List of practice Questions

Let
$β = \lim_{x →0} \frac{αx-(e^{3x}-1)}{αx(e^{3x}-1) }$for some$ α \in R.$
Then the value of α+β is

The value of $\log_e2\frac{d}{dx}(\log_{cos⁡ x}\cosec x) $ at $x=\frac{\pi}{4}$ is

$\int_{0}^{20\pi} (|\sin x| + |\cos x|)^2 \,dx$
is equal to

For any real number x, let [ x ] denote the largest integer less than equal to x Let f be a real valued function defined on the interval [-10,10] by $f(x)=\begin{cases} x-[x], & \text { if }(x) \text { is odd } \\ 1+[x]-x & \text { if }(x) \text { is even }\end{cases}$Then the value of$ \frac{\pi^2}{10} \int\limits_{-10}^{10} f(x) \cos \pi x d x$ is :

The slope of the tangent to a curve C : y=y(x) at any point [x, y) on it is $\frac{2 e ^{2 x }-6 e ^{- x }+9}{2+9 e ^{-2 x }}$ If C passes through the points $\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\right) $and $\left(\alpha, \frac{1}{2} e ^{2 \alpha}\right)$$ then $e ^\alpha$ is equal to :

The total number of functions, $f :\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\}$ such that $f (1)+ f (2)= f (3)$, is equal to :

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x ^4+ x ^3+ x ^2+ x +1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is :

For $n \in N$, let $S _{ n }=\left\{ z \in C :| z -3+2 i |=\frac{ n }{4}\right\}$ and $T_n=\left\{z \in C:|z-2+3 i|=\frac{1}{n}\right\}$ Then the number of elements in the set $\left\{ n \in N : S _{ n } \cap T _{ n }=\phi\right\}$ is :

The number of solutions of $|\cos x |=\sin x$, such that $-4 \pi \leq x \leq 4 \pi$ is :

Which of the following statements is a tautology?

Let a₁ = b₁ = 1, a_n = a_n_{– 1} + 2 and b_n = a_a + b_n_{– 1} for every natural number n ≥ 2. Then$\sum_{n=1}^{15}$ $a_n⋅b_n$ is equal to ______.

A man of 60 kg is running on the road and suddenly jumps into a stationary trolly car of mass 120 kg. Then, the trolly car starts moving with velocity 2ms^–1. The velocity of the running man was _____ ms^–1, when he jumps into the car.

As per the given figure, two blocks each of mass $250 \,g$ are connected to a spring of spring constant $2 \,Nm ^{-1}$ If both are given velocity $v$ in opposite directions, then maximum elongation of the spring is :