Ajaysaini231/JEE-Main-2025-Math huggingface dataset:
Question: Let \(f(x)=\begin{cases}(1+ax)^{1/x},&x<0,\\1+b,&x=0,\\\displaystyle\frac{\sqrt{x+4}-2}{\sqrt[3]{x+c}-2},&x>0\end{cases}\) be continuous at \(x=0\). Then \(e^a b c\) is equal to:
Options:
64
72
48
36
Answer: 48
Question: Line L1 passes through the point (1, 2, 3) and is parallel to the z-axis. Line L2 passes through the point (λ, 5, 6) and is parallel to the y-axis. If for λ = λ₁, λ₂ with λ₂ < λ₁ the shortest distance between these two lines is 3, then the square of the distance of the point (λ₁, λ₂, 7) from the line L1 is:
Options:
40
32
25
37
Answer: 25
Question: All five-letter words are made using all the letters A, B, C, D and E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by Wₙ. Let the probability P(Wₙ) of choosing the word Wₙ satisfy P(Wₙ) = 2·P(Wₙ₋₁) for n > 1. If P(CDBEA) = 2^α / (2^β − 1), with α, β ∈ ℕ, then α + β is equal to:
Options:
Answer: 183
Question: Let the product of the focal distances of the point P(4,2√3) on the hyperbola H: \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then p² + q² is equal to:
Options:
Answer: 120
Question: Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}+2\hat{j}-\hat{k}$, $\vec{c}=\lambda\hat{j}+\mu\hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a}\times\vec{c}=\vec{b}\times\hat{d}$ and $\vec{c}\cdot\hat{d}=1$. If $\vec{c}$ is perpendicular to $\vec{a}$, then $\lvert3\lambda\hat{d}+\mu\vec{c}\rvert^2$ is equal to:
Options:
Answer: 5
Question: If the number of seven‐digit numbers such that the sum of their digits is even is $m\cdot n\cdot10^n$, where $m,n\in\{1,2,3,\dots,9\}$, then $m+n$ is equal to:
Options:
Answer: 14
Question: The area of the region bounded by the curve \(y = \max\{|x|,\,|x-2|\}\), the x-axis, and the lines \(x=-2\) and \(x=4\) is equal to:
Options:
Answer: 12
Question: Let $a>0$. If the function $f(x)=6x^{3}-45a x^{2}+108a^{2}x+1$ attains its local maximum and minimum values at the points $x_{1}$ and $x_{2}$ respectively such that $x_{1}x_{2}=54$, then $a+x_{1}+x_{2}$ is equal to:
Options:
15
18
24
13
Answer: 18
Question: The sum of the infinite series $\cot^{-1}\left(\tfrac{7}{4}\right)+\cot^{-1}\left(\tfrac{19}{4}\right)+\cot^{-1}\left(\tfrac{39}{4}\right)+\cot^{-1}\left(\tfrac{67}{4}\right)+\ldots$ is:
Options:
\frac{\pi}{2}+\tan^{-1}\left(\frac{1}{2}\right)
\frac{\pi}{2}-\cot^{-1}\left(\frac{1}{2}\right)
\frac{\pi}{2}+\cot^{-1}\left(\frac{1}{2}\right)
\frac{\pi}{2}-\tan^{-1}\left(\frac{1}{2}\right)
Answer: \frac{\pi}{2}-\tan^{-1}\left(\frac{1}{2}\right)
Question: Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(2)=1$ and $f'(2)=4$. If \(\displaystyle\lim_{x\to0}\bigl(f(2+x)\bigr)^{3/x}=e^{\alpha}\), then the number of times the curve \(y=4x^{3}-4x^{2}-4(\alpha-7)x-\alpha\) meets the $x$-axis is:
Options:
2
1
0
3
Answer: 2
Question: Let $A = \{-3,-2,-1,0,1,2,3\}$ and $R$ be a relation on $A$ defined by $xRy$ iff $2x - y \in \{0,1\}$. Let $\ell$ be the number of elements in $R$. Let $m$ and $n$ be the minimum numbers of elements that must be added to $R$ to make it reflexive and symmetric, respectively. Then $\ell + m + n$ is equal to:
Options:
18
17
15
16
Answer: 17
Question: Let the product of \(\omega_1 = (8+i)\sin\theta + (7+4i)\cos\theta\) and \(\omega_2 = (1+8i)\sin\theta + (4+7i)\cos\theta\) be \(\alpha + i\beta\), where \(i=\sqrt{-1}\). Let \(p\) and \(q\) be the maximum and minimum values of \(\alpha+\beta\), respectively. Then \(p+q\) is equal to:
Options:
140
130
160
150
Answer: 130
Question: Let the values of $p$, for which the shortest distance between the lines $\displaystyle\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $\mathbf{r}=(p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$ is $\tfrac{1}{\sqrt6}$, be $a,b$ with $a<b$. Then the length of the latus rectum of the ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is:
Options:
9
\tfrac{3}{2}
\tfrac{2}{3}
18
Answer: \tfrac{2}{3}
Question: The axis of a parabola is the line \(y = x\) and its vertex and focus are in the first quadrant at distances \(\sqrt{2}\) and \(2\sqrt{2}\) from the origin, respectively. If the point \((1, k)\) lies on the parabola, then a possible value of \(k\) is:
Options:
4
9
3
8
Answer: 9
Question: Let the domains of the functions $f(x)=\log_{10}\log_{10}\bigl(8-\log_{2}(x^2+4x+5)\bigr)$ and $g(x)=\sin^{-1}\!\bigl(\tfrac{7x+10}{x-2}\bigr)$ be $(\alpha,\beta)$ and $[\gamma,\delta]$, respectively. Then $\alpha^2+\beta^2+\gamma^2+\delta^2$ is equal to:
Options:
15
13
16
14
Answer: 15
Question: A line passing through the point $A(-2,0)$ touches the parabola $P:\;y^2 = x - 2$ at the point $B$ in the first quadrant. The area of the region bounded by the line $AB$, the parabola $P$, and the $x$-axis is:
Options:
\tfrac{7}{3}
2
\tfrac{8}{3}
3
Answer: \tfrac{8}{3}
Question: Let the sum of the focal distances of the point P(4,3) on the hyperbola \(H:\;\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\tfrac{8\sqrt{5}}{3}\). If for \(H\), the length of the latus rectum is \(l\) and the product of the focal distances of the point \(P\) is \(m\), then \(9l^2+6m\) is equal to:
Options:
184
186
185
187
Answer: 185
Question: Let the matrix \(A=\begin{pmatrix}1&0&0\\1&0&1\\0&1&0\end{pmatrix}\) satisfy \(A^n = A^{n-2} + A^2 - I\) for \(n\ge3\). Then the sum of all the elements of \(A^{50}\) is:
Options:
53
52
39
44
Answer: 53
Question: If the sum of the first 20 terms of the series \(\displaystyle\frac{4\cdot1}{4+3\cdot1^2+1^4}+\frac{4\cdot2}{4+3\cdot2^2+2^4}+\frac{4\cdot3}{4+3\cdot3^2+3^4}+\dots\) is \(\tfrac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to:
Options:
423
420
421
422
Answer: 421
Question: If \(1^2\binom{15}{1}^2 + 2^2\binom{15}{2}^2 + 3^2\binom{15}{3}^2 + \dots + 15^2\binom{15}{15}^2 = 2^m\,3^n\,5^k\), where \(m,n,k\) are positive integers, then \(m + n + k\) is equal to:
Options:
19
21
18
20
Answer: 19
Question: Let for two distinct values of $p$ the lines $y = x + p$ touch the ellipse $E:\;\frac{x^2}{4}+\frac{y^2}{3}=1$ at the points $A$ and $B$. Let the line $y = x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $ABCD$ is equal to:
Options:
36
24
48
20
Answer: 24
Question: Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively, and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of the A.P. in A and B respectively such that D = d + 3, d > 0. If \(\frac{p+q}{p-q} = \frac{19}{5}\), then p - q is equal to:
Options:
600
450
630
540
Answer: 540
Question: If a curve $y=y(x)$ passes through the point $(1,\tfrac{\pi}{2})$ and satisfies the differential equation $\bigl(7x^4\cot y - e^x\csc y\bigr)\frac{dx}{dy} = x^5$, $x\ge1$, then at $x=2$, the value of $\cos y$ is:
Options:
\frac{2e^2 - e}{64}
\frac{2e^2 + e}{64}
\frac{2e^2 - e}{128}
\frac{2e^2 + e}{128}
Answer: \frac{2e^2 - e}{128}
Question: The centre of a circle $C$ is at the centre of the ellipse $E:\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $\triangle PF_1F_2$ is $30$ and the length of the major axis of $E$ is $17$, then the distance between the foci of $E$ is:
Options:
26
13
12
\frac{13}{2}
Answer: 13
Question: Let $f(x)+2f\bigl(\tfrac{1}{x}\bigr)=x^2+5$ and $2g(x)-3g\bigl(\tfrac{1}{x}\bigr)=x$, $x>0$. If $\alpha=\int_1^2 f(x)\,dx$ and $\beta=\int_1^2 g(x)\,dx$, then the value of $9\alpha+\beta$ is:
Options:
1
0
10
11
Answer: 11
Question: Let $A$ be the point of intersection of the lines
$$L_1:\;\frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1},$$
and
$$L_2:\;\frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}.$$Let $B$ and $C$ be points on $L_1$ and $L_2$ respectively such that $AB=AC=\sqrt{15}$. Then the square of the area of triangle $ABC$ is:
Options:
54
63
57
60
Answer: 54
Question: Let the mean and the standard deviation of the observations 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respectively. Then the mean deviation about the mode of these observations is:
Options:
1
3/4
2
1/2
Answer: 1
Question: If \(\alpha\) is a root of the equation \(x^2 + x + 1 = 0\) and \(\displaystyle\sum_{k=1}^n\bigl(\alpha^k + \alpha^{-k}\bigr)^2 = 20\), then \(n\) is equal to:
Options:
Answer: 11
Question: If \(\displaystyle\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^{9}}\,dx = \frac{1}{m}\bigl((\sqrt{1+x^2}+x)^{n}(\sqrt{1+x^2}-x)\bigr)+C\), where \(C\) is the constant of integration and \(m,n\in\mathbb{N}\), then \(m+n\) is equal to:
Options:
Answer: 379
Question: A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card being a spade is \(\tfrac{11}{50}\), then n is equal to:
Options:
Answer: 2
Question: Let $m$ and $n$, $(m<n)$ be two 2-digit numbers. Then the total number of pairs $(m,n)$ such that $\gcd(m,n)=6$ is:
Options:
Answer: 64
Question: Let the three sides of a triangle ABC be given by the vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let $G$ be the centroid of the triangle ABC. Then $$6\bigl(|\overrightarrow{AG}|^2+|\overrightarrow{BG}|^2+|\overrightarrow{CG}|^2\bigr)$$ is equal to:
Options:
Answer: 164
Question: Let f, g: (1, \infty) \to \mathbb{R} be defined as f(x) = \frac{2x+3}{5x+2} and g(x) = \frac{2-3x}{1-x}. If the range of the function f\circ g: [2,4] \to \mathbb{R} is [\alpha, \beta], then \frac{1}{\beta - \alpha} is equal to:
Options:
68
29
2
56
Answer: 56
Question: Consider the sets A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 \ge 25\}, B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}, C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : 2x^2 + y^2 \le 4\}, and D = A \cap B. The total number of one-one functions from the set D to the set C is:
Options:
15120
19320
17160
18290
Answer: 17160
Question: Let A = \{1, 6, 11, 16, \ldots\} and B = \{9, 16, 23, 30, \ldots\} be the sets consisting of the first 2025 terms of two arithmetic progressions. Then n(A \cup B) is:
Options:
3814
4027
3761
4003
Answer: 3761
Question: The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is:
Options:
129/182
103/182
17/26
19/26
Answer: 129/182
Question: For an integer n \ge 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)^{2n-3} is 16, then the distance of the point P(2n-1, n^2-4n) from the line x + y = 8 is:
Options:
\sqrt{2}
2\sqrt{2}
5\sqrt{2}
3\sqrt{2}
Answer: 3\sqrt{2}
Question: Let the shortest distance between the lines \(\frac{x-3}{3} = \frac{y-\alpha}{-1} = \frac{z-3}{1}\) and \(\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-\beta}{4}\) be 3\sqrt{30}. Then the positive value of 5\alpha + \beta is:
Options:
42
46
48
40
Answer: 46
Question: If \lim_{x\to1} \frac{(x-1)(6 + \lambda \cos(x-1)) + \mu \sin(1-x)}{(x-1)^3} = -1, where \lambda, \mu \in \mathbb{R}, then \lambda + \mu is equal to:
Options:
18
20
19
17
Answer: 18
Question: Let f: [0, \infty) \to \mathbb{R} be a differentiable function such that f(x) = 1 - 2x + \int_0^x e^{t-x} f(t) \, dt for all x \in [0, \infty). Then the area of the region bounded by y = f(x) and the coordinate axes is:
Options:
\sqrt{5}
1/2
\sqrt{2}
2
Answer: 1/2
Question: Let A and B be two distinct points on the line L: (x-6)/3 = (y-7)/2 = (z-7)/-2. Both A and B are at a distance 2\sqrt{17} from the foot of the perpendicular drawn from the point (1, 2, 3) to the line L. If O is the origin, then OA \cdot OB is equal to:
Options:
49
47
21
62
Answer: 47
Question: Let f: \mathbb{R} \to \mathbb{R} be a continuous function satisfying f(0) = 1 and f(2x) - f(x) = x for all x \in \mathbb{R}. If \lim_{n\to\infty} [f(x) - f(x/2^n)] = G(x), then \sum_{r=1}^{10} G(r^2) is equal to:
Options:
540
385
420
215
Answer: 385
Question: 1 + 3 + 5^2 + 7 + 9^2 + \dots up to 40 terms is equal to:
Options:
43890
41880
33980
40870
Answer: 41880
Question: In the expansion of \(\bigl(\frac{\sqrt{2}}{3} + \frac{1}{\sqrt{3}}\bigr)^n\), n \in \mathbb{N}, if the ratio of the 15th term from the beginning to the 15th term from the end is \tfrac{1}{6}, then the value of \binom{n}{3} is:
Options:
4060
1040
2300
4960
Answer: 2300
Question: Considering the principal values of the inverse trigonometric functions, \sin^{-1}\bigl(\tfrac{\sqrt{3}}{2}x + \tfrac{1}{2}\sqrt{1-x^2}\bigr), -\tfrac{1}{2} < x < \tfrac{1}{\sqrt{2}}, is equal to:
Options:
\frac{\pi}{4} + \sin^{-1} x
\frac{\pi}{6} + \sin^{-1} x
-\frac{5\pi}{6} - \sin^{-1} x
\frac{5\pi}{6} - \sin^{-1} x
Answer: \frac{\pi}{6} + \sin^{-1} x
Question: Consider two vectors \mathbf{u}=3\mathbf{i}-\mathbf{j} and \mathbf{v}=2\mathbf{i}+\mathbf{j}-\lambda\mathbf{k}, \lambda > 0. The angle between them is given by \cos^{-1}\bigl(\tfrac{\sqrt{5}}{2\sqrt{7}}\bigr). Let \mathbf{v}=\mathbf{v}_1+\mathbf{v}_2, where \mathbf{v}_1 is parallel to \mathbf{u} and \mathbf{v}_2 is perpendicular to \mathbf{u}. Then the value of |\mathbf{v}_1|^2 + |\mathbf{v}_2|^2 is:
Options:
\tfrac{23}{2}
14
\tfrac{25}{2}
10
Answer: 14
Question: Let the three sides of a triangle be on the lines 4x - 7y + 10 = 0, x + y = 5, and 7x + 4y = 15. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines x = 0, y = 0, and x + y = 1 is:
Options:
5
\sqrt{5}
\sqrt{20}
20
Answer: \sqrt{5}
Question: The value of \int_{-1}^{1} \frac{(1 + \sqrt{|x|} - x)e^x + (\sqrt{|x|} - x)e^{-x}}{e^x + e^{-x}} \, dx is equal to:
Options:
3 - \frac{2\sqrt{2}}{3}
2 + \frac{2\sqrt{2}}{3}
1 - \frac{2\sqrt{2}}{3}
1 + \frac{2\sqrt{2}}{3}
Answer: 1 + \frac{2\sqrt{2}}{3}
Question: The length of the latus-rectum of the ellipse whose foci are (2,5) and (2,-3) and eccentricity is \tfrac{4}{5} is:
Options:
\tfrac{6}{5}
\tfrac{50}{3}
\tfrac{10}{3}
\tfrac{18}{5}
Answer: \tfrac{18}{5}
Question: Consider the equation x^2 + 4x - n = 0, where n \in [20,100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to:
Options:
7
8
6
5
Answer: 6
Question: A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is:
Options:
\tfrac{11}{15}
\tfrac{28}{75}
\tfrac{2}{15}
\tfrac{3}{5}
Answer: \tfrac{28}{75}
Question: If 10\sin^4\theta + 15\cos^4\theta = 6, then the value of \displaystyle \frac{27\csc^6\theta + 8\sec^6\theta}{16\sec^8\theta} is:
Options:
\frac{2}{5}
\frac{3}{4}
\frac{3}{5}
\frac{1}{5}
Answer: \frac{2}{5}
Question: If the area of the region \{(x,y): |x-5| \le y \le 4\sqrt{x}\} is A, then 3A is equal to _____
Options:
Answer: 368
Question: Let A = \begin{pmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{pmatrix}. If for some \theta \in (0,\pi), A^2 = A^T, then the sum of the diagonal elements of the matrix (A + I)^3 + (A - I)^3 - 6A is equal to:
Options:
Answer: 6
Question: Let A = \{z \in \mathbb{C} : |z - 2 - i| = 3\}, B = \{z \in \mathbb{C} : \Re(z - 2) = 2\} and S = A \cap B. Then \sum_{z \in S} |z|^2 is equal to _____
Options:
Answer: 22
Question: Let C be the circle x^2 + (y-1)^2 = 2, E_1 and E_2 be two ellipses whose centres lie at the origin and major axes lie on the x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E_1 and E_2 at P(x_1,y_1), Q(x_2,y_2) and R(x_3,y_3) respectively. Given that P is the midpoint of the line segment QR and PQ = \frac{2\sqrt{2}}{3}, the value of 9(x_1y_1 + x_2y_2 + x_3y_3) is equal to _____
Options:
Answer: 46
Question: Let m and n be the number of points at which the function f(x)=\max\{x, x^3, x^5, \dots, x^{21}\}, x\in\mathbb{R}, is not differentiable and not continuous, respectively. Then m + n is equal to _____
Options:
Answer: 3
Question: If the orthocentre of the triangle formed by the lines \(y = x + 1\), \(y = 4x - 8\) and \(y = m x + c\) is at \((3, -1)\), then \(m - c\) is:
Options:
0
-2
4
2
Answer: 0
Question: Let the vectors \(a\) and \(b\) be of the same magnitude such that \(\frac{\lvert a + b\rvert + \lvert a - b\rvert}{\lvert a + b\rvert - \lvert a - b\rvert} = \sqrt{2} + 1\). Then \(\frac{\lvert a + b\rvert^2}{\lvert a\rvert^2}\) is:
Options:
2 + 4\sqrt{2}
1 + \sqrt{2}
2 + \sqrt{2}
4 + 2\sqrt{2}
Answer: 2 + \sqrt{2}
Question: Let \(A = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : |\alpha - 1| \le 4 \text{ and } |\beta - 5| \le 6\}\) and \(B = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \le 144\}\). Then:
Options:
\(B \subset A\)
\(A \cup B = \{(x, y) : -4 \le x \le 4,\; -1 \le y \le 11\}\)
neither \(A \subset B\) nor \(B \subset A\)
\(A \subset B\)
Answer: \(B \subset A\)
Question: If the range of the function \(f(x) = \frac{5 - x}{x^2 - 3x + 2}\), \(x \neq 1,2\), is \(( -\infty, \alpha) \cup [\beta, \infty)\), then \(\alpha^2 + \beta^2\) is equal to:
Options:
190
192
188
194
Answer: 194
Question: A bag contains 19 unbiased coins and one coin with heads on both sides. One coin drawn at random is tossed and a head turns up. If the probability that the drawn coin was unbiased is \(\frac{m}{n}\), with \(\gcd(m,n)=1\), then \(n^2 - m^2\) is equal to:
Options:
80
60
72
64
Answer: 80
Question: Let a random variable X take values 0, 1, 2 and 3 with P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) = \(\tfrac{1 - 2p}{2}\). If \(E(X^2) = 2E(X)\), then the value of \(8p - 1\) is:
Options:
0
2
1
3
Answer: 2
Question: If the area of the region \(\{(x,y): 1 + x^2 \le y \le \min\{x + 7,\;11 - 3x\}\}\) is \(A\), then \(3A\) is equal to:
Options:
50
49
46
47
Answer: 50
Question: Let \(f: \mathbb{R} \to \mathbb{R}\) be a polynomial function of degree four having extreme values at \(x = 4\) and \(x = 5\). If \(\displaystyle \lim_{x \to 0} \frac{f(x)}{x^2} = 5\), then \(f(2)\) is equal to:
Options:
12
10
8
14
Answer: 10
Question: The number of solutions of the equation \(\cos 2\theta\cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}\) in \([-\tfrac{\pi}{2},\tfrac{\pi}{2}]\) is:
Options:
7
5
6
9
Answer: 7
Question: Let \(a_n\) be the \(n\)th term of an A.P. If \(S_n = a_1 + a_2 + \dots + a_n = 700\) for some \(n\), \(a_6 = 7\) and \(S_7 = 7\), then \(a_n\) is equal to:
Options:
56
65
64
70
Answer: 64
Question: If the locus of \(z \in \mathbb{C}\), such that \(\Re\bigl(\tfrac{z-1}{2z + i}\bigr) + \Re\bigl(\tfrac{\overline{z}-1}{2\overline{z} - i}\bigr) = 2\), is a circle of radius \(r\) and center \((a,b)\), then \(\tfrac{15ab}{r^2}\) is equal to:
Options:
24
12
18
16
Answer: 18
Question: Let the length of a latus rectum of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) be 10. If its eccentricity is the minimum value of the function \(f(t) = t^2 + t + \tfrac{11}{12}\), \(t \in \mathbb{R}\), then \(a^2 + b^2\) is equal to:
Options:
125
126
120
115
Answer: 126
Question: Let \(y = y(x)\) be the solution of the differential equation \((x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x\), with \(y(0) = 1\). Then \(\displaystyle \int_{-3}^{3} y(x)\,dx\) is:
Options:
24
36
30
18
Answer: 24
Question: If the equation of the line passing through the point \((0, -\tfrac{1}{2}, 0)\) and perpendicular to the lines \(\mathbf{r} = \lambda(\hat{i} + a\hat{j} + b\hat{k})\) and \(\mathbf{r} = (\hat{i} - \hat{j} - 6\hat{k}) + \mu(-b\hat{i} + a\hat{j} + 5\hat{k})\) is \(\frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}\), then \(a + b + c + d\) is equal to:
Options:
10
14
13
12
Answer: 14
Question: Let \(p\) be the number of all triangles that can be formed by joining the vertices of a regular polygon \(P\) of \(n\) sides and \(q\) be the number of all quadrilaterals that can be formed by joining the vertices of \(P\). If \(p + q = 126\), then the eccentricity of the ellipse \(\frac{x^2}{16} + \frac{y^2}{n} = 1\) is:
Options:
\frac{3}{4}
\frac{1}{2}
\frac{\sqrt{7}}{4}
\frac{1}{\sqrt{2}}
Answer: \frac{1}{\sqrt{2}}
Question: Consider the lines \(L_1: x - 1 = y - 2 = z\) and \(L_2: x - 2 = y - 2 = z - 1\). Let the feet of the perpendiculars from the point \(P(5,1,-3)\) on the lines \(L_1\) and \(L_2\) be \(Q\) and \(R\) respectively. If the area of the triangle \(PQR\) is \(A\), then \(4A^2\) is equal to:
Options:
139
147
151
143
Answer: 147
Question: The number of real roots of the equation \(x\lvert x - 2\rvert + 3\lvert x - 3\rvert + 1 = 0\) is:
Options:
4
2
1
3
Answer: 1
