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

Answer: -1080

Choices:
- -1080
- -1020
- -1200
- -120


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

Answer: \(\frac{1}{2}\)

Choices:
- \(\frac{2}{3}\)
- \(\frac{1}{2}\)
- \(\frac{4}{9}\)
- \(\frac{3}{5}\)


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

Answer: \(\frac{1}{6}(7 \hat{\mathbf{i}} + 12 \hat{\mathbf{j}} + \hat{\mathbf{k}})\)

Choices:
- \(\frac{1}{12}(7 \hat{\mathbf{i}} + 4 \hat{\mathbf{j}} + 3 \hat{\mathbf{k}})\)
- \(\frac{1}{2}(\hat{\mathbf{i}} + 4 \hat{\mathbf{j}} + 7 \hat{\mathbf{k}})\)
- \(\frac{1}{6}(12 \hat{\mathbf{i}} + 12 \hat{\mathbf{j}} + \hat{\mathbf{k}})\)
- \(\frac{1}{6}(7 \hat{\mathbf{i}} + 12 \hat{\mathbf{j}} + \hat{\mathbf{k}})\)


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

Answer: \(\frac{1}{|\mathbf{A} \mathbf{B}|} (\operatorname{adj}(\mathbf{B}) + \operatorname{adj}(\mathbf{A}))\)

Choices:
- \(\mathbf{A} \mathbf{B}^{-1} + \mathbf{A}^{-1} \mathbf{B}\)
- \(\operatorname{adj}\left(\mathbf{B}^{-1}\right) + \operatorname{adj}\left(\mathbf{A}^{-1}\right)\)
- \(\frac{\mathbf{A} \mathbf{B}^{-1}}{|\mathbf{A}|} + \frac{\mathbf{B} \mathbf{A}^{-1}}{|\mathbf{B}|}\)
- \(\frac{1}{|\mathbf{A} \mathbf{B}|} (\operatorname{adj}(\mathbf{B}) + \operatorname{adj}(\mathbf{A}))\)


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

Answer: 44

Choices:
- 52
- 48
- 44
- 40


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

Answer: 8

Choices:
- 4
- 32
- 8
- 16


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

Answer: 10

Choices:
- 2
- 5
- 8
- 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

Answer: \(\tan^{-1} \left( \frac{9}{7} \right)\)

Choices:
- \(\tan^{-1} \left( \frac{4}{5} \right)\)
- \(\tan^{-1} \left( \frac{9}{7} \right)\)
- \(\tan^{-1} \left( \frac{11}{9} \right)\)
- \(\frac{\pi}{2} - \tan^{-1} \left( \frac{3}{2} \right)\)


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

Answer: \(3 \sqrt{30}\)

Choices:
- \(9 \sqrt{15}\)
- \(\sqrt{30}\)
- \(8 \sqrt{15}\)
- \(3 \sqrt{30}\)


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

Answer: \(2 - \sqrt{3}\)

Choices:
- \(2 \sqrt{3}\)
- \(2 - \sqrt{3}\)
- \(5 \sqrt{3}\)
- \(2 + \sqrt{3}\)


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

Answer: \(\mathbf{R}\)

Choices:
- \([0, \infty)\)
- \([1, \infty)\)
- \((0, \infty)\)
- \(\mathbf{R}\)


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

Answer: 12

Choices:
- 6
- 10
- 20
- 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

Answer: 36000

Choices:
- 36000
- 37000
- 34000
- 35000


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

Answer: 7

Choices:
- 10
- 7
- 8
- 9


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

Choices:
- \(\frac{8}{3}\)
- \(\frac{7}{3}\)
- 2
- 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

Choices:
- 2
- \(\log_e 2\)
- 1
- \(e^2\)


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

Answer: 100

Choices:
- 50
- 100
- \(\frac{81}{25}\)
- \(\frac{121}{25}\)


Question: The value of \(\left( \sin 70^\circ \right) \left( \cot 10^\circ \cot 70^\circ - 1 \right)\) is

Choices:
- \(\frac{2}{3}\)
- 1
- 0
- \(\frac{3}{2}\)


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

Answer: 39

Choices:
- 22
- 39
- 40
- 26


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

Answer: \(x - \tan^{-1} \frac{8}{12}\)

Choices:
- \(x - \tan^{-1} \frac{4}{3}\)
- \(x + \tan^{-1} \frac{4}{5}\)
- \(x - \tan^{-1} \frac{8}{12}\)
- \(x + \tan^{-1} \frac{4}{12}\)


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

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

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

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

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

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:

Answer: 280

Choices:
- 56
- 132
- 174
- 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:

Answer: 144

Choices:
- 96
- 144
- 120
- 72


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:

Answer: 5

Choices:
- 5
- 3
- 4
- 8


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:

Answer: 136

Choices:
- 148
- 136
- 144
- 140


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:

Answer: \frac{9}{4}

Choices:
- \frac{4}{5}
- \frac{7}{4}
- \frac{8}{3}
- \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:

Answer: \frac{\pi}{6} \sqrt{\frac{\pi}{3}}

Choices:
- \frac{\pi}{4} \sqrt{\frac{\pi}{3}}
- \frac{\pi}{6} \sqrt{\frac{\pi}{3}}
- \frac{\pi}{4} \sqrt{3}
- \frac{\pi}{6} \sqrt{3}


Question: The area of the region enclosed by the curves $y=x^2-4x+4$ and $y^2=16-8x$ is:

Answer: \frac{8}{3}

Choices:
- \frac{8}{3}
- \frac{4}{3}
- 8
- 5


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:

Answer: 2 and 3

Choices:
- 2 and 3
- 2 and 2
- 3 and 2
- 1 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:

Answer: \frac{343\sqrt{3}}{8}

Choices:
- 17\sqrt{3}
- \frac{263\sqrt{3}}{8}
- \frac{34\sqrt{3}}{3}
- \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:

Choices:
- 2
- 1
- 0
- 3


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:

Choices:
- e^{-1}
- e^2
- e^{-2}
- 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:

Answer: 151

Choices:
- 151
- 139
- 163
- 127


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:

Answer: 5

Choices:
- 6
- 5
- 8
- 4


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:

Answer: 3\sqrt{5}

Choices:
- 6
- 5\sqrt{2}
- 4\sqrt{3}
- 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:

Answer: 16

Choices:
- 16
- 12
- 22
- 9


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:

Answer: e^{\pi / 6}

Choices:
- e^{\pi / 12}
- e^{\pi / 4}
- e^{\pi / 3}
- 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:

Answer: 25

Choices:
- 24
- 25
- 17
- 27


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:

Answer: \pi

Choices:
- 4\pi
- \frac{5\pi}{6}
- \pi
- \frac{5}{2}


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:

Answer: 1 + \frac{\pi}{2}

Choices:
- 1 + \frac{\pi}{2}
- 1 + \frac{\pi}{3}
- 1 + \frac{\pi}{6}
- 1 + \frac{\pi}{4}


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:

Answer: 8

Choices:
- 10
- 9
- 8
- 7


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:

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:

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:

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:

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:

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

Choices:
- $3 + \sqrt{3}$
- $4$
- $4 - \sqrt{3}$
- $3$


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:

Answer: 21

Choices:
- 17
- 21
- 56
- 42


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

Answer: $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$

Choices:
- $\frac{1 - \sqrt{3}}{\sqrt{2}}$
- $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$
- $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$
- $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$


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

Answer: 57

Choices:
- 57
- 59
- 55
- 56


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:

Answer: 35

Choices:
- 20
- 10
- 35
- 70


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

Answer: 14

Choices:
- 14
- 21
- 28
- 7


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

Answer: $5\sqrt{5}$

Choices:
- 5
- $5\sqrt{5}$
- $5\sqrt{6}$
- 10


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

Answer: 91

Choices:
- 84
- 113
- 91
- 101


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

Answer: 441

Choices:
- 441
- 398
- 312
- 409


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

Answer: 7

Choices:
- 9
- 5
- 7
- 4


Question: The area of the region $\left\{(x, y): x^2 + 4x + 2 \leq y \leq |x + 2|\right\}$ is equal to

Answer: $\frac{20}{3}$

Choices:
- 7
- 5
- $\frac{24}{5}$
- $\frac{20}{3}$


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

Answer: 25

Choices:
- 20
- 90
- 45
- 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

Choices:
- 5
- 3
- 4
- 6


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

Answer: $I(9, 13)$

Choices:
- $I(19, 27)$
- $I(9, 1)$
- $I(1, 13)$
- $I(9, 13)$


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

Answer: $\frac{9}{19}$

Choices:
- $\frac{8}{17}$
- $\frac{9}{19}$
- $\frac{9}{17}$
- $\frac{8}{19}$


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

Answer: 118

Choices:
- 92
- 118
- 102
- 108


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

Answer: $\frac{5\sqrt{3}}{2}$

Choices:
- $\sqrt{\frac{15}{2}}$
- $\frac{5\sqrt{3}}{2}$
- $2\sqrt{2}$
- $\sqrt{\frac{14}{3}}$


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:

Answer: $-\frac{1}{2\sqrt{5}}$

Choices:
- 0
- $\frac{1}{\sqrt{15}}$
- $\frac{1}{2\sqrt{5}}$
- $-\frac{1}{2\sqrt{5}}$


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:

Answer: $\frac{567}{121}$

Choices:
- $\frac{730}{119}$
- $\frac{825}{111}$
- $\frac{821}{123}$
- $\frac{567}{121}$


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

Answer: $\sqrt{\frac{11}{6}}$

Choices:
- $\sqrt{\frac{11}{6}}$
- $\frac{1}{3\sqrt{2}}$
- 16
- 18


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

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

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

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

Answer: 19


Question: The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is

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:

Answer: 65

Choices:
- 127
- 258
- 65
- 2049


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:

Answer: 26

Choices:
- 26
- 18
- 2
- 16


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:

Choices:
- 2
- $\frac{9}{2}$
- 1
- 0


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:

Answer: 283

Choices:
- 283
- 287
- 295
- 299


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:

Answer: $(-\infty, \infty)$

Choices:
- $(-\infty, -1] \cup [0, \infty)$
- $(-\infty, -1] \cup [1, \infty)$
- $(-\infty, \infty)$
- $(-\infty, \infty) - \{0\}$


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:

Answer: $\frac{1}{2}$

Choices:
- $\frac{1}{2}$
- $\frac{1}{4}$
- $\frac{2}{3}$
- $\frac{1}{3}$


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:

Answer: 8

Choices:
- 10
- 4
- 2
- 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:

Answer: 310

Choices:
- 270
- 340
- 320
- 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:

Answer: 30

Choices:
- 29
- 30
- 31
- 36


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:

Answer: $\frac{4}{15}$

Choices:
- $\frac{4}{15}$
- $\frac{1}{3}$
- $\frac{2}{5}$
- $\frac{4}{3}$


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:

Answer: 11

Choices:
- 13
- 15
- 9


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:

Answer: 675

Choices:
- 540
- 675
- 1350
- 135


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:

Answer: 6

Choices:
- 7
- 6
- 1
- 9


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:

Answer: $6x - 9y = 20$

Choices:
- $x + 9y = 36$
- $4x - 9y = 12$
- $6x - 9y = 20$
- $9x - 9y = 32$


Question: If $f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx$, $f(0) = -6$, then $f(1)$ is equal to:

Answer: $4(\log_e 2 - 2)$

Choices:
- $4(\log_e 2 - 2)$
- $2 - \log_e 2$
- $\log_e 2 + 2$
- $4(\log_e 2 + 2)$


Question: The area of the region bounded by the curves $x(1 + y^2) = 1$ and $y^2 = 2x$ is:

Answer: $\frac{\pi}{2} - \frac{1}{3}$

Choices:
- $2\left(\frac{\pi}{2} - \frac{1}{3}\right)$
- $\frac{\pi}{2} - \frac{1}{3}$
- $\frac{\pi}{4} - \frac{1}{3}$
- $\frac{1}{2}\left(\frac{\pi}{2} - \frac{1}{3}\right)$


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:

Answer: 66

Choices:
- 54
- 44
- 41
- 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:

Answer: 22

Choices:
- 20
- 18
- 26


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:

Choices:
- -2
- 6
- -6
- 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:

Answer: 6

Choices:
- $-2\sqrt{10}$
- 1
- 6
- -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:

Answer: 14


Question: The number of natural numbers between 212 and 999, such that the sum of their digits is 15, is:

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:

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:

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:

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:

Answer: 100

Choices:
- 100
- 120
- 110
- 90


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:

Answer: 90

Choices:
- 90
- 84
- 122
- 108


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:

Choices:
- 2
- 3
- 1
- 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:

Answer: an equivalence relation

Choices:
- both reflexive and transitive but not symmetric
- an equivalence relation
- reflexive but neither symmetric nor transitive
- both reflexive and symmetric but not transitive


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:

Answer: 192

Choices:
- 392
- 384
- 192
- 96


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:

Answer: 161

Choices:
- 173
- 164
- 158
- 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:

Answer: $(8,26,12)$

Choices:
- $(5,17,4)$
- $(2,8,5)$
- $(8,26,12)$
- $(-1,-1,1)$


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:

Answer: 308

Choices:
- 462
- 77
- 154
- 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:

Answer: $4 \log_e 3$

Choices:
- $3 \log_e 4$
- $4 \log_e 3$
- $6 \log_e 4$
- $2 \log_e 3$


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:

Answer: $\frac{24 \sqrt{6}}{5}$

Choices:
- $\frac{12 \sqrt{6}}{5}$
- $6 \sqrt{6}$
- $\frac{18 \sqrt{6}}{5}$
- $\frac{24 \sqrt{6}}{5}$


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:

Answer: 242

Choices:
- 288
- 222
- 242
- 262


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:

Answer: 7

Choices:
- 13
- 10
- 3
- 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:

Answer: 25

Choices:
- 20
- 18
- 25
- 16


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:

Answer: 1280

Choices:
- 1280
- 1295
- 1215
- 1040


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:

Answer: 37

Choices:
- 47
- 37
- 36
- 40


Question: The value of $\lim_{n \to \infty} \left( \sum_{k=1}^n \frac{k^3 + 6k^2 + 11k + 6}{(k+3)!} \right)$ is:

Answer: $\frac{5}{3}$

Choices:
- $\frac{4}{3}$
- 2
- $\frac{7}{3}$
- $\frac{5}{3}$


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:

Answer: 2184

Choices:
- 2184
- 2196
- 2148
- 2172


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:

Answer: $\frac{1}{\log_e 3 - \log_e 4}$

Choices:
- $\frac{1}{\log_e 3 - \log_e 4}$
- $\frac{2}{\log_e 3 - \log_e 4}$
- $\frac{1}{\log_e 4 - \log_e 3}$
- $\frac{-1}{\log_e 4}$


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:

Answer: $2 \sqrt{14}$

Choices:
- $\sqrt{14}$
- $3 \sqrt{7}$
- $2 \sqrt{14}$
- $5 \sqrt{7}$


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:

Answer: 14

Choices:
- 16
- 12
- 14
- 18


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

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

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

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

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

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:

Answer: 157

Choices:
- 253
- 154
- 125
- 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:

Answer: $-\sqrt{6}$

Choices:
- $\sqrt{6}$
- $-\sqrt{6}$
- $-\sqrt{3}$
- $\sqrt{3}$


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:

Answer: $\frac{4-\sqrt{2}}{14}$

Choices:
- $\frac{\sqrt{3}+1}{10(4+\sqrt{3})}$
- $\frac{5-\sqrt{3}}{2\sqrt{2}}$
- $\frac{9\sqrt{3}+3}{10(4+\sqrt{3})}$
- $\frac{4-\sqrt{2}}{14}$


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:

Answer: 27

Choices:
- 25
- 19
- 29
- 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:

Answer: 224

Choices:
- 280
- 224
- 210
- 168


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:

Choices:
- 2
- $\frac{5}{2}$
- $\frac{1}{2}$
- $\frac{15}{6}$


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:

