Ajaysaini231/JEE-Main-2025-Math huggingface dataset:
Question: Let $[x]$ denote the greatest integer less than or equal to $x$. Then the domain of $f(x)=\sec^{-1}(2[x]+1)$ is:
Options:
$(-\infty, -1] \cup [0, \infty)$
$(-\infty, -1] \cup [1, \infty)$
$(-\infty, \infty)$
$(-\infty, \infty) - \{0\}$
Answer: $(-\infty, \infty)$
Question: Let $S$ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $S$, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:
Options:
$\frac{1}{2}$
$\frac{1}{4}$
$\frac{2}{3}$
$\frac{1}{3}$
Answer: $\frac{1}{2}$
Question: If $\sum_{r=1}^{13} \left\{\frac{1}{\sin \left(\frac{1}{4}+(r-1) \frac{2}{3}\right) \sin \left(\frac{7}{4}+\frac{2}{7}\right)}\right\}=a \sqrt{3}+b$, $a, b \in \mathbb{Z}$, then $a^2+b^2$ is equal to:
Options:
10
4
2
8
Answer: 8
Question: Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int_{0}^{x} t f(t) \, dt$. If $g(x^3)=x^6+x^7$, then value of $\sum_{r=1}^{15} f(r^3)$ is:
Options:
270
340
320
310
Answer: 310
Question: Let $f:[0,3] \rightarrow A$ be defined by $f(x)=2x^3-15x^2+36x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $g(x)=\frac{x^{200}}{x^{2005}+1}$. If both the functions are onto and $S=\{x \in \mathbb{Z} : x \in A \text{ or } x \in B\}$, then $n(S)$ is equal to:
Options:
29
30
31
36
Answer: 30
Question: Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag $B_2$ is:
Options:
$\frac{4}{15}$
$\frac{1}{3}$
$\frac{2}{5}$
$\frac{4}{3}$
Answer: $\frac{4}{15}$
Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=x f(x)$ for all $x \in \mathbb{R}$, $\int_{0}^{2} x F'(x) \, dx = 6$ and $\int_{0}^{2} x^2 F''(x) \, dx = 40$, then $F'(2) + \int_{0}^{2} F(x) \, dx$ is equal to:
Options:
13
15
9
Answer: 11
Question: For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left(\frac{1}{a_k}\right)$, then the value of $507 S_{2025}$ is:
Options:
540
675
1350
135
Answer: 675
Question: Let $f: \mathbb{R} - \{0\} \rightarrow (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is:
Options:
7
6
1
9
Answer: 6
Question: If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line:
Options:
$x + 9y = 36$
$4x - 9y = 12$
$6x - 9y = 20$
$9x - 9y = 32$
Answer: $6x - 9y = 20$
Question: If $f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx$, $f(0) = -6$, then $f(1)$ is equal to:
Options:
$4(\log_e 2 - 2)$
$2 - \log_e 2$
$\log_e 2 + 2$
$4(\log_e 2 + 2)$
Answer: $4(\log_e 2 - 2)$
Question: The area of the region bounded by the curves $x(1 + y^2) = 1$ and $y^2 = 2x$ is:
Options:
$2\left(\frac{\pi}{2} - \frac{1}{3}\right)$
$\frac{\pi}{2} - \frac{1}{3}$
$\frac{\pi}{4} - \frac{1}{3}$
$\frac{1}{2}\left(\frac{\pi}{2} - \frac{1}{3}\right)$
Answer: $\frac{\pi}{2} - \frac{1}{3}$
Question: The square of the distance of the point $\left(\frac{10}{7}, \frac{22}{7}, 7\right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is:
Options:
54
44
41
66
Answer: 66
Question: If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $\left(\sqrt{2}, \frac{4}{3}\right)$, and the length of the chord is $\frac{2\sqrt{6}}{3}$, then $\alpha$ is:
Options:
20
18
26
Answer: 22
Question: If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to:
Options:
-2
6
-6
2
Answer: 2
Question: Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum of all possible distinct values of $m$ is:
Options:
$-2\sqrt{10}$
1
6
-6
Answer: 6
Question: 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 $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is:
Options:
Answer: 14
Question: The number of natural numbers between 212 and 999, such that the sum of their digits is 15, is:
Options:
Answer: 64
Question: If $y = y(x)$ is the solution of the differential equation $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y\right) \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{\pi^2 - 8}{4}$, then $y^2(0)$ is equal to:
Options:
Answer: 4
Question: The interior angles of a polygon with $n$ sides are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to:
Options:
Answer: 20
Question: Let $f(x) = \lim_{n \to \infty} \sum_{t=0}^{n} \left(\frac{\tan \left(\frac{x}{2^{n+1}}\right) + \tan^3 \left(\frac{x}{2^{n+1}}\right)}{1 - \tan^3 \left(\frac{x}{2^{n+1}}\right)}\right)$. Then $\lim_{x \to 0} \frac{x}{(x - f(x))}$ is equal to:
Options:
Answer: 1
Question: Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\sum_{i=1}^{10}(x_i-2)=30$, $\sum_{i=1}^{10}(x_i-\beta)^2=98$, $\beta>2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1-1)+4\beta$, $2(x_2-1)+4\beta$, $\ldots$, $2(x_{10}-1)+4\beta$, then $\frac{\beta \mu}{\sigma^2}$ is equal to:
Options:
100
120
110
90
Answer: 100
Question: Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{\text{th}}$ term is:
Options:
90
84
122
108
Answer: 90
Question: The number of solutions of the equation $\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)=0$ is:
Options:
2
3
1
4
Answer: 4
Question: Define a relation $\mathrm{R}$ on the interval $\left[0, \frac{x}{y}\right)$ by $x \mathrm{R} y$ if and only if $\sec^2 x - \tan^2 y = 1$. Then $\mathrm{R}$ is:
Options:
both reflexive and transitive but not symmetric
an equivalence relation
reflexive but neither symmetric nor transitive
both reflexive and symmetric but not transitive
Answer: an equivalence relation
Question: Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
Options:
392
384
192
96
Answer: 192
Question: Let $\mathrm{P}$ be the set of seven-digit numbers with sum of their digits equal to 11. If the numbers in $\mathrm{P}$ are formed by using the digits 1, 2, and 3 only, then the number of elements in the set $P$ is:
Options:
173
164
158
161
Answer: 161
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\imath}+2\hat{\jmath}+\hat{k}$ and $\overrightarrow{\mathrm{b}}=2\hat{\imath}+7\hat{\jmath}+3\hat{k}$. Let $\mathrm{L}_1: \overrightarrow{\mathrm{r}}=(-\hat{\imath}+2\hat{\jmath}+\hat{k})+\lambda \overrightarrow{\mathrm{a}}, \lambda \in \mathbf{R}$ and $\mathrm{L}_2: \overrightarrow{\mathrm{r}}=(\hat{\jmath}+\hat{k})+\mu \overrightarrow{\mathrm{b}}, \mu \in \mathbf{R}$ be two lines. If the line $\mathrm{L}_3$ passes through the point of intersection of $\mathrm{L}_1$ and $\mathrm{L}_2$, and is parallel to $\vec{a}+\vec{b}$, then $\mathrm{L}_3$ passes through the point:
Options:
$(5,17,4)$
$(2,8,5)$
$(8,26,12)$
$(-1,-1,1)$
Answer: $(8,26,12)$
Question: Let $\overrightarrow{\mathrm{a}}=2\hat{\imath}-\hat{\jmath}+3\hat{k}$, $\overrightarrow{\mathrm{b}}=3\hat{\imath}-5\hat{\jmath}+\hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}$ and $(\vec{a}+\vec{c}) \cdot (\vec{b}+\vec{c})=168$. Then the maximum value of $|\vec{c}|^2$ is:
Options:
462
77
154
308
Answer: 308
Question: The integral $80 \int_0^{\frac{\pi}{2}} \left(\frac{\sin \theta + \cos \theta}{9 + 16 \sin 2 \theta}\right) d\theta$ is equal to:
Options:
$3 \log_e 4$
$4 \log_e 3$
$6 \log_e 4$
$2 \log_e 3$
Answer: $4 \log_e 3$
Question: Let the ellipse $\mathbf{E}_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \mathbf{a} > \mathbf{b}$ and $\mathbf{E}_2: \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, \mathbf{A} < \mathbf{B}$ have the same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$ meet at $A, B, C$ and $D$, then the area of the quadrilateral $ABCD$ equals:
Options:
$\frac{12 \sqrt{6}}{5}$
$6 \sqrt{6}$
$\frac{18 \sqrt{6}}{5}$
$\frac{24 \sqrt{6}}{5}$
Answer: $\frac{24 \sqrt{6}}{5}$
Question: Let $\mathbf{A}=\left[a_{ij}\right]=\left[\begin{array}{cc} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{array}\right]$. If $\mathbf{A}_{ij}$ is the cofactor of $\mathbf{a}_{ij}$, $\mathbf{C}_{ij}=\sum_{k=1}^2 \mathbf{a}_{ik} \mathbf{A}_{jk}, 1 \leq i, j \leq 2$, and $\mathbf{C}=\left[\mathbf{C}_{ij}\right]$, then $8|\mathbf{C}|$ is equal to:
Options:
288
222
242
262
Answer: 242
Question: Let $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbf{C}$. Then the minimum value of $|z_1 - z_2|$ is:
Options:
13
10
3
7
Answer: 7
Question: Let $\mathbf{L}_1: \frac{x-1}{1} = \frac{y-2}{1} = \frac{z-1}{2}$ and $\mathbf{L}_2: \frac{x+1}{1} = \frac{y-2}{2} = \frac{z}{4}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $L_1$, then $|5\alpha - 11\beta - 8\gamma|$ equals:
Options:
20
18
25
16
Answer: 25
Question: Let $M$ and $m$ respectively be the maximum and the minimum values of $\begin{vmatrix} 1 + \sin^2 x & \cos^2 x & 4 \sin 4x \\ 1 + \sin^2 x & \cos^2 x & 4 \sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 4x \end{vmatrix}, x \in \mathbf{R}$. Then $M^4 - m^4$ is equal to:
Options:
1280
1295
1215
1040
Answer: 1280
Question: Let $ABC$ be a triangle formed by the lines $7x - 6y + 3 = 0$, $x + 2y - 31 = 0$ and $9x - 2y - 19 = 0$. Let the point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3x + 6y - 53 = 0$. Then $h^2 + k^2 + hk$ is equal to:
Options:
47
37
36
40
Answer: 37
Question: The value of $\lim_{n \to \infty} \left( \sum_{k=1}^n \frac{k^3 + 6k^2 + 11k + 6}{(k+3)!} \right)$ is:
Options:
$\frac{4}{3}$
2
$\frac{7}{3}$
$\frac{5}{3}$
Answer: $\frac{5}{3}$
Question: The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7} + \sqrt[3]{11})^n$ is 183, is:
Options:
2184
2196
2148
2172
Answer: 2184
Question: Let $y = y(x)$ be the solution of the differential equation $\cos x (\log_e (\cos x))^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_e 2}$, then $y\left(\frac{\pi}{6}\right)$ is equal to:
Options:
$\frac{1}{\log_e 3 - \log_e 4}$
$\frac{2}{\log_e 3 - \log_e 4}$
$\frac{1}{\log_e 4 - \log_e 3}$
$\frac{-1}{\log_e 4}$
Answer: $\frac{1}{\log_e 3 - \log_e 4}$
Question: Let the line $x + y = 1$ meet the circle $x^2 + y^2 = 4$ at the points $A$ and $B$. If the line perpendicular to $AB$ and passing through the midpoint of the chord $AB$ intersects the circle at $C$ and $D$, then the area of the quadrilateral $ADBC$ is equal to:
Options:
$\sqrt{14}$
$3 \sqrt{7}$
$2 \sqrt{14}$
$5 \sqrt{7}$
Answer: $2 \sqrt{14}$
Question: Let the area of the region $\{(x, y): 2y \leq x^2 + 3, y + |x| \leq 3, y \geq |x-1|\}$ be $A$. Then $6A$ is equal to:
Options:
16
12
14
18
Answer: 14
Question: Let $\mathbf{S} = \{x: \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} (2x + 1)\}$. Then $\sum_{x \in \mathbf{S}} (2x - 1)^2$ is equal to
Options:
Answer: 5
Question: Let $f: (0, \infty) \to \mathbf{R}$ be a twice differentiable function. If for some $a \neq 0$, $\int_0^1 f(\lambda x) d\lambda = a f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f'\left(\frac{1}{16}\right)$ is equal to
Options:
Answer: 112
Question: The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is
Options:
Answer: 1405
Question: Let $\mathbf{S} = \{m \in \mathbf{Z}: \mathbf{A}^{m^2} + \mathbf{A}^m = 3 \mathbf{I} - \mathbf{A}^{-6}\}$, where $\mathbf{A} = \left[\begin{array}{cc} 2 & -1 \\ 1 & 0 \end{array}\right]$. Then $n(\mathbf{S})$ is equal to
Options:
Answer: 2
Question: Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbf{N}$ for which $\lim_{x \to 0^+} \left( x \left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \ldots + \left[\frac{p}{x}\right] \right) - x^2 \left( \left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \ldots + \left[\frac{p^2}{x^2}\right] \right) \right) \geq 1$ is equal to
Options:
Answer: 24
Question: Let $f(x)=\int_0^x t\left(t^2-9t+20\right) dt$, $1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha+\beta)$ equals:
Options:
253
154
125
157
Answer: 157
Question: Let $\hat{a}$ be a unit vector perpendicular to the vectors $\vec{b}=\hat{i}-2\hat{j}+3\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}-\hat{k}$, and makes an angle of $\cos^{-1}\left(-\frac{1}{3}\right)$ with the vector $\hat{i}+\hat{j}+\hat{k}$. If $\hat{a}$ makes an angle of $\frac{\pi}{3}$ with the vector $\hat{i}+\alpha\hat{j}+\hat{k}$, then the value of $\alpha$ is:
Options:
$\sqrt{6}$
$-\sqrt{6}$
$-\sqrt{3}$
$\sqrt{3}$
Answer: $-\sqrt{6}$
Question: If for the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+(\tan x)y=\frac{2+\sec x}{(1+2\sec x)^2}$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}$, then $f\left(\frac{\pi}{4}\right)$ is equal to:
Options:
$\frac{\sqrt{3}+1}{10(4+\sqrt{3})}$
$\frac{5-\sqrt{3}}{2\sqrt{2}}$
$\frac{9\sqrt{3}+3}{10(4+\sqrt{3})}$
$\frac{4-\sqrt{2}}{14}$
Answer: $\frac{4-\sqrt{2}}{14}$
Question: Let $P$ be the foot of the perpendicular from the point $(1,2,2)$ on the line $L: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$. Let the line $\vec{r}=(-\hat{i}+\hat{j}-2\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$, $\lambda \in \mathbb{R}$, intersect the line $L$ at $Q$. Then $2(PQ)^2$ is equal to:
Options:
25
19
29
27
Answer: 27
Question: Let $A=\left[a_{ij}\right]$ be a matrix of order $3 \times 3$, with $a_{ij}=(\sqrt{2})^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha+\beta\sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then $\alpha+\beta$ is equal to:
Options:
280
224
210
168
Answer: 224
Question: Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right-angled triangle $AMN$ is inscribed in the triangle $OAB$, where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$, respectively. If the area of the triangle $AMN$ is $\frac{4}{9}$ of the area of the triangle $OAB$ and $AN:NB=\lambda:1$, then the sum of all possible value(s) of $\lambda$ is:
Options:
2
$\frac{5}{2}$
$\frac{1}{2}$
$\frac{15}{6}$
Answer: 2
Question: If all the words with or without meaning made using all the letters of the word 'KANPUR' are arranged as in a dictionary, then the word at $440^{\text{th}}$ position in this arrangement is:
Options:
PRNAUK
PRKANU
PRKAUN
PRNAKU
Answer: PRKAUN
Question: If the set of all $a \in \mathbb{R}$, for which the equation $2x^2+(a-5)x+15=3a$ has no real root, is the interval $(\alpha, \beta)$, and $X=\{x \in \mathbb{Z}: \alpha < x < \beta\}$, then $\sum_{x \in X} x^2$ is equal to:
Options:
2109
2129
2119
2139
Answer: 2139
Question: Let $A=\left[a_{ij}\right]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0,1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:
Options:
$\frac{3}{4}$
$\frac{5}{8}$
$\frac{5}{8}$
$\frac{1}{4}$
Answer: $\frac{5}{8}$
Question: Let the function $f(x)=\left(x^2+1\right)\left|x^2-ax+2\right|+\cos|x|$ be not differentiable at the two points $x=\alpha=2$ and $x=\beta$. Then the distance of the point $(\alpha, \beta)$ from the line $12x+5y+10=0$ is equal to:
Options:
5
4
3
2
Answer: 3
Question: Let the area enclosed between the curves $|y|=1-x^2$ and $x^2+y^2=1$ be $\alpha$. If $9\alpha=\beta\pi+\gamma$, $\beta, \gamma$ are integers, then the value of $|\beta-\gamma|$ equals:
Options:
27
33
15
18
Answer: 33
Question: The remainder, when $7^{103}$ is divided by 23, is equal to:
Options:
6
17
9
14
Answer: 14
Question: If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose midpoint is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha+\beta$ is equal to:
Options:
58
46
37
72
Answer: 58
Question: If the domain of the function $\log_5\left(18x-x^2-77\right)$ is $(\alpha, \beta)$ and the domain of the function $\log_{(\alpha-1)}\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2+\beta^2+\gamma^2$ is equal to:
Options:
195
179
186
174
Answer: 186
Question: Let a circle $C$ pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3x+2y+2=0$. Then the length of the chord of the circle $C$, whose midpoint is $(1,2)$, is:
Options:
$\sqrt{3}$
$2\sqrt{2}$
$2\sqrt{3}$
$4\sqrt{2}$
Answer: $2\sqrt{3}$
Question: Let a straight line $L$ pass through the point $P(2,-1,3)$ and be perpendicular to the lines $\frac{x-1}{4}=\frac{y+1}{4}=\frac{z-3}{2}$ and $\frac{x-3}{4}=\frac{y-2}{3}=\frac{z+2}{4}$. If the line $L$ intersects the $yz$-plane at the point $Q$, then the distance between the points $P$ and $Q$ is:
Options:
$\sqrt{10}$
$2\sqrt{3}$
2
3
Answer: 3
Question: Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability that the ball drawn is white is $\frac{29}{45}$, then $n$ is equal to:
Options:
6
3
5
4
Answer: 6
Question: Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x+y+z=1$, $x+2y+4z=m$, and $x+4y+10z=m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to:
Options:
3080
560
3410
440
Answer: 440
Question: Let $S=\mathbb{N} \cup \{0\}$. Define a relation $R$ from $S$ to $\mathbb{R}$ by: $R=\left\{(x, y): \log_e y = x \log_e \left(\frac{x}{5}\right), x \in S, y \in \mathbb{R}\right\}$. Then, the sum of all the elements in the range of $R$ is equal to:
Options:
$\frac{10}{9}$
$\frac{3}{2}$
$\frac{5}{2}$
$\frac{5}{3}$
Answer: $\frac{5}{3}$
Question: If $\sin x + \sin^2 x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, then $\left(\cos^{12} x + \tan^{12} x\right) + 3\left(\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x\right) + \left(\cos^6 x + \tan^6 x\right)$ is equal to:
Options:
4
1
3
2
Answer: 2
Question: If $24 \int_0^{\frac{\pi}{4}} \left(\sin \left|4x - \frac{\pi}{12}\right| + |2 \sin x|\right) dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to:
Options:
Answer: 12
Question: Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + \left(a_5 + a_{10} + a_{15} + \ldots + a_{2020}\right) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to:
Options:
Answer: 11132
Question: If $\lim_{t \to 0} \left(\int_0^1 (3x + 5)^t dx\right)^{\frac{1}{t}} = \frac{2}{56} \left(\frac{8}{5}\right)^{\frac{2}{\alpha}}$, then $\alpha$ is equal to:
Options:
Answer: 64
Question: Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ) = \frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to:
Options:
Answer: 1328
