Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3} \). Then \( f \) is:

• (A) both one-one and onto
• (B) one-one but not onto
• (C) onto but not one-one
• (D) neither one-one nor onto

Consider the quadratic equation \( (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0, \; n \in \mathbb{R}. \)
Let \( \alpha \) be the minimum value of the product of its roots and \( \beta \) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \( \alpha \) and the common ratio is \( \dfrac{\alpha}{\beta} \), is:

• (A) \( \frac{61}{37} \)
• (B) \( \frac{121}{81} \)
• (C) \( \frac{364}{243} \)
• (D) \( \frac{1093}{729} \)

Let \( S = \{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0 \}. \) Then \( \displaystyle \sum_{z \in S} z^8 \) is equal to:

• (A) \(162\)
• (B) \(184\)
• (C) \(262\)
• (D) \(324\)

The sum of all possible values of \( \theta \in [0,2\pi] \), for which the system of equations : \[ x\cos3\theta - 8y - 12z = 0 \] \[ x\cos2\theta + 3y + 3z = 0 \] \[ x + y + 3z = 0 \]
has a non-trivial solution, is equal to :

• (A) \( \pi \)
• (B) \( 2\pi \)
• (C) \( 3\pi \)
• (D) \( 4\pi \)

Let \( A = \begin{bmatrix} 1 & 0 & 0
3 & 1 & 0
9 & 3 & 1 \end{bmatrix} \) and \( B = [b_{ij}], 1 \le i,j \le 3 \). If \( B = A^{99} - I \), then the value of \( \dfrac{b_{31}-b_{21}}{b_{32}} \) is:

• (A) \(99\)
• (B) \(199\)
• (C) \(149\)
• (D) \(159\)

The sum \( 1 + \frac{1}{2}(1^2+2^2) + \frac{1}{3}(1^2+2^2+3^2) + \ldots \) upto \(10\) terms is equal to:

• (A) \(130\)
• (B) \(155\)
• (C) \( \frac{315}{2} \)
• (D) \( \frac{325}{2} \)

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

• (A) \(2184\)
• (B) \(3064\)
• (C) \(7056\)
• (D) \(11340\)

Let the mean and the variance of seven observations \(2,4,\alpha,8,\beta,12,14\), \( \alpha < \beta \), be \(8\) and \(16\) respectively. Then the quadratic equation whose roots are \(3\alpha+2\) and \(2\beta+1\) is :

• (A) \(x^2-35x+306=0\)
• (B) \(x^2-41x+420=0\)
• (C) \(x^2-45x+506=0\)
• (D) \(x^2-37x+342=0\)

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue ball and one green ball is:

• (A) \( \frac{63}{925} \)
• (B) \( \frac{17}{231} \)
• (C) \( \frac{16}{231} \)
• (D) \( \frac{64}{925} \)

Let \(C\) be a circle having centre in the first quadrant and touching the \(x\)-axis at a distance of \(3\) units from the origin. If the circle \(C\) has an intercept of length \(6\sqrt{3}\) on \(y\)-axis, then the length of the chord of the circle \(C\) on the line \(x-y=3\) is:

• (A) \(8\)
• (B) \(6\)
• (C) \(6\sqrt{2}\)
• (D) \(8\sqrt{2}\)

The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \( \frac{\sqrt3}{2} \) and its directrices are \( x=\pm \frac{4\sqrt6}{3} \).
Let \( H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\), and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is :

• (A) \( \frac{4\sqrt2}{\sqrt7} \)
• (B) \( \frac{4\sqrt2}{7} \)
• (C) \( \frac{4}{\sqrt7} \)
• (D) \( \frac{8}{7} \)

Let \(x=-9\) be a directrix of an ellipse \(E\), whose centre is at the origin and eccentricity is \( \frac13 \). Let \(P(\alpha,0), \alpha>0\), be a focus of \(E\) and \(AB\) be a chord passing through \(P\). Then the locus of the mid point of \(AB\) is :

• (A) \(9y^2=8x(1-x)\)
• (B) \(3y^2=4x(1-x)\)
• (C) \(9y^2=8x(x-1)\)
• (D) \(3y^2=4x(x-1)\)

If \( \sin\!\left(\tan^{-1}(x\sqrt2)\right)=\cot\!\left(\sin^{-1}\!\sqrt{1-x^2}\right),\; x\in(0,1) \), then the value of \(x\) is :

• (A) \( \frac12 \)
• (B) \( \frac13 \)
• (C) \( \frac23 \)
• (D) \( \frac58 \)

The shortest distance between the lines \[ \frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} \]
and \[ \frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} \]
is :

• (A) \( \frac{5\sqrt6}{6} \)
• (B) \( 2\sqrt5 \)
• (C) \( 3\sqrt5 \)
• (D) \( 4\sqrt5 \)

Let \( \vec a = 2\hat i + 3\hat j + 3\hat k \) and \( \vec b = 6\hat i + 3\hat j + 3\hat k \). Then the square of the area of the triangle with adjacent sides determined by the vectors \( (2\vec a + 3\vec b) \) and \( (\vec a - \vec b) \) is :

• (A) \(450\)
• (B) \(900\)
• (C) \(1800\)
• (D) \(2400\)

Let \( \lim_{x \to 2} \frac{(\tan(x - 2))(rx^2 + (p - 2)x - 2p)}{(x - 2)^2} = 5 \) for some \( r, p \in \mathbb{R} \). If the set of all possible values of \( q \), such that the roots of the equation \( rx^2 - px + q = 0 \) lie in \( (0, 2) \), be the interval \( (\alpha, \beta] \), then \( 4(\alpha + \beta) \) equals :

• (A) 11
• (B) 13
• (C) 17
• (D) 21

Let \( A = \begin{bmatrix} 1 & 3 & -1
2 & 1 & \alpha
0 & 1 & -1 \end{bmatrix} \) be a singular matrix. Let \( f(x) = \int_{0}^{x} (t^2 + 2t + 3) dt, \; x \in [1, \alpha] \). If M and m are respectively the maximum and the minimum values of f in \( [1, \alpha] \), then \( 3(M - m) \) is equal to :

• (A) 64
• (B) 68
• (C) 72
• (D) 76

Let \( f: \mathbb{R} \to \mathbb{R} \) be such that \( f(xy) = f(x)f(y) \), for all \( x, y \in \mathbb{R} \) and \( f(0) \neq 0 \). Let \( g: [1, \infty) \to \mathbb{R} \) be a differentiable function such that \[ x^2 g(x) = \int_{1}^{x} (t^2 f(t) - tg(t))dt. \]
Then \( g(2) \) is equal to :

• (A) \( \frac{13}{8} \)
• (B) \( \frac{11}{16} \)
• (C) \( \frac{15}{32} \)
• (D) \( \frac{17}{64} \)

The area of the region \( \{(x, y): x^2 - 8x \le y \le -|x|\} \) is:

• (A) \( \frac{343}{6} \)
• (B) \( \frac{637}{6} \)
• (C) \( \frac{437}{6} \)
• (D) \( \frac{523}{6} \)

The value of the integral \( \int_{-1}^{1} \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1} dx \) is equal to :

• (A) \( 3 \log_e 2 \)
• (B) \( 2 \log_e 2 \)
• (C) \( 5 \log_e 3 \)
• (D) \( 3 \log_e 3 \)

Let \( R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_e(x + y) \le 2\} \). Then the minimum number of elements, required to be added in R to make it a transitive relation, is __________.

If \( (1-x^3)^{10} = \sum_{r=0}^{10} a_r x^r (1-x)^{30-2r} \), then \( \frac{9a_9}{a_{10}} \) is equal to __________.

Let the line \( x-y=4 \) intersect the circle \( C: (x-4)^2 + (y+3)^2 = 9 \) at the points Q and R. If \( P(\alpha, \beta) \) is a point on C such that PQ = PR, then \( (6\alpha + 8\beta)^2 \) is equal to __________.

Let the image of the point \( P(0, -5, 0) \) in the line \( \frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{-2} \) be the point R and the image of the point \( Q(0, -1/2, 0) \) in the line \( \frac{x-1}{1} = \frac{y+9}{4} = \frac{z+1}{1} \) be the point S. Then the square of the area of the parallelogram PQRS is __________.

Let \( f(x) = \begin{cases} x^3 + 8; & x<0,
x^2 - 4; & x \ge 0, \end{cases} \) and \( g(x) = \begin{cases} (x-8)^{1/3}; & x<0,
(x+4)^{1/2}; & x \ge 0. \end{cases} \) Then the number of points, where the function gof is discontinuous, is __________.