Answer: PRKAUN

Choices:
- PRNAUK
- PRKANU
- PRKAUN
- PRNAKU


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:

Answer: 2139

Choices:
- 2109
- 2129
- 2119
- 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:

Answer: $\frac{5}{8}$

Choices:
- $\frac{3}{4}$
- $\frac{5}{8}$
- $\frac{5}{8}$
- $\frac{1}{4}$


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:

Choices:
- 5
- 4
- 3
- 2


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:

Answer: 33

Choices:
- 27
- 33
- 15
- 18


Question: The remainder, when $7^{103}$ is divided by 23, is equal to:

Answer: 14

Choices:
- 6
- 17
- 9
- 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:

Answer: 58

Choices:
- 58
- 46
- 37
- 72


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:

Answer: 186

Choices:
- 195
- 179
- 186
- 174


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:

Answer: $2\sqrt{3}$

Choices:
- $\sqrt{3}$
- $2\sqrt{2}$
- $2\sqrt{3}$
- $4\sqrt{2}$


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:

Choices:
- $\sqrt{10}$
- $2\sqrt{3}$
- 2
- 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:

Answer: 6

Choices:
- 6
- 3
- 5
- 4


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:

Answer: 440

Choices:
- 3080
- 560
- 3410
- 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:

Answer: $\frac{5}{3}$

Choices:
- $\frac{10}{9}$
- $\frac{3}{2}$
- $\frac{5}{2}$
- $\frac{5}{3}$


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:

Choices:
- 4
- 1
- 3
- 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:

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:

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:

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:

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:

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

Answer: $ABO$ is an obtuse angled isosceles triangle

Choices:
- 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


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

Answer: 675

Choices:
- 545
- 715
- 735
- 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

Answer: $\frac{75}{8}$

Choices:
- $\frac{75}{8}$
- $\frac{25}{2}$
- $\frac{125}{8}$
- $\frac{75}{8}$


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

Answer: $\frac{127}{72}$

Choices:
- $\frac{127}{72}$
- $\frac{131}{72}$
- $\frac{171}{72}$
- $\frac{167}{72}$


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

Answer: 20

Choices:
- 6
- 18
- 8
- 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

Answer: $(\alpha, \beta^2 - 4\gamma)$

Choices:
- $(\gamma, \beta^2 - 4\alpha)$
- $(\alpha, \beta^2 + 4\gamma)$
- $(\gamma, \beta^2 + 4\alpha)$
- $(\alpha, \beta^2 - 4\gamma)$


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

Answer: $\frac{81}{2}$

Choices:
- $1.81 \sqrt{2}$
- 41
- 82
- $\frac{81}{2}$


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

Answer: $\frac{3}{4}$

Choices:
- $\frac{1}{2}$
- $\frac{3}{4}$
- $\frac{1}{4}$
- $\frac{2}{3}$


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

Answer: 9

Choices:
- 9
- 1
- 7
- 8


Question: $\cos \left( \sin^{-1} \frac{3}{8} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)$ is equal to:

Choices:
- 0
- $\frac{32}{65}$
- $\frac{33}{65}$


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 $\frac{9 \sqrt{2}}{2}$.

Answer: only (S1) is true

Choices:
- both are true
- only (S2) is true
- only (S1) is true
- both are false


Question: The area (in sq. units) of the region $\{(x, y): 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3\}$ is

Answer: $\frac{84}{3}$

Choices:
- $\frac{80}{3}$
- $\frac{84}{3}$
- $\frac{32}{3}$
- $\frac{17}{3}$


Question: The sum, of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$, is

Answer: $6(2 - \sqrt{2})$

Choices:
- $3(3 - \sqrt{2})$
- $6(3 - \sqrt{2})$
- $6(2 - \sqrt{2})$
- $3(2 - \sqrt{2})$


Question: Let $T_r$ be the $r^{\text{th}}$ term of an A.P. If for some $m$, $T_m = \frac{1}{25}$, $T_{25} = \frac{1}{20}$, and $20 \sum_{r=1}^{25} T_r = 13$, then $5m \sum_{r=m}^{2m} T_r$ is equal to

Answer: 126

Choices:
- 98
- 126
- 142
- 112


Question: Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denote the number of defective oranges, then the variance of $x$ is

Answer: $\frac{28}{75}$

Choices:
- $\frac{28}{75}$
- $\frac{18}{25}$
- $\frac{26}{75}$
- $\frac{14}{25}$


Question: Let for some function $y = f(x)$, $\int_0^x t f(t) \, dt = x^2 f(x)$, $x > 0$ and $f(2) = 3$. Then $f(6)$ is equal to

Choices:
- 3
- 6
- 2


Question: If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{25 x^2 \cos^2 x}{2(x + e)^2} \, dx = \pi (\alpha \pi^2 + \beta)$, $\alpha, \beta \in \mathbb{Z}$, then $(\alpha + \beta)^2$ equals

Answer: 100

Choices:
- 64
- 196
- 144
- 100


Question: Let $\langle a_n \rangle$ be a sequence such that $a_0 = 0$, $a_1 = \frac{1}{2}$ and $2 a_{n+2} = 5 a_{n+1} - 3 a_n$, $n = 0, 1, 2, 3, \ldots$. Then $\sum_{k=1}^{100} a_k$ is equal to

Answer: $3 a_{100} - 100$

Choices:
- $3 a_{99} - 100$
- $3 a_{100} - 100$
- $3 a_{99} + 100$
- $3 a_{100} + 100$


Question: The number of different 5-digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

