Download JEE Advanced 2008 Mathematics Question Paper - 2

Please wait! Loading...

 Download as PDF

2007:Paper-2

2009:Paper-2

All the Papers

Marking Scheme

• For each question in Section I, you will be awarded 3 Marks if you have darkened only 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 II, you will be awarded 3 Marks if you darken only 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 III, you will be awarded 4 Marks if you darken only 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 6 Marks if you have darken ALL the bubble corresponding ONLY to the correct answer or awarded 1 mark each for correct bubbling of answer in any row. No negative mark will be awarded for an incorrectly bubbled answer.

SECTION - I
(Single Correct Choice Type)

This section contains 9 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

1. A particle $P$ starts from the point $z_0$=1+$2i$, where $i$=$\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i}$+$\hat{j}$ and then it moves through an angle of $\frac{\pi}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point $z_2$. The point $z_2$ is given by
   1. $6+7i$
   2. $-7+6i$
   3. $7+6i$
   4. $-6+7i$
2. Let the function $g:(-\infty, \infty)$ $\to \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be given by $g(u)$=$2\tan^{-1}(e^u)-$$\frac{\pi}{2}$. Then $g$ is
   1. even and is strictly increasing in $(0, \infty)$
   2. odd and is strictly decreasing in $(-\infty, \infty)$
   3. odd and is strictly increasing in $(-\infty, \infty)$
   4. neither even nor odd, but is strictly increasing in $(-\infty, \infty)$
3. Consider a branch of the hyperbola $x^2-$$2y^2-$$2\sqrt{2}x-$$4\sqrt{2}y-$6=0 with vertex at the point $A$. Let $B$ be one of the end points of the latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $ABC$ is
   1. $1-\sqrt{\frac{2}{3}}$
   2. $\sqrt{\frac{3}{2}}-1$
   3. $1+\sqrt{\frac{2}{3}}$
   4. $\sqrt{\frac{3}{2}}+1$
4. The area of the region between the curves $y$=$\sqrt{\frac{1+sinx}{cosx}}$ and $y$=$\sqrt{\frac{1-sinx}{cosx}}$ bounded by the lines $x=0$ and $x$=$\frac{\pi}{4}$ is
   1. $\int \limits_0^{\sqrt{2}-1}\frac{t}{(1+t^2)\sqrt{1-t^2}}dt$
   2. $\int \limits_0^{\sqrt{2}-1}\frac{4t}{(1+t^2)\sqrt{1-t^2}}dt$
   3. $\int \limits_0^{\sqrt{2}+1}\frac{4t}{(1+t^2)\sqrt{1-t^2}}dt$
   4. $\int \limits_0^{\sqrt{2}+1}\frac{t}{(1+t^2)\sqrt{1-t^2}}dt$
5. Consider three points $P$=$(-sin (\beta -\alpha), -cos \beta)$, $Q$=$(cos (\beta - \alpha), sin \beta)$ and $R$=$(cos (\beta - \alpha + \theta), sin (\beta-\theta))$, where $0 < \alpha$, $\beta$, $\theta < \frac{\pi}{4}$. Then
   1. $P$ lies on the line segment $RQ$
   2. $Q$ lies on the line segment $PR$
   3. $R$ lies on the line segment $QP$
   4. $P$, $Q$, $R$ are non-collinear
6. An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is
   1. 2, 4 or 8
   2. 3, 6 or 9
   3. 4 or 8
   4. 5 or 10
7. Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\vec{OP}$ (where $O$ is the origin) is given by $\hat{a}cost$+$\hat{b}sint$. When $P$ is farthest from origin $O$, let $M$ be the length of $\vec{OP}$ and $\hat{u}$ be the unit vector along $\vec{OP}$. Then
   1. $\hat{u}$=$\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}$ and $M$=$(1+\hat{a}•\hat{b})^{\frac{1}{2}}$
   2. $\hat{u}$=$\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}$ and $M$=$(1+\hat{a}•\hat{b})^{\frac{1}{2}}$
   3. $\hat{u}$=$\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}$ and $M$=$(1+2\hat{a}•\hat{b})^{\frac{1}{2}}$
   4. $\hat{u}$=$\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}$ and $M$=$(1+2\hat{a}•\hat{b})^{\frac{1}{2}}$
8. Let $I$=$\int \frac{e^x}{e^{4x}+e^{2x}+1}dx$, $J$=$\int \frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx$. Then for an arbitrary constant $C$, the value of $J-I$ equals
   1. $\frac{1}{2}log\left(\frac{e^{4x}-e^{2x}+1}{e^{4x}+e^{2x}+1}\right) +C$
   2. $\frac{1}{2}log\left(\frac{e^{2x}+e^{x}+1}{e^{2x}-e^{x}+1}\right) +C$
   3. $\frac{1}{2}log\left(\frac{e^{2x}-e^{x}+1}{e^{2x}+e^{x}+1}\right) +C$
   4. $\frac{1}{2}log\left(\frac{e^{4x}+e^{2x}+1}{e^{4x}-e^{2x}+1}\right) +C$
9. Let $g(x)$=$log f(x)$ where $f(x)$ is twice differentiable positive function on $(0, \infty)$ such that $f(x+1)$=$xf(x)$. Then, for $N$=1, 2, 3, ... ,
   $g"\left(N+\frac{1}{2}\right)-$$g"\left(\frac{1}{2}\right)$=
   1. $-4 \left\{1+\frac{1}{9}+\frac{1}{25}\right.$$\left.+...+\frac{1}{(2N-1)^2}\right\}$
   2. $4 \left\{1+\frac{1}{9}+\frac{1}{25}\right.$$\left.+...+\frac{1}{(2N-1)^2}\right\}$
   3. $-4 \left\{1+\frac{1}{9}+\frac{1}{25}\right.$$\left.+...+\frac{1}{(2N+1)^2}\right\}$
   4. $4 \left\{1+\frac{1}{9}+\frac{1}{25}\right.$$\left.+...+\frac{1}{(2N+1)^2}\right\}$

SECTION - II
(Reasoning Type)

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

10. Suppose four distinct positive numbers $a_1$, $a_2$, $a_3$, $a_4$ are in G.P. Let $b_1$=$a_1$, $b_2$=$b_1$+$a_2$, $b_3$=$b_2$+$a_3$ and $b_4$=$b_3$+$a_4$
    STATEMENT-1: The numbers $b_1$, $b_2$, $b_3$, $b_4$ are neither in A.P. nor in G.P.
    and
    STATEMENT-2:The numbers $b_1$, $b_2$, $b_3$, $b_4$ are in H.P.
    1. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is a correct explanation for STATEMENT-1
    2. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is NOT a correct explanation for STATEMENT-1
    3. STATEMENT-1 is True, STATEMENT-2 is False
    4. STATEMENT-1 is False, STATEMENT-2 is True
11. Let $a$, $b$, $c$, $p$, $q$ be real numbers. Suppose $\alpha$, $\beta$ are the roots of the equation $z^2$+$2px$+$q$=0 and $\alpha$, $\frac{1}{\beta}$ are the roots of the equation $ax^2$+$2bx$+$c$, where $\beta^2 \notin \text{{-1, 0, 1}}$.
    STATEMENT-1: $(p^2-q)$$(b^2-ac) \geq 0$.
    and
    STATEMENT-2:$b \neq pa$ or $c \neq qa$
    1. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is a correct explanation for STATEMENT-1
    2. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is NOT a correct explanation for STATEMENT-1
    3. STATEMENT-1 is True, STATEMENT-2 is False
    4. STATEMENT-1 is False, STATEMENT-2 is True
12. Consider $L_1$:$2x$+$3y$+$p-$3=0
    $L_2$:$2x$+$3y$+$p$+3=0, where $p$ is a real number, and $C$:$x^2$+$y^2$+$6x$$-10y$+30=0.
    STATEMENT-1: If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$.
    and
    STATEMENT-2:If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not always a chord of circle $C$.
    1. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is a correct explanation for STATEMENT-1
    2. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is NOT a correct explanation for STATEMENT-1
    3. STATEMENT-1 is True, STATEMENT-2 is False
    4. STATEMENT-1 is False, STATEMENT-2 is True
13. Let a solution$y$=$y(x)$ of the differential equation $x\sqrt{x^2-1}dy-$$y\sqrt{y^2-1}dx=0$ satisfy $y(2)$=$\frac{2}{\sqrt{3}}$
    STATEMENT-1:$y(x)$=$\sec\left(\sec^{-1}x-\frac{\pi}{6}\right)$.
    and
    STATEMENT-2:$y(x)$ is given by $\frac{1}{y}$=$\frac{2\sqrt{3}}{x}-$$\sqrt{1-\frac{1}{x^2}}$
    1. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is a correct explanation for STATEMENT-1
    2. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is NOT a correct explanation for STATEMENT-1
    3. STATEMENT-1 is True, STATEMENT-2 is False
    4. STATEMENT-1 is False, STATEMENT-2 is True

SECTION - III
(Linked Comprehension Type)

This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for Questions 14 to 16

Consider the function $(-\infty,\infty)$$\to$$(-\infty, \infty)$ defined by $f(x)$=$\frac{x^2-ax+1}{x^2+ax+1}$, $0 < a < 2$.

14. Which of the following is true?
    1. $(2+a)^2$$f"(1)$+$(2-a)^2$$f"(-1)$=0
    2. $(2-a)^2$$f"(1)-$$(2+a)^2$$f"(-1)$=0
    3. $f'(1)$$f'(-1)$=$(2-a)^2$
    4. $f'(1)$$f'(-1)$=$-(2+a)^2$
15. Which of the following is true?
    1. $f(x)$ is decreasing on $(-1, 1)$ and has a local minimum at $x$=1
    2. $f(x)$ is increasing on $(-1, 1)$ and has a local maximum at $x$=1
    3. $f(x)$ is increasing on $(-1, 1)$ but has neither a local maximum nor a local minimum at $x$=1
    4. $f(x)$ is decreasing on $(-1, 1)$ but has neither a local maximum nor a local minimum at $x$=1
16. Let $g(x)$=$\int \limits_0^{e^{x}}\frac{f'(t)}{1+t^2}dt$. Which of the following is true?
    1. $g'(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$
    2. $g'(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
    3. $g'(x)$ changes sign on both $(-\infty, 0)$ and $(0, \infty)$
    4. $g'(x)$ does not change sign on $(-\infty, \infty)$

Paragraph for Questions 17 to 19

Consider the lines
$L_1$:$\frac{x+1}{3}$=$\frac{y+2}{1}$=$\frac{z+1}{2}$
$L_2$:$\frac{x-2}{1}$=$\frac{y+2}{2}$=$\frac{z-3}{3}$

17. The unit vector perpendicular to both $L_1$ and $L_2$ is
    1. $\frac{-\hat{i}+7\hat{j}+7\hat{k}}{\sqrt{99}}$
    2. $\frac{-\hat{i}-7\hat{j}+5\hat{k}}{5\sqrt{3}}$
    3. $\frac{-\hat{i}+7\hat{j}+5\hat{k}}{5\sqrt{3}}$
    4. $\frac{7\hat{i}-7\hat{j}-\hat{k}}{\sqrt{99}}$
18. The shortest distance between $L_1$ and $L_2$ is
    1. 0
    2. $\frac{17}{\sqrt{3}}$
    3. $\frac{41}{5\sqrt{3}}$
    4. $\frac{17}{5\sqrt{3}}$
19. The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines $L_1$ and $L_2$ is
    1. $\frac{2}{\sqrt{75}}$
    2. $\frac{7}{\sqrt{75}}$
    3. $\frac{13}{\sqrt{75}}$
    4. $\frac{23}{\sqrt{75}}$

SECTION - IV
(Matrix-Match Type)

This contains 3 questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in column I have to be matched with statements (p, q, r, s) in column II. The answers to these questions have to be appropriately bubbled as illustrated in the following example. If the correct match are A-p, A-s, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows:

20. Consider the lines given by
    $L_1$ : $x$+$3y-$5=0
    $L_2$ : $3x-$$ky-$1=0
    $L_3$ : $5x$+$2y-$12=0
    Match the Statements/Expressions in Column-I with the Statements/Expressions given in Column-II and indicate your answer by darkening the appropriate bubbles in the 4×4 matrix given in the ORS.                        

    	Column - I		Column - II
    (A)	$L_1$, $L_2$, $L_3$ are concurrent, if	(p)	$k=-9$
    (B)	One of $L_1$, $L_2$, $L_3$ is parallel to at least one of the other two, if	(q)	$k=-\frac{6}{5}$
    (C)	$L_1$, $L_2$, $L_3$ form a triangle, if	(r)	$k=\frac{5}{6}$
    (D)	$L_1$, $L_2$, $L_3$ do not form a triangle, if	(s)	$k=5$

21. Match the Statements/Expressions in Column-I with the Statements/Expressions given in Column-II and indicate your answer by darkening the appropriate bubbles in the 4×4 matrix given in the ORS.                        

    	Column - I		Column - II
    (A)	The minimum value of $\frac{x^2+2x+4}{x+2}$ is	(p)	0
    (B)	Let $A$ and $B$ be 3×3 matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric, and $(A+B)$$(A-B)$=$(A-B)$$(A+B)$. If $(AB)^t$=$(-1)^kAB$, where $(AB)^t$ is the transpose of the matrix $AB$, then the possible values of $k$ are	(q)	1
    (C)	Let $a$=$log_3 log_3 2$. An integer $k$ satisfying 1 < $2^{(-k+3^{-a})}$ < 2, must be less than	(r)	2
    (D)	If $\sin \theta$=$\cos \phi$, then the possible values of $\frac{1}{\pi}$$(\theta±\phi-\frac{\pi}{2})$ are	(s)	3

22. Consider all possible permutations of the letters of the word $ENDEANOEL$ Match the Statements/Expressions in Column-I with the Statements/Expressions given in Column-II and indicate your answer by darkening the appropriate bubbles in the 4×4 matrix given in the ORS.                        

    	Column - I		Column - II
    (A)	The number of permutations containing the word $ENDEA$ is	(p)	5!
    (B)	The number of permutations in which the letter $E$ occurs in the first and the last positions is	(q)	2×5!
    (C)	The number of permutations in which none of the letters $D$, $L$, $N$ occurs in the last five positions is	(r)	7×5!
    (D)	The number of permutations in which the letters $A$, $E$, $O$ occur only in odd positions is	(s)	21×5!

2008:Paper-1

2008:Ans-Keys

All the Papers Ans-Keys

Please wait! Loading...

 Download as PDF