List of top Mathematics Questions

The integrating factor of the differential equation \( \frac{dy}{dx} + y = \frac{1 + y}{x} \) is:
If \( \vec{\alpha} = \hat{i} - 4 \hat{j} + 9 \hat{k} \) and \( \vec{\beta} = 2\hat{i} - \hat{j} + \lambda \hat{k} \) are two mutually parallel vectors, then \( \lambda \) is equal to:
Let \( A = [a_{ij}] \) be a square matrix of order 3 such that \( a_{ij} = \hat{i} - 2 \hat{j} \). Then, which of the following is true?
Let \( A \) be a matrix of order \( m \times n \) and \( B \) be a matrix such that \( A^T B \) and \( B A^T \) are defined. Then, the order of \( B \) is:
During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by \( \vec{B} = 2\hat{i} + \hat{j} \), \( \vec{W} = 6\hat{i} + 12\hat{j} \) and \( \vec{F} = 12\hat{i} + 1\hat{j} \) respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.
If \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), then prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).
Find a point \( P \) on the line \( \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9} \) such that its distance from point \( Q(2, 4, -1) \) is 7 units. Also, find the equation of the line joining \( P \) and \( Q \).
Find the image \( A' \) of the point \( A(1, 6, 3) \) in the line \( \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \). Also, find the equation of the line joining \( A \) and \( A' \).
For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data, it was revealed that two-thirds of the total applicants were females and the other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in the written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.
Sketch the graph of \( y = |x + 3| \) and find the area of the region enclosed by the curve, x-axis, between \( x = -6 \) and \( x = 0 \), using integration.
If \( x = a \left( \cos \theta + \log \tan \frac{\theta}{2} \right) \) and \( y = \sin \theta \), then find \( \frac{d^2y}{dx^2} \) at \( \theta = \frac{\pi}{4} \).
Find the absolute maximum and absolute minimum of the function \( f(x) = 2x^3 - 15x^2 + 36x + 1 \) on \( [1, 5] \).
The probability distribution for the number of students being absent in a class on a Saturday is as follows: \[ \begin{array}{|c|c|} \hline X & P(X)
\hline 0 & p
2 & 2p
4 & 3p
5 & p
\hline \end{array} \] Where \( X \) is the number of students absent. (i) Calculate \( p \).
(ii) Calculate the mean of the number of absent students on Saturday.
Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors \( \mathbf{a} = 3 \hat{i} + \hat{j} + 2 \hat{k} \) and \( \mathbf{b} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k} \). Determine the angle formed between the kite strings. Assume there is no slack in the strings.
The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate is its area increasing when the side of the triangle is 15 cm?
Solve the following linear programming problem graphically: Maximise \( Z = x + 2y \) Subject to the constraints: \[ x - y \geq 0 \] \[ x - 2y \geq -2 \] \[ x \geq 0, \, y \geq 0 \]
Evaluate: $ \tan^{-1} \left[ 2 \sin \left( 2 \cos^{-1} \frac{\sqrt{3}}{2} \right) \right]$
The diagonals of a parallelogram are given by \( \mathbf{a} = 2 \hat{i} - \hat{j} + \hat{k} \) and \( \mathbf{b} = \hat{i} + 3 \hat{j} - \hat{k}\) . Find the area of the parallelogram.
Verify that lines given by \( \vec{r} = (1 - \lambda) \hat{i + (\lambda - 2) \hat{j} + (3 - 2\lambda) \hat{k} \) and \( \vec{r} = (\mu + 1) \hat{i} + (2\mu - 1) \hat{j} - (2\mu + 1) \hat{k} \) are skew lines. Hence, find shortest distance between the lines.}
Find : \[ I = \int \frac{x + \sin x}{1 + \cos x} \, dx \]
Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]
Find the intervals in which the function $f(x) = 5x^3 - 3x^2$ is (i) increasing (ii) decreasing.

Assertion (A):
\( f(x) = \begin{cases} 3x - 8, & x \leq 5 \\ 2k, & x > 5 \end{cases} \)
is continuous at \( x = 5 \) for \( k = \frac{5}{2} \).

Reason (R):
For a function \( f \) to be continuous at \( x = a \),
\[ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \]

The graph of a trigonometric function is as shown. Which of the following will represent the graph of its inverse?
graph of a trigonometric function
Differentiate $2\cos^2 x$ w.r.t. $\cos^2 x$.