Ajaysaini231/JEE-Main-2025-Math huggingface dataset:
Question: If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
Options:
-1080
-1020
-1200
-120
Answer: -1080
Question: One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is
Options:
\(\frac{2}{3}\)
\(\frac{1}{2}\)
\(\frac{4}{9}\)
\(\frac{3}{5}\)
Answer: \(\frac{1}{2}\)
Question: Let the position vectors of the vertices \(A\), \(B\) and \(C\) of a tetrahedron \(ABCD\) be \(\hat{\mathbf{i}} + 2 \hat{\mathbf{j}} + \hat{\mathbf{k}}\), \(\hat{\mathbf{i}} + 3 \hat{\mathbf{j}} - 2 \hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}} + \hat{\mathbf{j}} - \hat{\mathbf{k}}\) respectively. The altitude from the vertex \(D\) to the opposite face \(ABC\) meets the median line segment through \(A\) of the triangle \(ABC\) at the point \(E\). If the length of \(AD\) is \(\frac{\sqrt{110}}{3}\) and the volume of the tetrahedron is \(\frac{\sqrt{805}}{6 \sqrt{2}}\), then the position vector of \(E\) is
Options:
\(\frac{1}{12}(7 \hat{\mathbf{i}} + 4 \hat{\mathbf{j}} + 3 \hat{\mathbf{k}})\)
\(\frac{1}{2}(\hat{\mathbf{i}} + 4 \hat{\mathbf{j}} + 7 \hat{\mathbf{k}})\)
\(\frac{1}{6}(12 \hat{\mathbf{i}} + 12 \hat{\mathbf{j}} + \hat{\mathbf{k}})\)
\(\frac{1}{6}(7 \hat{\mathbf{i}} + 12 \hat{\mathbf{j}} + \hat{\mathbf{k}})\)
Answer: \(\frac{1}{6}(7 \hat{\mathbf{i}} + 12 \hat{\mathbf{j}} + \hat{\mathbf{k}})\)
Question: If \(\mathbf{A}\), \(\mathbf{B}\), and \(\left(\operatorname{adj}\left(\mathbf{A}^{-1}\right) + \operatorname{adj}\left(\mathbf{B}^{-1}\right)\right)\) are non-singular matrices of same order, then the inverse of \(\mathbf{A} \left(\operatorname{adj}\left(\mathbf{A}^{-1}\right) + \operatorname{adj}\left(\mathbf{B}^{-1}\right)\right)^{-1} \mathbf{B}\), is equal to
Options:
\(\mathbf{A} \mathbf{B}^{-1} + \mathbf{A}^{-1} \mathbf{B}\)
\(\operatorname{adj}\left(\mathbf{B}^{-1}\right) + \operatorname{adj}\left(\mathbf{A}^{-1}\right)\)
\(\frac{\mathbf{A} \mathbf{B}^{-1}}{|\mathbf{A}|} + \frac{\mathbf{B} \mathbf{A}^{-1}}{|\mathbf{B}|}\)
\(\frac{1}{|\mathbf{A} \mathbf{B}|} (\operatorname{adj}(\mathbf{B}) + \operatorname{adj}(\mathbf{A}))\)
Answer: \(\frac{1}{|\mathbf{A} \mathbf{B}|} (\operatorname{adj}(\mathbf{B}) + \operatorname{adj}(\mathbf{A}))\)
Question: Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is
Options:
52
48
44
40
Answer: 44
Question: Let a curve \(y = f(x)\) pass through the points \((0, 5)\) and \((\log_e 2, k)\). If the curve satisfies the differential equation \(2(3 + y) e^{2x} dx - (7 + e^{2x}) dy = 0\), then \(k\) is equal to
Options:
4
32
8
16
Answer: 8
Question: If the function \(f(x) = \begin{cases} \frac{2}{x} \left\{ \sin (k_1 + 1) x + \sin (k_2 - 1) x \right\}, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e \left( \frac{2 + k_1 x}{2 + k_2 x} \right), & x > 0 \end{cases}\) is continuous at \(x = 0\), then \(k_1^2 + k_2^2\) is
Options:
2
5
8
10
Answer: 10
Question: If the line \(3x - 2y + 12 = 0\) intersects the parabola \(4y = 3x^2\) at the points \(A\) and \(B\), then at the vertex of the parabola, the line segment \(AB\) subtends an angle equal to
Options:
\(\tan^{-1} \left( \frac{4}{5} \right)\)
\(\tan^{-1} \left( \frac{9}{7} \right)\)
\(\tan^{-1} \left( \frac{11}{9} \right)\)
\(\frac{\pi}{2} - \tan^{-1} \left( \frac{3}{2} \right)\)
Answer: \(\tan^{-1} \left( \frac{9}{7} \right)\)
Question: Let \(P\) be the foot of the perpendicular from the point \(Q(10, -3, -1)\) on the line \(\frac{x - 3}{7} = \frac{y - 2}{-1} = \frac{z + 2}{-2}\). Then the area of the right-angled triangle \(PQR\), where \(R\) is the point \((3, -2, 1)\), is
Options:
\(9 \sqrt{15}\)
\(\sqrt{30}\)
\(8 \sqrt{15}\)
\(3 \sqrt{30}\)
Answer: \(3 \sqrt{30}\)
Question: Let the arc \(AC\) of a circle subtend a right angle at the centre \(O\). If the point \(B\) on the arc \(AC\), divides the arc \(AC\) such that \(\frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5}\), and \(\overrightarrow{OC} = \alpha \overrightarrow{OA} + \beta \overrightarrow{OB}\), then \(\alpha + \sqrt{2}(\sqrt{3} - 1) \beta\) is equal to
Options:
\(2 \sqrt{3}\)
\(2 - \sqrt{3}\)
\(5 \sqrt{3}\)
\(2 + \sqrt{3}\)
Answer: \(2 - \sqrt{3}\)
Question: Let \(f(x) = \log_e x\) and \(g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1}\). Then the domain of \(f \circ g\) is
Options:
\([0, \infty)\)
\([1, \infty)\)
\((0, \infty)\)
\(\mathbf{R}\)
Answer: \(\mathbf{R}\)
Question: If the system of equations \((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5\), \(\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7\), \((\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9\) has infinitely many solutions, then \(\lambda^2 + \lambda\) is equal to
Options:
6
10
20
12
Answer: 12
Question: The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
Options:
36000
37000
34000
35000
Answer: 36000
Question: Let \(\mathbf{R} = \{(1, 2), (2, 3), (3, 3)\}\) be a relation defined on the set \(\{1, 2, 3, 4\}\). Then the minimum number of elements, needed to be added in \(\mathbf{R}\) so that \(\mathbf{R}\) becomes an equivalence relation, is
Options:
10
7
8
9
Answer: 7
Question: Let the area of a \(\triangle PQR\) with vertices \(P(5, 4)\), \(Q(-2, 4)\) and \(R(a, b)\) be 35 square units. If its orthocenter and centroid are \(O\left(2, \frac{14}{5}\right)\) and \(C(c, d)\) respectively, then \(c + 2d\) is equal to
Options:
\(\frac{8}{3}\)
\(\frac{7}{3}\)
2
3
Answer: 3
Question: The value of \(\int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left(\log_e x\right)^2 + 1}}{e^{\left(\log_e x\right)^2 + 1}} \right)^{-1} dx\) is
Options:
2
\(\log_e 2\)
1
\(e^2\)
Answer: 1
Question: Let \(\left| \frac{z - \overline{z}}{2z + \overline{z}} \right| = \frac{1}{3}\), \(z \in \mathbb{C}\), be the equation of a circle with center at \(C\). If the area of the triangle, whose vertices are at the points \((0, 0)\), \(C\) and \((\alpha, 0)\) is 11 square units, then \(\alpha^2\) equals
Options:
50
100
\(\frac{81}{25}\)
\(\frac{121}{25}\)
Answer: 100
Question: The value of \(\left( \sin 70^\circ \right) \left( \cot 10^\circ \cot 70^\circ - 1 \right)\) is
Options:
\(\frac{2}{3}\)
1
0
\(\frac{3}{2}\)
Answer: 1
Question: Let \(I(x) = \int \frac{dx}{(x - 11)^{\frac{a}{13}} (x + 15)^{\frac{c}{13}}}\). If \(I(37) - I(24) = \frac{1}{4} \left( \frac{1}{5^{\frac{1}{3}}} - \frac{1}{6^{\frac{1}{13}}} \right)\), \(b, c \in \mathbb{N}\), then \(3(b + c)\) is equal to
Options:
22
39
40
26
Answer: 39
Question: If \(\frac{\pi}{2} \leq x \leq \frac{3\pi}{4}\), then \(\cos^{-1} \left( \frac{12}{13} \cos x + \frac{5}{13} \sin x \right)\) is equal to
Options:
\(x - \tan^{-1} \frac{4}{3}\)
\(x + \tan^{-1} \frac{4}{5}\)
\(x - \tan^{-1} \frac{8}{12}\)
\(x + \tan^{-1} \frac{4}{12}\)
Answer: \(x - \tan^{-1} \frac{8}{12}\)
Question: Let the circle \(C\) touch the line \(x - y + 1 = 0\), have the centre on the positive \(x\)-axis, and cut off a chord of length \(\frac{4}{\sqrt{13}}\) along the line \(-3x + 2y = 1\). Let \(H\) be the hyperbola \(\frac{x^2}{\alpha^2} - \frac{y^2}{2\alpha} = 1\), whose one of the foci is the centre of \(C\) and the length of the transverse axis is the diameter of \(C\). Then \(2\alpha^2 + 3\beta^2\) is equal to
Options:
Answer: 19
Question: If the equation \(a(b - c)x^2 + b(c - a)x + c(a - b) = 0\) has equal roots, where \(a + c = 15\) and \(b = \frac{36}{5}\), then \(a^2 + c^2\) is equal to
Options:
Answer: 117
Question: If the set of all values of \(a\), for which the equation \(5x^3 - 15x - a = 0\) has three distinct real roots, is the interval \((\alpha, \beta)\), then \(\beta - 2\alpha\) is equal to
Options:
Answer: 30
Question: The sum of all rational terms in the expansion of \(\left(1 + 2^{1/2} + 3^{1/2}\right)^6\) is equal to
Options:
Answer: 612
Question: If the area of the larger portion bounded between the curves \(x^2 + y^2 = 25\) and \(y = |x - 1|\) is \(\frac{1}{4}(b \pi + c)\), \(b, c \in \mathbb{N}\), then \(b + c\) is equal to
Options:
Answer: 77
Question: For a $3 \times 3$ matrix $M$, let $\text{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A|=\frac{1}{2}$ and $\text{trace}(A)=3$. If $B=\text{adj}(\text{adj}(2 A))$, then the value of $|B|+\text{trace}(B)$ equals:
Options:
56
132
174
280
Answer: 280
Question: In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is:
Options:
96
144
120
72
Answer: 144
Question: Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1$. If $u$ and $v$ satisfy the equations $\begin{aligned} \alpha u+\beta v &= 18 \\ \gamma u+\delta v &= 20 \end{aligned}$ then $u+v$ equals:
Options:
5
3
4
8
Answer: 5
Question: Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3\hat{i}+2\hat{j}+2\hat{k}$. If the distance of the point $Q$ from the point $R(1,3,3)$ is 5, then the square of the area of $\triangle PQR$ is equal to:
Options:
148
136
144
140
Answer: 136
Question: If $A$ and $B$ are two events such that $P(A \cap B)=0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2-7x+1=0$, then the value of $\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}$ is:
Options:
\frac{4}{5}
\frac{7}{4}
\frac{8}{3}
\frac{9}{4}
Answer: \frac{9}{4}
Question: If $\int e^x \left( \frac{x \sin^{-1} x}{\sqrt{1-x^2}} + \frac{x \ln^{-1} x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g\left(\frac{1}{2}\right)$ equals:
Options:
\frac{\pi}{4} \sqrt{\frac{\pi}{3}}
\frac{\pi}{6} \sqrt{\frac{\pi}{3}}
\frac{\pi}{4} \sqrt{3}
\frac{\pi}{6} \sqrt{3}
Answer: \frac{\pi}{6} \sqrt{\frac{\pi}{3}}
Question: The area of the region enclosed by the curves $y=x^2-4x+4$ and $y^2=16-8x$ is:
Options:
\frac{8}{3}
\frac{4}{3}
8
5
Answer: \frac{8}{3}
Question: Let $f(x)=\int_0^{x^2} \frac{t^2-8t+15}{e^t} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are:
Options:
2 and 3
2 and 2
3 and 2
1 and 3
Answer: 2 and 3
Question: Let $P(4,4\sqrt{3})$ be a point on the parabola $y^2=4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral $PQMN$ is equal to:
Options:
17\sqrt{3}
\frac{263\sqrt{3}}{8}
\frac{34\sqrt{3}}{3}
\frac{343\sqrt{3}}{8}
Answer: \frac{343\sqrt{3}}{8}
Question: Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a} + 2 \vec{b}$ and $3 \vec{a} - \lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is:
Options:
2
1
0
3
Answer: 0
Question: If $\lim_{x \to \infty} \left( \left( \frac{e}{1-e} \right) \left( \frac{1}{e} - \frac{x}{1+e} \right) \right)^x = \alpha$, then the value of $\frac{\log_e \alpha}{1 + \log_e \alpha}$ equals:
Options:
e^{-1}
e^2
e^{-2}
e
Answer: e
Question: Let $\mathbf{A}=\{1,2,3,4\}$ and $\mathbf{B}=\{1,4,9,16\}$. Then the number of many-one functions $f: \mathbf{A} \to \mathbf{B}$ such that $1 \in f(\mathbf{A})$ is equal to:
Options:
151
139
163
127
Answer: 151
Question: Suppose that the number of terms in an A.P. is $2k, k \in \mathbb{N}$. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then $k$ is equal to:
Options:
6
5
8
4
Answer: 5
Question: The perpendicular distance, of the line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$ from the point $P(2,-10,1)$, is:
Options:
6
5\sqrt{2}
4\sqrt{3}
3\sqrt{5}
Answer: 3\sqrt{5}
Question: If the system of linear equations: $\begin{aligned} x + y + 2z &= 6 \\ 2x + 3y + az &= a + 1 \\ -x - 3y + bz &= 2b \end{aligned}$ where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ is equal to:
Options:
16
12
22
9
Answer: 16
Question: If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right) + \left(x - 2e^{\tan^{-1} y}\right) \frac{dy}{dx} = 0, y \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)=1$, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to:
Options:
e^{\pi / 12}
e^{\pi / 4}
e^{\pi / 3}
e^{\pi / 6}
Answer: e^{\pi / 6}
Question: Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos \theta) x - 1 = 0, \theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4 + \beta_\theta^4$, then $16(M + m)$ equals:
Options:
24
25
17
27
Answer: 25
Question: The sum of all values of $\theta \in [0, 2\pi]$ satisfying $2\sin^2 \theta = \cos 2\theta$ and $2\cos^2 \theta = 3\sin \theta$ is:
Options:
4\pi
\frac{5\pi}{6}
\pi
\frac{5}{2}
Answer: \pi
Question: Let the curve $z(1+i) + \bar{z}(1-i) = 4, z \in \mathbb{C}$, divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha - \beta|$ equals:
Options:
1 + \frac{\pi}{2}
1 + \frac{\pi}{3}
1 + \frac{\pi}{6}
1 + \frac{\pi}{4}
Answer: 1 + \frac{\pi}{2}
Question: Let $\mathbf{E}: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$ and $\mathbf{H}: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$. Let the distance between the foci of $\mathbf{E}$ and the foci of $\mathbf{H}$ be $2\sqrt{3}$. If $a - A = 2$, and the ratio of the eccentricities of $\mathbf{E}$ and $\mathbf{H}$ is $\frac{1}{3}$, then the sum of the lengths of their latus rectums is equal to:
Options:
10
9
8
7
Answer: 8
Question: If $\sum_{r=1}^{30} \frac{r^3 (^{30}C_r)^2}{^{30}C_{r-1}} = \alpha \times 2^{29}$, then $\alpha$ is equal to:
Options:
Answer: 465
Question: Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is:
Options:
Answer: 3
Question: Let $A(6,8)$, $B(10\cos\alpha, -10\sin\alpha)$ and $C(-10\sin\alpha, 10\cos\alpha)$, be the vertices of a triangle. If $L(a,9)$ and $G(h,k)$ be its orthocenter and centroid respectively, then $(5a - 3h + 6k + 100\sin 2\alpha)$ is equal to:
Options:
Answer: 145