Answer: 4607

Choices:
- 4608
- 5720
- 5719
- 4607


Question: The relation $R = \{(x, y): x, y \in \mathbb{Z} \text{ and } x + y \text{ is even}\}$ is:

Answer: an equivalence relation

Choices:
- reflexive and symmetric but not transitive
- an equivalence relation
- symmetric and transitive but not reflexive
- reflexive and transitive but not symmetric


Question: Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min \{1 + x + [x], x + 2[x]\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{cases}$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals

Answer: 5


Question: Let $M$ denote the set of all real matrices of order $3 \times 3$ and let $S = \{-3, -2, -1, 1, 2\}$. Let $S_1 = \{A = [a_{ij}] \in M: A = A^T \text{ and } a_{ij} \in S, \forall i, j\}$, $S_2 = \{A = [a_{ij}] \in M: A = -A^T \text{ and } a_{ij} \in S, \forall i, j\}$, $S_3 = \{A = [a_{ij}] \in M: a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j\}$. If $n(S_1 \cup S_2 \cup S_3) = 125 \alpha$, then $\alpha$ equals

Answer: 1613


Question: If $\alpha = 1 + \sum_{r=1}^6 (-3)^{r-1} {}^{12}C_{2r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x - \sqrt{3} y + 1 = 0$ is

Answer: 5


Question: Let $E_1: \frac{x^2}{5} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$ are constructed such that their centres and eccentricities are same as that of $E_1$, and the length of minor axis of $E_i$ is the length of major axis of $E_{i+1} (i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right)$, is equal to

Answer: 54


Question: Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2 \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{d} = \vec{a} \times \vec{b}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - 2 \vec{a}|^2 = 8$ and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$, then $|10 - 3 \vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2$ is equal to

Answer: 6


Question: Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to:

Answer: 8925

Choices:
- 8750
- 9100
- 8925
- 8575


Question: If the system of equations $x + 2y - 3z = 2$, $2x + \lambda y + 5z = 5$, $14x + 3y + \mu z = 33$ has infinitely many solutions, then $\lambda + \mu$ is equal to:

Answer: 12

Choices:
- 13
- 10
- 12
- 11


Question: Let $\mathbf{A} = \{x \in (0, \pi) - \{\frac{\pi}{2}\} : \log_{(2 / \pi)} |\sin x| + \log_{(2 / \pi)} |\cos x| = 2\}$ and $\mathbf{B} = \{x \geqslant 0 : \sqrt{x} (\sqrt{x} - 4) - 3 |\sqrt{x} - 2| + 6 = 0\}$. Then $n(\mathbf{A} \cup \mathbf{B})$ is equal to:

Answer: 8

Choices:
- 4
- 8
- 6
- 2


Question: The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$, and the $y$-axis is:

Answer: $1 - \log_e 2$

Choices:
- $1 - \log_e 2$
- $\log_e 2$
- $1 + \log_e 2$
- $2 \log_e 2 - 1$


Question: The equation of the chord of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is:

Answer: $48x + 25y = 169$

Choices:
- $48x + 25y = 169$
- $5x + 16y = 31$
- $25x + 101y = 176$
- $4x + 122y = 134$


Question: Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$, and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:

Answer: 33

Choices:
- 44
- 55


Question: Let $f : (0, \infty) \to \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2 f(2)$ is equal to:

Answer: 39

Choices:
- 39
- 19
- 29
- 23


Question: If $7 = 5 + \frac{1}{7} (5 + \alpha) + \frac{1}{77} (5 + 2\alpha) + \frac{1}{77} (5 + 3\alpha) + \infty$, then the value of $\alpha$ is:

Answer: 6

Choices:
- $\frac{6}{7}$
- 6
- $\frac{1}{7}$
- 1


Question: Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of the points where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:

Answer: 8

Choices:
- 6
- 8
- 9
- 7


Question: Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $\mathbf{P}(\mathbf{E})$ is:

Answer: $\frac{3}{8}$

Choices:
- $\frac{3}{16}$
- $\frac{5}{8}$
- $\frac{3}{8}$
- $\frac{1}{8}$


Question: Let the position vectors of three vertices of a triangle be $4 \vec{p} + \vec{q} - 3 \vec{r}$, $-5 \vec{p} + \vec{q} + 2 \vec{r}$, and $2 \vec{p} - \vec{q} + 2 \vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p} + \vec{q} + \vec{r}}{2}$ and $\alpha \vec{p} + \beta \vec{q} + \gamma \vec{r}$ respectively, then $\alpha + 2 \beta + 5 \gamma$ is equal to:

Choices:
- 3
- 4
- 1
- 6


Question: Let $\vec{a} = 3 \hat{i} - \hat{j} + 2 \hat{k}$, $\vec{b} = \vec{a} \times (\hat{i} - 2 \hat{k})$, and $\vec{c} = \vec{b} \times \hat{k}$. Then the projection of $\vec{c} - 2 \hat{j}$ on $\vec{a}$ is:

Answer: $2 \sqrt{14}$

Choices:
- $2 \sqrt{14}$
- $\sqrt{14}$
- $3 \sqrt{7}$
- $2 \sqrt{7}$


Question: The number of real solution(s) of the equation $x^2 + 3x + 2 = \min \{|x - 3|, |x + 2|\}$ is:

Choices:
- 0
- 2
- 3


Question: The function $f : (-\infty, \infty) \to (-\infty, 1)$, defined by $f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}$, is:

