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List of top Mathematics Questions

Let A be a $ 2 \times 2 $ matrix with real entries such that $ A^T = \alpha A + I $, where $ \alpha \in \mathbb{R} \setminus \{-1, 1\} $. If $ \text{det}(A^2 - A) = 4 $, then the sum of all possible values of $ \alpha $ is equal to:

Let $ f: [2, 4] \to \mathbb{R} $ be a differentiable function such that $ (x \log x) f'(x) + (\log x) f(x) \geq 1 $, $ x \in [2, 4] $ with $ f(2) = \frac{1}{2} $ and $ f(4) = \frac{1}{4} $. Consider the following two statements:
$ (A) \quad f(x) \geq 1 \quad \text{for all} \quad x \in [2, 4] $
$ (B) \quad f(x) \leq \frac{1}{8} \quad \text{for all} \quad x \in [2, 4] $ Then,

The value of the integral \[ \int_{\log_2}^{-\log_2} e^x \left( \log \left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx \] is equal to:

The number of elements in the set $S = \{ \theta \in [0, 2\pi] : 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 = 0 \}$ is:

Let $x_1, x_2, \dots, x_{100}$ be in an arithmetic progression, with $x_1 = 2$ and their mean equal to 200. If $y_i = (i \cdot x_i)$, then the mean of $y_1, y_2, \dots, y_{100}$ is:

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is______

Statement (P $\Rightarrow$ Q) $\land$ (R $\Rightarrow$ Q) is logically equivalent to :

From the top A of a vertical wall AB of height 30 m, the angles of depression of the top P and bottom Q of a vertical tower PQ are 15$^\circ$ and 60$^\circ$ respectively. B and Q are on the same horizontal level. If C is a point on AB such that CB = 15 m, then the area (in m$^2$) of the quadrilateral BCPQ is equal to :

Let $A = \{x \in \mathbb{R}: |x+3| + |x+4| \le 3\}$,} \[ B = \left\{ x \in \mathbb{R} : 3 \cdot \sum_{r=1}^{\infty} \frac{3^{x-3}}{10^r} < 3^{-3x} \right\}, \] where $[x]$ denotes the greatest integer function. Then,

Let the number $(22)^{2022}$ + $(2022)^{22}$ leave the remainder $ \alpha $ when divided by 3 and $ \beta $ when divided by 7. Then $ (\alpha^2 + \beta^2) $ is equal to:}

If $(20)^{19}$ + $2(21)(20)^{18} + 3(21)^2 (20)^{17} +$ $\cdots + 20(21)^{19} =$ $k(20)^{19}$, then $k$ is equal to _____ :

Let the quadratic curve passing through the point $ (-1, 0) $ and touching the line $ y = x $ at $ (1, 1) $ be $ y = f(x) $. Then the x-intercept of the normal to the curve at the point $ (\alpha, \alpha + 1) $ in the first quadrant is:

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q - p$ is equal to:

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is:

If the coefficients of $ x $ and $ x^2 $ in $ (1 + x)^p(1 - x)^q $ are 4 and -5 respectively, then $ 2p + 3q $ is equal to:

Let the function $f(x)=2 x^3+(2 p-7) x^2+3(2 p-9) x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$ Then, the set of all values of $p$ is

Let $[t]$ denote the greatest integer less than or equal to t Then the value of the integral $\int\limits_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to

The plane $2x−y+z=4$ intersects the line segment joining the points $A(a,−2,4)$ and $B(2,b,−3)$ at the point C in the ratio $2:1$ and the distance of the point C from the origin is $\sqrt5$. If $ab<0 $ and $P$ is the point $(a−b,b,2b−a)$. Then $CP^2 $ is equal to

Let the line $\frac{x}{1}=\frac{6−y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{1}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3−y}{3}=\frac{z-6}{1}$ at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane 2x – 2y + z = 14 is

Let the plane P pass through the intersection of the planes $2 x+3 y-z=2$and $x+2 y+3 z=6,$ and be perpendicular to the plane $2 x+y-z+1=0$If d is the distance of P from the point (-7,1,1), then $d^2$ is equal to :

The distance of the point $(7,-3,-4)$ from the plane passing through the points $(2,-3,1),(-1,1,-2)$ and $(3,-4,2)$ is :

Let f and g be twice differentiable functions on R such that
f''(x)=g''(x)+6x
f''(1)=4g'(1)-3=9
f(2)=3g(2)=12.
Then which of the following is NOT true?

Let $M$ be the maximum value of the product of two positive integers when their sum is 66 Let the sample space $S =\left\{x \in Z : x(66-x) \geq \frac{5}{9} M\right\}$ and the event $A =\{x \in S : x$ is a multiple of 3$\}$ Then $P ( A )$ is equal to

Let f be a continuous function satisfying $∫^{t ^2}_ 0 ( f ( x ) + x ^2 ) d x = \frac{4}{3} t^3 , ∀ t > 0.$ Then $f(\frac{π^2}{4})$ is equal to

Let a circle $C_1$ be obtained on rolling the circle $x^2+y^2-4 x-6 y+11=0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$ Let $C_2$ be the image of $C_1$ in $T$ Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is: