Ajaysaini231/JEE-Main-2025-Math huggingface dataset:
Question: If the image of the point $P(1, 0, 3)$ in the line joining the points $A(4, 7, 1)$ and $B(3, 5, 3)$ is $Q(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to:
Options:
\frac{47}{3}
\frac{46}{3}
18
13
Answer: \frac{46}{3}
Question: Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $\int_1^x f(t)\,dt = 5x f(x) - x^5 - 9$ for all $x \geq 1$, then the value of $f(3)$ is:
Options:
18
32
22
26
Answer: 32
Question: The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is:
Options:
4
10
6
8
Answer: 4
Question: Let $A = \{1, 2, 3, \ldots, 10\}$ and $R$ be a relation on $A$ such that $R = \{(a, b) : a = 2b + 1\}$. Let $(a_1, a_2), (a_2, a_3), (a_3, a_4), \ldots, (a_k, a_{k+1})$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$, for which such a sequence exists, is equal to:
Options:
6
7
5
8
Answer: 5
Question: If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:
Options:
\frac{4}{\sqrt{17}}
\frac{\sqrt{3}}{16}
\frac{3}{\sqrt{19}}
\frac{\sqrt{5}}{7}
Answer: \frac{4}{\sqrt{17}}
Question: The line $L_1$ is parallel to the vector $\vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k}$ and passes through the point $(7, 6, 2)$ and the line $L_2$ is parallel to the vector $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ and passes through the point $(5, 3, 4)$. The shortest distance between the lines $L_1$ and $L_2$ is:
Options:
\frac{23}{\sqrt{38}}
\frac{21}{\sqrt{57}}
\frac{23}{\sqrt{57}}
\frac{21}{\sqrt{38}}
Answer: \frac{23}{\sqrt{38}}
Question: Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_a^b \frac{9x^2}{1 + 5^x} \, dx$ is equal to:
Options:
24
27
18
21
Answer: 24
Question: If the system of equations \\ 
$2x + \lambda y + 3z = 5$ \\ 
$3x + 2y - z = 7$ \\ 
$4x + 5y + \mu z = 9$ \\ 
 has infinitely many solutions, then $(\lambda^2 + \mu^2)$ is equal to:
Options:
22
18
26
30
Answer: 26
Question: If $\theta \in \left[-\frac{7\pi}{6}, \frac{4\pi}{3}\right]$, then the number of solutions of \\ 
$\sqrt{3}\csc^2\theta - 2(\sqrt{3} - 1)\csc\theta - 4 = 0$, is equal to:
Options:
6
8
10
7
Answer: 6
Question: Given three identical bags each containing 10 balls, whose colours are as follows: \\ 
\textbf{Bag I:} Red = 3, Blue = 2, Green = 5 \\ 
\textbf{Bag II:} Red = 4, Blue = 3, Green = 3 \\ 
\textbf{Bag III:} Red = 5, Blue = 1, Green = 4 \\ 
A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is $p$ and if the ball is Green, the probability that it is from bag III is $q$, then the value of $\left( \frac{1}{p} + \frac{1}{q} \right)$ is:
Options:
6
9
7
8
Answer: 7
Question: If the mean and the variance of $6,\ 4,\ a,\ 8,\ b,\ 12,\ 10,\ 13$ are $9$ and $9.25$ respectively, then $a + b + ab$ is equal to:
Options:
105
103
100
106
Answer: 103
Question: If the domain of the function \\ 
$f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$ is $(a,\ b)$, then $(1 + a)^2 + b^2$ is equal to:
Options:
26
29
25
30
Answer: 26
Question: $4\int_0^1 \left(\frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}}\right) dx - 3 \log_e(\sqrt{3})$ is equal to:
Options:
$2 + \sqrt{2} + \log_e(1 + \sqrt{2})$
$2 - \sqrt{2} - \log_e(1 + \sqrt{2})$
$2 + \sqrt{2} - \log_e(1 + \sqrt{2})$
$2 - \sqrt{2} + \log_e(1 + \sqrt{2})$
Answer: $2 - \sqrt{2} - \log_e(1 + \sqrt{2})$
Question: If $\lim\limits_{x \to 0} \frac{\cos(2x) + a\cos(4x) - b}{x^4}$ is finite, then $(a + b)$ is equal to:
Options:
$\frac{1}{2}$
0
$\frac{3}{4}$
$-1$
Answer: $\frac{1}{2}$
Question: If $\sum_{r=0}^{10} \left(\frac{10^{r+1} - 1}{10^r}\right) \cdot {}^{11}C_{r+1} = \frac{\alpha^{11} - 11^{11}}{10^{10}}$, then $\alpha$ is equal to:
Options:
15
11
24
20
Answer: 20
Question: The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is: \\ 
(Figure: 3 rows of boxes, arranged in a T-shape, with a total of 8 boxes)
Options:
5880
960
840
5760
Answer: 5760
Question: Let the point $P$ of the focal chord $PQ$ of the parabola $y^2 = 16x$ be $(1, -4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m : n$, $\gcd(m, n) = 1$, then $m^2 + n^2$ is equal to:
Options:
17
10
37
26
Answer: 17
Question: Let $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{b} \times \vec{c} = \vec{d}$, then $|\vec{a} \cdot \vec{d}|$ is equal to:
Options:
18
12
9
15
Answer: 15
Question: Let the area of the triangle formed by a straight line $L : x + by + c = 0$ with coordinate axes be $48$ square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of $45^\circ$ with the positive x-axis, then the value of $b^2 + c^2$ is:
Options:
90
93
97
83
Answer: 97
Question: Let $A$ be a $3 \times 3$ real matrix such that $A^2(A - 2I) - 4(A - I) = O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5 = \alpha A^2 + \beta A + \gamma I$, where $\alpha$, $\beta$ and $\gamma$ are real constants, then $\alpha + \beta + \gamma$ is equal to:
Options:
12
20
76
4
Answer: 12
Question: Let $y = y(x)$ be the solution of the differential equation 
$\frac{dy}{dx} + 2y \sec^2 x = 2\sec^2 x + 3 \tan x \cdot \sec^2 x$, such that $y(0) = \frac{5}{4}$. Then $12 \left( y\left(\frac{\pi}{4}\right) - e^{-2} \right)$ is equal to:
Options:
Answer: 21
Question: If the sum of the first 10 terms of the series 
\( \frac{4 \cdot 1}{1 + 4 \cdot 1^4} + \frac{4 \cdot 2}{1 + 4 \cdot 2^4} + \frac{4 \cdot 3}{1 + 4 \cdot 3^4} + \ldots \) is \( \frac{m}{n} \), where gcd(m, n) = 1, then m + n is equal to:
Options:
Answer: 441
Question: If $y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right)$, then $(x - y)^2 + 3y^2$ is equal to:
Options:
Answer: 3
Question: Let $A(4, -2)$, $B(1, 1)$ and $C(9, -3)$ be the vertices of a triangle $ABC$. Then the maximum area of the parallelogram $AFDE$, formed with vertices $D$, $E$, and $F$ on the sides $BC$, $CA$ and $AB$ of the triangle $ABC$ respectively, is:
Options:
Answer: 3
Question: If the set of all $a \in \mathbb{R} \setminus \{1\}$, for which the roots of the equation $(1 - a)x^2 + 2(a - 3)x + 9 = 0$ are positive is $(-\infty, -\alpha] \cup [\beta, \gamma)$, then $2\alpha + \beta + \gamma$ is equal to:
Options:
Answer: 7
Question: The largest $n\in\mathbb{N}$ such that $3^n$ divides $50!$ is:
Options:
21
22
20
23
Answer: 22
Question: The number of sequences of ten terms, whose terms are either $0$, $1$, or $2$, that contain exactly five $1$’s and exactly three $2$’s is equal to:
Options:
360
45
2520
1820
Answer: 2520
Question: Let one focus of the hyperbola $H:\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$ be at $(\sqrt{10},0)$ and the corresponding directrix be $x=\tfrac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9\,(e^2+l)$ is equal to:
Options:
14
15
16
12
Answer: 16
Question: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function such that
$(\sin x\cos y)\bigl[f(2x+2y)-f(2x-2y)\bigr]=(\cos x\sin y)\bigl[f(2x+2y)+f(2x-2y)\bigr]$
for all $x,y\in\mathbb{R}$. If $f'(0)=\tfrac12$, then the value of $24\,f''\bigl(\tfrac{5\pi}{3}\bigr)$ is:
Options:
2
-3
3
-2
Answer: -3
Question: Let $A=\begin{pmatrix}\alpha-1 & -1\\6 & \beta\end{pmatrix}$, $\alpha>0$, such that $\det(A)=0$ and $\alpha+\beta=1$. If $I$ denotes the $2\times2$ identity matrix, then the matrix $(I+A)^8$ is:
Options:
\begin{pmatrix}4 & -1\\6 & -1\end{pmatrix}
\begin{pmatrix}257 & -64\\514 & -127\end{pmatrix}
\begin{pmatrix}1025 & -511\\2024 & -1024\end{pmatrix}
\begin{pmatrix}766 & -255\\1530 & -509\end{pmatrix}
Answer: \begin{pmatrix}766 & -255\\1530 & -509\end{pmatrix}
Question: The term independent of $x$ in the expansion of
$\displaystyle\bigl(\tfrac{x+1}{x^{2/3}+1-x^{1/3}} - \tfrac{x+1}{x-x^{1/2}}\bigr)^{10}$,
for $x>1$, is:
Options:
210
150
240
120
Answer: 210
Question: If $\theta \in [-2\pi, 2\pi]$, then the number of solutions of $2\sqrt{2}\cos^2\theta + (2-\sqrt{6})\cos\theta - \sqrt{3} = 0$ is equal to:
Options:
12
6
8
10
Answer: 8
Question: Let $a_1,a_2,a_3,\dots$ be in an A.P. such that $\displaystyle\sum_{k=1}^{12}a_{2k-1}=-\tfrac{72}{5}a_1$, $a_1\neq0$. If $\displaystyle\sum_{k=1}^n a_k=0$, then $n$ is:
Options:
11
10
18
17
Answer: 11
Question: If the function $f(x)=2x^3 - 9a x^2 + 12a^2 x + 1$, where $a>0$, attains its local maximum and local minimum at $p$ and $q$ respectively, such that $p^2 = q$, then $f(3)$ is equal to:
Options:
55
10
23
37
Answer: 37
Question: Let $z$ be a complex number such that $|z|=1$. If $\displaystyle\frac{2 + k^2 z}{k + \overline{z}} = k z$, $k\in\mathbb{R}$, then the maximum distance of $k + i k^2$ from the circle $|z - (1 + 2i)| = 1$ is:
Options:
\sqrt{5} + 1
2
3
\sqrt{3} + 1
Answer: \sqrt{5} + 1
Question: If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-2\hat{k}$, and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is:
Options:
\frac{1}{\sqrt{155}}(-7\hat{i} + 9\hat{j} + 5\hat{k})
\frac{1}{\sqrt{155}}(-7\hat{i} + 9\hat{j} - 5\hat{k})
\frac{1}{\sqrt{155}}(7\hat{i} + 9\hat{j} + 5\hat{k})
\frac{1}{\sqrt{155}}(7\hat{i} + 9\hat{j} - 5\hat{k})
Answer: \frac{1}{\sqrt{155}}(7\hat{i} + 9\hat{j} + 5\hat{k})
Question: Let $A$ be the set of all functions $f: \mathbb{Z} \to \mathbb{Z}$ and $R$ be a relation on $A$ such that $R = \{(f,g): f(0)=g(1) \text{ and } f(1)=g(0)\}$. Then $R$ is:
Options:
Symmetric and transitive but not reflective
Symmetric but neither reflective nor transitive
Reflexive but neither symmetric nor transitive
Transitive but neither reflexive nor symmetric
Answer: Symmetric but neither reflective nor transitive
Question: For \(\alpha,\beta,\gamma\in\mathbb{R}\), if \(\displaystyle\lim_{x\to0}\frac{x^2\sin(\alpha x)+(\gamma-1)e^{x^2}}{\sin(2x)-\beta x}=3\), then \(\beta+\gamma-\alpha\) is equal to:
Options:
7
4
6
-1
Answer: 7
Question: Let \(P_n=\alpha^n+\beta^n,\ n\in\mathbb{N}\). If \(P_{10}=123,\ P_9=76,\ P_8=47\) and \(P_1=1\), then the quadratic equation having roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) is:
Options:
x^2 - x + 1 = 0
x^2 + x - 1 = 0
x^2 - x - 1 = 0
x^2 + x + 1 = 0
Answer: x^2 + x - 1 = 0
Question: If the system of linear equations $3x + y + \beta z = 3$, $2x + \alpha y - z = -3$, $x + 2y + z = 4$ has infinitely many solutions, then the value of $22\beta - 9\alpha$ is:
Options:
49
31
43
37
Answer: 31
Question: If $S$ and $S'$ are the foci of the ellipse $\frac{x^2}{18} + \frac{y^2}{9} = 1$ and $P$ is a point on the ellipse, then $\min\bigl(SP\cdot S'P\bigr) + \max\bigl(SP\cdot S'P\bigr)$ is equal to:
Options:
3(1+\sqrt{2})
3(6+\sqrt{2})
9
27
Answer: 27
Question: Let the vertices Q and R of the triangle PQR lie on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\), QR = 5, and the coordinates of the point P be \((0,2,3)\). If the area of the triangle PQR is \(\tfrac{m}{n}\), then:
Options:
m - 5\sqrt{21}\,n = 0
2m - 5\sqrt{21}\,n = 0
5m - 2\sqrt{21}\,n = 0
5m - 21\sqrt{2}\,n = 0
Answer: 2m - 5\sqrt{21}\,n = 0
Question: Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of ABCD is equal to:
Options:
\sqrt{340}
12
\sqrt{110}
7\sqrt{3}
Answer: \sqrt{110}
Question: Let $a\in\mathbb{R}$ and $A$ be a matrix of order $3\times3$ such that $\det(A)=-4$ and
$$A+I=\begin{pmatrix}1 & a & 1\\2 & 1 & 0\\a & 1 & 2\end{pmatrix},$$
where $I$ is the $3\times3$ identity. If
$$\det\bigl((a+1)\adj((a-1)A)\bigr)=2^m3^n,\quad m,n\in\{0,1,2,\dots,20\},$$
then $m+n$ is equal to:
Options:
14
17
15
16
Answer: 16
Question: Let the focal chord $PQ$ of the parabola $y^2 = 4x$ make an angle of $60^\circ$ with the positive x-axis, where $P$ lies in the first quadrant. If the circle, whose one diameter is $PS$, $S$ being the focus of the parabola, touches the y-axis at the point $(0,\alpha)$, then $5\alpha^2$ is equal to:
Options:
15
25
30
20
Answer: 15
Question: Let ⌊·⌋ denote the greatest integer function. If
\[
\displaystyle\int_{0}^{e^3} \Bigl\lfloor\frac{1}{e^{x-1}}\Bigr\rfloor\,dx = \alpha - \ln 2,
\]
then \(\alpha^3\) is equal to:
Options:
Answer: 8
Question: Let \(f:\mathbb{R}\to\mathbb{R}\) be a thrice‐differentiable odd function satisfying \(f''(x)=f(x)\), \(f(0)=0\), and \(f'(0)=3\). Then \(9f(\ln 3)\) is equal to:
Options:
Answer: 36
Question: If the area of the region \(\{(x,y):\;4 - x^2 \le y \le x^2,\;y \le 4,\;x \ge 0\}\) is \(\tfrac{80\sqrt{2}}{\alpha} - \beta\), where \(\alpha,\beta\in\mathbb{N}\), then \(\alpha + \beta\) is equal to:
Options:
Answer: 22
Question: Three distinct numbers are selected randomly from the set {1,2,3,…,40}. If the probability that the selected numbers are in an increasing G.P. is m/n, gcd(m,n)=1, then m+n is equal to:
Options:
Answer: 2477
Question: The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$ is equal to:
Options:
Answer: 768
Question: Let $f: \mathbb{R}\to\mathbb{R}$ be a function defined by $$f(x)=\lvert x+2\rvert - 2\lvert x\rvert.$$ If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m+n$ is:
Options:
5
3
2
4
Answer: 3
Question: Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$- and $z$-axes, respectively, is half of the angle that this line makes with the positive $x$-axis. Then the sum of all possible values of the angle $\beta$ is:
Options:
$\tfrac{3\pi}{4}$
$\pi$
$\tfrac{\pi}{2}$
$\tfrac{3\pi}{2}$
Answer: $\tfrac{3\pi}{4}$
Question: If the four distinct points $(4,6)$, $(-1,5)$, $(0,0)$ and $(k,3k)$ lie on a circle of radius $r$, then $10k + r^2$ is equal to:
Options:
32
33
34
35
Answer: 35
Question: Let the mean and variance of five observations $x_1=1$, $x_2=3$, $x_3=a$, $x_4=7$ and $x_5=b$, with $a>b$, be $5$ and $10$ respectively. Then the variance of the observations $n + x_n$ for $n=1,2,\dots,5$ is:
Options:
17
16.4
17.4
16
Answer: 16
Question: Let $A=\{-2,-1,0,1,2,3\}$. Define a relation $R$ on $A$ by $xRy$ if and only if $y=\max(x,1)$. If $l$ is the number of elements in $R$, and $m$ and $n$ are the minimum numbers of ordered pairs that must be added to $R$ to make it reflexive and symmetric respectively, then $l+m+n$ is equal to:
Options:
12
11
13
14
Answer: 12
Question: Consider the lines $x(3\lambda+1)+y(7\lambda+2)=17\lambda+5$, $\lambda$ being a parameter, all passing through a point $P$. One of these lines (say $L$) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$, then the value of $d^2$ is:
Options:
20
30
10
15
Answer: 20
Question: Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $(k,\tfrac{k}{2})$ from the line $3x+4y+5=0$ is:
Options:
15
5\sqrt{3}
15\sqrt{5}
12
Answer: 15
Question: Line $L_1$ of slope $2$ and line $L_2$ of slope $\tfrac12$ intersect at the origin $O$. In the first quadrant, $P_1,P_2,\dots,P_{12}$ are 12 points on $L_1$ and $Q_1,Q_2,\dots,Q_9$ are 9 points on $L_2$. Then the total number of triangles that can be formed with vertices chosen from the 22 points $O,P_1,\dots,P_{12},Q_1,\dots,Q_9$ is:
Options:
1080
1134
1026
1188
Answer: 1134
Question: Let $f$ be a function such that $f(x) + 3\,f\bigl(24/x\bigr) = 4x$, $x\neq0$. Then $f(3) + f(8)$ is equal to:
Options:
11
10
12
13
Answer: 11
Question: The integral \(\displaystyle\int_{0}^{\pi}\frac{8x}{4\cos^2x + \sin^2x}\,dx\) is equal to:
Options:
$2\pi^2$
$4\pi^2$
$\pi^2$
$\tfrac{3\pi^2}{2}$
Answer: $2\pi^2$
Question: The area of the region $\{(x,y):\lvert x-y\rvert \le y \le 4\sqrt{x}\}$ is:
Options:
512
$\tfrac{1024}{3}$
$\tfrac{512}{3}$
$\tfrac{2048}{3}$
Answer: $\tfrac{1024}{3}$
Question: If the probability that the random variable $X$ takes the value $x$ is given by
$$P(X=x)=k\,(x+1)3^{-x},\quad x=0,1,2,\dots,$$
where $k$ is a constant, then $P(X\ge3)$ is equal to:
Options:
7/27
4/9
8/27
1/9
Answer: 1/9
Question: Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 3(\tan^2x)\,y + 3y = \sec^2x$, with $y(0)=\tfrac{1}{3}+e^3$. Then $y\bigl(\tfrac{\pi}{4}\bigr)$ is equal to:
Options:
\tfrac{2}{3}
\tfrac{4}{3}
\tfrac{4}{3} + e^3
\tfrac{2}{3} + e^3
Answer: \tfrac{4}{3}
Question: If $z_1, z_2, z_3\in\mathbb{C}$ are the vertices of an equilateral triangle whose centroid is $z_0$, then $\displaystyle\sum_{k=1}^3 (z_k - z_0)^2$ is equal to:
Options:
0
1
i
-i
Answer: 0
Question: The number of solutions of the equation $$(4-\sqrt{3})\sin x - 2\sqrt{3}\cos^2x = -\frac{4}{1+\sqrt{3}},\quad x\in[-2\pi,\tfrac{5\pi}{2}]$$ is equal to:
Options:
4
3
6
5
Answer: 5
Question: The shortest distance between the curves $$y^2 = 8x$$ and $$x^2 + y^2 + 12y + 35 = 0$$ is:
Options:
2\sqrt{3} - 1
\sqrt{2}
3\sqrt{2} - 1
2\sqrt{2} - 1
Answer: 2\sqrt{2} - 1
Question: Let $C$ be the circle of minimum area enclosing the ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\tfrac12$ and foci $(\pm2,0)$. Let $PQR$ be a variable triangle whose vertex $P$ lies on $C$ and whose side $QR$ of length $29$ is parallel to the major axis of the ellipse and passes through the point where the ellipse intersects the negative $y$–axis. Then the maximum area of $\triangle PQR$ is:
Options:
$6(3+\sqrt{2})$
$8(3+\sqrt{2})$
$6(2+\sqrt{3})$
$8(2+\sqrt{3})$
Answer: $8(2+\sqrt{3})$
Question: The distance of the point $(7,10,11)$ from the line $\displaystyle\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3}$ along the line $\displaystyle\frac{x-9}{2}=\frac{y-13}{-3}=\frac{z-17}{6}$ is:
Options:
18
14
12
16
Answer: 14
Question: The sum \(1 + \tfrac{1+3}{2!} + \tfrac{1+3+5}{3!} + \tfrac{1+3+5+7}{4!} + \dots\) up to infinitely many terms is equal to:
Options:
6e
4e
3e
2e
Answer: 2e
Question: If the domain of the function $f(x)=\log_7\bigl(1-\log_4(x^2-9x+18)\bigr)$ is $(\alpha,\beta)\cup(\gamma,\delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to:
Options:
18
16
15
17
Answer: 18
Question: Let $I$ be the identity matrix of order $3\times3$ and let
$$A=\begin{pmatrix} \lambda & 2 & 3\\4 & 5 & 6\\7 & -1 & 2\end{pmatrix}$$
with $|A|=-1$. Let $B$ be the inverse of the matrix $\mathrm{adj}(A)\,\mathrm{adj}(A^2)$. Then $|\lambda B + I|$ is equal to:
Options:
Answer: 38
Question: Let \((1+x+x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \dots + a_{20} x^{20}\). If \((a_1 + a_3 + a_5 + \dots + a_{19}) - 11a_2 = 121k\), then \(k\) is equal to:
Options:
Answer: 239
Question: If \(\displaystyle\lim_{x\to0}\bigl(\tfrac{\tan x}{x}\bigr)^{1/x^2}=p\), then \(96\log_e p\) is equal to:
Options:
Answer: 32
Question: Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}-3\hat{j}+3\hat{k}$, $\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{d}$ be a vector such that $\vec{b}\times\vec{d}=\vec{c}\times\vec{d}$ and $\vec{a}\cdot\vec{d}=4$. Then $|\vec{a}\times\vec{d}|^2$ is equal to:
Options:
Answer: 128
Question: If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $$3x^2 - y^2 - ax + by + \gamma = 0,$$ then $a + b + \gamma$ is equal to:
Options:
Answer: 141
Question: Let $A$ be a matrix of order $3\times3$ with $\lvert A\rvert=5$. If \(\bigl\lvert 2\,\mathrm{adj}\bigl(3A\,\mathrm{adj}(2A)\bigr)\bigr\rvert=2^{\alpha}3^{\beta}5^{\gamma}\), where $\alpha,\beta,\gamma\in\mathbb{N}$, then $\alpha+\beta+\gamma$ is equal to:
Options:
25
26
27
28
Answer: 27
Question: Let a line passing through the point $(4,1,0)$ intersect the line $L_1:\;\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ at the point $A(\alpha,\beta,\gamma)$ and the line $L_2:\;x-6 = y - z + 4$ at the point $B(a,b,c)$. Then $\det\begin{pmatrix}1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c\end{pmatrix}$ is equal to:
Options:
8
16
12
6
Answer: 8
Question: Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}\,x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then
$$\frac{P_{25} + \sqrt{3}\,P_{24}}{2P_{23}} \;+\; \frac{Q_{25} - Q_{23}}{Q_{24}}$$
is equal to:
Options:
3
4
5
7
Answer: 5
Question: The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is:
Options:
16923
3763
33845
18817
Answer: 18817
Question: Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $0 \le x^2 + 2y \le 4$. Let $\ell$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added to $R$ to make it a reflexive relation. Then $\ell + m$ is equal to:
Options:
19
20
17
18
Answer: 18
Question: A line passing through the point $P(\sqrt{5},\sqrt{5})$ intersects the ellipse $\dfrac{x^2}{36}+\dfrac{y^2}{25}=1$ at points $A$ and $B$ such that $(PA)\cdot(PB)$ is maximum. Then $5\bigl(PA^2+PB^2\bigr)$ is equal to:
Options:
218
377
290
338
Answer: 338
Question: The sum $1 + 3 + 11 + 25 + 45 + 71 + \dots$ up to 20 terms is equal to:
Options:
7240
7130
6982
8124
Answer: 7240
Question: If the domain of the function \(f(x)=\log_e\bigl(\tfrac{2x-3}{5+4x}\bigr)+\sin^{-1}\bigl(\tfrac{4+3x}{2-x}\bigr)\) is $[\alpha,\beta]$, then $\alpha^2+4\beta$ is equal to:
Options:
5
4
3
7
Answer: 4
Question: If \(\sum_{r=1}^9 \binom{r+3}{2} C_r = \alpha\bigl(\tfrac{3}{2}\bigr)^9 - \beta\), where \(\alpha,\beta\in\mathbb{N}\), then \((\alpha+\beta)^2\) is equal to:
Options:
27
9
81
18
Answer: 81
Question: If \(y(x)=\begin{vmatrix} \sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix},\;x\in\mathbb{R},\) then \(\frac{d^2y}{dx^2}+y\) is equal to:
Options:
-1
28
27
1
Answer: -1
Question: The number of solutions of the equation $2x + 3\tan x = \pi$, $x\in[-2\pi,2\pi]\setminus\{\pm\tfrac{\pi}{2},\pm\tfrac{3\pi}{2}\}$ is:
Options:
6
5
4
3
Answer: 5
Question: Let $g$ be a differentiable function such that $\displaystyle\int_0^x g(t)\,dt = x - \int_x^\pi g(t)\,dt$, $x\ge0$, and let $y=y(x)$ satisfy the differential equation $\frac{dy}{dx} - y\tan x = 2(x+1)\sec x\,g(x)$ for $x\in[0,\tfrac{\pi}{2}]$. If $y(0)=0$, then $y\bigl(\tfrac{\pi}{3}\bigr)$ is equal to:
Options:
\frac{2\pi}{3\sqrt{3}}
\frac{4\pi}{3}
\frac{2\pi}{3}
\frac{4\pi}{\sqrt{3}}
Answer: \frac{4\pi}{3}
Question: A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_1:\;2x+y+6=0$ and $L_2:\;4x+2y-p=0,\;p>0$, at the points $A$ and $B$, respectively. If $AB=\tfrac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$, then $\tfrac{AM}{BM}$ is equal to:
Options:
5
4
2
3
Answer: 3
Question: Let $z\in\mathbb{C}$ be such that \(\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i\). Then the sum of all possible values of $z^2$ is:
Options:
19 - 2i
-19 - 2i
19 + 2i
-19 + 2i
Answer: -19 - 2i
Question: Let f(x)=\int x^3\sqrt{3 - x^2}\,dx. If 5f(\sqrt{2}) = -4, then f(1) is equal to:
Options:
-2\sqrt{2}/5
-8\sqrt{2}/5
-4\sqrt{2}/5
-6\sqrt{2}/5
Answer: -6\sqrt{2}/5
Question: Let $a_1, a_2, a_3, \dots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \tfrac{111}{4}$, then $24(a_1 + a_2 + a_3)$ is equal to:
Options:
131
130
129
128
Answer: 129
Question: Let the domain of the function \(f(x)=\log_2\bigl(\log_4\bigl(\log_6(3+4x-x^2)\bigr)\bigr)\) be \((a,b)\). If \(\displaystyle\int_0^a \lfloor x^2\rfloor\,dx = p - \sqrt{q} - \sqrt{r}\), with \(p,q,r\in\mathbb{N}\) and \(\gcd(p,q,r)=1\), where \(\lfloor\cdot\rfloor\) is the greatest integer function, then \(p+q+r\) is equal to:
Options:
10
8
11
9
Answer: 10
Question: The radius of the smallest circle which touches the parabolas $y = x^2 + 2$ and $x = y^2 + 2$ is:
Options:
\tfrac{7\sqrt{2}}{2}
\tfrac{7\sqrt{2}}{16}
\tfrac{7\sqrt{2}}{4}
\tfrac{7\sqrt{2}}{8}
Answer: \tfrac{7\sqrt{2}}{8}