Answer: One-one but not onto

Choices:
- Neither one-one nor onto
- Onto but not one-one
- Both one-one and onto
- One-one but not onto


Question: In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to:

Answer: 515

Choices:
- 525
- 510
- 515
- 505


Question: Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1 + x)^{2n - 1}$. If $2A = 5B$, then $n$ is equal to:

Answer: 21

Choices:
- 22
- 20
- 21
- 19


Question: Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2 \log_e (x - 2) - x^2 + a x + 1$ is strictly increasing and $(b, c)$ be the largest open interval in which the function $g(x) = (x - 1)^3 (x + 2 - a)^2$ is strictly decreasing. Then $100(a + b - c)$ is equal to:

Answer: 360

Choices:
- 420
- 360
- 160
- 280


Question: For some $a$, $b$, let $f(x) = \left| \begin{array}{ccc} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to:

Answer: 16

Choices:
- 16
- 25
- 9
- 36


Question: If the equation of the parabola with vertex $\mathbf{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to:

Answer: 9

Choices:
- 7
- 9
- 8
- 6


Question: If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{\beta + \frac{(1 + \beta^5)}{(\alpha - \beta)}\right\} + \cot^{-1} \left\{\gamma + \frac{(1 + \gamma^2)}{(\beta - \gamma)}\right\} + \cot^{-1} \left\{\alpha + \frac{(1 + \alpha^2)}{(\gamma - \alpha)}\right\}$ is equal to:

Answer: $\pi$

Choices:
- $\pi$
- 0
- $\frac{\pi}{2} - (\alpha + \beta + \gamma)$
- $3 \pi$


Question: Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is:

Answer: 957


Question: If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| + C$, where $C$ is the constant of integration, then $\alpha + 2 \beta$ is equal to:

Answer: 16


Question: Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right) = 0$, then $y'\left(\frac{\pi}{4}\right) + y\left(\frac{\pi}{4}\right)$ is equal to:

Answer: 1


Question: Number of functions $f : \{1, 2, \ldots, 100\} \to \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:

Answer: 392


Question: Let $H_1: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2: -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$, then $25 e_2^2$ is equal to:

Answer: 55


Question: Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to:

Answer: 784

Choices:
- 628
- 812
- 526
- 784


Question: Let $x = x(y)$ be the solution of the differential equation $y^2 \mathrm{d}x + \left(x - \frac{1}{y}\right) \mathrm{d}y = 0$. If $x(1) = 1$, then $x\left(\frac{1}{y}\right)$ is:

Answer: $3 - e$

Choices:
- $\frac{1}{2} + e$
- $3 + e$
- $3 - e$
- $\frac{3}{2} + e$


Question: Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to:

Answer: 14

Choices:
- 4
- 14
- 13
- 11


Question: The product of all solutions of the equation $e^{5(\log_e x)^2 + 3} = x^8$, $x > 0$, is:

Answer: $e^{8/5}$

Choices:
- $e^{8/5}$
- $e^{5/5}$
- $e^2$
- $e$


Question: Let the triangle PQR be the image of the triangle with vertices $(1,3)$, $(3,1)$, and $(2,4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to:

Answer: 22

Choices:
- 19
- 24
- 21
- 22


Question: Let $f(x) = 7 \tan^8 x + 7 \tan^8 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_0^{\pi/4} f(x) \mathrm{d}x$ and $I_2 = \int_0^{\pi/4} x f(x) \mathrm{d}x$. Then $7 I_1 + 12 I_2$ is equal to:

Choices:
- 2
- 1
- $2\pi$
- $\pi$


Question: Let the parabola $y = x^2 + p x - 3$ meet the coordinate axes at the points $P$, $Q$, and $R$. If the circle $C$ with centre at $(-1, -1)$ passes through the points $P$, $Q$, and $R$, then the area of $\triangle PQR$ is:

Answer: 6

Choices:
- 7
- 4
- 6
- 5


Question: Let $L_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{6}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$?

Answer: $\left(\frac{14}{3}, -3, \frac{22}{3}\right)$

Choices:
- $\left(\frac{14}{3}, -3, \frac{22}{3}\right)$
- $\left(-\frac{5}{3}, -7, 1\right)$
- $\left(2, 3, \frac{1}{3}\right)$
- $\left(\frac{5}{3}, -1, \frac{1}{3}\right)$


Question: Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x) f'(y) + f'(x) f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_e f(n)$ is equal to:

Answer: 2525

Choices:
- 2525
- 5220
- 2384
- 2406


Question: From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:

Answer: 5148

Choices:
- 5148
- 6084
- 4356
- 14950


Question: Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left((\sec^{-1} x)^2 + (\csc^{-1} x)^2\right)$ is:

Answer: $22\pi^2$

Choices:
- $24\pi^2$
- $22\pi^2$
- $31\pi^2$
- $18\pi^2$


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(x + y) = f(x) f(y)$ for all $x, y \in \mathbb{R}$. If $f'(0) = 4a$ and $f$ satisfies $f''(x) - 3a f'(x) - f(x) = 0$, $a > 0$, then the area of the region $\mathbb{R} = \{(x, y) \mid 0 \leq y \leq f(a x), 0 \leq x \leq 2\}$ is:

Answer: $e^2 - 1$

Choices:
- $e^2 - 1$
- $e^2 + 1$
- $e^4 + 1$
- $e^4 - 1$


Question: The area of the region, inside the circle $(x - 2\sqrt{3})^2 + y^2 = 12$ and outside the parabola $y^2 = 2\sqrt{3} x$ is:

