List of top Mathematics Questions

The distance of the point \( (7, 10, 11) \) from the line \( \frac{x-9}{2} = \frac{y-13}{3} = \frac{z-17}{6} \) along the line is:

A rectangle has a length \(L\) and a width \(W\), where \(L > W\). If the width, \(W\), is increased by 10%, which one of the following statements is correct for all values of \(L\) and \(W\)? 
Select the most appropriate option to complete the above sentence.

Let \( y = f(x) \) be the solution of the differential equation

\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]

such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:

If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32

If \( (1 + x + x^2)^{10} = 1 + a_1 x + a_2 x^2 + \dots \), then \( (a_1 + a_3 + a_5 + \dots + a_{19}) - 11a_2 \) equals to:
Let a circle \( C \) with radius \( r \) passes through four distinct points \( (0, 0), (k, 3k), (2, 3), (-1, 5) \), such that \( k \neq 0 \), then \( (10k + 2r^2) \) is equal to:
Evaluate the integral \( I = \int_0^\pi \frac{4 \cos^2 x + \sin^2 x}{8x} \, dx \).

If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and

and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to:

For the matrix [A] given below, the transpose is __________. 

\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \]

Integration of \(\ln(x)\) with \(x\), i.e. \(\int \ln(x)dx =\) __________.
 

If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
The domain of the function \[ f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}} \] is \( (a, b) \). Then \( (1 + a^2) + b^2 \) is:
If \( \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of the equation \[ \sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0 \] is:
If \[ \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} \] is finite, then \( a + b = \) __.
Total number of terms in an A.P. are even. Sum of odd terms is 24 and sum of even terms is 30. Last term exceeds the first term by \( \frac{21}{2} \). Find the total number of terms.
Let \( a_1, a_2, a_3, \dots \) be an A.P. and \( \sum_{k=1}^{12} a_{2k-1} = -\frac{72}{5} a_1 \) and \( \sum_{k=1}^{n} a_k = 0 \). Then the value of \( n \) is:
Let \( f: (0, \infty) \to \mathbb{R} \) be a function which is differentiable at all points of its domain and satisfies the condition \( x^2 f'(x) = 2f(x) + 3 \), with \( f(1) = 4 \). Then \( 2f(2) \) is equal to:
If \( \sin x + \sin^2 x = 1 \), \( x \in \left(0, \frac{\pi}{2} \right) \), then the expression \[ (\cos^2 x + \tan^2 x) + 3(\cos^4 x + \tan^4 x + \cos^4 x + \tan^4 x) + (\cos^6 x + \tan^6 x) \] is equal to:
Let \(E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1\) be an ellipse. Ellipses \(E_i\) are constructed such that their centers and eccentricities are the same as that of \(E_1\), and the length of the minor axis of \(E_{i+1}\) is the length of the major axis of \(E_i\). If \(A_i\) is the area of the ellipse \(E_i\), then \(\frac{5}{\pi} \sum_{i=1}^{\infty} A_i\) is equal to:
Let \( A = \{1,2,3\} \). The number of relations on \( A \), containing \( (1,2) \) and \( (2,3) \), which are reflexive and transitive but not symmetric, is ________.
The equation of the chord of the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \), whose mid-point is \( (3, 1) \) is:
Let \( \alpha_1 \) and \( \beta_1 \) be the distinct roots of \( 2x^2 + (\cos\theta)x - 1 = 0, \ \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha_1 + \beta_1 \), then \( 16(M + m) \) equals:
For a \( 3 \times 3 \) matrix \( M \), let trace(M) denote the sum of all the diagonal elements of \( M \). Let \( A \) be a \( 3 \times 3 \) matrix such that \( |A| = \frac{1}{2} \) and \( \text{trace}(A) = 3 \). If \( B = \text{adj}(\text{adj}(2A)) \), then the value of \( |B| + \text{trace}(B) \) equals:
Let \[ L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{-1} \quad {and} \quad L_2 : \frac{x+1}{1} = \frac{y-2}{2} = \frac{z-2}{2} \] be two lines. Let \( L_3 \) be a line passing through the point \( (\alpha, \beta, \gamma) \) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \) where \( 5x - 11y - 8z = 1 \), then \( 5x - 11y - 8z \) equals:

The integral is given by:

\[ 80 \int_{0}^{\frac{\pi}{4}} \frac{\sin\theta + \cos\theta}{9 + 16 \sin 2\theta} d\theta \]

is equals to?