List of top Mathematics Questions

Find the position vector of point \( C \) which divides the line segment joining points \( A \) and \( B \) having position vectors \( \hat{i} + 2\hat{j} - \hat{k} \) and \( -\hat{i} + \hat{j} + \hat{k} \), respectively, in the ratio 4:1 externally. Further, find \( | \overrightarrow{AB} | : | \overrightarrow{BC} | \).
Find: \[ \int \frac{2 + \sin 2x}{1 + \cos 2x} e^x \, dx \]
If \( x \cos(p + y) + \cos p \sin(p + y) = 0 \), prove that \( \cos p \frac{dy}{dx} = -\cos^2(p + y) \), where \( p \) is a constant.
Evaluate: \[ \int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} \, dx \]
Assertion (A): Projection of \( \vec{a} \) on \( \vec{b} \) is the same as the projection of \( \vec{b} \) on \( \vec{a} \).
Reason (R): The angle between \( \vec{a} \) and \( \vec{b} \) is the same as the angle between \( \vec{b} \) and \( \vec{a} \) numerically.
Let \( E \) and \( F \) be two events such that \( P(E) = 0.1, P(F) = 0.3, P(E \cup F) = 0.4 \). Then \( P(F \,|\, E) \) is:
Evaluate: \[ \int_{0}^{\pi/2} \sin 2x \cos 3x \, dx \]
The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called:
If \( \alpha, \beta \), and \( \gamma \) are the angles which a line makes with the positive directions of \( x, y, z \) axes respectively, then which of the following is not true?
If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - 2\hat{j} + \hat{k} \), then \( \vec{a} \) and \( \vec{b} \) are:
\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:
A function \( f(x) = |1 - x + |x|| \) is:
If a line makes an angle of \( \frac{\pi}{4} \) with the positive directions of both \( x \)-axis and \( z \)-axis, then the angle which it makes with the positive direction of \( y \)-axis is:
If \( |a| = 2 \) and \( -3 \leq k \leq 2 \), then \( |a| |k| \in: \)
Given \[ \frac{d}{dx} F(x) = \frac{1}{\sqrt{2x - x^2}} \] and \( F(1) = 0 \), find \( F(x) \).
Let \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] be a square matrix such that \[ \text{adj } A = A. \] Then, \( (a + b + c + d) \) is equal to:
If \[ \begin{bmatrix} 1 & 3 & 1 \\ k & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix} \] has a determinant of \( \pm 6 \), then the value of \( k \) is:
Of the following, which group of constraints represents the feasible region given below?
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If \[ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}, \] then \( A^{-1} \) is:
If \( x = e^{y^2} \), prove that \[ \frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2}. \]
Check the differentiability of \( f(x) \) at \( x = 1 \), where: \[ f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1, \\ 3 - x, & 1 \leq x \leq 2. \end{cases} \]
Find the value of \( a \) and \( b \) so that the function \( f \) defined as: \[ f(x) = \begin{cases} \frac{x - 2}{|x - 2|} + a, & x < 2, \\ a + b, & x = 2, \\ \frac{x - 2}{|x - 2|} + b, & x > 2, \end{cases} \] is a continuous function.
Find the intervals in which the function \[ f(x) = \frac{\log x}{x} \] is strictly increasing or strictly decreasing.
Find the absolute maximum and absolute minimum values of the function \[ f(x) = \frac{x^2}{2} + \frac{2}{x} \] on the interval \( [1, 2] \).
Find the equation of the line passing through the point of intersection of the lines: \[ \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 2}{3}, \quad \frac{x - 1}{0} = \frac{y - 3}{-3} = \frac{z - 7}{2}, \] and perpendicular to these given lines.