List of top Mathematics Questions

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 \( A' \).
Solve the quadratic equation \(\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0\) using the quadratic formula.
Differentiate: \[ \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \quad \text{w.r.t.} \quad \cos^{-1}(2x\sqrt{1 - x^2}), \quad x \in \left(\frac{1}{\sqrt{2}}, 1\right) \]

Let \( y^2 = 12x \) be the parabola and \( S \) its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ) = \frac{147}{4} \). Let \( C \) be the circle described by taking \( PQ \) as a diameter. If the equation of the circle \( C \) is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then \( \beta - \alpha \) is equal to: 

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 the area of a triangle \( \triangle PQR \) with vertices \( P(5, 4) \), \( Q(-2, 4) \), and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O(2, \frac{14}{5}) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to:

Let  R = {(1, 2), (2, 3), (3, 3)}} be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is:

Let \( A = \{1, 2, 3, \dots, 10\} \) and \( B = \left\{ \frac{m}{n} : m, n \in A, m <n \text{ and } \gcd(m, n) = 1 \right\} \). Then \( n(B) \) is equal to :
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
Solve the following linear programming problem graphically: Maximise \( Z = 20x + 30y \) Subject to the constraints: \[ x + y \leq 0, \quad 2x + 3y \geq 100, \quad x \geq 14, \quad y \geq 14. \]
Let \( \hat{a} \) be a unit vector perpendicular to the vectors \[ \mathbf{b} = \hat{i} - 2\hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{c} = 2\hat{i} + 3\hat{j} - \hat{k}, \] \(\text{and makes an angle of}\) \( \cos\left( -\frac{1}{3} \right) \) \(\text{with the vector}\) \( \hat{i} + \alpha \hat{j} + \hat{k} \). \(\text{If}\) \( \hat{a} \) \(\text{makes an angle of}\) \( \frac{\pi}{3} \) \(\text{with the vector}\) \( \hat{i} + \alpha \hat{j} + \hat{k} \), then the value of \( \alpha \) \(\text{is:}\)
Let the function \(f(x) = (x^2 - 1)|x^2 - ax + 2| + \cos|x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to:
Let $ A = [a_{ij}] $ be a $3 \times 3 $ matrix such that: $$ A = \begin{bmatrix} 0 & 0 & 4 \\ 1 & 0 & 0 \\ 0 & 1 & 3 \\ \end{bmatrix}, A^{-1} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & 0 \\ 2 & 1 & 0 \end{bmatrix}. $$ Then $ a_{23} $ equals:
The particular integral of differential equation \( \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = e^{-x}\log x \) is:
Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I (Curve)} & \textbf{List-II (Orthogonal trajectory)} \\ \hline (A) \; xy = c & (I) \; \tfrac{y^2}{2} + x^2 = c \\ (B) \; e^x + e^{-y} = c & (II) \; y(y^2 + 3x^2) = c \\ (C) \; y^2 = cx & (III) \; y^2 - x^2 = 2c \\ (D) \; x^2 - y^2 = cx & (IV) \; e^y - e^{-x} = c \\ \hline \end{array} \]
Choose the correct answer from the options given below:
The integral equation corresponding to the boundary value problem \[ \frac{d^2y}{dx^2} + \lambda y(x) = 0; \quad y(0) = 0; \quad y(1) = 0 \] is

where \[ k(x,t) = \begin{cases} t(1-x), & \text{if } t < x \\ x(1-t), & \text{if } t > x \end{cases} \]
The general solution of differential equation \( \frac{d^2y}{dx^2} + 9y = \cos(3x) \) is:
The Laplace transform of \( \cos\sqrt{t} \) is:
Which one of the following statement is not correct?
Let \( \vec{F} \) be the vector valued function and f be a scalar function. Let \( \nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z} \) then,
(A) div (grad f) = \( \nabla^2 f \)
(B) curl curl \( \vec{F} \) = grad curl \( \vec{F} \) - \( \nabla^2 \vec{F} \)
(C) div curl \( \vec{F} \) = \( \vec{0} \)
(D) curl grad f = \( \vec{0} \)
(E) div (\(f\vec{F}\)) = f div \( \vec{F} \) + (grad f) \( \times \vec{F} \)
Choose the correct answer from the options given below:
The directional derivative of \( \nabla \cdot (\nabla f) \) at the point (1, -2, 1) in the direction of the normal to the surface \( xy^2z = 3x + z^2 \) where \( f = 2x^3y^2z^4 \) and \( \nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z} \) is
The value of \( \oint_S \vec{F} \cdot d\vec{s} \) where \( \vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k} \) taken over the cylinder \( x^2+y^2=4, z=0 \) and \( z=3 \) is:
The surface area of the plane \(x + 2y + 2z = 12\) cut off by \(x=0, y=0\) and \(x^2+y^2=16\) is
If R is a closed region in the xy-plane bounded by a simple closed curve C and if M(x, y) and N(x, y) are continuous functions of x and y having continuous derivative in R, then
The value of \( \int_2^3 \vec{A} \cdot \frac{d\vec{A}}{dt} dt \) if \( \vec{A}(2) = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{A}(3) = 4\hat{i} - 2\hat{j} + 3\hat{k} \) is