Question: Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $\left|\frac{x+1}{x+b}\right| = 1$ and $\left|\begin{array}{ccc} x+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{array}\right| = 1$, $z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to:
Options:
Answer: 10
Question: Let $O$ be the origin, the point $A$ be $z_1 = \sqrt{3} + 2 \sqrt{2} i$, the point $B(z_2)$ be such that $\sqrt{3} |z_2| = |z_1|$ and $\arg(z_2) = \arg(z_1) + \frac{\pi}{2}$. Then
Options:
area of triangle $ABO$ is $\frac{11}{\sqrt{3}}$
$ABO$ is an obtuse angled isosceles triangle
area of triangle $ABO$ is $\frac{11}{4}$
$ABO$ is a scalene triangle
Answer: $ABO$ is an obtuse angled isosceles triangle
Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = (2 + 3a) x^2 + \left( \frac{a + 2}{a - 1} \right) x + b, a \neq 1$. If $f(x + y) = f(x) + f(y) + 1 - \frac{2}{7} xy$, then the value of $28 \sum_{i=1}^{5} |f(i)|$ is
Options:
545
715
735
675
Answer: 675
Question: Let $ABCD$ be a trapezium whose vertices lie on the parabola $y^2 = 4x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to $y$-axis. If the diagonal $AC$ is of length $\frac{25}{4}$ and it passes through the point $(1,0)$, then the area of $ABCD$ is
Options:
$\frac{75}{8}$
$\frac{25}{2}$
$\frac{125}{8}$
$\frac{75}{8}$
Answer: $\frac{75}{8}$
Question: The sum of all local minimum values of the function $f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x - 4)(x - 5), & x > 2 \end{cases}$ is
Options:
$\frac{127}{72}$
$\frac{131}{72}$
$\frac{171}{72}$
$\frac{167}{72}$
Answer: $\frac{127}{72}$
Question: Let ${}^n C_{r-1} = 28$, ${}^n C_r = 56$ and ${}^n C_{r+1} = 70$. Let $A(4 \cos t, 4 \sin t)$, $B(2 \sin t, -2 \cos t)$ and $C(3r - n, r^2 - n - 1)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x - 1)^2 + (3y)^2 = \alpha$, is the locus of the centroid of triangle $ABC$, then $\alpha$ equals
Options:
6
18
8
20
Answer: 20
Question: Let the equation of the circle, which touches $x$-axis at the point $(a, 0)$, $a > 0$ and cuts off an intercept of length $b$ on $y$-axis be $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$. If the circle lies below $x$-axis, then the ordered pair $(2a, b^2)$ is equal to
Options:
$(\gamma, \beta^2 - 4\alpha)$
$(\alpha, \beta^2 + 4\gamma)$
$(\gamma, \beta^2 + 4\alpha)$
$(\alpha, \beta^2 - 4\gamma)$
Answer: $(\alpha, \beta^2 - 4\gamma)$
Question: If $f(x) = \frac{x^2}{2^2 + \sqrt{2}}$, $x \in \mathbb{R}$, then $\sum_{k=1}^{81} f\left(\frac{x}{82}\right)$ is equal to
Options:
$1.81 \sqrt{2}$
41
82
$\frac{81}{2}$
Answer: $\frac{81}{2}$
Question: Two numbers $\mathbf{k}_1$ and $\mathbf{k}_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_1} + i^{k_2}$, $(i = \sqrt{-1})$ is non-zero, equals
Options:
$\frac{1}{2}$
$\frac{3}{4}$
$\frac{1}{4}$
$\frac{2}{3}$
Answer: $\frac{3}{4}$
Question: If the image of the point $(4, 4, 3)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to
Options:
9
1
7
8
Answer: 9
Question: $\cos \left( \sin^{-1} \frac{3}{8} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)$ is equal to:
Options:
0
$\frac{32}{65}$
$\frac{33}{65}$
Answer: 0
Question: Let $\mathbf{A}(x, y, z)$ be a point in $xy$-plane, which is equidistant from three points $(0, 3, 2)$, $(2, 0, 3)$ and $(0, 0, 1)$. Let $\mathbf{B} = (1, 4, -1)$ and $\mathbf{C} = (2, 0, -2)$. Then among the statements (S1): $\triangle ABC$ is an isosceles right angled triangle, and (S2): the area of $\triangle ABC$ is $\frac{9 \sqrt{2}}{2}$.
Options:
both are true
only (S2) is true
only (S1) is true
both are false
Answer: only (S1) is true