List of top Mathematics Questions

The value of the double integral \( \iint_R e^{x^2} dxdy \), where R is a region given by \( 2y \le x \le 2 \) and \( 0 \le y \le 1 \), is:
If p is a prime number and O(G) denotes the order of a group G and p divides O(G), then group G has an element of order p. Then, this is a statement of
Let A be a \( 2 \times 2 \) matrix with \( \det(A) = 4 \) and \( \text{trace}(A) = 5 \). Then the value of \( \text{trace}(A^2) \) is:
A complete solution of \( y'' + a_1 y' + a_2 y = 0 \) is \( y = b_1 e^{-x} + b_2 e^{-3x} \), where \( a_1, a_2, b_1 \) and \( b_2 \) are constants, then the respective values of \( a_1 \) and \( a_2 \) are:
If \( \vec{F} \) be the force and C is a non-closed arc, then \( \int_C \vec{F} \cdot d\vec{r} \) represents:
If C is a triangle with vertices (0,0), (1,0) and (1,1) which are oriented counter clockwise, then \( \oint_C 2xydx + (x^2+2x)dy \) is equal to:
For the given linear programming problem,
Minimum Z = 6x + 10y
subject to the constraints
\( x \ge 6 \); \( y \ge 2 \); \( 2x+y \ge 10 \); \( x,y \ge 0 \),
the redundant constraints are:
The value of \( v_3 \) for which the vector \( \vec{v} = e^y \sin x \hat{i} + e^y \cos x \hat{j} + v_3 \hat{k} \) is solenoidal, is:
If C is the positively oriented circle represented by \( |z|=2 \), then \( \int_C \frac{e^{2z}}{z-4} dz \) is:

Match List-I with List-II and choose the correct option:

LIST-I (Differential)LIST-II (Order/degree / nature)
(A) \( \left(y + x\left(\frac{dy}{dx}\right)^2\right)^{5/3} = x \frac{d^2y}{dx^2} \)(I) order = 2, degree = 2, non-linear
(B) \( \left(\frac{d^2y}{dx^2}\right)^{1/3} = \left(y + \frac{dy}{dx}\right)^{1/2} \)(III) order = 2, degree = 3, non-linear
(C) \( y = x \frac{dy}{dx} + \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{1/2} \)(IV) order = 1, degree = 2, non-linear
(D) \( (2 + x^3) \frac{dy}{dx} = \left(e^{\sin x}\right)^{1/2} + y \)(II) order = 1, degree = 1, linear


Choose the correct answer from the options given below:

Consider the function \( f(x, y) = x^2 + xy^2 + y^4 \), then which of the following statement is correct:
The solution of the differential equation \( \frac{xdy - ydx}{xdx + ydy} = \sqrt{x^2+y^2} \) is:
The solution of the differential equation \( (x^2 - 4xy - 2y^2)dx + (y^2 - 4xy - 2x^2)dy = 0 \), is
Which of the following are correct:
A. Every infinite bounded set of real number has a limit point
B. The set \( S = \{x : 0<x \le 1, x \in \mathbb{R}\} \) is a closed set
C. The set of whole real numbers is open as well closed set
D. The set \( S = \{1, -1, \frac{1}{2}, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{3}, ... \} \) is neither open set nor closed set
The value of the integral \( \int_0^\infty \int_0^x x e^{-x^2/y} dy \, dx \) is:
The system of linear equations \( x+y+z=6 \), \( x+2y+5z=10 \), \( 2x+3y+\lambda z=\mu \) has a unique solution, if
Let A and B be two symmetric matrices of same order, then which of the following statement are correct:
A. AB is symmetric
B. A+B is symmetric
C. \( A^T B = AB^T \)
D. \( BA = (AB)^T \)

The locus of point \( z \) which satisfies:

\[ \arg\left( \frac{z - 1}{z + 1} \right) = \frac{\pi}{3} \] is:

If \( \vec{F} = x^2 \hat{i} + z \hat{j} + yz \hat{k} \), for \( (x, y, z) \in \mathbb{R}^3 \), then:

Evaluate \( \oiint_S \vec{F} \cdot d\vec{S} \), where \( S \) is the surface of the cube formed by \( x = \pm 1, y = \pm 1, z = \pm 1 \):

Let \( b_1 = 3, b_2, b_3, \ldots \) be a geometric progression of increasing positive numbers. Let \[ \sum_{n=1}^{20} b_{3n} = 4\sum_{n=1}^{20} b_{3n-2}. \] Then, the sum of the first ten terms of the G.P. is:
Solve the following pair of linear equations by graphical method : \(2x + y = 9\) and \(x - 2y = 2\).
In a trapezium \(ABCD\), \(AB \parallel DC\) and its diagonals intersect at \(O\). Prove that \[ \frac{OA}{OC} = \frac{OB}{OD} \]
The probability that a student buys a colouring book is 0.7, and a box of colours is 0.2. The probability that she buys a colouring book, given that she buys a box of colours, is 0.3. Find:
(i) The probability that she buys both the colouring book and the box of colours.
(ii) The probability that she buys a box of colours given she buys the colouring book.
Find length and breadth of a rectangular park whose perimeter is \(100 \, \text{m}\) and area is \(600 \, \text{m}^2\).
Using prime factorisation, find the HCF of 144, 180 and 192.