List of practice Questions

A block of mass $m$ is kept on a platform which starts from rest with constant acceleration $\dfrac{g}{2}$ upward, as shown in the figure. Work done by the normal reaction on the block in time $t$ is:
Expression for time in terms of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light) is proportional to:
In a car race on a straight road, car A takes a time $t$ less than car B at the finish and passes the finishing point with a speed $v$ more than that of car B. Both the cars start from rest and travel with constant accelerations $a_1$ and $a_2$ respectively. Then $v$ is equal to
If the midpoints of the sides $BC$, $CA$ and $AB$ of a triangle $ABC$ are respectively $(2,1)$, $(-1,-2)$ and $(3,3)$, then the equation of the side $BC$ is
Two vertices of a triangle are $(5,-1)$ and $(-2,3)$. If the origin is the orthocentre of this triangle, then the coordinates of the third vertex of that triangle are
For the following question, enter the correct numerical value upto TWO decimal places. If the numerical value has more than two decimal places, round-off the value to TWO decimal places. (For example: Numeric value 5 will be written as 5.00 and 2.346 will be written as 2.35) If \( z \), \( iz \) and \( z+iz \) are the vertices of a triangle and if \( |z| = 4 \), then the area (in sq. units) of that triangle is ______.
For the following question, enter the correct numerical value upto TWO decimal places. If the numerical value has more than two decimal places, round-off the value to TWO decimal places. (For example: Numeric value 5 will be written as 5.00 and 2.346 will be written as 2.35) There are 10 points in a plane out of which 6 are collinear. The number of straight lines formed by joining all these points is ____.
For the following question, enter the correct numerical value upto TWO decimal places. If the numerical value has more than two decimal places, round-off the value to TWO decimal places. (For example: Numeric value 5 will be written as 5.00 and 2.346 will be written as 2.35) Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is ____.
For the following question, enter the correct numerical value upto TWO decimal places. If the numerical value has more than two decimal places, round-off the value to TWO decimal places. (For example: Numeric value 5 will be written as 5.00 and 2.346 will be written as 2.35) An envelope is known to have come from either `LONDON` OR `CLIFTON`. On the postal card only two successive letters `ON` are visible. The probability that the envelope comes from LONDON is \( \dfrac{12}{__} \).
Let $f$ be a polynomial function such that $f'(x)=f(x)\,f''(x)$ for all $x\in\mathbb{R}$. Then:
The minimum distance of a point on the curve $y=x^2-4$ from the origin is
Evaluate: \[ \int \frac{dx}{\cos x+\sqrt{3}\sin x} \]
If \[ \int \frac{1}{(x+100)\sqrt{x+99}}\,dx = f(x)+c \] then $f(x)=$
Evaluate: \[ \int_{0}^{\pi} \frac{x\sin x}{1+\cos^2 x}\,dx \]
If $y=y(x)$ is the solution of the differential equation \[ x\frac{dy}{dx}+2y=x^2 \] satisfying $y(1)=1$, then $y\!\left(\dfrac{1}{2}\right)$ is equal to:
A square matrix (non-singular) satisfies $A^2 - A + 2I = 0$. Then $A^{-1}=$
Consider the following statements \[ P:\ \text{Suman is brilliant} \] \[ Q:\ \text{Suman is rich} \] \[ R:\ \text{Suman is honest} \] The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as:
10 men and 6 women are to be seated in a row so that no two women sit together. The number of ways they can be seated is
$1^3 - 2^3 + 3^3 - 4^3 + \cdots + 9^3 =$
Universal set, \[ U = \{x \mid x^5 - 6x^4 + 11x^3 - 6x^2 = 0\} \] \[ A = \{x \mid x^2 - 5x + 6 = 0\} \] \[ B = \{x \mid x^2 - 3x + 2 = 0\} \] What is $(A \cap B)$ equal to?
If $R$ is an equivalence relation on a set $A$, then $R \circ R$ is
Evaluate the limit: \[ \lim_{x \to \infty} \left(\frac{x^2 + 5x + 3}{x^2 + x + 2}\right)^x \]
If one root of the quadratic equation $ax^2+bx+c=0$ is $3-4i$, then $31a+b+c=$
If \[ A(\alpha)= \begin{pmatrix} \cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{pmatrix} \] then $A(\alpha)\,A(\beta)=$
Aniline on heating with ‘X’ in the presence of KOH gives a product with very bad smell. Which of the following is ‘X’?