List of top Mathematics Questions

Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.

A box contains 5 red balls and 7 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?

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 $ y=\sin(\cos(x^2)) $. Find $ \frac{dy}{dx} $ at $ x=\frac{\sqrt{\pi}}{2} $.

The value of $ (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) $ is:

If $ \mathbf{p} $ and $ \mathbf{q} $ are unit vectors, then which of the following values of $ \mathbf{p} \cdot \mathbf{q} $ is not possible?

In a group of 50 family members, three medicines are given to fight a virus.
• 40 members received medicine A
• 30 received medicine B
• 20 received medicine C
• 10 received all three medicines How many members received at least two medicines?

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

If \[ A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \] are three matrices, then find $ABC$.

A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector $3\hat{i} + 15\hat{j} + 6\hat{k}$ and the other is along the vector $2\hat{i} + 10\hat{j} + \lambda \hat{k}$, then the value of $\lambda$ is:

Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

If $ f(x) = x^{2}+bx+c $ and $ f(1+k) = f(1-k) $ $ \forall K \in \mathbb{R} $, for two real numbers b and c, then

Suppose that the number of terms in an A.P. is $ 2k, k \in \mathbb{N} $. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then $ k $ is equal to:

If $A = \left[ \begin{array}{cc} 1 & \cot \frac{\theta}{2} \\ -\cot \frac{\theta}{2} & 1 \end{array} \right]$ then $A^{-1} =$

Find the local maxima and local minima of the function \[ f(x) = \frac{8}{3} x^3 - 12x^2 + 18x + 5. \]

Let $ a_n $ be the $ n $-th term of an A.P. If $ S_n = a_1 + a_2 + a_3 + \cdots + a_n = 700 $, $ a_6 = 7 $, and $ S_7 = 7 $, then $ a_n $ is equal to:

Let $m, n \in \mathbb{N}$ such that $m<n$ and $P_{m \times n}(\mathbb{R})$ and $Q_{n \times m}(\mathbb{R})$ are matrices over real numbers and let $\rho(V)$ denotes the rank of the matrix V. Then, which of the following are NOT possible.
A. $ \rho(PQ) = n $
B. $ \rho(QP) = m $
C. $ \rho(PQ) = m $
D. $ \rho(QP) = \lfloor(m+n)/2\rfloor $, where $\lfloor \rfloor$ is the greatest integer function

Suppose that the number of terms in an A.P. is $ 2k, k \in \mathbb{N} $. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then $ k $ is equal to:

The distance between the lines represented by \[ 16x^2 + 9y^2 + 48x - 24xy - 36y + 35 = 0 \] is ......... units

If $y = a \cos(\log x) + b \sin(\log x)$, then $x^2y'' + xy'1$ is:

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on $3x + 2y + 2 = 0$. Then the length of the chord of the circle C, whose midpoint is (1, 2), is:

Population of Town A and B was 20,000 in 1985. In 1989, the population of Town A was 25,000, and Town B had 28,000. What will be the difference in population between the two towns in 1993?

If $A$ is a square matrix of order 3 such that $|A| = 2$, then $|\operatorname{adj}(\operatorname{adj}(A))|$ is: