List of top Mathematics Questions asked in KEAM

Let \[ f(x) = \frac{|5 - x|(x + 5)}{\tan(x - 5)} \quad {for} \quad x \neq 5. \] Then \[ \lim_{x \to 5} f(x) { is equal to:} \]

Let \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\tan^2 x}{1+5^x} \, dx \). Then:

The first term and the 6th term of a G.P. are 2 and \( \frac{64}{243} \) respectively. Then the sum of first 10 terms of the G.P. is:

The period of the function \( f(x) = \sin\left( \frac{3x}{2} \right) \) is equal to:
If \( \cos \theta = \frac{2 \cos \alpha + 1}{2 + \cos \alpha} \), then \( \tan^2 \left( \frac{\theta}{2} \right) \) is equal to:
If \( x_1, i=2, 3, \ldots, n \) are \( n \) observations such that \( \sum_{i=1}^{n} x_i^2 = 550 \), mean \( \bar{x} = 5 \) and variance is zero, then the number of observations is equal to:
If \( f(x) = \int_0^{x^3} (t + 4)^2 \, dt \), then \( f'(2) \) is equal to:
A line makes angles \( \alpha, \beta, \gamma \) with \( x, y \), and \( z \)-axes respectively. Then the value of \( \sin^2\alpha + \sin^2\beta - \cos^2\gamma \) is:
The area of the triangle formed by the coordinate axes and a line whose perpendicular from the origin makes an angle of 45° with the x-axis is 50 square units. Then the equation of the line is:
If \( \tan \left( \frac{\pi}{12} + 2x \right) = \cot 3x \), where \( 0<x<\frac{\pi}{2} \), then the value of \( x \) is:
The number of positive integers that have at most seven digits and contain only the digits 0 and 9 is
If \[ \int \frac{dx}{\sqrt{16 - 9x^2}} = A \sin^{-1}(Bx) + C, { where } C { is an arbitrary constant, then } A + B = \]
The function \[ f(x) = x^{3/5}(5x - 12) \] is increasing in the set:
If \( \vec{a} \) and \( \vec{b} \) are non-collinear unit vectors and \( |\vec{a} + \vec{b}|^2 = 3 \), then \( (3\vec{a} + 2\vec{b}) \cdot (3\vec{a} - \vec{b}) \) is equal to:
The value of \( \sin^2 \left( \frac{3\pi}{8} \right) + \sin^2 \left( \frac{7\pi}{8} \right) \) is:

Two circles \(C_1\) and \(C_2\) have radii 18 and 12 units, respectively. If an arc of length \( \ell \) of \(C_1\) subtends an angle 80° at the centre, then the angle subtended by an arc of same length \( \ell \) of \(C_2\) at the centre is:

If \( 2 \) is a solution of the inequality \( \frac{x-a}{a-2x}<-3 \), then \( a \) must lie in the interval:

The centre of a square is at the origin of the complex plane. If one of the vertices is at \( -3i \), then the area of the square is:
Number of integers greater than 7000 can be formed using the digits 2, 4, 5, 7, 8 is (Repetition of digits is not allowed):
cos \( A \cos 2A \) is equal to:
The coordinates of the vertex of the parabola \[ y = 2x^2 - 12x + 26 \] are

Let \[ A = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1 \end{pmatrix} \] and let \( B = \frac{1}{|A|} A \). Then the value of \( |B| \) is equal to:

If \( f'(x) = 4x\cos^2(x) \sin\left(\frac{x}{4}\right) \), then \( \lim_{x \to 0} \frac{f(\pi + x) - f(\pi)}{x} \) is equal to:

The expression
\[ \frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right)} \] is equal to:

The integral \(\int e^x \sqrt{e^x} \, dx\) equals: