List of top Mathematics Questions

If the system of linear equations
x + y + z = 2, 2x + y − z = 3, 3x + 2y + kz = 4
has a unique solution, then:

Minimize $Z = -50x + 20y$ subject to the constraints: \[ 2x - y \geq -5, \quad 3x + y \geq 3, \quad 2x - 3y \leq 12, \quad x \geq 0, \quad y \geq 0. \] Then which of the following is/are true:
(A) Feasible region is unbounded.
(B) $Z$ has no minimum value.
(C) The minimum value of $Z$ is 100.
(D) The minimum value of $Z$ is -300.

Optimize $Z = 3x + 9y$ subject to the constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]

The integrating factor of the differential equation \[ (y \log_e y) \frac{dx}{dy} + x = 2 \log_e y \] is:

Given \[ A = \begin{bmatrix} 0 & \alpha & \beta \\ -\alpha & 0 & \gamma \\ -\beta & -\gamma & 0 \end{bmatrix}, \] the matrix $A$ is a:
(A) square matrix
(B) diagonal matrix
(C) symmetric matrix
(D) skew-symmetric matrix .
Choose the correct answer from the options given below:

The equation of a line passing through the origin and parallel to the line \[ \vec{r} = 3\hat{i} + 4\hat{j} - 5\hat{k} + t(2\hat{i} - \hat{j} + 7\hat{k}), \] where $t$ is a parameter, is:
  (A) $\frac{x}{2} = \frac{y}{-1} = \frac{z}{7}$  (B) $\vec{r} = m(12\hat{i} - 6\hat{j} + 42\hat{k});$ where $m$ is the parameter  (C) $\vec{r} = (12\hat{i} - 6\hat{j} + 42\hat{k}) + s(0\hat{i} - 0\hat{j} + 0\hat{k});$ where $s$ is the parameter  (D) $\frac{x - 3}{3} = \frac{y - 4}{-4} = \frac{z + 5}{0}$  (E) $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$
Choose the correct answer from the options given below:

For \[ f(x) = \int \frac{e^x}{\sqrt{4 - e^{2x}}} \, dx, \] if the point $\left(0, \frac{\pi}{2}\right)$ satisfies $y = f(x)$, then the constant of integration of the given integral is:

Let $A$ be a matrix such that $A^2 = I$, where $I$ is an identity matrix. Then $(I + A)^4 - 8A$ is equal to:

A die is thrown three times. Events A and B are defined as below:
A: 6 on the third throw
B: 4 on the first and 5 on the second throw
The probability of A given that B has already occurred, is:

Match List-I with List-II:

List-I	List-II
(A) 4î − 2ĵ − 4k̂	(I) A vector perpendicular to both î + 2ĵ + k̂ and 2î + 2ĵ + 3k̂
(B) 4î − 4ĵ + 2k̂	(II) Direction ratios are −2, 1, 2
(C) 2î − 4ĵ + 4k̂	(III) Angle with the vector î − 2ĵ − k̂ is cos⁻¹(1/√6)
(D) 4î − ĵ − 2k̂	(IV) Dot product with −2î + ĵ + 3k̂ is 10

Choose the correct answer from the options given below:

If the solution of the differential equation \[ \frac{dy}{dx} = \frac{ax + 3}{2y + 5} \] represents a circle, then $a$ is equal to:

The area (in square units) enclosed between the curve $x^2 = 4y$ and the line $x = y$ is:

The area (in square units) bounded by the curve y = |x−2| between x = 0, y = 0, and x = 5 is:

The value of \[ \int_{-1}^{1} \tan^{-1} x \, dx \] is:

If \[ y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}}, \] then $\frac{dy}{dx}$ is:

If \[ A = \begin{bmatrix} 2 & 4 \\ x & 2 \end{bmatrix} \] and $A$ is singular, then $x$ is equal to:

For \( a, b > 0 \), if \( P = \begin{bmatrix} 0 & -a \\ 2a & b \end{bmatrix} \) and \( Q = \begin{bmatrix} b & a \\ -b & 0 \end{bmatrix} \) are two matrices such that \( PQ = \begin{bmatrix} 2 & 0 \\ 3 & 8 \end{bmatrix} \), then the value of \( (a + b)^{ab} \) is:

The domain of \( f(x) = \cos^{-1}(7x) \) is:

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 10 - x^2 \), then:

If \( P(A) = 0.4 \), \( P(B) = 0.8 \) and \( P(A | B) = 0.6 \), then \( P(A \cup B) \) is:

The angle between the lines \(\vec{r} = 3\hat{i} - 2\hat{j} + 1\hat{k} + \mu (4\hat{i} + 6\hat{j} + 12\hat{k})\) and \(\vec{r} = 7\hat{i} - 3\hat{j} + 9\hat{k} + \lambda (5\hat{i} + 8\hat{j} - 4\hat{k})\) is:

The area of the parallelogram, whose adjacent sides are given by the vectors \(\vec{a} = 2\hat{i} - \hat{j} + 5\hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{j} + 2\hat{k}\), is:

A particle moves along the curve \(6x = y^3 + 2\). The points on the curve at which the \(x\) coordinate is changing 8 times as fast as \(y\) coordinate are:

If \( (\cos x)^y = (\sin y)^x \) then \( \frac{dy}{dx} \) is:

If \( A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix} \), then \( B^{-1} A^{-1} \) is equal to: