List of top Mathematics Questions

A box with a square base is to have an open top. The surface area of the box is 147 cm\(^2\). What should its dimensions be in order that the volume is largest?
If \( x = f(t) \) and \( y = g(t) \) are differentiable functions of \( t \), so that \( y \) is a function of \( x \) and \( \frac{dx}{dt} \neq 0 \), then prove that: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \] Hence find \( \frac{dy}{dx} \), if \( x = at^2 \), \( y = 2at \).

Solve the following L.P.P. by graphical method: 
Maximize: 
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]

Find the volume of the parallelepiped whose vertices are
\( A(3,2,-1), B(-2,2,-3), C(3,5,-2) \) and \( D(-2,5,4) \).
If the equation of the transverse common tangent of the circles \(x^2 + y^2 - 4x + 6y + 4 = 0\) and \(x^2 + y^2 + 2x - 2y - 2 = 0\) is \(ax + by + c = 0\), then \(\frac{a}{c} =\)
A vector of magnitude \(\sqrt{2}\) units along the internal bisector of the angle between the vectors \(2\mathbf{i} - 2\mathbf{j} + \mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\) is:
If \( y = 44x^{45} + 45x^{44} \), then \( y'' \) is:
If \(z = 1 - \sqrt{3}\,i\), then \(z^3 - 3z^2 + 3z = \;?\)

The sum of the order and degree of the differential equation: \[ \frac{d^y}{dx^t} = c + \left( \frac{d^y}{dx^t} \right)^{\frac{3}{2}} \] is: 

If \( f(x) = 5x \csc(\sqrt{x}) - (x-2) \csc(\sqrt{x}) \), then \( \lim\limits_{x \to \infty} f(x^2) \) is:
Let \( \pi \) be the plane that passes through the point \( (-2,1,-1) \) and is parallel to the plane \( 2x - y + 2z = 0 \). Then the foot of the perpendicular drawn from the point \( (1,2,1) \) to the plane \( \pi \) is:
If \( P(2, \beta, \alpha) \) lies on the plane \( x + 2y - z - 2 = 0 \) and \( Q (\alpha, -1, \beta) \) lies on the plane \( 2x - y + 3z + 6 = 0 \), then the direction cosines of the line \( PQ \) are:
The equation of one of the tangents drawn from the point \( (0,1) \) to the hyperbola \( 45x^2 - 4y^2 = 5 \) is:
The equation of the pair of asymptotes of the hyperbola \( 4x^2 - 9y^2 - 24x - 36y - 36 = 0 \) is:
If \( A_1, A_2, A_3 \) are the areas of the ellipse \( x^2 + 4y^2 = 4 \), its director circle, and auxiliary circle respectively, then \( A_2 + A_3 - A_1 \) is:
The equation of the normal drawn to the parabola \( y^2 = 6x \) at the point \( (24,12) \) is:

The equations \( 2x - 3y + 1 = 0 \) and \( 4x - 5y - 1 = 0 \) are the equations of two diameters of the circle \( S = x^2 + y^2 + 2gx + 2fy - 11 = 0 \) and \( R \) are the points of contact of the tangents drawn from the point \( P(-2, -2) \) to this circle. If \( C \) is the centre of the circle, \( S = 0 \) is the equation of the circle, then the area (in square units) of the quadrilateral \( PQCR \) is:

When the origin is shifted to \( (h, k) \) by translation of axes, the transformed equation of \( x^2 + 2x + 2y - 7 = 0 \) does not contain \( x \) and constant terms. Then \( (2h + k) = \):
A manufacturing company noticed that 1% of its products are defective. If a dealer orders 300 items from this company, then the probability that the number of defective items is at most one is:
7 coins are tossed simultaneously and the number of heads turned up is denoted by the random variable \( X \). If \( \mu \) is the mean and \( \sigma^2 \) is the variance of \( X \), then \( \frac{\mu^2}{P(X = 3)} \) is:
If a number is drawn at random from the set \( \{1, 3, 5, 7, \dots, 59\} \), then the probability that it lies in the interval in which the function \( f(x) = x^3 - 16x^2 + 20x - 5 \) is strictly decreasing is:
If 12 dice are thrown at a time, then the probability that a multiple of 3 does not appear on any die is:
Mean deviation about the mean for the following data is: \[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-6 & 1 \\ 6-12 & 2 \\ 12-18 & 3 \\ 18-24 & 2 \\ 24-30 & 1 \\ \hline \end{array} \]
A(1, 2, 1), B(2, 3, 2), C(3, 1, 3) and D(2, 1, 3) are the vertices of a tetrahedron. If \( \theta \) is the angle between the faces ABC and ABD then \( \cos \theta \) is:
If \( \vec{c} \) is a vector along the bisector of the internal angle between the vectors \( \vec{a} = 4\hat{i} + 7\hat{j} - 4\hat{k} \) and \( \vec{b} = 12\hat{i} - 3\hat{j} + 4\hat{k} \), and the magnitude of \( \vec{c} \) is \( 3\sqrt{13} \), then \( \vec{c} = \):