List of practice Questions

Value of : $\sum_{k=1}^{\infty} \frac{(-1)^k \cdot k(k+1)}{k!}$

In $\Delta ABC$ if $\frac{\tan(A-B)}{\tan A} + \frac{\sin^2 C}{\sin^2 A} = 1$ where $A, B, C \in (0, \frac{\pi}{2})$ then

Let x be the number of 9 digit numbers formed by taking digits from first 9 natural numbers, where only one digit is repeated twice & y be the number of 9 digit numbers formed from first 9 natural numbers, such that exactly 2 digits repeated twice then

Consider the 10 observations 2, 3, 5, 10, 11, 13, 15, 21, a and b such that mean of observation is a and variance is 34.2. Then the mean deviation about median, is :

Let $\tan \left( \frac{\pi}{4} + \frac{1}{2} \cos^{-1} \frac{2}{3} \right) + \tan \left( \frac{\pi}{4} - \frac{1}{2} \sin^{-1} \frac{2}{3} \right) = k$. Then number of solution of the equation $\sin^{-1}(kx - 1) = \sin x - \cos^{-1} x$ is/are :

If $f(x^2 + 1) = x^4 + 5x^2 + 1$, then find $\int_{0}^{3} f(x) dx$ :

The value of $\lim_{x \to 0} \frac{\ln(\sec(ex) \cdot \sec(e^2x) \dots \sec(e^{10}x))}{e^2 - e^{2\cos x}}$ is :

Let $x \frac{dy}{dx} - \sin 2y = x^3(2 - x^3) \cos^2 y ; y(2) = 0$, then find $\tan(y(1))$ :

Given $\frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}$ such that roots of the quadratic equation $\lambda x^2 + (\lambda+1)x + 3 = 0$ are $\alpha$ & $\beta$, then sum of values of $\lambda$ is equal to :

The value of $S = \sum_{r=1}^{20} \sqrt{\pi \int_{0}^{r} x |\sin \pi x| dx}$ is :

Let $\int \frac{1 - 5 \cos^2 x}{\sin^5 x \cos^2 x} dx = f(x) + c$ then find $f\left(\frac{\pi}{4}\right) - f\left(\frac{\pi}{6}\right)$ :

If $\vec{a}, \vec{b} \& \vec{c}$ are unit vectors such that $(\vec{a}-\vec{b})^2 + (\vec{b}-\vec{c})^2 + (\vec{c}-\vec{a})^2 = 9$. Find positive k if $|2\vec{a} + k\vec{b} + k\vec{c}| = 3$ :

Product of first 3 terms of a G.P. is 27 and sum is $R - \{a,b\}$, then $a^2 + b^2$ is equal to :

If $g(x) = 3x^2 + 2x - 3, f(0) = -3, 4g(f(x)) = 3x^2 - 32x + 72$ then find $f(g(2))$ where $f(x)>0$ for all valid x:

Find possible no. of triplets (b, c, d), such that $x^2 + 2$ is divisor of $x^3 + bx^2 + cx + d$ & b, c, d $ \le $ 20 & b, c, d $ \in $ N :

Maximum value of $n$ for which $40^n$ divides $60!$ is

Let $z = (1+i)(1+2i)(1+3i)\cdots(1+ni)$ and $|z|^2 = 44200$. Find the value of $n$.

Let $f(x)$ be a differentiable function satisfying \[ f(x)=e^x+\int_0^1 (y+xe^x)f(y)\,dy \] Find $f(0)+e$, where $e$ is Napier's constant.

Let 4 integers $a_1, a_2, a_3, a_4$ are in A.P. with integral common difference $d$ such that $a_1+a_2+a_3+a_4=48$ and $a_1a_2a_3a_4+d^4=361$. Then the greatest term in this A.P. is

Evaluate \[ \left(\frac{4}{7}+\frac{1}{3}\right)+\left(\frac{4}{7}+\frac{4}{3}\right)\left(\frac{1}{3}\right) +\left(\frac{4}{7}+\frac{4}{3}\right)^2\left(\frac{1}{3}\right)^2+\cdots \]

If domain of $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty)$, then $(\alpha+\beta+\gamma+\delta)$ is

Let

be a continuous function at $x=0$, then the value of $(a^2+b^2)$ is (where $[\ ]$ denotes greatest integer function).

If the equation \[ x^4-ax^2+9=0 \] has four real and distinct roots, then the least possible integral value of $a$ is

If dataset $A=\{1,2,3,\ldots,19\}$ and dataset $B=\{ax+b;\,x\in A\}$. If mean of $B$ is $30$ and variance of $B$ is $750$, then sum of possible values of $b$ is

If $f(a)$ is the area bounded in the first quadrant by $x=0$, $x=1$, $y=x^2$ and $y=|ax-5|-|1-ax|+ax^2$, then find $f(0)+f(1)$.