Answer: $6\pi - 16$

Choices:
- $3\pi + 8$
- $6\pi - 16$
- $3\pi - 8$
- $6\pi - 8$


Question: Let the foci of a hyperbola be $(1, 14)$ and $(1, -12)$. If it passes through the point $(1, 6)$, then the length of its latus-rectum is:

Answer: $\frac{288}{5}$

Choices:
- $\frac{24}{5}$
- $\frac{25}{6}$
- $\frac{144}{5}$
- $\frac{288}{5}$


Question: If $\sum_{r=1}^n T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\lim_{n \to \infty} \sum_{r=1}^n \left(\frac{1}{T_r}\right)$ is equal to:

Answer: $\frac{2}{3}$

Choices:
- 0
- $\frac{2}{3}$
- 1
- $\frac{1}{3}$


Question: A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of $X$, then the value of $64(\mu + \sigma^2)$ is:

Answer: 48

Choices:
- 51
- 64
- 32
- 48


Question: The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ is:

Answer: 5

Choices:
- 6
- 5
- 7
- 4


Question: A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $(2, 5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of $r$ is the interval $(\alpha, \beta)$, then $3\beta - 2\alpha$ is equal to:

Answer: 15

Choices:
- 10
- 15
- 12
- 14


Question: Let $A = \{1, 2, 3, \ldots, 10\}$ and $B = \left\{\frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1\right\}$. Then $n(B)$ is equal to:

Answer: 31

Choices:
- 36
- 31
- 37
- 29


Question: Let $z_1, z_2$, and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}$, $\arg(z_2) = 0$, and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is:

Answer: 29

Choices:
- 24
- 29
- 41
- 31


Question: Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \operatorname{adj}(-6 \operatorname{adj}(3 A))) = 2^{m+n} \cdot 3^{mn}$, $m > n$. Then $4m + 2n$ is equal to:

Answer: 34


Question: If $\sum_{x=0}^5 \frac{{ }^{11} C_{2x}}{2x + 2} = \frac{m}{n}$, $\gcd(m, n) = 1$, then $m - n$ is equal to:

Answer: 2035


Question: Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4 \hat{k}$, $\lambda > 0$, on the vector $\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is:

Answer: 16


Question: Let the function, $f(x) = \begin{cases} -3 a x^2 - 2, & x < 1 \\ a^2 + b x, & x \geqslant 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1$, $b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is:

Answer: 34


Question: Let $L_1: \frac{x-1}{3} = \frac{y-1}{4} = \frac{z+1}{0}$ and $L_2: \frac{x-2}{2} = \frac{y}{0} = \frac{z+4}{0}$, $\alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26 \alpha (PB)^2$ is:

Answer: 216


Question: The distance of the line $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the line $\frac{x}{4}=\frac{y-2}{2}=\frac{z+3}{3}$ is:

Answer: $\sqrt{14}$

Choices:
- $\sqrt{17}$
- $\sqrt{15}$
- $\sqrt{14}$
- $\sqrt{13}$


Question: Let $\mathbf{A}=\{(x, y) \in \mathbf{R} \times \mathbf{R} : |x+y| \geqslant 3\}$ and $\mathbf{B}=\{(x, y) \in \mathbf{R} \times \mathbf{R} : |x|+|y| \leq 3\}$. If $\mathbf{C}=\{(x, y) \in \mathbf{A} \cap \mathbf{B} : x=0 \text{ or } y=0\}$, then $\sum_{(x, y) \in \mathbf{C}}|x+y|$ is:

Answer: 12

Choices:
- 15
- 24
- 18
- 12


Question: Let $\boldsymbol{X}=\mathbf{R} \times \mathbf{R}$. Define a relation $R$ on $X$ as: $\left(a_1, b_1\right) R \left(a_2, b_2\right) \Leftrightarrow b_1=b_2$. Statement I: $\mathbf{R}$ is an equivalence relation. Statement II: For some $(a, b) \in X$, the set $S=\{(x, y) \in X : (x, y) R (a, b)\}$ represents a line parallel to $y=x$. In the light of the above statements, choose the correct answer from the options given below:

Answer: Statement I is true but Statement II is false

Choices:
- Both Statement I and Statement II are false
- Statement I is true but Statement II is false
- Both Statement I and Statement II are true
- Statement I is false but Statement II is true


Question: Let $\int x^3 \sin x \, dx = g(x) + C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma$, $\alpha, \beta, \gamma \in \mathbb{Z}$, then $\alpha + \beta - \gamma$ equals:

Answer: 55

Choices:
- 48
- 55
- 62
- 47


Question: A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the rod $AB$ internally in the ratio $2:1$ is $9\left(x^2 + \alpha y^2 + \beta xy + \gamma x + 28 y\right) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to:

Answer: 23

Choices:
- 22
- 21
- 23
- 24


Question: If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{5}$ is $\frac{m}{n}$, where $m, n$ are coprime numbers, then $m+n$ is equal to:

Answer: 14

Choices:
- 21
- 14
- 6
- 10


Question: $\lim_{x \to \infty} \frac{(2x^2 - 3x + 5)(3x - 1)^{\frac{2}{x}}}{(3x^2 + 5x + 4) \sqrt{(3x + 2)^2}}$ is equal to:

Answer: $\frac{2}{3\sqrt{e}}$

Choices:
- $\frac{2}{\sqrt{3x}}$
- $\frac{2e}{\sqrt{3}}$
- $\frac{2}{3\sqrt{e}}$
- $\frac{2e}{3}$