Question: Let \(e_1\) and \(e_2\) be the eccentricities of the ellipse \(\tfrac{x^2}{a^2}+\tfrac{y^2}{25}=1\) and the hyperbola \(\tfrac{x^2}{16}-\tfrac{y^2}{b^2}=1\), respectively. If \(b<5\) and \(e_1e_2=1\), then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:
Options:
\tfrac{4}{5}
\tfrac{3}{5}
\tfrac{\sqrt{7}}{4}
\tfrac{\sqrt{3}}{2}
Answer: \tfrac{3}{5}
Question: Let the system of equations \(x + 5y - z = 1\), \(4x + 3y - 3z = 7\), and \(24x + y + \lambda z = \mu\), with \(\lambda,\mu \in \mathbb{R}\), have infinitely many solutions. Then the number of solutions of this system, if \(x,y,z\) are integers and satisfy \(7 \le x+y+z \le 77\), is:
Options:
3
6
5
4
Answer: 3
Question: If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is:
Options:
760
755
750
757
Answer: 757
Question: If the function \(f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}\) is continuous at \(x = 0\), then \(f(0)\) is equal to:
Options:
Answer: 2
Question: If 
\[ \int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt{3x^{-24} + x^{-26}} \right) dx \] 
is equal to 
\[ \frac{\alpha}{3(\alpha + 1)} \left(3x^{\beta} + x^{\gamma}\right)^{\frac{\alpha + 1}{\alpha}} + C,\ x > 0, \] 
where \(\alpha, \beta, \gamma \in \mathbb{Z}\), and \(C\) is the constant of integration, then \(\alpha + \beta + \gamma\) is equal to:
Options:
Answer: 19
Question: For \( t > -1 \), let \( \alpha_t \) and \( \beta_t \) be the roots of the equation 
\[ \left((t + 2)^{\frac{1}{7}} - 1\right)x^2 + \left((t + 2)^{\frac{1}{6}} - 1\right)x + \left((t + 2)^{\frac{1}{21}} - 1\right) = 0. \] 
If \( \lim_{t \to -1^+} \alpha_t = a \) and \( \lim_{t \to -1^+} \beta_t = b \), then \( 72(a + b)^2 \) is equal to:
Options:
Answer: 98
Question: Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be (-5, 0) and 5x + 9 = 0, respectively. If the product of the focal distances of a point \( (\alpha, 2\sqrt{5}) \) on the hyperbola is p, then 4p is equal to:
Options:
Answer: 189
Question: The sum of the series 
\[ 2 \times 1 \times \binom{20}{4} - 3 \times 2 \times \binom{20}{5} + 4 \times 3 \times \binom{20}{6} - 5 \times 4 \times \binom{20}{7} + \ldots + 18 \times 17 \times \binom{20}{20} \] 
is equal to:
Options:
Answer: 34
Question: Evaluate the limit: \( \lim_{x \to \infty} \frac{\tan\left(5x^{1/3}\right) \log_e(1 + 3x^2)}{\left(\tan^{-1}(3\sqrt{x})\right)^2 \left(e^{5x^{4/3}} - 1\right)} \) is equal to:
Options:
1/15
1
1/3
5/3
Answer: 1/3
Question: If the shortest distance between the lines \(\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}\) and \(\frac{x}{1} = \frac{y}{\alpha} = \frac{z - 5}{1}\) is \(\frac{5}{\sqrt{6}}\), then the sum of all possible values of \(\alpha\) is:
Options:
3/2
-3/2
3
-3
Answer: -3
Question: Let \(x = -1\) and \(x = 2\) be the critical points of the function \(f(x) = x^3 + ax^2 + b \log_2|x| + 1,\ x \ne 0\). Let \(m\) and \(M\) respectively be the absolute minimum and the absolute maximum values of \(f\) in the interval \(\left[-2, -\frac{1}{2}\right]\). Then \(|M + m|\) is equal to: (Take \(\log_2 2 = 0.7\))
Options:
21.1
19.8
22.1
20.9
Answer: 21.1
Question: The remainder when \( (64^{64})^{64} \) is divided by 7 is equal to:
Options:
4
1
3
6
Answer: 1
Question: Let P be the parabola, whose focus is \((-2, 1)\) and directrix is \(2x + y + 2 = 0\). Then the sum of the ordinates of the points on P, whose abscissa is \(-2\), is:
Options:
3/2
5/2
1/4
3/4
Answer: 3/2