List of top Questions asked in KEAM- 2018

If the sum of the coefficients in the expansion of \( (a^2x^2 - 2ax + 1)^{51} \) is zero, then \( a \) is equal to

The sum of odd integers from 1 to 2001 is

The foci of the hyperbola \( 16x^2 - 9y^2 - 64x + 18y - 90 = 0 \) are

The value of \( \sin \left(\frac{31\pi}{3}\right) \) is

If \( y = \frac{\sin^2 x}{1+\cot x} + \frac{\cos^2 x}{1+\tan x} \), then \( y'(x) \) is equal to

Sum of last 30 coefficients in the binomial expansion of \( (1+x)^{59} \) is

\( (\sqrt{3} + \sqrt{2})^4 - (\sqrt{3} - \sqrt{2})^4 = \)

If \( z_1 = 2 - i \) and \( z_2 = 1 + i \), then \( \left|\frac{z_1 + z_2 + 1}{z_1 - z_2 + i}\right| \) is

If \( f(x) = \sqrt{\frac{x - \sin x}{x + \cos^2 x}} \), then \( \lim_{x \to \infty} f(x) \) is equal to

Three players A, B and C play a game. The probability that A, B and C finish the game are respectively \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \). The probability that the game is finished is

The focus of the parabola \( (y + 1)^2 = -8(x + 2) \) is

Which of the following is the equation of a hyperbola?

Let \( f(x) = px^2 + qx + r \), where \( p,q,r \) are constants and \( p \neq 0 \). If \( f(5) = -3f(2) \) and \( f(-4) = 0 \), then the other root of \( f \) is

Let \( f:\mathbb{R} \to \mathbb{R} \) satisfy \( f(x)f(y) = f(xy) \) for all real numbers \( x \) and \( y \). If \( f(2) = 4 \), then \( f\left(\frac{1}{2}\right) \) is

The focus of the parabola \( y^2 - 4y - x + 3 = 0 \) is

Eccentricity of the ellipse \( 4x^2 + y^2 - 8x + 4y - 8 = 0 \) is

If \( f:\mathbb{R} \to (0,\infty) \) is an increasing function and if \( \lim_{x \to 2018} \frac{f(3x)}{f(x)} = 1 \), then \( \lim_{x \to 2018} \frac{f(2x)}{f(x)} \) is equal to

If \( f \) is differentiable at \( x = 1 \), then \( \lim_{x \to 1} \frac{x^2 f(1) - f(x)}{x - 1} \) is

The value of \( \left( i^{18} + \left(\frac{1}{i}\right)^{25} \right)^3 \) is equal to

The modulus of \( \frac{1+i}{1-i} - \frac{1-i}{1+i} \) is

If \( z = e^{i4\pi/3} \), then \( (z^{192} + z^{194})^3 \) is equal to

If \( a \) and \( b \) are real numbers and \( (a + ib)^{11} = 1 + 3i \), then \( (b + ia)^{11} \) is equal to

If \( \alpha \neq \beta \), \( \alpha^2 = 5\alpha - 3 \), \( \beta^2 = 5\beta - 3 \), then the equation having \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \) as its roots is

If the conjugate of a complex number \( z \) is \( \frac{1}{i - 1} \), then \( z \) is

The value of \( \frac{2(\cos 75^\circ + i \sin 75^\circ)}{0.2(\cos 30^\circ + i \sin 30^\circ)} \) is