Question: Let the point $A$ divide the line segment joining the points $P(-1,-1,2)$ and $Q(5,5,10)$ internally in the ratio $r:1$ ($r>0$). If $O$ is the origin and $(\overrightarrow{OQ} \cdot \overrightarrow{OA} - \frac{1}{5}|\overrightarrow{OP} \times \overrightarrow{OA}|^2 = 10)$, then the value of $r$ is:

Answer: 7

Choices:
- $\sqrt{7}$
- 14
- 3
- 7


Question: The length of the chord of the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$, whose mid-point is $\left(1, \frac{1}{2}\right)$, is:

Answer: $\frac{2}{3}\sqrt{15}$

Choices:
- $\frac{5}{3}\sqrt{15}$
- $\frac{1}{3}\sqrt{15}$
- $\frac{2}{3}\sqrt{15}$
- $\sqrt{15}$


Question: The system of equations $x + y + z = 6$, $x + 2y + 5z = 9$, $x + 5y + \lambda z = \mu$ has no solution if:

Answer: $\lambda = 17, \mu \neq 18$

Choices:
- $\lambda = 15, \mu \neq 17$
- $\lambda \neq 17, \mu \neq 18$
- $\lambda = 17, \mu \neq 18$
- $\lambda = 17, \mu = 18$


Question: Let the range of the function $f(x) = 6 + 16 \cos x \cdot \cos \left(\frac{\pi}{3} - x\right) \cdot \cos \left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x$, $x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is:

Answer: 11

Choices:
- 11
- 8
- 10
- 9


Question: Let $x = x(y)$ be the solution of the differential equation $y = \left(x - y \frac{dx}{dy}\right) \sin \left(\frac{x}{y}\right)$, $y > 0$ and $x(1) = \frac{\pi}{2}$. Then $\cos (x(2))$ is equal to:

Answer: $2(\log_e 2)^2 - 1$

Choices:
- $1 - 2(\log_e 2)^2$
- $1 - 2(\log_e 2)$
- $2(\log_e 2) - 1$
- $2(\log_e 2)^2 - 1$


Question: A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is $1 \, \text{cm}$, the ice-cream melts at the rate of $81 \, \text{cm}^3 / \text{min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4\pi} \, \text{cm} / \text{min}$. The surface area (in $\text{cm}^2$) of the chocolate ball (without the ice-cream layer) is:

Answer: $256\pi$

Choices:
- $196\pi$
- $256\pi$
- $225\pi$
- $128\pi$


Question: The number of complex numbers $z$, satisfying $|z| = 1$ and $\left|\frac{z}{2} + \frac{2}{z}\right| = 1$, is:

Answer: 8

Choices:
- 4
- 8
- 10
- 6


Question: Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$, $A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ and $A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, then $a_{23}$ equals:

Answer: -1

Choices:
- -1
- 2
- 1
- 0


Question: If $I = \int_0^{\frac{\pi}{2}} \frac{\sin \frac{1}{2} x}{\sin^2 x + \cos^2 x} \, dx$, then $\int_0^{2I} \frac{x \sin x \cos x}{\sin^2 x + \cos^4 x} \, dx$ equals:

Answer: $\frac{\pi^2}{16}$

Choices:
- $\frac{\pi^2}{12}$
- $\frac{\pi^2}{4}$
- $\frac{\pi^2}{16}$
- $\frac{\pi^2}{8}$


Question: A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:

Answer: $\frac{4}{5}$

Choices:
- $\frac{7}{10}$
- $\frac{4}{5}$
- $\frac{23}{30}$
- $\frac{3}{5}$


Question: Let the shortest distance from $(a, 0)$, $a > 0$, to the parabola $y^2 = 4x$ be 4. Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is:

Answer: $x^2 + y^2 - 6x + 5 = 0$

Choices:
- $x^2 + y^2 - 10x + 9 = 0$
- $x^2 + y^2 - 6x + 5 = 0$
- $x^2 + y^2 - 4x + 3 = 0$
- $x^2 + y^2 - 8x + 7 = 0$


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

Answer: 13

Choices:
- 18
- 13
- 8
- 20


Question: If the area of the region $\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{|x|} - e^{-x}, a > 0\}$ is $\frac{\pi^2 + 8e + 1}{e}$, then the value of $a$ is:

Answer: 5

Choices:
- 8
- 7
- 5
- 6


Question: The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is:

Answer: 8788


Question: The roots of the quadratic equation $3x^2 - px + q = 0$ are $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88, then $q - 2p$ is equal to:

Answer: 474


Question: The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is:

Answer: 17280


Question: The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle $C$ of radius 5. If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$, are $\lambda_1$ and $\lambda_2$, $\lambda_1 < \lambda_2$, then $12\lambda_1 + 29\lambda_2$ is equal to:

Answer: 15


Question: Let $\alpha, \beta$ be the roots of the equation $x^2 - ax - b = 0$ with $\operatorname{Im}(\alpha) < \operatorname{Im}(\beta)$. Let $P_n = \alpha^n - \beta^n$. If $\mathbf{P}_3 = -5\sqrt{7}i$, $\mathbf{P}_4 = -3\sqrt{7}i$, $\mathbf{P}_5 = 11\sqrt{7}i$ and $\mathbf{P}_6 = 45\sqrt{7}i$, then $|\alpha^4 + \beta^4|$ is equal to:

Answer: 31