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List of top Mathematics Questions

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 \[ f(x) = \begin{vmatrix} 2 \cos^4 x & 2 \sin^4 x & 3 + \sin^2 2x \\ 3 + 2 \cos^4 x & 2 \sin^4 x & \sin^2 2x \\ 2 \cos^4 x & 3 + 2 \sin^4 x & \sin^2 2x \end{vmatrix} \] then $ \frac{1}{5} f'(0) $ is equal to

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 $\frac{a}{c} =$

The solution of the differential equation $ x \cos y \, dy = (x e^x \log x + e^x) \, dx $ is:

Suppose AB is a focal chord of the parabola $ y^2 = 12x $ of length $ l $ and slope $ m<\sqrt{3} $. If the distance of the chord AB from the origin is $ d $, then $ ld^2 $ is equal to _________.

The 20^th term from the end of the progression \[ 20, 19, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \ldots, -129 \frac{1}{4} \] is:

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 $ X $ is a random variable with the probability mass function (p.m.f.) as follows: \[ P(X = x) = \begin{cases} \frac{5}{16}, & x = 0, \\ \frac{kx}{48}, & x = 1, \\ \frac{1}{4}, & x = 2, \\ \frac{1}{4}, & x = 3, \end{cases} \] then find $ E(X) $:

If $ y = 44x^{45} + 45x^{44} $, then $ y'' $ is:

Let $ A $ and $ B $ be two independent events such that $ P(A) = \frac{3}{5} $ and $ P(B) = \frac{4}{9} $.

List-I (Sentences)	List-II (Adverbs)
(A) $ P(A \cap B) $	(I) $\frac{2}{5}$
(B) $ P(A \| B) $	(II)$ \frac{4}{15}$
(C) $ P(A' \| B) $	(III) $\frac{3}{5}$
(D) $ P(A' \cap B') $	(IV) $\frac{2}{9}$

Choose the correct answer from the options given below :

The value of $ I = \int_{0}^{1.5} \left\lfloor x^2 \right\rfloor dx $, where [ ] denotes the greatest integer function, is:

If \[ y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}}, \] then $\frac{dy}{dx}$ is:

The value of \[ \int_{-1}^{1} \tan^{-1} x \, dx \] is:

For \[ f(x) = \int \frac{e^x}{\sqrt{4 - e^{2x}}} \, dx, \] if the point $\left(0, \frac{\pi}{2}\right)$ satisfies $y = f(x)$, then the constant of integration of the given integral is:

Let $ A (a, b), B(3, 4) $ and $ (-6, -8) $ respectively denote the centroid, circumcentre, and orthocentre of a triangle. Then, the distance of the point $ P(2a + 3, 7b + 5) $ from the line $ 2x + 3y - 4 = 0 $ measured parallel to the line $ x - 2y - 1 = 0 $ is

The coefficient of $x^n$ in the expansion of \[\frac{e^{7x} + e^x}{e^{3x}}\] is:

If $z = 1 - \sqrt{3}\,i$, then $z^3 - 3z^2 + 3z = \;?$

\[ \lim_{x \to \frac{\pi}{2}} \frac{\int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \left( \sin\left(2t^{1/3}\right) + \cos\left(t^{1/3}\right) \right) \, dt}{\left( x - \frac{\pi}{2} \right)^2} \] is equal to:

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:

List-I (Sentences)	List-II (Adverbs)
(A) \( P(A \cap B) \)	(I) \(\frac{2}{5}\)
(B) \( P(A \| B) \)	(II)\( \frac{4}{15}\)
(C) \( P(A' \| B) \)	(III) \(\frac{3}{5}\)
(D) \( P(A' \cap B') \)	(IV) \(\frac{2}{9}\)