$ is also a unit vector. Then which of the following are TRUE?} (A) $|\vec{a}-\vec{b}|=0$
(B) $|\vec{a}-\vec{b}|=\sqrt{3}$
(C) Angle between $\vec{a}$ and $\vec{b}=\frac{2\pi}{3}$
(D) Angle between $\vec{a}$ and $\vec{b}=\frac{\pi}{3}$

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}| = 2 \), \( |\vec{b}| = 1 \) and \( \vec{a} \cdot \vec{b} = \sqrt{3} \) then the angle between \( 2\vec{b} \) and \( -\vec{a} \) is:

If $A=\begin{bmatrix} a & b \\ b & a \end{bmatrix}$ and $A^{2}=\begin{bmatrix} \alpha & \beta \\ \beta & a \end{bmatrix}$, then $(a-b)$ is:

Area bounded by the curve $y=x^{3}$ and line $y=4x$ is:

If $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2\hat{j}+\hat{k}$ then Match List-I with List-II:}

If vertices A and C of a $\Delta ABC$ lie along a line and the line segment AC has length 3, then the area of $\Delta ABC$ is:

In a sphere, the rate of change of volume is:

If the lines $x=ay+b, z=cy+d$ and $x=a'y+b', z=c'y+d'$ are perpendicular, then:

Match List-I (Inverse Trigonometric function Principal values) with List-II:

If $\int_{0}^{a}\sqrt{x}dx = \frac{4a}{3}$, then $\int_{a}^{a+1}x\,dx$ is:

If $a_{ij}=\begin{cases} 0,& i\ne j\\ 2i-j,& i=j \end{cases}$ then matrix A is:

If a unit vector makes equal acute angles with the coordinate axes, then the projection of this vector on $-5\mathbf{i}+7\mathbf{j}-\mathbf{k}$ is:

Match List-I (Matrix expressions) with List-II (Properties).

If $x=t^{2}, y=t^{3}$, then $\frac{d^{2}y}{dx^{2}}$ is:

If A is a matrix of order $3\times4$ and B is a matrix such that AB and BA are both defined, then order of B is:

If the function $f(x)=x^{3}-kx$ is increasing for all real x, then:

Area of the region bounded by the curves $x=y^{2},\; y=-1,\; y=2$ and y-axis is:

Match List-I (Differential equations) with List-II (Order and Degree).

Evaluate \[ \int \frac{1+x+\sqrt{x+x^{2}}}{\sqrt{1+x}+\sqrt{x}}dx \]

Which of the following set of constraints represents the feasible region (shaded portion) in the figure given below?

Value of the determinant \(\begin{vmatrix} \log_{3}512 & \log_{4}3 \\ \log_{3}8 & \log_{4}9 \end{vmatrix}\) is:

General solution of the differential equation \[ \frac{dy}{dx}=e^{x-y}+3x^{2}e^{-y} \] is:

Particular solution of the differential equation \[ \frac{dy}{dx}+2y^{2}=0,\quad y(1)=1 \] is: