List of top Mathematics Questions

List-I	List-II
(A) Absolute maximum value	(I) 3
(B) Absolute minimum value	(II) 0
(C) Point of maxima	(III) -5
(D) Point of minima	(IV) 4

Let for any three distinct consecutive terms $ a, b, c $ of an A.P., the lines $ ax + by + c = 0 $ be concurrent at the point $ P $ and $ Q (\alpha, \beta) $ be a point such that the system of equations \[x + y + z = 6,\]\[2x + 5y + \alpha z = \beta,\]\[x + 2y + 3z = 4,\]has infinitely many solutions. Then $ (PQ)^2 $ is equal to ______.

If in a G.P. of64terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:

The numbers $1, 2, 3, \ldots, m$ are arranged in random order. The number of ways this can be done, so that the numbers $1, 2, \ldots, r \, (r < m)$ appear as neighbours is:

If \[ S(x) = (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \ldots + 60(1 + x)^{60}, \, x \neq 0, \] and \[ (60)^2 S(60) = a(b)^b + b, \] where $a, b \in \mathbb{N}$, then $(a + b)$ is equal to ________.

Let $\text{P}(x, y, z)$ be a point in the first octant, whose projection in the xy-plane is the point $\text{Q}$. Let $\text{OP} = \gamma$; the angle between $\text{OQ}$ and the positive x-axis be $\theta$; and the angle between $\text{OP}$ and the positive z-axis be $\phi$, where $\text{O}$ is the origin. Then the distance of $\text{P}$ from the x-axis is:

If $ 2 \sin^3 x + \sin 2x \cos x + 4 \sin x - 4 = 0 $ has exactly 3 solutions in the interval $ \left[ 0, \frac{n \pi}{2} \right] $, $ n \in \mathbb{N} $, then the roots of the equation $ x^2 + nx + (n - 3) = 0 $ belong to:

For a differentiable function $ f : \mathbb{R} \to \mathbb{R} $, suppose \[ f'(x) = 3f(x) + \alpha, \] where $ \alpha \in \mathbb{R} $, $ f(0) = 1 $, and \[ \lim_{x \to -\infty} f(x) = 7. \] Then $ 9f(-\log_2 3) $ is equal to __________ .

Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]

If $f(x) = \begin{cases} 2+2x & ;x∈(-1,0)\\ 1-\frac x3 & ;x∈[0,3) \end{cases}$ and $g(x) = \begin{cases} x & ;x∈[0,1)\\ -x & ;x∈(-3,0) \end{cases}$. Then range of $fog(x)$ is

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = 6x^2 + 11x - 10$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

If the foot is perpendicular from (1, 2, 3) to the line $\frac{x+1}{2} = \frac{y-2}{5} = \frac{z-1}{1}$ is $( a, \beta, \gamma)$, then find $a + \beta + \gamma$

The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, $x$ m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$.
On the basis of the above information, answer the following questions:
(i)Express $\theta$ in terms of the height of the camera installed on the pole and x.
(ii) Find $\frac{d\theta}{dx}$.
(iii) (a) Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.
(iii) (b) If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is $\frac{3}{101} \, \text{rad/s}$, then find the speed of the car.

If $ x \cos(p + y) + \cos p \sin(p + y) = 0 $, prove that $ \cos p \frac{dy}{dx} = -\cos^2(p + y) $, where $ p $ is a constant.

Let $ f(x) = 3\sqrt{x - 2} + \sqrt{4 - x} $ be a real-valued function. If $ \alpha $ and $ \beta $ are respectively the minimum and the maximum values of $ f $, then $ \alpha^2 + 2\beta^2 $ is equal to:

The monthly incomes of A and B are in the ratio 8 : 7 and their expenditures are in the ratio 19 : 16. If each saves Rs 2500 per month, find the monthly income of each

If the origin is shifted to remove the first degree terms from the equation $2x^2 - 3y^2 + 4xy + 4x + 4y - 14 = 0$, then with respect to this new co-ordinate system, the transformed equation of $x^2 + y^2 - 3xy + 4y + 3 = 0$ is

Show that $11 \times 19 \times 23 + 3 \times 11$ is not a prime number.

Let PQR be a triangle with R (-1,4, 2). Suppose M(2, 1, 2) is the mid point of PQ. The distance of the centroid of $\triangle PQR$ from the point of intersection of the line$\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}$ and $\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}$ is

The combined equation of a possible pair of adjacent sides of a square with area 16 square units whose centre is the point of intersection of the lines \[ x + 2y - 3 = 0 \quad \text{and} \quad 2x - y - 1 = 0 \] is: \

(d) If vectors $ \mathbf{5i - \lambda j + 2k} $ and $ \mathbf{2i + 3j + 4k} $ are perpendicular, find $ \lambda $:

Let $ y = y(x) $ be the solution of the differential equation\[\frac{dy}{dx} + \frac{2x}{\left( 1 + x^2 \right)^2} y = x e^{\frac{1}{1+x^2}}, \quad y(0) = 0. \] Then the area enclosed by the curve \[ f(x) = y(x) e^{\frac{1}{1+x^2}} \]and the line $ y - x = 4 $ is _______.