Let \([\cdot]\) denote the greatest integer function. If the domain of the function \(f(x) = \cos^{-1}\left(\frac{4x+2[x]}{3}\right)\) is \([\alpha, \beta]\), then \(12(\alpha+\beta)\) is equal to:

• (A) 6
• (B) 8
• (C) 9
• (D) 4

If the set of all solutions of \(|x^2+x-9| = |x| + |x^2-9|\) is \([\alpha, \beta] \cup [\gamma, \infty)\), then \((\alpha^2+\beta^2+\gamma^2)\) is equal to:

• (A) 9
• (B) 18
• (C) 36
• (D) 72

Let \(z\) be a complex number such that \(|z+2| = |z-2|\) and \(\arg\left(\frac{z+3}{z-i}\right) = \frac{\pi}{4}\). Then \(|z|^2\) is equal to:

• (A) 9
• (B) 4
• (C) 5
• (D) 1

The number of functions \(f : \{1, 2, 3, 4\} \to \{a, b, c\}\), which are not onto, is:

• (A) 48
• (B) 45
• (C) 51
• (D) 35

Let \(S = \left\{ A = \begin{bmatrix} a & b
c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} and A^2 - 4A + 3I = 0 \right\}\) be a set of \(2 \times 2\) matrices. Then the number of matrices in \(S\), for which the sum of the diagonal elements is equal to 4, is:

• (A) 20
• (B) 17
• (C) 21
• (D) 19

Let \(A = \begin{bmatrix} 1 & 1 & 2
-2 & 0 & 1
1 & 3 & 5 \end{bmatrix}\). Then the sum of all elements of the matrix \(adj(adj(2(adjA)^{-1}))\) is equal to:

• (A) 3
• (B) 4
• (C) -4
• (D) -3

The first term of an A.P. of 30 non-negative terms is \(\frac{10}{3}\). If the sum of this A.P. is the cube of its last term, then its common difference is:

• (A) \(\frac{5}{87}\)
• (B) \(\frac{25}{83}\)
• (C) \(\frac{15}{29}\)
• (D) \(\frac{5}{29}\)

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

• (A) 5040
• (B) 3050
• (C) 3410
• (D) 4320

Let the smallest value of \(k \in \mathbb{N}\), for which the coefficient of \(x^3\) in \((1+x)^3 + (1+x)^4 + (1+x)^5 + \dots + (1+x)^{99} + (1+kx)^{100}, x \neq 0\), is \(\left( 43n + \frac{101}{4} \right) \binom{100}{3}\) for some \(n \in \mathbb{N}\), be \(p\). Then the value of \(p + n\) is:

• (A) 10
• (B) 11
• (C) 12
• (D) 13

Suppose that the mean and median of the non-negative numbers \(21, 8, 17, a, 51, 103, b, 13, 67, (a > b)\), are 40 and 21, respectively. If the mean deviation about the median is 26, then \(2a\) is equal to:

• (A) 109
• (B) 117
• (C) 161
• (D) 131

Let the line \(L_1 : x + 3 = 0\) intersect the lines \(L_2 : x - y = 0\) and \(L_3 : 3x + y = 0\) at the points A and B, respectively. Let the bisector of the obtuse angle between the lines \(L_2\) and \(L_3\) intersect the line \(L_1\) at the point C. Then \(BC^2 : AC^2\) is equal to:

• (A) 5:1
• (B) 1:5
• (C) 2:3
• (D) 3:2

Let the vertex A of a triangle ABC be \((1, 2)\), and the mid-point of the side AB be \((5, -1)\). If the centroid of this triangle is \((3, 4)\) and its circumcenter is \((\alpha, \beta)\), then \(21(\alpha + \beta)\) is equal to:

• (A) 309
• (B) 403
• (C) 497
• (D) 524

Suppose that two chords, drawn from the point \((1, 2)\) on the circle \(x^2 + y^2 + x - 3y = 0\) are bisected by the y-axis. If the other ends of these chords are R and S, and the mid point of the line segment RS is \((\alpha, \beta)\), then \(6(\alpha + \beta)\) is equal to:

• (A) 1
• (B) 3
• (C) 4
• (D) 6

A line with direction ratios \(1, -1, 2\) intersects the lines \(\frac{x}{2} = \frac{y}{3} = \frac{z+1}{3}\) and \(\frac{x+1}{-1} = \frac{y-2}{1} = \frac{z}{4}\) at the points P and Q, respectively. If the length of the line segment PQ is \(\alpha\), then \(225\alpha^2\) is equal to:

• (A) 1024
• (B) 1014
• (C) 1104
• (D) 1204

The square of the distance of the point \((-2, -8, 6)\) from the line \(\frac{x-1}{1} = \frac{y-1}{2} = \frac{z}{-1}\) along the line \(\frac{x+5}{1} = \frac{y+5}{-1} = \frac{z}{2}\) is equal to:

• (A) 3
• (B) 6
• (C) 8
• (D) 12

If \(y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1 + \sqrt{1 - x^2}}\right)\), then \(\frac{dy}{dx}\) at \(x = \frac{\sqrt{3}}{2}\) is equal to:

• (A) 3
• (B) -1
• (C) 1
• (D) 2

Let \(f\) be a real polynomial of degree \(n\) such that \(f(x) = f'(x)f''(x)\), for all \(x \in \mathbb{R}\). If \(f(0) = 0\), then \(36\left( f'(2) + f''(2) + \int_0^2 f(x) dx \right)\) is equal to:

• (A) 42
• (B) 46
• (C) 56
• (D) 66

The area of the region \(\{(x, y): y \le \pi - |x|, y \le |x \sin x|, y \ge 0\}\) is:

• (A) \(1+\frac{\pi^2}{8}\)
• (B) \(2+\frac{\pi^2}{4}\)
• (C) \(\frac{\pi^2}{8}-1\)
• (D) \(4+\frac{\pi^2}{2}\)

Let \(\int_{-2}^{2} (|\sin x| + [x \sin x]) dx = 2(3 - \cos 2) + \beta\), where \([\cdot]\) is the greatest integer function. Then \(\beta \sin\left(\frac{\beta}{2}\right)\) equals:

• (A) 1
• (B) 2
• (C) 4
• (D) 8

Let \(y = y(x)\) be the solution of the differential equation \(\frac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)\), \(y(0) = \frac{1}{2}\). Then \((2y(1) - 1)\) is equal to:

• (A) \(\sqrt{3}\tan\left(\frac{11\sqrt{3}}{6}\right)\)
• (B) \(\frac{\sqrt{3}}{2}\tan\left(\frac{11\sqrt{3}}{12}\right)\)
• (C) \(\sqrt{3}\tan\left(\frac{11\sqrt{3}}{12}\right)\)
• (D) \(\frac{\sqrt{3}}{2}\tan\left(\frac{11\sqrt{3}}{6}\right)\)

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is \(p\), then \(96p\) is equal to __________.

Consider the parabola \(P : y^2 = 4kx\) and the ellipse \(E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is \(e\), then \(e^2 + 2\sqrt{2}\) is equal to __________.

If \(A = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}\) and \(B = \tan 81^\circ - \tan 3^\circ\), then \(\frac{B}{A}\) is equal to __________.

Let \(\vec{a}_k = (\tan \theta_k)\hat{i} + \hat{j}\) and \(\vec{b}_k = \hat{i} - (\cot \theta_k)\hat{j}\), where \(\theta_k = \frac{2^{k-1}\pi}{2^n+1}\), for some \(n \in \mathbb{N}, n > 5\). Then the value of \(\frac{\sum_{k=1}^n |\vec{a}_k|^2}{\sum_{k=1}^n |\vec{b}_k|^2}\) is __________.

The number of points, at which the function \(f(x) = \max \{6x, 2+3x^2\} + |x-1| \cos \left| x^2 - \frac{1}{4} \right|, x \in (-\pi, \pi)\), is not differentiable, is __________.