Download JEE Advanced 2012 Mathematics Question Paper - 2

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Marking Scheme

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

Section I (Single Correct Answer Type)

Section I contains 8 multiple choice questions, Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

1. The equation of a plane passing through the line of intersection of the planes $x$+$2y$+$3z$=2 and $x-$$y$+$z$=3 and at a distance $\frac{2}{\sqrt{3}}$ from the point (3, 1, - 1) is
   1. $5x$$-11y$+$z$=17
   2. $\sqrt{2}x$+$y$=$3\sqrt{2} - 1$
   3. $x$+$y$+$z$=$\sqrt{3}$
   4. $x$$-\sqrt{2}y$=$1-\sqrt{2}$
      
2. Let $PQR$ be a triangle of area $\Delta$ with $a=2$, $b=\frac{7}{2}$ and $c=\frac{5}{2}$, where $a$, $b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P$, $Q $ and $R$ respectively. Then $\frac{2sinP-sin2P} {2sinP+sin2P} $ equals
   1. $\frac{3}{4\Delta}$
   2. $\frac{45}{4\Delta}$
   3. $\left(\frac{3}{4\Delta}\right)^2$
   4. $\left(\frac{45}{4\Delta}\right)^2$
      
3. If $\vec{a} $ and $\vec{b} $ are vectors such that $|\vec{a} +\vec{b} |$=$\sqrt{29}$ and $\vec{a} $×$(2\hat{i}+3\hat{j}+4\hat{k})$=$(2\hat{i}+3\hat{j}+4\hat{k})$×$\vec{b} $, then a possible value of $(\vec{a} +\vec{b}) $. $(-7\hat{i}+2\hat{j}+3\hat{k})$ is    
   1. 0
   2. 3
   3. 4
   4. 8
      
4. If $P$ is a 3×3 matrix such that $P^T$= 2$P$+$I$, where $P^T$ is the transpose of $P$ and $I$ is the 3×3 identity matrix, then there exists a column matrix $X$=$\begin{equation*} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{equation*}$ such that
   1. $\begin{equation*}PX=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{equation*}$
   2. $PX=X$
   3. $PX=2X$
   4. $PX= - X$
      
5. Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation
   $(\sqrt[3]{1+a}-1)x^2$+$(\sqrt{1+a}-1)x$+$(\sqrt[6]{1+a}-1)$=0 where $a$ >-1. Then $\lim \limits_{a \to 0^+} \alpha(a)$ and $\lim \limits_{a \to 0^+} \beta(a)$ are
   1. $-\frac{5}{2}$, 1
   2. $-\frac{1}{2}$, -1
   3. $-\frac{7}{2}$, 1
   4. $-\frac{9}{2}$, 3
6. Four fair dice $D_1$, $D_2$, $D_3$ and $D_4$, each having six faces numbered 1, 2, 3, 4, 5 and 6, are rolled simultaneously. The probability that $D_4$ shows a number appearing on one of $D_1$, $D_2$ and $D_3$ is
   1. $\frac{91}{216}$
   2. $\frac{108}{216}$
   3. $\frac{125}{216}$
   4. $\frac{127}{216}$
7. The value of the integral $\int \limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^2+ln\frac{\pi+x}{\pi - x}\right)cosxdx$ is
   1. 0
   2. $\frac{\pi^2}{2}$ -4
   3. $\frac{\pi^2}{2}$+4
   4. $\frac{\pi^2}{2}$
8. Let $a_1$, $a_2$, $a_3$, ... be in harmonic progression with $a_1$ =5 and $a_{20}$=25. The least positive integer $n$ for which $a_n$ < 0 is
   1. 22
   2. 23
   3. 24
   4. 25

Section II (Paragraph Type)

This section contains 6 multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Questions 9 and 10

Let $a_n$ denote the number of all $n$-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let $b_n$ = the number of such $n$-digit integers ending with digit 1 and $c_n$ = the number of such $n$-digit integers ending with digit 0.

9. The value of $b_6$ is
   1. 7
   2. 8
   3. 9
   4. 11
10. Which of the following is correct?
    1. $a_{17}=a_{16}+a_{15}$
    2. $c_{17} \neq c_{16}+c_{15}$
    3. $b_{17} \neq b_{16}+c_{16}$
    4. $a_{17}=c_{16}+b_{16}$

Paragraph for Questions 11 and 12

Let $f(x)$=$(1-x)^2sin^x$+$x^2$ for all $x \in R$, let $g(x)$=$\int \limits_{1}^{x}\left(\frac{2(t-1)}{t+1}-lnt\right)f(t)dt$ for all $x \in (1, \infty)$

11. Which of the following is true?
    1. $g$ is increasing on $(1, \infty)$
    2. $g$ is decreasing on $(1, \infty)$
    3. $g$ is increasing on $(1, 2)$ and decreasing on $(2, \infty)$
    4. $g$ is decreasing on $(1, 2)$ and increasing on $(2, \infty)$
12. Consider the statement s:
    P : There exists some $x \in R$ such that $f(x)$+$2x$=$2(1+x^2)$
    Q : There exists some $x \in R$ such that $2f(x)$+1=$2x(1+x)$
    Then
    1. both P and Q are true
    2. P is true and Q is false
    3. P is false and Q is true
    4. both P and Q are false

Paragraph for Questions 13 and 14

A tangent $PT$ is drawn to the circle $x^2$+$y^2$=4 at the point $P(\sqrt{3}, 1)$. A straight line $L$, perpendicular to the $PT$ is a tangent to the circle $(x-3)^2$+$y^2$=1.

13. A possible equation of $L$ is
    1. $x-$$\sqrt{3}y$=1
    2. $x$+$\sqrt{3}y$=1
    3. $x-$$\sqrt{3}y$=-1
    4. $x$+$\sqrt{3}y$=5
14. A common tangent of the two circles is
    1. $x=4$
    2. $y=2$
    3. $x$+$\sqrt{3}y$=4
    4. $x$+$2\sqrt{2}y$=6

Section III (Multiple Correct Answer(s) Type)

This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

15. For every integer $n$, let $a_n$ and $b_n$ be real numbers. Let function$f : R \to R$ be given by $\left\{\begin{array}{ll} a_n+sin \pi x & \text { for } x \in [2n, 2n+1], \\ b_n + cos \pi x & \text { for } x \in (2n-1, 2n) .\end{array}\right.$, for all integers $n$. If $f$ is continuous, then which of the following hold(s) for all $n$?
    1. $a_{n-1}-b_{n-1}=0$
    2. $a_{n}-b_{n}=1$
    3. $a_{n}-b_{n+1}=1$
    4. $a_{n-1}-b_{n}=-1$
16. If $f(x)$=$\int \limits_0^x e^{t^2}(t-2)(t-3)dt$ for all $x \in (0, \infty)$, then
    1. $f$ has a local maximum àt $x=2$
    2. $f$ is decreasing on (2,3)
    3. there exists some $c \in (0, \infty)$ such that $f"(c)=0$
    4. $f$ has a local minimum àt $x=3$
17. If the straight lines $\frac{x-1}{2}$=$\frac{y+1}{k}$=$\frac{z}{2}$ and $\frac{x+1}{5}$=$\frac{y+1}{2}$=$\frac{z}{k}$ are coplanar, then the plane(s) containing these two lines is(are)
    1. $y+2z=-1$
    2. $y+z=-1$
    3. $y-z=-1$
    4. $y-2z=-1$
18. Let $X$ and $Y$ be two events such that $P(X|Y)$=$\frac{1}{2}$, $P(Y|X)$=$\frac{1}{3}$ and $P(X \cap Y)$=$\frac{1}{6}$. Which of the following is (are) correct?
    1. $P(X \cup Y)=\frac{2}{3}$
    2. $X$ and $Y$ are independent
    3. $X$ and $Y$ are not independent
    4. $P(X^c \cap Y)=\frac{1}{3}$
19. If the adjoint of a 3×3 mattix $P$ is $\begin{equation*} \begin{bmatrix} 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{bmatrix} \end{equation*},$ then the possible value(s) of the determinant of $P$ is(are)
    1. $-2$
    2. $-1$
    3. 1
    4. 2
20. Let $f(-1,1) \to R$ be such that $f(cos 4 \theta)$=$\frac{2}{2-sec^2 \theta}$ for $\theta \in \left(0, \frac{\pi}{4}\right)\cup$$\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then the value(s) of $f\left(\frac{1}{3}\right)$ is (are)
    1. $1-\sqrt{\frac{3}{2}}$
    2. $1+\sqrt{\frac{3}{2}}$
    3. $1-\sqrt{\frac{2}{3}}$
    4. $1+\sqrt{\frac{2}{3}}$

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