List of top Questions asked in KEAM

The value of $\dfrac{(1+i)^n}{(1-i)^{n-4}}$, where $i=\sqrt{-1}$ and $n$ is an integer, is:

Let $x$ and $y$ be real numbers. If $(3+i)x + y + (1-i)y + 3i - 4 = (2x+1)i + (x-y+2)i$, where $i=\sqrt{-1}$, then the pair $(x,y)$ is equal to:

Let $z_1 = \dfrac{5+7i}{7-5i}, \, z_2 = \dfrac{3+2i}{3-2i}$ and $z_3 = \dfrac{1+11i}{11-i}$. Then $z_1\overline{z_1} + z_2\overline{z_2} + z_3\overline{z_3}$ is equal to:

Let $t_1, t_2, t_3, \ldots, t_{2n}$ be in G.P. with common ratio $r$. Then:

If $\dfrac{4^{n+1} + 16^{n+1}}{4^n + 16^n}$ is the Geometric Mean between $4$ and $16$, then the value of $n$ is:

If $4\sin^2 x - 2(1+\sqrt{3})\sin x + \sqrt{3} = 0$ and $15^\circ<x<150^\circ$, then the values of $x$ are:

Let $x$ be a real number such that $\dfrac{x-3}{x-2} \geq 1$. Then the solution set of the inequality is:

There are two main entrances to a building with five floors. Each entrance leads to three lifts and each lift can stop at all the five floors. A person enters the building and reaches a floor. The number of possible ways that the person can reach the floor, is

If $\sin \theta \cos \theta>0$, then $\theta$ lies

The sum of all 3-digit numbers that can be formed using $1,2,3,4$ without repetitions is

If ${}^9P_5 = (504)({}^6P_r)$, then the value of $r$ is equal to:

A box contains 24 identical balls of which one ball is black and the remaining balls are green. Three balls are taken simultaneously and randomly. The number of ways of getting only green balls, is

Let $x$ be a real number such that $5<|x - 1|<15$. Then

The coefficient of $\dfrac{1}{x^2}$ in the binomial expansion of $\left(3x - \dfrac{1}{3x}\right)^4$ is:

If $(x \;\; 3 \;\; -1)\begin{pmatrix} 1 & 1 & 1 \\ -1 & 0 & 1 \\ 1 & 0 & -1 \end{pmatrix}\begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} = 0$, then the values of $x$ are:

Evaluate the determinant $\begin{vmatrix} 11 & 1 & 1 \\ 1 & 21 & 1 \\ 1 & 1 & 31 \end{vmatrix}$:

Let $P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 10 & 100 & -1 \end{pmatrix}$. Then $P^{4052}$ is equal to:

If $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $(\alpha I + \beta A)^2 = A$, where $I$ is $2 \times 2$ unit matrix, then $\alpha^2 - \beta^2 =$:

$\displaystyle \int \frac{x^4 - 1}{x + 1} \, dx$ is equal to:

$\displaystyle \int_{-6}^{0} \left[t^3 + 9t^2 + 27t + 29 + (t+3)\cos(t+3)\right] dt$ is equal to:

If $I = \displaystyle \int_{-1}^{1} \frac{x^4}{1 - x^4} \cos^{-1}\left(\frac{2x}{1+x^2}\right) dx$, then $2I$ is equal to:

$\displaystyle \int \frac{\sin t + \cos t}{13 + 36\sin^2 t} \, dt$ is equal to:

$\displaystyle \int_{0}^{1} \left[\tan^{-1}\left(\frac{1}{1+x+x^2+x^3}\right) + \tan^{-1}(1+x+x^2+x^3)\right] dx$ is equal to:

The physical quantity that doesn’t have appropriate unit is

Which one of the following statements is CORRECT?