List of top Mathematics Questions asked in KEAM

Solve the equation: \[ \frac{(3x - 4)^2}{9} - \frac{(4x - 3)^2}{8} = 1 \quad \text{(length of the latus rectum)} \]

The end-points of a diameter of a circle are \( (-1, 4) \) and \( (5, 4) \). Then the equation of the circle is

A 3-digit number greater than 500 is to be made with the numbers 3, 4, 5, and 7. Find the number of possible ways.

The equation of perpendicular bisector of the line segment joining the points (10, 0) and (0, -4) is

Given: \( P(B) = 0.8 \) and \( P(A \cup B) = 0.8 \), find \( P(A' \cup B') \).

The 1st and 20th term of a GP are 512 and \( \frac{1}{1024} \), respectively. Find the common ratio.

Evaluate the integral: \[ \int_0^1 \frac{t}{(t + 1)^3} \, dt \]

Given that \( |AB| = 21 \) and \( |A^{-1}| = 7 \), find \( |B| \).

Evaluate the integral: \[ \int 16x^3 \log_e x \, dx \]

If \( ay = x + b \) is the equation of the line passing through the points (-5, -2) and (4, 7), then the value of \( 2a + b \) is equal to:

Find the number of combinations of 3-lettered numbers that can be created by 3, 4, 5, and 7 with repeating. The three digits cannot be equal to zero.

Evaluate the integral: \[ \int \left( 27x^3 (1 - x^3)^{\frac{1}{3}} \right) dx \]

The range of the function \( f(x) = \frac{1}{7 + 4\sin x + 3\cos x} \).

Evaluate \( \int \frac{x^2 + 6x + 1}{(x+3)^2} \, dx \).

If \( \alpha^2 - \frac{1}{\alpha^2} = 2 \), find \( \left( \alpha + \frac{1}{\alpha} \right)^{16} \).

If \( 2 \cot^{-1} \left( \frac{4}{3} \right) = \cos^{-1} \left( \frac{x}{5} \right) \), find \( x \).

Find the equation of the curve \( (x, y) \) if \( \cos^{-1}(x-2) = \sin^{-1}(y+1) \).

If the sum of the first two terms of a G.P. is 12 and the third term is 16, find the common ratio \( r \).

If \( y = \frac{1}{3\sqrt{x}} \left( \frac{2}{x} - 3 \right) \), find the interval in which \( y \) is strictly decreasing.

If \( Z = 1 + i \) and \( Z - 24\overline{Z} = \lambda Z^2 \), find \( \lambda \).

Find the locus of \( Z = x + iy \) satisfying \[ \frac{\text{Re}(z)}{2 + i} + \frac{\text{Im}(z)}{1 + 2i} = \frac{3}{1 - 2i} \]

Find the value of \( \cot 10^\circ \times \cot 30^\circ \times \cot 45^\circ \times \cot 60^\circ \times \cot 80^\circ \)

If \( f(x) = x^2 - 10x \), \( g(x) = e^x + 5 \), find \( g(2x) - f(g(x)) \).

Max of \( Z = 7x + 10y \) subject to \( x + y \geq 3 \), \( x + 2y \geq 4 \), \( x, y \geq 0 \).

If \( |\mathbf{a} - \mathbf{b}| = \frac{\sqrt{3}}{2} \) where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).