Download JEE Advanced 2011 Mathematics Question Paper - 1

Please wait! Loading...

 Download as PDF

2010:Paper-1

2012:Paper-1

All the Papers

Marking Scheme

• In Section I (Total marks : 21), for each question 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 all other cases, minus one (-1) mark will be awarded.
• In Section II (Total marks : 16), for each question you will be awarded 4 marks If you darken All the bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise. There are No negative marks in this section.
• In Section III (Total marks : 15), for each question 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 all other cases, minus one (-1) mark will be awarded.
• In Section IV (Total marks : 28), for each question you will be awarded 4 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks otherwise. There are No negative marks in this section.

SECTION - I (Total Marks : 21)
(Single Correct Answer Type)

This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

1. Let $(x_0, y_0)$ be the solution of the following equations
   $(2x)^{ln2}$=$(3y)^{ln3}$
   $3^{lnx}$=$2^{lny}$.
   Then $x_0$ is
   1. $\frac{1}{6}$
   2. $\frac{1}{3}$
   3. $\frac{1}{2}$
   4. 6
      
2. The value of $\int \limits_{\sqrt{ln2}}^{\sqrt{ln3}} \frac{x sinx^2}{sinx^2+sin(ln6-x^2)}dx$ is  
   1. $\frac{1}{4}ln\frac{3}{2}$
   2. $\frac{1}{2}ln\frac{3}{2}$
   3. $ln\frac{3}{2}$
   4. $\frac{1}{6}ln\frac{3}{2}$
      
3. Let $\vec{a}$=$\hat{i}$+$\hat{j}$+$\hat{k}$, $\vec{b}$=$\hat{i}$$-\hat{j}$+$\hat{k}$ and $\vec{c}$=$\hat{i}$$-\hat{j}$$-\hat{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$, whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$, is given by
   1. $\hat{i}$$-3\hat{j}$+$3\hat{k}$
   2. $-3\hat{i}$$-3\hat{j}$$-\hat{k}$
   3. $3\hat{i}$$-\hat{j}$+$3\hat{k}$
   4. $\hat{i}$+$3\hat{j}$$-3\hat{k}$
      
4. Let $P$ ={$\theta : sin \theta - cos \theta=\sqrt{2}cos\theta$} and $Q $={$\theta : sin \theta + cos \theta=\sqrt{2}sin \theta$} be two sets. Then
   1. $P \subset Q$ and $Q-P$$\neq \phi$
   2. $Q \not\subset P$
   3. $P \not\subset Q$
   4. $P=Q$
      
5. Let the straight line $x=b$ divide the area enclosed by $y=(1-x^2)$, y=0, and x=0 into two parts $R_1(0 \leq x \leq b)$ and $R_2(b \leq x \leq 1)$ such that $R_1-R_2$=$\frac{1}{4}$. Then b equals  
   1. $\frac{3}{4}$
   2. $\frac{1}{2}$
   3. $\frac{1}{3}$
   4. $\frac{1}{4}$
      
6. Let $\alpha$ and $\beta$ be the roots of $x^2$$-6x-2$=0 with $\alpha > \beta$. If $a_n$=$\alpha^n-\beta^n$ for $n \geq 1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is  
   1. 1
   2. 2
   3. 3
   4. 4
      
7. A straight line $L$ through the point (3, - 2) is inclined at an angle 60° to the line $\sqrt{3}x$+$y$=1. If $L$ also intersects the $x-$axis, then the equation of $L$ is  
   1. $y$+$\sqrt{3}x$+2-$3\sqrt{3}$=0
   2. $y$$-\sqrt{3}x$+2+$3\sqrt{3}$=0
   3. $\sqrt{3}y$$-x$+3+$2\sqrt{3}$=0
   4. $\sqrt{3}y$+$x$-3+$2\sqrt{3}$=0
      

SECTION - II (Total Marks : 16)
(Multiple Correct Answer 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.

8. The vector(s) which is/are coplanar with vectors $\hat{i} +\hat{j} +2\hat{k}$ and $\hat{i} +2\hat{j}+\hat{k}$, and perpendicular to the vector $\hat{i} +\hat{j} +\hat{k} $ is/are
   1. $\hat{j} - \hat{k}$
   2. $-\hat{i} + \hat{j}$
   3. $\hat{i} - \hat{j}$
   4. $-\hat{j} + \hat{k}$
      
9. Let $M$ and $N$ be two 3×3 non-singular skew symmetric matrices such that $MN$=$NM$. If $P^T$ denotes the transpose of $P$, then $M^2N^2$$(M^TN)^{-1}$$(MN^{-1})^T$ is equal to
   1. $M^2$
   2. $-N^2$
   3. $-M^2$
   4. $MN$
10. Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 be the reciprocal to that of the ellipse $x^2$+4$y^2$=4. If the hyperbola passes through a focus of the ellipse, then
    1. the equation of the hyperbola is $\frac{x^2} {3}-\frac{y^2}{2}$=1
    2. a focus of the hyperbola is (2, 0)
    3. the eccentricity of the hyperbola is $\sqrt{\frac{5}{3}}$
    4. the equation of the hyperbola is $x^2-3y^2$=3
11. Let $f:R \to R$ be a function such that
    $f(x+y)$=$f(x)$+$f(y)$, $\forall x, y \in R$.
    If $f(x)$ is differentiable at $x$=0, then
    1. $f(x)$ is differentiable only in a finite interval containing zero.
    2. $f(x)$ is continuous $\forall x \in R$
    3. $f'(x)$ is constant $\forall x \in R$
    4. $f(x)$ is differentiable except at finitely many points

SECTION - III (Total Marks : 15)
(Paragraph Type)

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

Paragraph for Question Nos. 12 to 14

Let $a$, $b$ and $c$ be three real numbers satisfying $\begin{equation*} \begin{bmatrix} a & b & c \end{bmatrix} \end{equation*}$$\begin{equation*} \begin{bmatrix} 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} 0 & 0 & 0 \end{bmatrix} \end{equation*}$ $\hspace{2cm} ...(E)$

12. If the point $P(a,b,c)$, with reference to $(E)$, lies on the plane $2x$+$y$+$z$=1, then the value of $7a$+$b$+$c$ is
    1. 0
    2. 12
    3. 7
    4. 6
13. Let $\omega$ be a solution of $x^3-1$ with $Im(\omega)$ > 0. If $a$ =2 with $b$ and $c$ satisfying $(E)$, the the value of $\frac{3}{\omega^a}$+$\frac{1}{\omega^b}$+$\frac{3}{\omega^c}$ is equal to
    1. -2
    2. 2
    3. 3
    4. -3
14. Let $b$=6, with $a$ and $c$ satisfying $(E)$. If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2$+$bx$+$c$, then $\sum \limits_{n=0}^{\infty}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n$ is
    1. 6
    2. 7
    3. $\frac{6}{7}$
    4. $\infty$

Paragraph for Question Nos. 15 to 16

Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls, and $U_2$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U_1$ and put into $U_2$. However, if tail appears then 2 balls are drawn at random from $U_1$ and put into $U_2$. Now 1 ball is drawn at random from $U_2$ .

15. The probability of the drawn ball from $U_2$ being white is
    1. $\frac{13}{30}$
    2. $\frac{23}{30}$
    3. $\frac{19}{30}$
    4. $\frac{11}{30}$
16. Given that the drawn ball from $U_2$ is white, the probability that head appeared on the coin is
    1. $\frac{17}{23}$
    2. $\frac{11}{23}$
    3. $\frac{15}{23}$
    4. $\frac{12}{23}$

SECTION - IV (Total Marks : 28)
(Integer Answer Type)

This section contains 7 questions. The answer to each of these questions is a single-digit integer, ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.

17. Consider the parabola $y^2$ =8$x$. Let $\Delta_1$ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac{1}{2},2\right)$ on the parabola and $\Delta_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latus rectum. Then $\frac{\Delta_1}{\Delta_2}$ is
18. Let $a_1$, $a_2$, $a_3$, ..., $a_{100}$ be an arithmetic progression with $a_1$=3 and $S_p$ =$\sum \limits_{i=1}^{p}a_i$, $1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$, let $m$=$5n$. If $\frac{S_m}{S_n}$ does not depend on $n$, then $a_2$ is
19. The positive integer value of $n$ > 3 satisfying the equation $\frac{1}{sin \left(\frac{\pi}{n}\right)}$=$\frac{1}{sin \left(\frac{2\pi}{n}\right)}$+$\frac{1}{sin \left(\frac{3\pi}{n}\right)}$ is
20. Let $f:[1, \infty) \to [2, \infty)$ be a differentiable function such that $f(1)=2$. If 6$\int \limits_1^x f(t)dt$=$3xf(x)-$$x^3$ for all $x \geq 1$, then the value of $f(2)$ is
21. If $z$ is any complex number satisfying $|z-3-2i| \leq$2, then the minimum value of $|2z-6+5i|$ is
22. The minimum value of the sum of real numbers $a^{-5}$, $a^{-4}$, $3a^{-3}$, 1, $a^{8}$ and $a^{10}$ with $a$ >0 is
23. Let $f(\theta)$=$\sin \left(tan^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right)$, where $-\frac{\pi}{4}<\theta<\frac{\pi}{4}$. Then the value of $\frac{d}{d(tan \theta)}(f(\theta))$ is

2011:Paper-2

2011:Ans-Keys

All the Papers Ans-Keys

Please wait! Loading...

 Download as PDF