Download JEE Advanced 2011 Mathematics Question Paper - 2

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Marking Scheme

• In Section I (Total marks : 24), for each question you will be awarded 3 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other cases, minus one (-1) mark will be awarded.
• In Section II (Total marks : 16), for each question you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise. There are No negative marks in this section.
• In Section III (Total marks : 24), for each question you will be awarded 4 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks otherwise. There are No negative marks in this section.
• In Section IV (Total marks : 16), for each question you will be awarded 2 marks for each row in which you have darkened ALL the bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise. Thus each question in this section carries a maximum of 8 marks. There are No negative marks in this section.

SECTION - I (Total Marks : 24)
(Single Correct Answer Type)

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

1. Let $P(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at the point $P$ intersects the $x$-axis at (9, 0), then the ceecentricity of the hyperbola is
   1. $\sqrt{\frac{5}{2}}$
   2. $\sqrt{\frac{3}{2}}$
   3. $\sqrt{2}$
   4. $\sqrt{3}$
2. A value of $b$ for which the equations
   $x^2$+$bx-$1=0
   $x^2$+$x$+$b$=0,
   have one root in common is
   1. $-\sqrt{2}$
   2. $-i\sqrt{3}$
   3. $i\sqrt{5}$
   4. $\sqrt{2}$
3. Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form \begin{equation*} \begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix} \end{equation*}, where each of $a$, $b$ and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is
   1. 2
   2. 6
   3. 4
   4. 8
4. The circle passing through the point (-1, 0) and touching the $y$-axis at (0, 2) also passes through the point
   1. $\left(-\frac{3}{2},0\right)$
   2. $\left(-\frac{5}{2},2\right)$
   3. $\left(-\frac{3}{2}, \frac{5}{2}\right)$
   4. (-4, 0)
5. If
   $\lim \limits_{x \to 0}[1+x(ln1+b^2)]^{\frac{1}{x}}$=$2bsin^2 \theta$, $b$>0 and $\theta \in (-\pi, \pi]$,
   then the value of $\theta$ is
   1. $±\frac{\pi}{4}$
   2. $±\frac{\pi}{3}$
   3. $±\frac{\pi}{6}$
   4. $±\frac{\pi}{2}$
6. Let $f:[-1, 2] \to [0, \infty)$ be a continuous function such that $f(x)$=$f(1-x)$ for all $x \in [-1, 2]$. Let $R_1$=$\int \limits_{-1}^{2}x f(x)dx$, and $R_2$ be the area of the region bounded by $y$=$f(x)$, $x$=$-1$, $x$=2, and the $x$-axis. Then
   1. $R_1=2R_2$
   2. $R_1=3R_2$
   3. $2R_1=R_2$
   4. $3R_1=R_2$
7. Let $f(x)=x^2$ and $g(x)$=$sinx$ for all $x \in R$. Then the set of all $x$ satisfying $(f∘g∘g∘f)(x)$=$(g∘g∘f)(x)$, where $(f∘g)(x)$=$f(g(x))$, is
   1. $±\sqrt{n \pi}$, $n \in${0, 1, 2, ...}
   2. $±\sqrt{n \pi}$, $n \in${1, 2, ...}
   3. $\frac{\pi}{2}+2n \pi$, $n \in${..., -2, -1, 0, 1, 2, ...}
   4. $2n \pi$, $n \in${..., -2, -1, 0, 1, 2, ...}
8. Let $(x, y)$ be any point on the parabola $y^2$=$4x$. Let $P$ be the point that divides the line segment from (0, 0) to $(x, y)$ in the ratio 1:3. Then the locus of $P$ is
   1. $x^2=y$
   2. $y^2=2x$
   3. $y^2=x$
   4. $x^2=2y$

SECTION - II (Total Marks : 16)
(Multiple Correct Answer(s) Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE may be correct.

9. If
   $f(x)$=$\left\{\begin{array}{lr}-x-\frac{\pi}{2}, & x \leq -\frac{\pi}{2} \\ -cosx, & -\frac{\pi}{2} < x \leq 0 \\ x-1, & 0< x \leq 1 \\ lnx, & x>1, \end{array}\right.$
   then
   1. $f(x)$ is continuous at $x=-\frac{\pi}{2}$
   2. $f(x)$ is not differentiable at $x=0$
   3. $f(x)$ is differentiable at $x=1$
   4. $f(x)$ is differentiable at $x=-\frac{3}{2}$
10. Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occuring is $\frac{2}{25}$. If $P(T)$ denotes the probability of the occurrence of the event $T$, then
    1. $P(E)=\frac{4}{5}$,$P(F)=\frac{3}{5}$
    2. $P(E)=\frac{1}{5}$,$P(F)=\frac{2}{5}$
    3. $P(E)=\frac{2}{5}$,$P(F)=\frac{1}{5}$
    4. $P(E)=\frac{3}{5}$,$P(F)=\frac{4}{5}$
11. Let $L$ be a normal to the parabola $y^2$=$4x$. If $L$ passes through the point (9, 6), then $L$ is given by
    1. $y-$$x$+3=0
    2. $y$+$3x-$33=0
    3. $y$+$x-$15=0
    4. $y-$$2x$+12=0
12. Let $f:(0, 1) \to R$ be defined by
    $f(x)$=$\frac{b-x}{1-bx}$,
    where $b$ is a constant such that $0 < b < 1$. Then
    1. $f$ is not invertible on (0, 1)
    2. $f \neq f^{-1}$ on (0, 1) and $f'(b)$=$\frac{1}{f'(0)}$
    3. $f = f^{-1}$ on (0, 1) and $f'(b)$=$\frac{1}{f'(0)}$
    4. $f^{-1}$ is differentiable on (0, 1)

SECTION - III (Total Marks : 24)
(Integer Answer Type)

This section contains 6 questions. The answer to each of these questions is a single-digit integer, ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.

13. Let $\omega$=$e^{i \pi/3}$, and $a$, $b$, $c$, $x$, $y$, $z$ be non -zero complex numbers such that
    $a$+$b$+$c$=$x$
    $a$+$b \omega$+$c \omega^2$=$y$
    $a$+$b \omega^2$+$c \omega$=$z$.
    Then the value of $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is
14. The number of distinct real roots of $x^4-$$4x^3$+$12x^2$+$x-$1=0 is
15. Let $y'(x)$+$y(x)g'(x)$=$g(x)g'(x)$, $y(0)$=0, $x \in R$, $f'(x)$ denotes $\frac{d f(x)}{dx}$ and $g(x)$ is a given non-constant differentiable function on $R$ with $g(0)$=$g(2)$=0. Then the value of $y(2)$ is
16. Let $M$ be a 3×3 matrix satisfying
    $ \begin{equation*} M \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix} \end{equation*},$$ \begin{equation*} M \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \end{equation*},$ and $\begin{equation*} M \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} 0 \\ 0 \\ 12 \end{bmatrix} \end{equation*}.$ Then the sum of the diagonal entries of $M$ is
17. Let $\vec{a}$=$-\hat{i}-\hat{k}$, $\vec{b}$=$-\hat{i}+\hat{j}$ and $\vec{c}$=$\hat{i}$+$2\hat{j}$+$3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}$×$\vec{b}$=$\vec{c}$×$\vec{b}$ and $\vec{r}$•$\vec{a}$=0, then the value of $\vec{r}$•$\vec{b}$ is
18. The straight line $2x-$ $3y$=1 divides the circular region $x^2$+$y^2 \leq $6 into two parts. If
    $S$=$\left\{ \left(2, \frac{3}{4} \right), \left(\frac{5}{2}, \frac{3}{4} \right),\right.$$\left. \left(\frac{1}{4}, -\frac{1}{4} \right), \left(\frac{1}{8}, \frac{1}{4} \right) \right\}$,
    then the number of point(s) in $S$ lying inside the smaller part is

SECTION - IV (Total Marks : 16)
(Matrix-Match Type)

This section contains 2 questions. Each question has four statements(A, B, C and D) given in Column I and five statements(p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching with ONE or MORE statement(s) in given Column II . For example, if for a given question , statement B matches with the statements given in q and r , then for the particular question, against statement B, darken the bubbles corresponding to q and r in the ORS.

19. Match the statements given in Column-I with the values given in Column-II.                          

    	Column - I		Column - II
    (A)	If $\vec{a}$=$\hat{j}$+$\sqrt{3}\hat{k}$, $\vec{b}$=$-\hat{j}$+$\sqrt{3}\hat{k}$ and $\vec{c}$=$2\sqrt{3}\hat{k}$ form a triangle, then the internal angle of the triangle between $\vec{a}$ and $\vec{b}$ is	(p)	$\frac{\pi}{6}$
    (B)	If $\int \limits_a^b(f(x)-3x)dx$=$a^2-b^2$, then the value of $f\left(\frac{\pi}{6}\right)$ is	(q)	$\frac{2 \pi}{3}$
    (C)	The value of $\frac{\pi ^2}{ln 3} \int \limits_{7/6}^{5/6} \sec(\pi x)dx$ is	(r)	$\frac{\pi}{3}$
    (D)	The maximum value of $\left| Arg\left(\frac{1}{2-z}\right) \right|$ for $|z|$=1, $z \neq 1$ is given by	(s)	$\pi$
    		(t)	$\frac{\pi}{2}$

20. Match the statements given in Column-I with the intervals/union of intervals given in Column-II.                          

    	Column - I		Column - II
    (A)	The set $\left\{Re\left(\frac{2iz}{1-z^2}\right)\right.$ : $z$ is a complex number, $|z|$=1, $\left. z \neq ±1 \right\}$	(p)	$(-\infty, -1)$$ \cup$$ (1, \infty)$
    (B)	The domain of the function $f(x)$=$\sin^{-1}\left(\frac{8(3)^{x-2}}{1-3^{2(x-1)}}\right)$ is	(q)	$(-\infty, 0) $$\cup$$ (0, \infty)$
    (C)	If $f(\theta)$=$\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$, then the set $\left \{f(\theta):0 \leq \theta < \frac{\pi}{2} \right \}$ is	(r)	$[2, \infty)$
    (D)	If $f(x)$=$x^{3/2}(3x-10)$, $x \geq 0$, then $f(x)$ is increasing in	(s)	$(-\infty, -1] $$\cup$$ [1, \infty)$
    		(t)	$(-\infty, 0]$$ \cup$$ [2, \infty)$

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