Download JEE Advanced 2009 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 ONLY the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In 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 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.
• 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 marking for incorrect answer(s) for this section.

SECTION - I
(Single Correct Choice 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 $P(3, 2, 6)$ be a point in a space and $ Q$ be a point on the line $\vec{r}$ =$(\hat{i}-\hat{j}+2\hat{k})$+$\mu(-3\hat{i}+\hat{j}+5\hat{k})$. Then the value of $\mu$ for which the vector $\vec{PQ}$ is parallel to the plane $x-$$4y$+$3z$=1 is
   1. $\frac{1}{4}$
   2. $-\frac{1}{4}$
   3. \frac{1}{8}$
   4. $-\frac{1}{8}$
2. Tangents drawn from the point $P(1, 8)$ to the circle $x^2$+$y^2$$-6x$$-4y-$11=0 touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $PAB$ is
   1. $x^2$+$y^2$+$4x$$-6y$+19=0
   2. $x^2$+$y^2$$-4x$$-10y$+19=0
   3. $x^2$+$y^2$$-2x$+$6y-$29=0
   4. $x^2$+$y^2$$-6x$$-4y$+19=0
3. Let $f$ be a non-negative function defined on the interval [0,1]. If $\int \limits_0^x \sqrt{1-(f'(t))^2}dt$=$\int \limits_0^x f(t)dt$, $0 \leq x \leq 1$ and $f(0)$=0, then
   1. $f\left(\frac{1}{2}\right)$ < $\frac{1}{2}$ and $f\left(\frac{1}{3}\right)$ > $\frac{1}{3}$
   2. $f\left(\frac{1}{2}\right)$ > $\frac{1}{2}$ and $f\left(\frac{1}{3}\right)$ > $\frac{1}{3}$
   3. $f\left(\frac{1}{2}\right)$ < $\frac{1}{2}$ and $f\left(\frac{1}{3}\right)$ < $\frac{1}{3}$
   4. $f\left(\frac{1}{2}\right)$ > $\frac{1}{2}$ and $f\left(\frac{1}{3}\right)$ < $\frac{1}{3}$
4. Let $z$=$x$+$iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation $z\bar{z}^3$+$\bar{z}z^3$=350 is
   1. 48
   2. 32
   3. 40
   4. 80
5. The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^2$+$9y^2$=9 meets its auxiliary circle at the point $M$. Then the area of the triangle with the vertices at $A$, $M$ and the origin $O$ is
   1. $\frac{31}{10}$
   2. $\frac{29}{10}$
   3. $\frac{21}{10}$
   4. $\frac{27}{10}$
6. Let $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$ are unit vectors such that $\left(\vec{a}×\vec{b}\right)$•$\left(\vec{c}×\vec{d}\right)$=1 and $\vec{a}$•$\vec{c}$=$\frac{1}{2}$, then
   1. $\vec{a}$, $\vec{b}$, $\vec{c}$ are non-coplanar
   2. $\vec{b}$, $\vec{c}$, $\vec{d}$ are non-coplanar
   3. $\vec{b}$, $\vec{d}$ are non-parallel
   4. $\vec{a}$, $\vec{d}$ are parallel and $\vec{b}$, $\vec{c}$ are parallel
7. Let $z$=$\cos \theta$+$i \sin \theta$. Then the value of $\sum \limits_{m=1}^{15} Im(z^{2m-1})$ at $\theta$=2° is
   1. $\frac{1}{sin2°}$
   2. $\frac{1}{3sin2°}$
   3. $\frac{1}{2sin2°}$
   4. $\frac{1}{4sin2°}$
8. The number of seven digit integers, with the sum of digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
   1. 55
   2. 66
   3. 77
   4. 88

SECTION - II
(Multiple Correct Choice 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. Area of the region bounded by the curve $y=e^x$ and the lines $x$=0 and $y$=$e$ is
   1. $e-$1
   2. $\int \limits_1^e ln(e+1-y)dy$
   3. $e-$$\int \limits_0^1 e^x dx$
   4. $\int \limits_1^e ln y dy$
10. Let $L$=$\lim \limits_{x \to 0} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}$, $a$ > 0. If $L$ is finite, then
    1. $a$=2
    2. $a$=1
    3. $L$=$\frac{1}{64}$
    4. $L$=$\frac{1}{32}$
11. In a triangle $ABC$ with fixed base $BC$, the vertex $A$ moves such that $\cos B$+$\cos C$=4 $\sin^2 \frac{A}{2}$. If $a$, $b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A$, $B$ and $C$, respectively, then
    1. $b$+$c$=$4a$
    2. $b$+$c$=$2a$
    3. locus of the point $A$ is an ellipse
    4. locus of the point $A$ is a pair of straight lines
12. If $\frac{sin^4x}{2}$+$\frac{cos^4x}{3}$=$\frac{1}{5}$, then
    1. $tan^2x$=$\frac{2}{3}$
    2. $\frac{sin^8x}{8}$+$\frac{cos^8x}{27}$=$\frac{1}{125}$
    3. $tan^2x$=$\frac{1}{3}$
    4. $\frac{sin^8x}{8}$+$\frac{cos^8x}{27}$=$\frac{2}{125}$

SECTION - III
(Comprehension Type)

This section contains 2 groups of questions . Each group has 3 multiple choice questions based on a paragraph. Each of these question has four choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct.

Paragraph for Questions 13 to 15

Let $\mathcal{A}$ be the set of all 3×3 symmetric matrices all of whose entries are either 0 and 1. Five of these entries are 1 and four of them are 0.

13. The number of matrices in $\mathcal{A}$ is
    1. 12
    2. 6
    3. 9
    4. 3
14. The number of matrices $A$ in $\mathcal{A}$ for which the system of linear equations $\begin{equation*} A \begin{bmatrix} x \\ y \\ z \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \end{equation*}$ has a unique solution, is
    1. less than 4
    2. at least 4 but less than 7
    3. at least 7 but less than 10
    4. at least 10
15. The number of matrices $A$ in $\mathcal{A}$ for which the system of linear equations $\begin{equation*} A \begin{bmatrix} x \\ y \\ z \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \end{equation*}$ is inconsistent, is
    1. 0
    2. more than 2
    3. 2
    4. 1

Paragraph for Questions 16 to 18

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.

16. The probability that $X$=3 equals
    1. $\frac{25}{216}$
    2. $\frac{25}{36}$
    3. $\frac{5}{36}$
    4. $\frac{125}{216}$
17. The probability that $X \geq$3 equals
    1. $\frac{125}{216}$
    2. $\frac{25}{36}$
    3. $\frac{5}{36}$
    4. $\frac{25}{216}$
18. The conditional probability that $X \geq$6 given $X$ > 3 equals
    1. $\frac{125}{216}$
    2. $\frac{25}{216}$
    3. $\frac{5}{36}$
    4. $\frac{25}{36}$

SECTION - IV
(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 :

19. Match the statements/expressions in Column-I with the open intervals in Column-II.                            

    	Column - I		Column - II
    (A)	Interval contained in the domain of definition of non-zero solutions of the differential equation $(x-3)^2y'$+$y$=0	( p)	$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
    (B)	Interval containing the value of the integral $\int_1^5 (x-1)$$(x-2)(x-3)$$(x-4)(x-5)dx$	(q)	$\left(0, \frac{\pi}{2}\right)$
    (C)	Interval in which at least one of the points of local maximum of $\cos^2x$+$\sin x$ lies	(r)	$\left(\frac{\pi}{8}, \frac{5\pi}{4}\right)$
    (D)	Interval in which $\tan ^{-1}$$(sinx+cosx)$ is increasing	(s)	$\left(0, \frac{\pi}{8}\right)$
    		(t)	$(-\pi, \pi)$

20. Match the conics in Column-I with the statements/expressions in Column-II.                            

    	Column - I		Column - II
    (A)	Circle	(p)	The locus of the point $(h, k)$ for which the line $hx$+$ky$=1 touches the circle $x^2$+$y^2$=4
    (B)	Parabola	(q)	Points $z$ in the complex plane satisfying $|z+2|-$$|z-2|$=±3
    (C)	Ellipse	(r)	Points of the conic have parametric representation $x$=$\sqrt{3}\left(\frac{1-t^2}{1+t^2}\right)$, $y$=$\frac{2t}{1+t^2}$
    (D)	Hyperbola	(s)	The eccentricity of the conic lies in the interval $1 \leq x < \infty$
    		(t)	Points $z$ in the complex plane satisfying $Re(z+1)^2$=$|z|^2$+1

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