Download JEE Advanced 2007 Mathematics Question Paper - 1

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Marking Scheme

• Section I contains 9 multiple choice questions which have only one correct answer. Each question carries +3 marks each for correct answer and – 1 Mark for each wrong answer.
• Section II contains 4 questions. Each question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason).
  Bubble (A) if both the statements are TRUE and STATEMENT-2 is the correct explanation of STATEMENT-1
  Bubble (B) if both the statements are TRUE but STATEMENT-2 is NOT the correct explanation of STATEMENT- 1
  Bubble (C) if STATEMENT-1 is TRUE and STATEMENT-2 is FALSE.
  Bubble (D) if STATEMENT-1 is FALSE and STATEMENT-2 is TRUE.
  carries +3 marks each for correct answer and – 1 mark for each wrong answer.
• Section III contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has only one correct answer and carries +4 marks for correct answer and – 1 mark for wrong answer.
• Section IV contains 3 questions. Each question contains statements given in 2 columns. Statements in the first column have to be matched with statements in the second column and each question carries +6 marks and marks will be awarded if all the four parts are correctly matched. No marks will be given for any wrong match in any question. There is no negative marking.

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. Let $\alpha$, $\beta$ be the roots of the equation $x^2-$$px$+$r$=0 and $\frac{\alpha}{2}$, $2\beta$ be the roots of the equation $x^2-$$qx$+$r$=0. Then the value of $r$ is
   
   1. $\frac{2}{9}$$(q-p)$$(2p-q)$
   2. $\frac{2}{9}$$(p-q)$$(2q-p)$
   3. $\frac{2}{9}$$(q-2p)$$(2q-p)$
   4. $\frac{2}{9}$$(2p-q)$$(2q-p)$
2. Let $f(x)$ be differentiable on the interval $(0, \infty)$ such that $f(1)$=1, and $\lim \limits_{t \to x}\frac{t^2f(x)-x^2f(t)}{t-x}$=1 for each $x$>0. Then $f(x)$ is
   
   1. $\frac{1}{3x}$+$\frac{2x^2}{3}$
   2. $\frac{-1}{3x}$+$\frac{4x^2}{3}$
   3. $\frac{-1}{x}$+$\frac{2}{x^2}$
   4. $\frac{1}{x}$
3. One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that Indian men is seated adjacent to his wife given that each American man is seated adjacent to his wife is
   
   1. $\frac{1}{2}$
   2. $\frac{1}{3}$
   3. $\frac{2}{5}$
   4. $\frac{1}{5}$
4. The tangent to the curve $y$=$e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$
   
   1. on the left of $x=c$
   2. on the right of $x=c$
   3. at no point
   4. at all points
5. $\lim \limits_{x \to \frac{\pi}{4}}$ $\frac{\int \limits_2^{\sec^2x}f(t)dt}{x^2-\frac{\pi^2}{16}}$ equals
   
   1. $\frac{8}{\pi}f(2)$
   2. $\frac{2}{\pi}f(2)$
   3. $\frac{2}{\pi}f\left(\frac{1}{2}\right)$
   4. $4f(2)$
6. A hyperbola, having the transverse axis of the length $2\sin \theta$, is confocal with the ellipse $3x^2$+$4y^2$=12. Then its equation is
   
   1. $x^2 cosec^2\theta-$$y^2\sec^2\theta$=1
   2. $x^2\sec^2\theta-$$y^2 cosec^2\theta$=1
   3. $x^2\sin^2\theta-$$y^2\cos^2\theta$=1
   4. $x^2\cos^2\theta-$$y^2\sin^2\theta$=1
7. The number of distinct real values of $\lambda$, for which the vectors $-\lambda^2 \hat{i}$+$\hat{j}$+$\hat{k}$, $\hat{i}-$$\lambda^2 \hat{j}$+$\hat{k}$ and $\hat{i}$+$\hat{j}-$$\lambda^2\hat{k}$ are coplanar, is
   
   1. zero
   2. one
   3. two
   4. three
8. A man walks a distance of 3 units from the origin towards the north-east $(N 45° E)$ direction. From there, he walks a distance of 4 units towards the north-west $(N 45° W)$ direction to reach a point $P$. Then the position of $P$ in the Argand plane is
   
   1. $3e^{\frac{i\pi}{4}}$+$4i$
   2. $(3-4i)e^{\frac{i\pi}{4}}$
   3. $(4+3i)e^{\frac{i\pi}{4}}$
   4. $(3+4i)e^{\frac{i\pi}{4}}$
9. The number of solutions of the pair of equations
   $2 \sin^2\theta -$$\cos 2 \theta$=0
   $2 \cos^2\theta -$$3\sin \theta$=0
   in the interval $[0, 2 \pi]$ is
   
   1. zero
   2. one
   3. two
   4. four

SECTION - II
(Assertion-Reason Type)

This section contains 4 questions numbered 10 to 13. Each question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

10. Let $H_1$, $H_2$, ... , $H_n$ be mutually exclusive and exhaustive events with $P(H_i)$ > 0, $i$=1, 2, ..., $n$. Let $ E$ be any other event with $0 < P(E) < 1$.
    STATEMENT-1: $P(H_i|E)$ > $P(E|H_i)•P(H_i)$ > for $i$=1, 2, ..., $n$.
    because
    STATEMENT-2:$\sum \limits_{i=1}^{n}P(H_i)$=1.
    
