List of top Mathematics Questions asked in KEAM

Given the line equation \( Ax + By + C = 0 \), which passes through the point \( (-10, 7) \) and is perpendicular to the line \( 11x - 8y - 16 = 0 \), find \( C \).

Given: \[ \sum_{k=0}^{5} \binom{10}{2k} = \alpha \quad \text{and} \quad \sum_{k=0}^{4} \binom{10}{2k+1} = \beta \] \text{Find the value of} \( \alpha - \beta \).

Find the domain of the composite function \( f \circ g(x) \) where \( f(x) = \log(5x) \) and \( g(x) = \cos(x) \).

If \[ \lim_{x \to 9} f(x) = 6 \quad {and} \quad \lim_{x \to 9} g(x) = 3, \] then \[ \lim_{x \to 9} \frac{f(x) - 2g(x)}{g(x)} \] is equal to

cos \( A \cos 2A \) is equal to:

Let a relation \( R \) on the set of natural numbers be defined by \( (x, y) \in R \) if and only if \( x^2 - 4xy + 3y^2 = 0 \) for all \( x, y \in \mathbb{N} \). Then the relation is:

If \( f(x) = \int_0^{x^3} (t + 4)^2 \, dt \), then \( f'(2) \) is equal to:

Given the Linear Programming Problem:
Maximize \( z = 11x + 7y \) subject to the constraints: \( x \leq 3 \), \( y \leq 2 \), \( x, y \geq 0 \).
Then the optimal solution of the problem is:

Evaluate \( \int \frac{e^{6 \log x} - e^{5 \log x}}{e^{4 \log x} - e^{3 \log x}} \, dx \):

If \[ \int \frac{dx}{\sqrt{16 - 9x^2}} = A \sin^{-1}(Bx) + C, { where } C { is an arbitrary constant, then } A + B = \]

The minimum value of \( f(x) = 7x^4 + 28x^3 + 31 \) is:

Evaluate the integral: \[ \int \frac{x^2 - 1}{x^4 + 3x^2 + 1} \, dx \]

If \( \tan \left( \frac{\pi}{12} + 2x \right) = \cot 3x \), where \( 0<x<\frac{\pi}{2} \), then the value of \( x \) is:

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

The common ratio of a G.P. is \( \frac{1}{2} \). If the product of the first three terms is 64, then the sum of the first 10 terms is:

The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:

The symmetric equation of the straight line passing through the points \( (-1, 4, 2) \) and \( (-3, 0, 5) \) is

For the curve \[ y = \alpha x^2 + \cos y + \beta, \] the value of \( \frac{dy}{dx} \) at \( (1,0) \) is 2. Then the value of \( \alpha \beta \) is equal to

Let \( S \) denote the set of all subsets of integers containing more than two numbers. A relation \( R \) on \( S \) is defined by:

\[ R = \{ (A, B) : \text{the sets } A \text{ and } B \text{ have at least two numbers in common} \}. \]

Then the relation \( R \) is:

The centre of the hyperbola \(16x^2 - 4y^2 + 64x - 24y - 36 = 0\) is at the point:

The axis of a parabola is parallel to the y-axis and its vertex is at \((5, 0)\). If it passes through the point \((2, 3)\), then its equation is: