Download JEE Advanced 2012 Mathematics Question Paper - 1

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Marking Scheme

• For each question in Section I, you will be awarded 3 marks if you darken the bubble corresponding to the correct answer ONLY and zero marks if no bubbles are darkened. In all other cases, minus one (-1) mark will be awarded in this section.
• For each question in Section II, you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero (0) marks will be awarded. No negative marks will be awarded for incorrect answers in this section.
• For each question in Section III, you will be awarded 4 marks if you darken the bubble corresponding to the correct answer ONLY. In all other cases zero (0) marks will be awarded. No negative marks will be awarded for incorrect answers in this section.

SECTION - I (Single Correct Answer Type)

Section I contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

1. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is
   1. 75
   2. 150
   3. 210
   4. 243
      
2. Let $f(x)$=$\left\{\begin{array}{lr}x^2\left|cos\frac{\pi}{x}\right| x \neq 0 \\ 0, x=0 \end{array}\right.$, $x \in R$, then $f$ is    
   1. differentiable both at $x$=0 and at $x$=2
   2. differentiable at $x$=0 but not differentiable at $x$=2
   3. not differentiable at $x$ =0 but differentiable at $x$=2.
   4. differentiable neither at $x$=0 nor at $x$=2.
      
3. The function $f:[0, 3] \to [1, 29]$, defined by $2x^3-$$15x^2$+$36x$+1, is  
   1. one-one and onto
   2. onto but not one-one
   3. one-one but not onto
   4. neither one-one nor onto
      
4. If $\lim \limits_{x \to \infty} \left(\frac{x^2+x+1}{x+1}-ax-b\right) $=4, then  
   1. $a$=1, $b$=4
   2. $a$=1, $b$=-4
   3. $a$ =2, $b$=-3.
   4. $a$=2, $b$=3.
      
5. Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a$=$z^2$+$z$+1 is real. Then $a$ cannot take the value  
   1. -1
   2. $\frac{1}{3}$
   3. $\frac{1}{2}$
   4. $\frac{3}{4}$
      
6. The ellipse $E_1$=$\frac{x^2}{9}$+$\frac{y^2}{4}$=1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse $E_2$ is  
   1. $\frac{\sqrt{2}}{2}$
   2. $\frac{\sqrt{3}}{2}$
   3. $\frac{1}{2}$
   4. $\frac{3}{4}$
      
7. Let P=$[a_{ij}]$ be a 3×3 matrix and let Q=$[b_{ij}]$, where $b_{ij}$=$2^{i+j}a_{ij}$ for $1 \leq i, j \leq 3$. If the determinant of P is 2, then the determinant of the matrix Q is
   1. $2^{10}$
   2. $2^{11}$
   3. $2^{12}$
   4. $2^{13}$
8. The integral $\int \frac{sec^2x}{(secx+tanx)^{9/2}}dx$ equals (for some arbitrary constant K).
   1. $-\frac{1}{(secx+tanx)^{11/2}}$$\left\{\frac{1}{11}-\frac{1}{7}(secx+tanx)^2\right\}$+K
   2. $\frac{1}{(secx+tanx)^{11/2}}$$\left\{\frac{1}{11}-\frac{1}{7}(secx+tanx)^2\right\}$+K
   3. $-\frac{1}{(secx+tanx)^{11/2}}$$\left\{\frac{1}{11}+\frac{1}{7}(secx+tanx)^2\right\}$+K
   4. $\frac{1}{(secx+tanx)^{11/2}}$$\left\{\frac{1}{11}+\frac{1}{7}(secx+tanx)^2\right\}$+K
9. The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, -1, 4) with the plane $5x-4y-z$=1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS is
   1. $\frac{1}{\sqrt{2}}$
   2. $\sqrt{2}$
   3. 2
   4. $2\sqrt{2}$
10. The locus of the mid-point of the chord of contact of tangents drawn from the points lying on the straight line $4x-5y$=20 to the circle $x^2+y^2$=9 is
    1. $20(x^2+y^2)$$-36x$+$45y$=0
    2. $20(x^2+y^2)$$+36x$$-45y$=0
    3. $36(x^2+y^2)$$-20x$+$45y$=0
    4. $36(x^2+y^2)$$+20x$+$-45y$=0

SECTION - II (Multiple Correct Answer(s) Type)

This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

11. Let $\theta$, $\phi \in [0, 2\pi]$ be such that 2$\cos \theta$ $(1-\sin \phi)$=$sin^2 \theta$ $\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right)$ $\cos \phi -$ 1, $\tan (2\phi - \theta)$ > 0 and -1 <$\sin \theta < -\frac{\sqrt{3}}{2}$. Then $\phi$ cannot satisfy
    1. $a < \phi < \frac{\pi}{2} $
    2. $\frac{\pi}{2} < \phi < \frac{4\pi}{3} $
    3. $\frac{4\pi}{3} < \phi < \frac{3\pi}{2} $
    4. $\frac{3\pi}{2} < \phi < 2\pi$
12. Let $S$ be the area of the region enclosed by $y=e^{-x^2}$, $y$=0, $x$=0, and $x$=1. Then
    1. $S \geq \frac{1}{e} $
    2. $S \geq 1-\frac{1}{e} $
    3. $S \leq \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$
    4. $S \leq \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{e}}(1-\frac{1}{\sqrt{2}})$
13. A ship is fitted with three engines $E_1$, $E_2$ and $E_3$. The engines function independently of each other with respective probabilities $\frac{1}{2}$, $\frac{1}{4}$ and $\frac{1}{4}$. For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let $X_1$, $X_2$ and $X_3$ denote respectively the events that engines $E_1$, $E_2$ and $E_3$ are functioning. Which of the following is(are) true?
    1. $P[X_1^c|X]\frac{3}{16}$
    2. $P$[exactly two engines of the ship are functioning|$X$]=$ \frac{7}{8}$
    3. $P[X|X_2]=\frac{5}{16}$
    4. $P[X|X_1]=\frac{7}{16}$
14. Tangents are drawn to the hyperbola $\frac{x^2}{9}-$$\frac{y^2}{4}$=1, parallel to the straight line $2x-$$y$=1. The points of contact of the tangents on the hyperbola are
    1. $\left(\frac{9}{2\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
    2. $\left(-\frac{9}{2\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)$
    3. $(3\sqrt{3}, -2\sqrt{2})$
    4. $(-3\sqrt{3}, 2\sqrt{2})$
15. If $y(x)$ satisfies the differential equation $y'-$$ytanx$=$2x secx$ and $y(0)$=0, then
    1. $y\left(\frac{\pi}{4}\right)$=$\frac{\pi^2}{8\sqrt{2}}$
    2. $y'\left(\frac{\pi}{4}\right)$=$\frac{\pi^2}{18}$
    3. $y\left(\frac{\pi}{3}\right)$=$\frac{\pi^2}{9}$
    4. $y'\left(\frac{\pi}{3}\right)$=$\frac{4\pi}{3}$+$\frac{2\pi^2}{3\sqrt{3}}$

SECTION - III (Integer Answer Type)

This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).

16. Let $f : R \to R$ be defined as $f(x)$=$|x|$+$|x^2-1|$. The total number of points at which $f$ attains either a local maximum or a local minimum is
17. The value of 6+$log_{\frac{3}{2}}$$\left(\frac{1}{3\sqrt{2}}\right.$$\left.\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}...}}}\right)$ is
18. Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x$=1 and a local minimum at $x$=3. $p(1)$=6 and $p(3)$=2, then $p'(0)$ is
19. If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a}-\vec{b}|^2$+$|\vec{b}-\vec{c}|^2$+$|\vec{c}-\vec{a}|^2$=9, then $|2\vec{a}+5\vec{b}+5\vec{c}|$ is
20. Let S be the focus of the parabola $y^2=8x$ and let PQ be the common chord of the circle $x^2$+$y^2$$-2x$$-4y$=0 and the given parabola. The area of the triangle PQS is

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