Download JEE Advanced 2008 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 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 4 marks if you have darkened the bubble(s) corresponding to the correct answer and zero mark for all other cases. It may be noted that there is no negative marking for wrong answer.
• For each question in Section III , 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 IV, you will be awarded 4 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.

SECTION - I
(Single Correct Choice Type)

This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

1. Consider the two curves
   $C_1$ : $y^2$=$4x$
   $C_2$ : $x^2$+$y^2$$-6x$+1=0
   Then,
   1. $C_1$ and $C_2$ touch each other only at one point
   2. $C_1$ and $C_2$ touch each other exactly at two points
   3. $C_1$ and $C_2$ intersect (but do not touch) exactly at two points
   4. $C_1$ and $C_2$ neither intersect nor touch each other
2. If $0 < x <1$, then $\sqrt{1+x^2}$[{$xcos(cot^{-1}x)$+$sin(cot^{-1}x)$}$^2-$1]$^{\frac{1}{2}}$=
   1. $\frac{x}{\sqrt{1+x^2}}$
   2. $x$
   3. $x\sqrt{1+x^2}$
   4. $\sqrt{1+x^2}$
3. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat{a}$, $\hat{b}$, $\hat{c}$ such that $\hat{a}$•$\hat{b}$=$\hat{b}$•$\hat{c}$=$\hat{c}$•$\hat{a}$=$\frac{1}{2}$. Then, the volume of parallelopiped is
   1. $\frac{1}{\sqrt{2}}$
   2. $\frac{1}{2\sqrt{2}}$
   3. $\frac{\sqrt{3}}{2}$
   4. $\frac{1}{\sqrt{3}}$
4. Let $a$ and $b$ be non-zero real numbers. Then, the equation
   ($ax^2$+$by^2$+$c$)($x^2$$-5xy$+$6y^2$)=0 represents
   1. four straight lines, when $c$=0 and $a$, $b$ are of the same sign
   2. two straight lines and a circle, when $a$=$b$, and $c$ is of sign opposite to that of $a$
   3. two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $ c$ is of sign opposite to that of $a$
   4. a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
5. Let $g(x)$=$\frac{(x-1)^n}{log cos^m(x-1)}$; $0 < x < 2$, $m$ and $n$ are integers, $m \neq 0$, $n > 0$, and let $p$ be the left hand derivative of $|x-1|$ at $x$=1. If $\lim \limits_{x \to 1+}g(x)$=$p$, then
   1. $n=1$, $m=1$
   2. $n=1$, $m=-1$
   3. $n=2$, $m=2$
   4. $n > 2$, $m=n$
6. The total number of local maxima and local minima of the function
   $f(x)$=$\left\{\begin{array}{ll}(2+x)^3, & -3 < x \leq -1, \\ x^{2/3} & -1 < x < 2 .\end{array}\right.$
   is
   1. 0
   2. 1
   3. 2
   4. 3

SECTION - II
(Multiple Correct Choice Type)

This section contains 4 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct.

7. A straight line through the vertex $P$ of a triangle $PQR$ intersects the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T$. If $S$ is not the centre of the circumcircle, then
   1. $\frac{1}{PS}$+$\frac{1}{ST}$ < $\frac{2}{\sqrt{QS×SR}}$
   2. $\frac{1}{PS}$+$\frac{1}{ST}$ > $\frac{2}{\sqrt{QS×SR}}$
   3. $\frac{1}{PS}$+$\frac{1}{ST}$ < $\frac{4}{QR}$
   4. $\frac{1}{PS}$+$\frac{1}{ST}$ > $\frac{4}{QR}$
8. Let $P(x_1, y_1)$ and $Q(x_2, y_2)$, $y_1$ < 0, $y_2$ < 0, be the end points of the latus rectum of the ellipse $x^2$+$4y^2$=4. The equation of the parabolas with the latus rectum $PQ$ are
   1. $x^2$+$2\sqrt{3}y$ = 3+$\sqrt{3}$
   2. $x^2-$$2\sqrt{3}y$ = 3+$\sqrt{3}$
   3. $x^2$+$2\sqrt{3}y$ = 3$-\sqrt{3}$
   4. $x^2-$$2\sqrt{3}y$ = 3$-\sqrt{3}$
9. Let $S_n$ =$\sum \limits_{k=1}^{n} \frac{n}{n^2+kn+k^2}$ and $T_n$=$\sum \limits_{k=0}^{n-1} \frac{n}{n^2+kn+k^2}$, for $n$=1, 2, 3, .... Then,
   1. $S_n$ < $\frac{\pi}{3\sqrt{3}}$
   2. $S_n$ > $\frac{\pi}{3\sqrt{3}}$
   3. $T_n$ < $\frac{\pi}{3\sqrt{3}}$
   4. $T_n$ > $\frac{\pi}{3\sqrt{3}}$
10. Let $f(x)$ be a non-constant twice differentiable function defined on $(-\infty, \infty)$ such that $f(x)$=$f(1-x)$ and $f'\left(\frac{1}{4}\right)$=0. Then,
    1. $f"(x)$ vanishes at least twice on [0, 1]
    2. $f'\left(\frac{1}{2}\right)$=0
    3. $\int \limits_{-1/2}^{1/2}f\left(x+\frac{1}{2}\right)sinxdx$=0
    4. $\int \limits_0^{1/2}f(t)e^{sin \pi t}dt$=$\int \limits_{1/2}^{1}f(1-t)e^{sin \pi t}dt$

SECTION - III
(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.

11. Let $f$ and $g$ be real valued functions defined on interval $(-1, 1)$ such that $g"(x)$ is continuous, $g(0) \neq 0$, $g'(0)$=0, $g"(0) \neq 0$, and $f(x)$=$g(x)sinx$
    STATEMENT-1: $\lim \limits_{x \to 0}[g(x)cotx-g(0)cosecx]$=$f"(0)$.
    and
    STATEMENT-2:$f'(0)$=$g(0)$
    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
12. Consider three planes
    $P_1$:$x-$$y$+$z$=1
    $P_2$:$x$+$y$$-z$=1
    $P_3$:$x-$$3y$+$3z$=2.
    Let $L_1$, $L_2$, $L_3$ be the lines of intersection of the planes $P_2$ and $P_3$, $P_3$ and $P_1$, and $P_1$ and $P_2$, respectively.
    STATEMENT-1: At least two of the lines $L_1$, $L_2$ and $L_3$ are non-parallel.
    and
    STATEMENT-2:The three planes do not have a common point.
    1. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is a correct explanation for STATEMENT-1
    2. STATEMENT-1 is True, STATEMENT-2 is True, STATEMENT-2 is NOT a correct explanation for STATEMENT-1
    3. STATEMENT-1 is True, STATEMENT-2 is False
    4. STATEMENT-1 is False, STATEMENT-2 is True
13. Consider the system of equations
    $x-$$2y$+$3z$=-1
    $-x$+$y$$-2z$=$k$
    $x-$$3y$+$4z$=1
    STATEMENT-1: The system of equation has no solution for $k \neq 3$
    and
    STATEMENT-2:The determinant $\left|\begin{array}{ccc}1 & 3 & -1 \\ -1 & -2 & k \\ 1 & 4 & 1\end{array}\right|$ $\neq 0$, for $k \neq 3$.
    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
14. Consider the system of equations
    $ax$+$by$=0,
    $cx$+$dy$=0, where $a$, $b$, $c$, $d$ $\in \text{{0,1}}$
    STATEMENT-1: The probability that the system of equations has a unique solution is $\frac{3}{8}$
    and
    STATEMENT-2:The probability that the system of equations has a solution is1.
    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 - IV
(Linked Comprehension Type)

This section contains 3 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 15 to 17

A circle $C$ of radius 1 is inscribed in an equilateral triangle $PQR$. The points of contact of $C$ with the sides $PQ$, $QR$, $RP$ are $D$, $E$, $F$, respectively. The line $PQ$ is given by the equation $\sqrt{3}x$+$y-$6=0 and the point $D$ is $\left(\frac{3\sqrt{3}}{2}, \frac{3}{2}\right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $PQ$.

15. The equation of the circle $C$ is
    1. $(x-2\sqrt{3})^2$+$(y-1)^2$=1
    2. $(x-2\sqrt{3})^2$+$(y+\frac{1}{2})^2$=1
    3. $(x-\sqrt{3})^2$+$(y+1)^2$=1
    4. $(x-\sqrt{3})^2$+$(y-1)^2$=1
16. Points $E$ and $F$ are given by
    1. $\left(\frac{\sqrt{3}}{2},\frac{3}{2}\right)$, $(\sqrt{3}, 0)$
    2. $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$, $(\sqrt{3}, 0)$
    3. $\left(\frac{\sqrt{3}}{2},\frac{3}{2}\right)$, $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$
    4. $\left(\frac{3}{2},\frac{\sqrt{3}}{2}\right)$, $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$
17. Equations of the sides $QR$, $RP$ are
    1. $y$=$\frac{2}{\sqrt{3}}x+1$, $y$=$-\frac{2}{\sqrt{3}}x-1$
    2. $y$=$\frac{1}{\sqrt{3}}x$, $y$=0
    3. $y$=$\frac{\sqrt{3}}{2}x+1$, $y$=$-\frac{\sqrt{3}}{2}x-1$
    4. $y$=$\sqrt{3}x$, $y$=0

Paragraph for Questions 18 to 20

Consider the functions defined implicitly by the equation $y^3-$$3y$+$x$=0 on various intervals in the real line. If $x \in (-\infty, -2)$ $\cup (2, \infty)$, the equation implicitly defines a unique real valued differentiable function $y$=$f(x)$. If $x \in (-2,2)$ ,the equation implicitly defines a unique real valued differentiable function $y$=$g(x)$ satisfying $g(0)$=0

18. If $f(-10 \sqrt{2})$=$2 \sqrt{2}$, then $f"(-10 \sqrt{2})$=
    1. $\frac{4\sqrt{2}}{7^33^2}$
    2. $-\frac{4\sqrt{2}}{7^33^2}$
    3. $\frac{4\sqrt{2}}{7^33}$
    4. $-\frac{4\sqrt{2}}{7^33}$
19. The area of the region bounded by $y$=$f(x)$, the $x-$axis, and the lines $x=a$ and $x=b$, where $-\infty < a < b < -2$, is
    1. $\int \limits_a^b \frac{x}{3\left((f(x))^2-1\right)}dx$+$bf(b)-$$af(a)$
    2. $-\int \limits_a^b \frac{x}{3\left((f(x))^2-1\right)}dx$+$bf(b)-$$af(a)$
    3. $\int \limits_a^b \frac{x}{3\left((f(x))^2-1\right)}dx$$-bf(b)$+$af(a)$
    4. $-\int \limits_a^b \frac{x}{3\left((f(x))^2-1\right)}dx$$-bf(b)$+$af(a)$
20. $\int \limits_{-1}^1g'(x)dx$
    1. $2g(-1)$
    2. 0
    3. $-2g(1)$
    4. $2g(1)$

Paragraph for Questions 21 to 23

Let $A$, $B$, $C$ be three sets of complex numbers as defined below
$A$={$z: Im z \geq 1$}
$B$={$z: |z-2-i|$=3}
$C$={$z: Re((1-i) z)$=$\sqrt{2}$ }.

21. The number of elements in the set $A \cap B \cap C$ is
    1. 0
    2. 1
    3. 2
    4. $\infty$
22. Let $z$ be any point in $A \cap B \cap C$. Then, $|z+1-i|^2$+$|z-5-i|^2$ lies between
    1. 25 and 29
    2. 30 and 34
    3. 35 and 39
    4. 40 and 44
23. Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $|w-2-i|$ < 3. Then $|z|-$$|w|$+3 lies between
    1. -6 and 3
    2. -3 and 6
    3. -6 and 6
    4. -3 and 9

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