Question: Let $y=f(x)$ be the solution of the differential equation $\frac{dy}{dx} + \frac{xy}{x^2-1} = \frac{x^4 + 4x}{\sqrt{1-x^2}}, -1 < x < 1$ such that $f(0)=0$. If $6 \int_{-1/2}^{1/2} f(x) dx = 2\pi - \alpha$ then $\alpha^2$ is equal to:
Options:
Answer: 27
Question: Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $(QR)^2$ is equal to:
Options:
Answer: 28
Question: Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{\text{th}}$ of the perimeter of $C$, then $\beta - \sqrt{3} \alpha$ is equal to
Options:
$3 + \sqrt{3}$
$4$
$4 - \sqrt{3}$
$3$
Answer: 4
Question: Let in a $\triangle ABC$, the length of the side $AC$ be 6, the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is:
Options:
17
21
56
42
Answer: 21
Question: Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is
Options:
$\frac{1 - \sqrt{3}}{\sqrt{2}}$
$\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$
$\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$
$\frac{1 - 2\sqrt{2}}{\sqrt{3}}$
Answer: $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$
Question: If the system of equations $2x - y + z = 4$, $5x + \lambda y + 3z = 12$, $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to
Options:
57
59
55
56
Answer: 57
Question: For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is:
Options:
20
10
35
70
Answer: 35
Question: The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to
Options:
14
21
28
7
Answer: 14
Question: Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z-4}{3}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is
Options:
5
$5\sqrt{5}$
$5\sqrt{6}$
10
Answer: $5\sqrt{5}$
Question: Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{87}{13}, \frac{-60}{13}\right)$, then $|\alpha \lambda|$ is equal to
Options:
84
113
91
101
Answer: 91
Question: If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{18} + \beta^{18} + \alpha^{13} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{18} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right)$ is equal to
Options:
441
398
312
409
Answer: 441
Question: For a statistical data $x_1, x_2, \ldots, x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} x_i^2 = 371$. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is
Options:
9
5
7
4
Answer: 7
Question: The area of the region $\left\{(x, y): x^2 + 4x + 2 \leq y \leq |x + 2|\right\}$ is equal to
Options:
7
5
$\frac{24}{5}$
$\frac{20}{3}$
Answer: $\frac{20}{3}$
Question: Let $S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ up to $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is
Options:
20
90
45
25
Answer: 25
Question: Let $f: \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6 f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}$. If the $\lim_{x \rightarrow 0} \left(\frac{1}{\alpha x} + f(x)\right) = \beta$; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to
Options:
5
3
4
6
Answer: 4
Question: If $I(m, n) = \int_0^1 x^{m-1} (1 - x)^{n-1} dx$, $m, n > 0$, then $I(9, 14) + I(10, 13)$ is
Options:
$I(19, 27)$
$I(9, 1)$
$I(1, 13)$
$I(9, 13)$
Answer: $I(9, 13)$
Question: $A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of 5 before $B$ throws a sum of 8, and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5. The probability, that $A$ wins if $A$ makes the first throw, is
Options:
$\frac{8}{17}$
