Download JEE Advanced 2016 Mathematics Question Paper - 1

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SECTION 1 (Maximum Marks:15)

• This section contains FIVE questions.
• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
• For each question, darken the bubble corresponding to the correct option in the ORS.
• For each question, marks will be awarded in one of the following categories:
  
  • Full Marks: +3 If only the bubble corresponding to the correct option is darkened;
  • Zero Marks: 0 If none of the bubbles is darkened;
  • Negative Marks: -1 In all other cases.

1. Let $-\frac{\pi}{6} < \theta < -\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2-2xsec \theta +1$=0 and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2+2xtan \theta -1$=0. If $\alpha_1 > \beta_1$ and $\alpha_2 > \beta_2$, then $\alpha_1+\beta_2$ equals
   1. $2sec \theta - tan \theta$
   2. $2 sec \theta$
   3. $-2 tan \theta$
   4. 0
2. A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
   1. 380
   2. 320
   3. 260
   4. 95
3. Let $S$ = $\left\{x \in (-\pi, \pi):x \neq 0,±\frac{\pi}{2} \right\}$. The sum of all distinct solutions of the equation $\sqrt{3} sec x$ + $cosec x$ + 2($tan x - $$cot x$) =0 in the set $S$ is equal to
   1. $- \frac{7 \pi}{9}$
   2. $- \frac{2 \pi}{9}$
   3. 0
   4. $ \frac{5 \pi}{9}$
4. A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces 20% and plant $T_2$ produces 80% of the total computers produced. 7% of produced computers in the factory turn out to be defective. It is known that
   $P$(computers turn out to be defective given that it is produced in plant $T_1$)
   =$10P$(computers turn out to be defective given that it is produced in plant $T_2$), where $P(E)$ denotes the probability of an event $E$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_2$ is
   1. $ \frac{36}{73}$
   2. $\frac{47}{79}$
   3. $\frac{78}{93}$
   4. $\frac{75}{83}$
5. The least value of $\alpha \in R$ for which $4 \alpha x^2 + \frac{1}{x} \geq 1$, for all $x$>0, is
   1. $ \frac{1}{64}$
   2. $\frac{1}{32}$
   3. $\frac{1}{27}$
   4. $\frac{1}{25}$

SECTION 2 (Maximum Marks:32)

• This section contains EIGHT questions.
• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s).
• For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened;
  • Partial Marks: +1 For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened;
  • Zero Marks: 0 If none of the bubbles is darkened;
  • Negative Marks: -2 In all other cases.
• For example, if (A), (C) and (D) are all the correct options for a question, darkening all these three will result in +4 marks; darkening only (A) and (D) will result in +2 marks; and darkening (A) and (B) result in -2 marks, as a wrong option is also darkened.

