Download JEE Advanced 2009 Mathematics Question Paper - 2

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Marking Scheme

• For each question in Section I, 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 case of bubbling of incorrect answer, minus one (-1) mark will be awarded.
• For each question in Section II, you will be awarded 4 marks if you darken the bubble(s) corresponding to the correct choice(s) for the answer and zero mark if no bubble is darkened. In all other cases minus one (-1) mark will be awarded.
• For each question in Section III, 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 marking for incorrect answer(s) for this section.
• For each question in Section IV you will be awarded 4 marks if you darken the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases minus one (-1) mark will be awarded.

SECTION - I
(Single Correct Choice Type)

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

1. If the sum of first $n$ terms of an A.P. is $cn^2$, then the sum of the squares of these $n$ terms is
   1. $\frac{n(4n^2-1)c^2}{6}$
   2. $\frac{n(4n^2+1)c^2}{3}$
   3. $\frac{n(4n^2-1)c^2}{3}$
   4. $\frac{n(4n^2+1)c^2}{6}$
2. A line with positive direction cosines passes through the point $P(2, -1, 2)$ and makes equal angles with the coordinate axes. The line meets the plane $2x$+$y$+$z$=9 at point $Q$. The length of the line segment $PQ$ equals
   1. 1
   2. $\sqrt{2}$
   3. $\sqrt{3}$
   4. 2
3. The normal at a point $P$ on the ellipse $x^2$+$4y^2$=16 meets the $x-$axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points
   1. $\left(±\frac{3\sqrt{5}}{2}, ± \frac{2}{7}\right)$
   2. $\left(±\frac{3\sqrt{5}}{2}, ±\frac{\sqrt{19}}{4}\right)$
   3. $\left(±2\sqrt{3}, ± \frac{1}{7}\right)$
   4. $\left(±2\sqrt{3}, ± \frac{4\sqrt{3}}{7}\right)$
4. The locus of the orthocentre of the triangle formed by the lines
   $(1+p)x$$-py$+$p(1+p)$=0,
   $(1+q)x$$-qy$+$q(1+q)$=0,
   and $y$=0 where $p \neq q$, is
   1. a hyperbola
   2. a parabola
   3. an ellipse
   4. a straight line

SECTION - II
(Multiple Correct Choice Type)

This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which One OR MORE may be correct.

5. If $I_n$=$\int \limits_{-\pi}^{\pi}\frac{sinnx}{(1+\pi^x)sinx}dx$, $n$=0, 1, 2, ... , then
1. $I_n$=$I_{n+2}$
2. $\sum \limits_{m=1}^{10}I_{2m+1}$=$10 \pi$
3. $\sum \limits_{m=1}^{10}I_{2m}$=0
4. $I_n$=$I_{n+1}$
6. An ellipse intersects the hyperbola $2x^2$$-2y^2$=1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then
   1. Equation of the ellipse is $x^2$+$2y^2$=2
   2. The focii of the ellipse are (±1,0)
   3. Equation of the ellipse is $x^2$+$2y^2$=4
   4. The focii of the ellipse are $(±\sqrt{2},0)$
7. For the function $f(x)$=$x$$\cos \frac{1}{x}$, $x \geq 1$,
   1. for at least one $x$ in the interval $[1, \infty)$, $f(x+2)$$-f(x)$ < 2
   2. $\lim \limits_{x \to \infty} f'(x)$=1
   3. for all $x$ in the interval $[1, \infty)$, $f(x+2)$$-f(x)$ > 2
   4. $f'(x)$ is strictly decreasing in the interval $[1, \infty)$
8. The tangent $PT$ and the normal $PN$ to the parabola $y^2$=$4ax$ at a point $P$ on it meet its axis at $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
   1. vertex is $\left(\frac{2a}{3}, 0\right)$
   2. directrix is $x$=0
   3. latus rectum is $\frac{2a}{3}$
   4. focus is $(a, 0)$
9. For $0 < \theta < \frac{\pi}{2}$, the solution(s) of $\sum \limits_{m=1}^{6}cosec \left(\theta+\frac{(m-1)\pi}{4}\right)$$cosec\left(\theta +\frac{m\pi}{4}\right)$=$4\sqrt{2}$ is(are)
   1. $\frac{\pi}{4}$
   2. $\frac{\pi}{6}$
   3. $\frac{\pi}{12}$
   4. $\frac{5\pi}{12}$

SECTION - III
(Matrix-Match Type)

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are lebelled A, B, C and D, while statements in Column II are levelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE or MORE statement(s) in given Column II .The appropriate bubbles corresponding to the correct answers to these questions have to be darkened as illustrated in the following example :
If the correct matches are A - p, s and t; B - q and r; C - p and q; D - s and t; then the correct darkening of the bubbles will look like the following :

10. Match the statements/expressions in Column-I with the values given in Column-II.                            

    	Column - I		Column - II
    (A)	Root(s) of the equation $2sin^2\theta$+$sin^22\theta$=2	(p)	$\frac{\pi}{6}$
    (B)	Points of discontinuity of the function $f(x)$=$\left[\frac{6x}{\pi}\right]$$\cos\left[\frac{3x}{\pi}\right]$, where $[y]$ denotes the largest integer less than or equal to $y$	(q)	$\frac{\pi}{4}$
    (C)	Volume of the parallelopiped with its edges represented by the vectors $\hat{i}+\hat{j}$, $\hat{i}+2\hat{j}$ and $\hat{i}+\hat{j}+\pi\hat{k}$	(r)	$\frac{\pi}{3}$
    (D)	Angle between vectors $\vec{a}$ and $\vec{b}$ where $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors satisfying $\vec{a}$+$\vec{b}$+$\sqrt{3}\vec{c}$=$\vec{0}$	(s)	$\frac{\pi}{2}$
    		(t)	$\pi$

11. Match the statements/expressions in Column-I with the values given in Column-II.                            

    	Column - I		Column - II
    (A)	The number of solutions of the equation $xe^{sinx}-$$\cos x$=0 in the interval $\left(0, \frac{\pi}{2}\right)$	(p)	1
    (B)	Value(s) of $k$ for which the planes $kx$+$4y$+$z$=0, $4x$+$ky$+$2z$=0 and $2x$+$2y$+$z$=0 intersect in a straight line.	(q)	2
    (C)	Value(s) of $k$ for which $|x-1|$+$|x-2|$+$|x+1|$+$|x+2|$=$4k$ has integer solution(s)	(r)	3
    (D)	If $y'$=$y$+1 and $y(0)$=1, then value(s) of $y(ln2)$	(s)	4
    		(t)	5

SECTION - IV
(Integer Answer Type)

This section contains 8 questions. The answer to each of these questions is a single-digit integer, ranging from 0 to 9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened. For example, if the correct answer to the question numbers X, Y, Z and W (say) are 6, 0, 9 and 2, respectively, then the correct darkening of the bubbles will look like the following:

12. The maximum value of the function $f(x)$=$2x^3$$-15x^2$+$36x-$48 on the set $A$={$x|x^2+20 \leq 9x$} is
13. Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations:
    $3x-$$y-$$z$=0
    $-3x$+$z$=0
    $-3x$+$2y$+$z$=0.
    Then the number of such points for which $x^2$+$y^2$+$z^2 \leq 100$ is
14. Let $ABC$ and $ABC'$ be two non-congruent triangles with sides $AB$=4, $AC$=$AC'$=$2\sqrt{2}$ and angle $B$=30°. The absolute value of the difference between the areas of these triangles is
15. Let $P(x)$ be a polynomial of degree 4 having extremum at $x$=1, 2 and $\lim \limits_{x \to 0} \left(1+\frac{p(x)}{x^2}\right)$=2. Then the value of $P(2)$ is
16. Let $f: R \to R $ be a continuous function which satisfies $f(x)$ =$\int \limits_{0}^{x} f(t)dt$. Then the value of $f(ln5)$ is
17. The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ paasing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is
18. The smallest value of $k$, for which the both the roots of the equation $x^2-$$8kx$+16$(k^2+k+1)$=0 are real, distinct and have values at least 4, is
19. If the function $f(x)$=$x^3$+$e^{\frac{x}{2}}$ and $g(x)$=$f^{-1}(x)$, then the value of $g'(1)$ is

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