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List of top Mathematics Questions

A drone is flying at a height of $h$ metres. At an instant it observes the angle of elevation of the top of an industrial turbine as $60^\circ$ and the angle of depression of the foot of the turbine as $30^\circ$. If the height of the turbine is $200\, \text{metres}$, find the value of $h$ and the distance of the drone from the turbine.
(Use $\sqrt{3} = 1.73$)

If $\begin{vmatrix} 2x & 3 \\ x & -8 \\ \end{vmatrix} = 0$, then the value of $x$ is:

Evaluate the following integrals: \[ \int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx \] and \[ \int \frac{1}{x^4 + 5x^2 + 6} \, dx \]

The variance of the numbers 8, 21, 34, 47, \dots, 320, is:

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

Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.

The domain and range of a real valued function $ f(x) = \cos (x-3) $ are respectively.

The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)

Find the value of $ x $ in the equation $ 4(x - 2) = 3(x + 5) $.

Differentiate $\frac{\sin x}{\sqrt{\cos x}}$ with respect to x.

Let A and B be the two points of intersection of the line $ y + 5 = 0 $ and the mirror image of the parabola $ y^2 = 4x $ with respect to the line $ x + y + 4 = 0 $. If $ d $ denotes the distance between A and B, and $ a $ denotes the area of $ \Delta SAB $, where $ S $ is the focus of the parabola $ y^2 = 4x $, then the value of $ (a + d) $ is:

The points $ (1,5), (2,3) $ and $ (-2, -11) $ form a:

The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate is its area increasing when the side of the triangle is 15 cm?

Given that $ \vec{a} \parallel \vec{b} $, $ \vec{a} \cdot \vec{b} = \frac{49}{2} $, and $ |\vec{a}| = 7 $, find $ |\vec{b}| $.

The integrating factor of the differential equation $ (x + 2y^3) \frac{dy}{dx} = 2y $ is:

If $ \int \frac{1}{2x^2} \, dx = k \cdot 2x + C $, then $ k $ is equal to:

The locus of point $ z $ which satisfies:

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

Let $ f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} $ be defined as:

\[ f(x, y) = \begin{cases} \frac{x}{\sqrt{x^2 + y^2}} & \text{if } (x, y) \neq (0, 0) \\ 1 & \text{if } (x, y) = (0, 0) \end{cases} \] Then, which of the following statements is true?

The value of the integral:

\[ \oint_C \frac{z^3 - z}{(z - z_0)^3} \, dz \] where $ z_0 $ is outside the closed curve $ C $ described in the positive sense, 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:

Find the residue of $ (67 + 89 + 90 + 87) \pmod{11} $:

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 $:

If, $ I_n = \int_{-\pi}^{\pi} \frac{\cos(nx)(1+2^x)}{dx} $, where $ n = 0, 1, 2, \dots $, then which of the following are correct?

A. $ I_n = I_{n+2} $, for all $ n = 0, 1, 2, \dots $
B. $ I_n = 0 $, for all $ n = 0, 1, 2, \dots $
C. $ \sum_{n=1}^{10} I_n = 2^{10} $
D. $ \sum_{n=1}^{10} I_n = 0 $

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:

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

LIST-I (Infinite Series)	LIST-II (Nature of Series)
(A) $ 12 - 7 - 3 - 2 + 12 - 7 - 3 - 2 + \dots $	(II) oscillatory
(B) $ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots $	(IV) conditionally convergent
(C) $ \sum_{n=0}^{\infty} \left( (n^3+1)^{1/3} - n \right) $	(I) convergent
(D) $ \sum_{n=1}^{\infty} \frac{1}{n \left( 1 + \frac{1}{n} \right)} $	(III) divergent

Choose the correct answer from the options given below:

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) \( \text{grad(div } \vec{F}) - \nabla^2 \vec{F} \)

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

LIST-I (Infinite Series)	LIST-II (Nature of Series)
(A) \( 12 - 7 - 3 - 2 + 12 - 7 - 3 - 2 + \dots \)	(II) oscillatory
(B) \( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \)	(IV) conditionally convergent
(C) \( \sum_{n=0}^{\infty} \left( (n^3+1)^{1/3} - n \right) \)	(I) convergent
(D) \( \sum_{n=1}^{\infty} \frac{1}{n \left( 1 + \frac{1}{n} \right)} \)	(III) divergent