List of top Questions asked in KEAM- 2019

Let \( f(x)=|x-2| \) and \( g(x)=f(f(x)) \). Then derivative of \( g \) at the point \( x=5 \) is

The value of the definite integral \( \int_0^{2\pi} \sqrt{1 + \sin \frac{x}{2}}\, dx \) is

The value of \( \lim_{n\to\infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{6n} \right] \) is

Suppose $\alpha \hat{i} + \alpha \hat{j} + \gamma \hat{k}$, $\hat{i} + \hat{k}$ and $\gamma \hat{i} + \gamma \hat{j} + \beta \hat{k}$ are coplanar where $\alpha, \beta, \gamma$ are positive constants. Then the product $\alpha\beta$ is

A unit vector $\vec{b}$ is coplanar with $\hat{i} + \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} + \hat{k}$ and is perpendicular to $\hat{i} + \hat{j} + \hat{k}$. Then $\vec{b} \cdot \hat{i}$ equals

The area of the triangle whose vertices are $A(1,-1,2)$, $B(2,1,-1)$ and $C(3,-1,2)$ is

Let $R = \{(a,b): a \leq b^2\}$ be a relation on the set of all real numbers. Then $R$ is

The domain of definition of the function $f(x) = \frac{\log_3(x+7)}{x^2 - 5x + 6}$ is

Let $f(x) = 3x - 5$. The inverse of $f$ is given by

If $^{n}C_{2017} = {}^{n}C_{2016}$, then $^{n}C_{4033}$ equals

The equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$ is

The foci of the hyperbola $\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1$ are

The image of the point $P(2,1)$ on the straight line $2x - 3y + 1 = 0$ is

If the centre of the circle inscribed in a square formed by the lines $x^2 - 8x + 12 = 0$ and $y^2 - 14y + 45 = 0$ is $(a,b)$, then $a + b$ is

If $^{5}P_r = {}^{6}P_{r-1}$, then the value of $r$ is

Given that the equation $x^2 - (2a + b)x + \left(2a^2 + b^2 - b + \frac{1}{2}\right) = 0$ has two real roots. The value of $b$ is

If A.M. and G.M. of the roots of a quadratic equation are 8 and 5 respectively, then the quadratic equation is

The value of \( \tan \frac{\pi}{8} \) is

In a G.P., $1, \frac{1}{2}, \frac{1}{4}, \ldots$, when the first $n$ number of terms are added, the sum is $\frac{1023}{512}$. Then the value of $n$ is

If \( A \) and \( B \) are two events associated with an experiment such that \( P(A \cup B) = P(A \cap B) \), and \( P(A) = 1/3 \), then \( P(B) \) equals

Three identical fair dice are rolled. The probability that the same number appears on each of them is

Let \( \omega \ne 1 \) be a cube root of unity and \( (1+\omega)^7 = a + \omega \). Then the value of \( a \) is

Let \( w = \frac{1-iz}{z-i} \). If \( |w| = 1 \), which of the following must be true?

For \( |z| \ge 2 \), if \( \left|z + \frac{1}{2}\right| \ge k \), the minimum possible value of \(k\) is

Let \( \cot \theta = -5/12 \) where \( \frac{\pi}{2} < \theta < \pi \). Then the value of \( \sin \theta \) is