    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. Tangents are drawn from the point $(17,7)$ to the circle $x^2$+$y^2$=169.
    STATEMENT-1: The tangents are mutually perpendicular.
    and
    STATEMENT-2:The locus of the points from which mutually perpendicular tangents can be drawn to the given circle $x^2$+$y^2$=338.
    
    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. Let the vectors $\vec{PQ}$, $\vec{QR}$, $\vec{RS}$, $\vec{ST}$, $\vec{TU}$ and $\vec{UP}$ represent the sides of a regular hexagon.
    STATEMENT-1: $\vec{PQ}$×$(\vec{RS}+\vec{ST})$$\neq$$\vec{0}$
    because
    STATEMENT-2:$\vec{PQ}×\vec{RS}$=$\vec{0}$ and $\vec{PQ}$×$\vec{ST}$$\neq$$\vec{0}$.
    
    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 $F(x)$ be an indefinite integral of $sin^2x$
    STATEMENT-1: The function $F(x)$ satisfies $F(x+\pi)$=$F(x)$ for all real $x$.
    because
    STATEMENT-2:$sin^2(x+\pi)$=$sin^2x$ for all real $x$.
    
    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 Compression Type)

This section contains 2 paragraphs P14-16 and P17-19. 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

Let $V_r$ denote the sum of the first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2r-1)$. Let $T_r$=$V_{r+1}-V_r-2$ and $Q_r$=$T_{r+1}-T_r$ for $r$=1, 2,...

14. The sum of $V_1$+$V_2$+...+$V_n$ is
    
    1. $\frac{1}{12}n(n+1)$$(3n^2-n+1)$
    2. $\frac{1}{12}n(n+1)$$(3n^2+n+2)$
    3. $\frac{1}{2}n$$(2n^2-n+1)$
    4. $\frac{1}{3}(2n^3-2n+3)$
15. $T_r$ is always
    
    1. an odd number
    2. an even number
    3. a prime number
    4. a composite number
16. Which of the following is a correct statement?
    
    1. $Q_1$, $Q_2$, $Q_3$, ... are in A.P. with common difference 5.
    2. $Q_1$, $Q_2$, $Q_3$, ... are in A.P. with common difference 6.
    3. $Q_1$, $Q_2$, $Q_3$, ... are in A.P. with common difference 11.
    4. $Q_1$=$Q_2$=$Q_3$=....

Paragraph for Questions 17 to 19

Consider the circle $x^2$+$y^2$=9 and the parabola $y^2$=$8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x-$axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x-$axis at $S$.

17. The ratio of the areas of the triangles $PQS$ and $PQR$ is
    
    1. $1:\sqrt{2}$
    2. 1:2
    3. 1:4
    4. 1:8
18. The radius of the circumcircle of the triangle $PRS$ is
    
    1. 5
    2. $3\sqrt{3}$
    3. $3\sqrt{2}$
    4. $2\sqrt{3}$
19. The radius of the incircle of the triangle $PQR$ is
    
    1. 4
    2. 3
    3. $\frac{8}{3}$
    4. 2

SECTION - IV
(Matrix-Match Type)

This section 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 matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows:

20. Consider the following linear equations
    $ax$+$by$+$cz$=0
    $bx$+$cy$+$az$=0
    $cx$+$ay$+$bz$=0
    Match the conditions/expressions in Column-I with the statements 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)	$a$+$b$+$c \neq 0$ and $a^2$+$b^2$+$c^2$=$ab$+$bc$+$ca$	(p)	The equations represent planes meeting only at a single point.
    (B)	$a$+$b$+$c$=0 and $a^2$+$b^2$+$c^2 \neq$$ab$+$bc$+$ca$	(q)	The equations represent the line $x$=$y$=$z$.
    (C)	$a$+$b$+$c \neq 0$ and $a^2$+$b^2$+$c^2 \neq$$ab$+$bc$+$ca$	(r)	The equations represent identical planes.
    (D)	$a$+$b$+$c$=0 and $a^2$+$b^2$+$c^2$=$ab$+$bc$+$ca$	(s)	The equations represent the whole of the three dimensional space.

21. In the following $[x]$ denotes the greatest integer less than or equal to $x$.
    Match the functions in Column-I with the properties 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)	$x|x|$	(p)	continuous in $(-1,1)$
    (B)	$\sqrt{|x|}$	(q)	differentiable in $(-1,1)$
    (C)	$x+[x]$	(r)	strictly increasing in $(-1,1)$
    (D)	$|x-1|$+$|x+1|$	(s)	not differentiable at least at one point in $(-1,1)$

22. Match the integrals in Column-I with the values 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)	$\int \limits_{-1}^{1}\frac{dx}{1+x^2}$	(p)	$\frac{1}{2}log\left(\frac{2}{3}\right)$
    (B)	$\int \limits_{0}^{1}\frac{dx}{1-x^2}$	(q)	$2 log \left(\frac{2}{3}\right)$
    (C)	$\int \limits_{2}^{3}\frac{dx}{1-x^2}$	(r)	$\frac{\pi}{3}$
    (D)	$\int \limits_1^2 \frac{dx}{x \sqrt{x^2-1}}$	(s)	$\frac{\pi}{2}$

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