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2006:Paper-2

2008:Paper-2

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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 $O(0,0)$, $P(3,4)$, $Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR$, $PQR$, $OQR$ are of equal area. The coordinates of $R$ are
   1. $\left(\frac{4}{3}, 3\right)$
   2. $\left(\frac{2}{3}, 3\right)$
   3. $\left(3, \frac{4}{3}\right)$
   4. $\left(\frac{4}{3}, \frac{2}{3}\right)$
2. If $|z|$=1 and $z \neq $±1, then all the values of $\frac{z}{1-z^2}$ lie on
   1. a line not passing through the origin
   2. $|z|=\sqrt{2}$
   3. the x-axis
   4. the y-axis
3. Let $E^c$ denote the complement of an event $E$. Let $E$, $F$, $G$ be pairwise independent events with $P(G)$ > 0 and $P( E \cap F \cap G)$=0. Then $P(E^c \cap F^c |G)$ equals
   1. $P(E^c)$+$P(F^c)$
   2. $P(E^c)-$$P(F^c)$
   3. $P(E^c)-$$P(F)$
   4. $P(E)-$$P(F^c)$
4. $\frac{d^2x}{dy^2}$ equals
   1. $\left(\frac{d^2y}{dx^2}\right)^{-1}$
   2. $-\left(\frac{d^2y}{dx^2}\right)^{-1}$$\left(\frac{dy}{dx}\right)^{-3}$
   3. $\left(\frac{d^2y}{dx^2}\right)$$\left(\frac{dy}{dx}\right)^{-2}$
   4. $-\left(\frac{d^2y}{dx^2}\right)$$\left(\frac{dy}{dx}\right)^{-3}$
5. The differential equation $\frac{dy}{dx}$=$\frac{\sqrt{1-y^2}}{y}$ determines a family of circles with
   1. variable radii and a fixed centre at $(0,1)$
   2. variable radii and a fixed centre at $(0,-1)$
   3. fixed radius 1 and variable centres along the x-axis
   4. fixed radius 1 and variable centres along the y-axis
6. Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be unit vectors such that $\vec{a}$+$\vec{b}$+$\vec{c}$=0. Which one of the following is correct?
   1. $\vec{a}×\vec{b}$=$\vec{b}×\vec{c}$=$\vec{c}×\vec{a}$=$\vec{0}$
   2. $\vec{a}×\vec{b}$=$\vec{b}×\vec{c}$=$\vec{c}×\vec{a} \neq \vec{0}$
   3. $\vec{a}×\vec{b}$=$\vec{b}×\vec{c}$=$\vec{a}×\vec{c} \neq \vec{0}$
   4. $\vec{a}×\vec{b}$, $\vec{b}×\vec{c}$, $\vec{c}×\vec{a}$ are mutually perpendicular
7. Let $ABCD$ be a quadrilateral with area 18, with side $AB$ parallel to the side $CD$ and $AB$=$2CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides, then its radius is
   1. 3
   2. 2
   3. $\frac{3}{2}$
   4. 1
8. Let $f(x)$ =$\frac{x}{(1+x^n)^{1/n}}$ for $n \geq 2$ and $g(x)$=$\underbrace{(f \circ f \circ \cdots \circ f)}_{f \text { occurs } n \text { times }}(x)$. Then $\int x^{n-2}g(x)dx$ equals
   1. $\frac{1}{n(n-1)}$$(1+nx^n)^{1-\frac{1}{n}}$+$K$
   2. $\frac{1}{n-1}$$(1+nx^n)^{1-\frac{1}{n}}$+$K$
   3. $\frac{1}{n(n+1)}$$(1+nx^n)^{1+\frac{1}{n}}$+$K$
   4. $\frac{1}{n+1}$$(1+nx^n)^{1+\frac{1}{n}}$+$K$
9. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The numbers of words that appear before the word COCHIN is
   1. 360
   2. 192
   3. 96
   4. 48

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. Consider the planes $3x-$$6y-$$2z$=15 and $2x$+$y-$$2z$=5
    STATEMENT-1: The parametric equations of the line of the intersection of the given planes are $x$=3+$14t$, $y$=1+$2t$, $z$=$15t$.
    because
    STATEMENT-2:The vector $14\hat{i}$+$2\hat{j}$+$15\hat{k}$ is parallel to the line of intersection of the given planes.
    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. STATEMENT-1: The curve $y$=$\frac{-x^2}{2}$+$x$+1 is symmetric with respect to the line $x$=1.
    because
    STATEMENT-2:A parabola is symmetric about its axis.
    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 $f(x)$=2+$\cos x$ for all real $x$
    STATEMENT-1: For each real $t$, there exists a point $c$ in $[t, t+\pi]$ such that $f'(c)$=0.
    because
    STATEMENT-2:$f(t)$=$f(t+2\pi)$ for each real $t$.
    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. Lines $L_1$:$y-x$=0 and $L_2$=$2x$+$y$=0 intersect the line $L_3$: $y$+2=0 at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$.
    STATEMENT-1: The ratio $PR : RQ$ equals $2\sqrt{2}$:$\sqrt{5}$
    because
    STATEMENT-2:In any triangle, bisector of an angle divides the triangle into two similar triangles.
    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 $A_1$, $G_1$, $H_1$ denote the arithmetic, geometric and harmonic means, respectively of two distinct positive numbers. For $n \geq 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_n$, $G_n$, $H_n$ respectively.

14. Which one of the following statements is correct?
    1. $G_1$ > $G_2$ > $G_3$ > ...
    2. $G_1$ < $G_2$ < $G_3$ < ...
    3. $G_1$ = $G_2$ = $G_3$ = ...
    4. $G_1$ < $G_3$ < $G_5$ < ... and $G_2$ > $G_4$ > $G_6$ > ...
15. Which one of the following statements is correct?
    1. $A_1$ > $A_2$ > $A_3$ > ...
    2. $A_1$ < $A_2$ < $A_3$ < ...
    3. $A_1$ > $A_3$ > $A_5$ > ... and $A_2$ < $A_4$ < $A_6$ < ...
    4. $A_1$ < $A_3$ < $A_5$ < ... and $A_2$ > $A_4$ > $A_6$ > ...
16. Which one of the following statements is correct?
    1. $H_1$ > $H_2$ > $H_3$ > ...
    2. $H_1$ < $H_2$ < $H_3$ < ...
    3. $H_1$ > $H_3$ > $H_5$ > ... and $H_2$ < $H_4$ < $H_6$ < ...
    4. $H_1$ < $H_3$ < $H_5$ < ... and $H_2$ > $H_4$ > $H_6$ > ...

Paragraph for Questions 17 to 19

If a continuous function $f$ defined on the real line R, assumes positive and negative values in R then the equation $f(x)$=0 has a root in R. For example, if it is known that a continuous function $f$ on R is positive at some point and its minimum value is negative then the equation $f(x)$=0 has a root in R.
Consider $f(x)$=$ke^x-$$x$ for all real $x$ where $k$ is a real constant.

17. The line $y$=$x$ meets $y$=$ke^x$ for $k \leq 0$ at
    1. no point
    2. one point
    3. two points
    4. more than two points
18. The positive value of $k$ for which $ke^x-x$=0 has only one root is
    1. $\frac{1}{e}$
    2. 1
    3. $e$
    4. $log_e 2$
19. For $k$>0, the set of all values of $k$ for which $ke^x-x$=0 has two distinct roots is
    1. $\left(0, \frac{1}{e}\right)$
    2. $\left(\frac{1}{e}, 1 \right)$
    3. $\left(\frac{1}{e}, \infty\right)$
    4. (0, 1)

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. Let $f(x)$=$\frac{x^2-6x+5}{x^2-5x+6}$ Match the expressions/statements in Column-I with the expressions/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)	If $-1 < x < 1$, then $f(x)$ satisfies	(p)	$0 < f(x) < 1$
    (B)	If $1 < x < 2$, then $f(x)$ satisfies	(q)	$f(x) < 0$
    (C)	If $3 < x < 5$, then $f(x)$ satisfies	(r)	$f(x)>0$
    (D)	If $x > 5$, then $f(x)$ satisfies	(s)	$f(x) < 1$

21. Let $(x, y)$ be such that $\sin ^{-1}(a x)$+$\cos^{-1}(y)$+$\cos^{-1}(bxy)$=$\frac{\pi}{2}$ Match the statements given in Column-I with 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)	If $a$=1 and $b$=0, then $(x, y)$	(p)	lines on $x^2$+$y^2$=1
    (B)	If $a$=1 and $b$=1, then $(x, y)$	(q)	lies on $(x^2-1)$/$(y^2-1)$=0
    (C)	If $a$=1 and $b$=2, then $(x, y)$	(r)	lies on $y=x$
    (D)	If $a$=2 and $b$=2, then $(x, y)$	(s)	lies on $(4x^2-1)$/$(y^2-1)$=0

22. Match the statements given 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)	Two intersecting circles	(p)	have a common tangent
    (B)	Two mutually external circles	(q)	have a common normal
    (C)	Two circles, one strictly inside the other	(r)	do not have a common tangent
    (D)	Two branches of a hyperbola	(s)	do not have a common normal

2007:Paper-1

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