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List of top Mathematics Questions

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 given series \( 1 - \frac{1}{2^p} + \frac{1}{3^p} - \frac{1}{4^p} + ... (p>0) \) is conditionally convergent, if 'p' lies in the interval:

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 \( (x^2 - 4xy - 2y^2)dx + (y^2 - 4xy - 2x^2)dy = 0 \), is

If \( \vec{F} \) is a vector point function and \( \phi \) is a scalar point function, then match List-I with List-II and choose the correct option:

LIST-I	LIST-II
(A) \( \text{div (grad } \phi) \)	(IV) \( \nabla \cdot \nabla \phi \)
(B) \( \text{curl (grad } \phi) \)	(III) \( \vec{0} \)
(C) \( \vec{F} \times \text{curl } \vec{F} \)	(I) \( \frac{1}{2} \nabla F^2 - (\vec{F} \cdot \nabla) \vec{F} \)
(D) \( \text{curl (curl } \vec{F}) \)	(II) \( \text{grad(div } \vec{F}) - \nabla^2 \vec{F} \)

Choose the correct answer from the options given below:

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 \( \vec{F} = x^2 \hat{i} + z \hat{j} + yz \hat{k} \), 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 \( R \) be the planar region bounded by the lines \( x = 0 \), \( y = 0 \) and the curve \( x^2 + y^2 = 4 \) in the first quadrant. Let \( C \) be the boundary of \( R \), oriented counter clockwise. Then, the value of:

\[ \oint_C x(1 - y) \, dx + (x^2 - y^2) \, dy \] is equal to:

The number of maximum basic feasible solution of the system of equations AX = b, where A is m \( \times \) n matrix, b is n \( \times \) 1 column matrix and rank of A is \(\rho(A) = m\), is:

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.

If the line \( \arg(z) = \frac{\pi}{3} \) intersects the curve \( |z - 2\sqrt{3}i| = 2 \), \( z \in \mathbb{C} \), at two distinct points \(A\) and \(B\), then \(AB\) equals:

If the system of linear equations : \[ x+y+z=4,\quad x+2y+3z=6,\quad 4x+5y+\lambda z=\mu \] has more than one solution, then the value of \( \lambda+\mu \) is equal to:

Two persons \(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of \(4\) before \(B\) throws a sum of \(9\), and \(B\) wins if he throws a sum of \(9\) before \(A\) throws a sum of \(4\). The probability that \(A\) wins if \(B\) makes the first throw is:

Let the distance between the foci of an ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b) \] be \(4\) and the distance between its directrices be \(10\). Then the length of its latus rectum is: