Download JEE Advanced 2010 Mathematics Question Paper - 2

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Marking Scheme

• For each question in Section-I : you will be awarded 5 marks if you have darkened only the bubble corresponding to the correct answer and zero mark if no bubbles are darkened. In all other cases, minus two (-2) mark will be awarded.
• For each question in Section-II : you will be awarded 3 marks if you darken the bubble corresponding to the correct answer and zero mark if no bubble is darkened. No negative marks will be awarded for incorrect answers in this Section.
• For each question in Section-III : you will be awarded 3 marks if you darken only the bubble corresponding to the correct answer and zero mark if no bubbles are darkened. In all other cases, minus one (-1) mark will be awarded.
• For each question in Section-IV : you will be awarded 2 marks for each row in which you have darkened the bubble(s) corresponding to the correct answer. Thus, each question in this section carries a maximum of 8 marks. There is no negative marks awarded for incorrect answer(s) in this Section

SECTION - I (Single Correct Choice Type)

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

1. For $r$=0, 1, ... , 10, let $A_r$, $B_r$ and $C_r$ denote, respectively, the coefficient of $x^r$ in the expansion of $(1+x)^{10}$, $(1+x)^{20}$ and $(1+x)^{30}$. Then
   $\sum \limits_{r=1}^{10}A_r(B_{10}B_r-C_{10}A_r)$
   is equal to
   1. $B_{10}-C_{10}$
   2. $A_{10}(B^2_{10}-C_{10}A_{10})$
   3. 0
   4. $C_{10}-B_{10}$
2. Let $S$={1, 2, 3, 4}. The total number of unordered pair of disjoint subsets of $S$ is equal to
   1. 25
   2. 34
   3. 42
   4. 41
3. Let $f$ be a real valued function defined on the interval (-1, 1) such that $e^{-x}f(X)$=2+$\int \limits_0^x \sqrt{t^4+1}dt$, for all $X \in (-1, 1)$ and let $f^{-1}$ be the inverse function of $f$. The $(f^{-1})'$(2) is equal to
   1. 1
   2. $\frac{1}{3}$
   3. $\frac{1}{2}$
   4. $\frac{1}{e}$
4. If the distance of the point $P(1, -2, 1)$ from the plane $x$+$2y$$-2z$=$\alpha$, where $\alpha >0$, is 5, then the foot of perpendicular from $P$ to the plane is
   1. $\left(\frac{8}{3}, \frac{4}{3}, -\frac{7}{3}\right)$
   2. $\left(\frac{4}{3}, -\frac{4}{3}, \frac{1}{3}\right)$
   3. $\left(\frac{1}{3}, \frac{2}{3}, \frac{10}{3}\right)$
   4. $\left(\frac{2}{3}, -\frac{1}{3}, \frac{5}{2}\right)$
5. Two adjacent sides of a parallelogram $ABCD$ are given by $\vec{AB}$=$2 \hat{i}$+$10\hat{j}$+$11\hat{k}$ and $\vec{AD}$=$- \hat{i}$+$2\hat{j}$+$2\hat{k}$. The side $AD$ is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that $AD$ becomes$AD'$. If $AD'$ makes a right angle with ths side $AB$, then the cosine of the angle $\alpha$ is given by
   1. $\frac{8}{9}$
   2. $\frac{\sqrt{17}}{9}$
   3. $\frac{1}{9}$
   4. $\frac{4\sqrt{5}}{9}$
6. A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively, is received by station A and then transmitted to station B. The probability of each station recieving the signal correctly is $\frac{3}{4}$. If the signal recieved at station B is green, then the probability that the original signal was green is
   1. $\frac{3}{5}$
   2. $\frac{6}{7}$
   3. $\frac{20}{23}$
   4. $\frac{9}{20}$

SECTION - II (Integer Type)

This section contains 5 questions. The answer to each question is a single-digit integer, ranging from 0 to 9. The correct digit below the question number in the ORS is to be bubbled.

7. Two parallel chords of a circle of a radius 2 are at a distance $\sqrt{3}$+1 apart. If the chords subtend at the centre, angles of $\frac{\pi}{k}$ and $\frac{2 \pi}{k}$, where $k$>0, then the value of $[k]$ is
   [Note : $[k]$ denotes the largest integer less than or equal to $k$]
8. Consider a triangle $ABC$ and let $a$, $b$ and $c$ denote the lengths of the sides opposite to the vertices $A$, $B$ and $C$ respectively. Suppose $a$=6, $b$=10 and the area of the triangle is $15 \sqrt{3}$. If $\angle{ACB}$ is obtuse and if $r$ denotes the radius of the incircle of the triangle, then $r^2$ is equal to
9. Let $f$ be a function defined on R (the set of all real numbers) such that
   $f'(x)$=2010$(x-2009)$$(x-2010)^2$$(x-2011)^3$$(x-2012)^4$, for all $x \in $R.
   If $g$ is a function defined on R with values in the interval $(0, \infty)$ such that
   $f(x)$=$ln(g(x))$, for all $x \in $R,
   then the number of points in R at which $g$ has a local maximum is
10. Let $a_1$, $a_2$, $a_3$, ... , $a_{11}$ be real numbers satisfying $a_1$=15, 27$-2a_2$ > 0 and $a_k$=$2a_{k-1}-a_{k-2}$ for $k$=3, 4, ... , 11. If $\frac{a_1^2+a_2^2+...a_{11}^2}{11}$=90, then the value of $\frac{a_1+a_2+...a_{11}}{11}$ is equal to
11. Let $k$ be a positive real number and let $A$=$\begin{equation*} \begin{bmatrix} 2k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2k \\ -2 \sqrt{k} & 2k & -1 \end{bmatrix} \end{equation*}$ and $B$=$\begin{equation*} \begin{bmatrix} 0 & 2k-1 & \sqrt{k} \\ 1-2k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2\sqrt{k} & 0 \end{bmatrix} \end{equation*}$. If det $(adj A)$+det $(adj B)$=$10^6$, then $[k]$ is equal to
    [Note : $adj M$ denotes the adjoint of a square matrix $M$ and $[k]$ denotes the largest integer less than or equal to $k$]

Section III (Paragraph Type)

This section contains 2 paragraphs. Based upon each paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.

Paragraph for Questions 12 to 14

Consider the polynomial $f(x)$=1+$2x$+$3x^2$+$4x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t$=$|s|$

12. The real number $s$ lies in the interval
    1. $\left(-\frac{1}{4}, 0\right)$
    2. $\left(-11, -\frac{3}{4}\right)$
    3. $\left(-\frac{3}{4}, -\frac{1}{2}\right)$
    4. $\left(0, \frac{1}{4}\right)$
13. The area bounded by the curve $y$=$f(x)$ and the lines $x$=0, $y$=0 and $x$=$t$, lies in the interval
    1. $\left(-\frac{3}{4}, 3\right)$
    2. $\left(\frac{21}{64}, \frac{11}{16}\right)$
    3. $\left(9, 10\right)$
    4. $\left(0, \frac{21}{64}\right)$
14. The function $f'(x)$ is
    1. increasing in $\left(-t, -\frac{1}{4}\right)$ and decreasing in $\left(-\frac{1}{4}, t\right)$
    2. decreasing in $\left(-t, -\frac{1}{4}\right)$ and increasing in $\left(-\frac{1}{4}, t\right)$
    3. increasing in $(-t, t)$
    4. decreasing in $(-t, t)$

Paragraph for Questions 15 to 17

Tangents are drawn from the point $P(3, 4)$ to the ellipse $\frac{x^2}{9}$+$\frac{y^2}{4}$=1 touching the ellipse at points A and B.

15. The coordinates of the point A and B are
    1. (3, 0) and (0, 2)
    2. $\left(-\frac{8}{5}, \frac{2\sqrt{161}}{15}\right)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$
    3. $\left(-\frac{8}{5}, \frac{2\sqrt{161}}{15}\right)$ and (0, 2)
    4. (3, 0) and $\left(-\frac{9}{5}, \frac{8}{5}\right)$
16. The orthocentre of the triangle $PAB$ is
    1. $\left(5, \frac{8}{7}\right)$
    2. $\left(\frac{7}{5}, \frac{25}{8}\right)$
    3. $\left(\frac{11}{5}, \frac{8}{5}\right)$
    4. $\left(\frac{8}{25}, \frac{7}{5}\right)$
17. The equation of the locus of the point whose distances from the point $P$ and the line $AB$ are equal, is
    1. $9x^2$+$y^2$$-6xy$$-54x$$-62y$+241=0
    2. $x^2$+$9y^2$+$6xy$$-54x$+$62y-$241=0
    3. $9x^2$+$9y^2$$-6xy$$-54x$$-62y-$241=0
    4. $x^2$+$y^2$$-2xy$$-27x$+$31y-$120=0

SECTION - 3 (Matrix-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 Column-II. For example, if for a given question, statement B matches with the statements given in q and r, then for that particular question against statement B, darken the bubbles corresponding to q and r in the ORS.

(A)	(p)	(q)	(r)	(s)	(t)
(B)	(p)	(q)	(r)	(s)	(t)
(C)	(p)	(q)	(r)	(s)	(t)
(D)	(p)	(q)	(r)	(s)	(t)

18. Match the statements in Column-I with those in Column-II.
    [Note : Here $z$ takes values in the complex plane and $Im z$ and $Re z$ denote, respectively, the imaginary part and real part of $z$].                            

    	Column - I		Column - II
    A)	The set of points $z$ satisfying $|z-i|z||$=$|z+i|z||$ is contained in or equal to	p)	an ellipse with eccentricity $\frac{4}{5}$
    B)	The set of points $z$ satisfying $|z+4|$+$|z-4|$=10 is contained in or equal to	q)	The set of points $z$ satisfying $Im z$=0
    C)	If $|w|$=2, then the set of points $z$=$w-$$\frac{1}{w}$ is contained in or equal to	r)	The set of points $z$ satisfying $|Im z| \leq$ 1
    D)	If $|w|$=1, then the set of points $z$=$w$+$\frac{1}{w}$ is contained in or equal to	s)	The set of points $z$ satisfying $|Re z| \leq$ 2
    		t)	The set of points $z$ satisfying $|z| \leq$ 3

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

    	Column - I		Column - II
    A)	A line from the origin meets the lines $\frac{x-2}{1}$=$\frac{y-1}{-2}$=$\frac{z+1}{1}$ and $\frac{x-\frac{8}{3}}{2}$=$\frac{y+3}{-1}$=$\frac{z-1}{1}$ at $P$ and $Q$ respectively. If length $PQ$=$d$, then $d^2$ is	p)	-4
    B)	The values of $x$ satisfying $tan^{-1}(x+3)-$$tan^{-1}(x-3)$=$sin^{-1}\left(\frac{3}{5}\right)$ are	q)	0
    C)	Non-zero vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ satisfy $\vec{a}$•$\vec{b}$=0, $(\vec{b}-\vec{a})$•$(\vec{b}+\vec{c})$=0 and 2$|\vec{b}+\vec{c}|$=$|\vec{b}-\vec{a}|$. If $\vec{a}$=$\mu \vec{b}+4\vec{c}$, then the possible values of $\mu$ are	r)	4
    D)	Let $f$ be the function on $[-\pi, \pi]$ given by $f(0)$=9 and $f(x)$=$sin\left(\frac{9x}{2}\right)/sin\left(\frac{x}{2}\right)$ for $x \neq 0$. The value of $\frac{2}{\pi}$$\int \limits_{-\pi}^{\pi}f(x)dx$ is	s)	5
    		t)	6

    
    
    

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