$\frac{9}{19}$
$\frac{9}{17}$
$\frac{8}{19}$
Answer: $\frac{9}{19}$
Question: Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8\left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)$ is equal to
Options:
92
118
102
108
Answer: 118
Question: Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2 \sqrt{1 + x^2}\right) dx + \left(1 + x^2\right) dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to
Options:
$\sqrt{\frac{15}{2}}$
$\frac{5\sqrt{3}}{2}$
$2\sqrt{2}$
$\sqrt{\frac{14}{3}}$
Answer: $\frac{5\sqrt{3}}{2}$
Question: $\lim_{x \rightarrow 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is:
Options:
0
$\frac{1}{\sqrt{15}}$
$\frac{1}{2\sqrt{5}}$
$-\frac{1}{2\sqrt{5}}$
Answer: $-\frac{1}{2\sqrt{5}}$
Question: Consider the region $R = \left\{(x, y): x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0\right\}$. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is:
Options:
$\frac{730}{119}$
$\frac{825}{111}$
$\frac{821}{123}$
$\frac{567}{121}$
Answer: $\frac{567}{121}$
Question: Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to
Options:
$\sqrt{\frac{11}{6}}$
$\frac{1}{3\sqrt{2}}$
16
18
Answer: $\sqrt{\frac{11}{6}}$
Question: Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that $x$ divides $y$, is
Options:
Answer: 5120
Question: If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1} \alpha) + \csc^2(\cot^{-1} \beta) = 36$, then $\alpha^2 + \beta$ is
Options:
Answer: 14
Question: Let $A$ be a $3 \times 3$ matrix such that $X^T A X = 0$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}$, $A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}$, and $\det(\text{adj}(2(A + I))) = 2^\alpha 3^\beta 5^\gamma$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is
Options:
Answer: 44
Question: Let $f$ be a differentiable function such that $2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt$, $x \geq 0$. Then $f(2)$ is equal to
Options:
Answer: 19
Question: The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is
Options:
Answer: 125
Question: Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $P=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0$. If $B=PAP^{T}$, $C=P^{T} B^{10} P$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m,n)=1$, then $m+n$ is:
Options:
127
258
65
2049
Answer: 65
Question: If the components of $\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ along and perpendicular to $\vec{b}=\hat{i}+\hat{j}-\hat{k}$ respectively, are $\frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})$ and $\frac{1}{11}(-4 \hat{i}-5 \hat{j}-17 \hat{k})$, then $\alpha^2+\beta^2+\gamma^2$ is equal to:
Options:
26
18
2
16
Answer: 26
Question: Let $A$, $B$, $C$ be three points in $xy$-plane, whose position vectors are given by $\sqrt{3} \hat{i}+\hat{j}$, $\hat{i}+\sqrt{3} \hat{j}$ and $a \hat{i}+(1-a) \hat{j}$ respectively with respect to the origin $O$. If the distance of the point $C$ from the line bisecting the angle between the vectors $\vec{OA}$ and $\vec{OB}$ is $\frac{\theta}{\sqrt{2}}$, then the sum of all the possible values of $a$ is:
Options:
2
$\frac{9}{2}$
1
0
Answer: 1
Question: Let the coefficients of three consecutive terms $T_r$, $T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt[4]{3}+\sqrt[4]{4})^{12}$. Then $p+q$ is equal to:
Options:
283
287
295
299
Answer: 283