List of practice Questions

The most expensive Indian painting sold in 1961 was painted by:

Japan's former longest serving prime minister who was shot dead while giving a compaign speech. His name was ___ .

Let S = $\{z \in C-\{i,2i\}:\frac{z^2+8iz-15}{z^2-3iz-2}\in R\}$ if $\alpha-\frac{13}{11}i\in S, a \in R-{0}$, then $242\alpha^2$ is equal to ____.

Let $a_n=\sin(\frac{1}{n^3})$ and $b_n=\sin(\frac{1}{n})$ for n ∈ $\N$. Then

Consider the following statements :
I. There exists a linear transformation from $\R^3$ to itself such that its range space and null space are the same.
II. There exists a linear transformation from $\R^2$ to itself such that its range space and null space are the same.
Then

Let
$\begin{pmatrix} 1 & -1 & 0 \\ 0 & 0 & 0 \\ -2 & 2 & 2 \end{pmatrix}$
and B = A⁵ + A⁴ + I₃. Which of the following is NOT an eigenvalue of B ?

The system of linear equations in x₁, x₂, x₃
$\begin{pmatrix} 1 & 1 & 1 \\ 0 & -1 & 1 \\ 2 & 3 & \alpha \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix} 3 \\ 1 \\ \beta \end{pmatrix}$
where α, β ∈ R, has

Let f(x, y) = $e^{x^2}+y^2$ for (x, y) ∈ $\R^2$ , and a_n be the determinant of the matrix
$\begin{pmatrix} \frac{∂^2f}{∂x^2} & \frac{∂^2f}{∂x∂y} \\ \frac{∂^2f}{∂y∂x} & \frac{∂^2f}{∂y^2} \end{pmatrix}$
evaluated at the point (cos(n),sin(n)). Then the limit $\lim\limits_{n \rightarrow \infin}$ a_n is

Let f(x, y) = ln(1 + x² + y² ) for (x, y) ∈ $\R^2$. Define
$\begin{matrix} P=\frac{∂^2f}{∂x^2}|_{(0,0)} & Q=\frac{∂^2f}{∂x∂y}|_{(0,0)} \\ R=\frac{∂^2f}{∂y∂x}|_{(0,0)} & S=\frac{∂^2f}{∂y^2}|_{(0,0)} \end{matrix}$
Then

The area of the curved surface
$S = \left\{ (x, y, z) ∈ \R^3 : z^2 = (x − 1)^2 + (y − 2)^2 \right\}$
lying between the planes z = 2 and z = 3 is

Let $a_n=\frac{1+2^{-2}+...+n^{-2}}{n}$ for n ∈ $\N$. Then

Let (a_n) be a sequence of real numbers such that the series $\sum\limits_{n=0}^{\infin}a_n(x-2)^n$ converges at x = −5. Then this series also converges at

From the additive group Q to which one of the following groups does there exist a non-trivial group homomorphism ?

If all the particles have same kinetic energy, The relation between the wavelengths of alpha particle, electron and proton is:

Let the shortest distance between the lines $L: \frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1}, \lambda \geq 0$ and $L_1: x+1=y-1=4-z$ be $2 \sqrt{6}$ If $(\alpha, \beta, \gamma)$ lies on $L$, then which of the following is NOT possible?

The number of turns of the coil of a moving coil galvanometer is increased in order to increase current sensitivity by $50 \%$ The percentage change in voltage sensitivity of the galvanometer will be:

Given below are two statements :
Statement I : In a typical transistor, all three regions emitter, base and collector have same doping level.
Statement II : In a transistor, collector is the thickest and base is the thinnest segment.
In the light of the above statements, choose the most appropriate answer from the options given below.

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to:

Let $a,b,c>1,a^3,b^3 \text{ and } c^3$ be in A.P., and $log_ab, log_ca \text{ and } log_bc$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a+4b+c}{3}$ and the common difference is $\frac{a−8b+c}{10}$ is −444, then abc is equal to:

Let $N$ denote the number that turns up when a fair die is rolled If the probability that the system of equations$ x+y+z=1 $ $ 2 x + Ny +2 z=2 $ $ 3 x +3 y+ N z=3$ has unique solution is $\frac{k}{6}$, then the sum of value of $k$ and all possible values of $N$ is

Choose the correct representation of conductometric titration of benzoic acid vs sodium hydroxide

For an object placed at a distance $24\, m$ from a lens, a sharp focused image is observed on a screen placed at a distance $12\, cm$ from the lens A glass plate of refractive index $15$ and thickness $1 \,cm$ is introduced between lens and screen such that the glass plate plane faces parallel to the screen By what distance should the object be shifted so that a sharp focused image is observed again on the screen?

Let $P$ be the plane, passing through the point $(1,-1,-5)$ and perpendicular to the line joining the points $(4,1,-3)$ and $(2,4,3)$. Then the distance of $P$ from the point $(3,-2,2)$ is

Electron beam used in an electron microscope, when accelerated by a voltage of $20 kV$, has a de-Broglie wavelength of $\lambda_0$. If the voltage is increased to $40 kV$, then the de-Broglie wavelength associated with the electron beam would be:

As shown in the figure a block of mass $10\, kg$ lying on a horizontal surface is pulled by a force $F$acting at an angle $30^{\circ}$, with horizontal For $\mu_{ s }=0.25$, the block will just start to move for the value of $F$ : [Given g=10ms^-2]