6. Consider a pyramid $OPQRS$ located in the first octant $(x \geq 0, y \geq 0, z \geq 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x-$axis and the $y -$ axis, respectively. The base $OPQR$ of the pyramid is a square with $OP$=3. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS$=3. Then
   1. The acute angle between $OQ$ and $OS$ is $ \frac{\pi}{3}$
   2. The equation of the plane containing the triangle $OQS$ is $x - y$ =0
   3. The length of the perpendicular from $P$ to the plane containing the triangle $OQS$ is $\frac{3}{\sqrt{2}}$
   4. The perpendicular distance from $O$ to the straight line containing $RS$ is $\sqrt{\frac{15}{2}}$
7. Let $f:(0, \infty) \to R$ be a differentiable function such that $f'(x)$=$2- \frac{f(x)}{x}$ for all $x \in (0, \infty)$ and $f(1) \neq 1$. Then
   1. $\lim \limits_{x \to 0+}f'(\frac{1}{x})$=1
   2. $\lim \limits_{x \to 0+}x f(\frac{1}{x})$=2
   3. $\lim \limits_{x \to 0+}x^2f'(x)$=0
   4. $|f(x)| \leq 2$ for all $x \in (0,2)$
8. Let $P$= $\begin{equation*} \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix} \end{equation*},$ where $ \alpha \in R$. Suppose $Q= [q_{ij}]$ is a matrix such that $PQ$ = $k I$, where $k \in R, k \neq 0$ and $I$ is the identity matrix of order 3. If $q_{23}=- \frac{k}{8}$ and det(Q)=$\frac{k^2}{2}$, then
   1. $\alpha =0$, $k$=8
   2. $ 4 \alpha $ - $k$ + 8=0
   3. det($P $adj($Q$))=$2^9$
   4. det($Q $adj($P$))=$2^{13}$
9. In a triangle $XYZ$, let $x$, $y$, $z$ be the lengths of the sides opposite to the angles $X$, $Y$, $Z$, respectively and $2s$ = $x+y+z$. If $ \frac{s-x}{4}$=$ \frac{s-y}{3}$=$ \frac{s-z}{2}$ and area of incircle of the triangle $XYZ$ is $\frac{8 \pi}{3}$, then
   1. area of the triangle $XYZ$ is $6\sqrt{6}$
   2. the radius of the circumcircle of the triangle $XYZ$ is $ \frac{35}{6}\sqrt{6}$
   3. $\frac{sin X}{2}$$\frac{sin Y}{2}$$\frac{sin Z}{2}$=$\frac{4}{35}$
   4. $sin^2\frac{X+Y}{2}$=$\frac{3}{5}$
10. A solution curve of the differentiable equation ($x^2$+$xy$+$4x$+$2y$+4)$\frac{dy}{dx}-y^2$=0, $x$>0 passes through the point (1, 3). Then the solution curve
    1. intersects $y$=$x$ +2 exactly at one point.
    2. intersects $y$=$x$ +2 exactly at two points.
    3. intersects $y$=$(x+2)^2$
    4. does NOT intersect $y$=$(x+3)^2$
11. Let $f:R \to R$, $g:R \to R$ and $h:R \to R$ be differentiable functions such that $f(x)$=$x^3$ +$3x$+2, $g(f(x))$=$x$ and $h(g(g(x)))$=$x$ for all $x \in R$. Then
    1. $g'(2)$=$\frac{1}{15}$
    2. $h'(1)$=666
    3. $h(0)$=16
    4. $h(g(3))$=36
12. The circle $C_1:x^2+y^2=3$, with center at $O$, intersects the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_2$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centers $Q_2$ and $Q_3$, respectively. If $Q_1$ and $Q_2$ lie on the $y-$axis, then
    1. $Q_2Q_3$=12
    2. $R_2R_3=4\sqrt{6}$
    3. area of the triangle $OR_2R_3=6\sqrt{2}$
    4. The area of the triangle $PQ_2Q_3=4\sqrt{2}$
13. Let $RS$ be the diameter of the circle $x^2+y^2=1$, where $S$ is the point (1, 0). Let $P$ be a variable point ( other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $RS$ at point $E$. Then the locus of $E$ passes through the point(s)
    1. $\left(\frac{1}{3}, \frac{1}{\sqrt{3}} \right)$
    2. $\left(\frac{1}{4}, \frac{1}{2} \right)$
    3. $\left(\frac{1}{3}, -\frac{1}{\sqrt{3}} \right)$
    4. $\left(\frac{1}{4}, -\frac{1}{2} \right)$

SECTION 3 (Maximum Marks:15)

• This section contains FIVE questions.
• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9, BOTH INCLUSIVE.
• For each question, darken the bubble corresponding to the correct integer in the ORS.
• Marking scheme:
  
  • Full Marks: +3 If the bubble corresponding to the answer is darkened
  • Zero Marks: 0 In all other cases.

14. The total number of distinct $ x \in R$ for which $ \left|\begin{array}{ccc}x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{array}\right|$ =10 is
15. Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2$+$(1+x)^3$+...+$(1+x)^{49}$+$(1+mx)^{50}$ is $(3n+1)$ $^{51}C_3$ for some positive integer $n$. Then the value of $n$ is....
16. The total number of distinct $x \in [0, 1]$ for which $\int \limits_0^x \frac{t^2}{1+t^4}dt$=$2x-$1 is
17. Let $\alpha, \beta \in R$ be such that $\lim \limits_{x \to 0}\frac{x^2 sin (\beta x)}{\alpha x - sin x}$=1. Then $6(\alpha + \beta)$ equals
18. Let $z$= $\frac{-1+\sqrt{3}i}{2}$, where $i=\sqrt{-1}$ and $r, s \in [1, 2, 3]$. Let $P$= $\begin{equation*} \begin{bmatrix} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{bmatrix} \end{equation*}$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r, s)$ for which $P^2$= $-I$ is

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