Download JEE Advanced 2017 Mathematics Question Paper - 2

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SECTION 1 (Maximum Marks:21)

• This section contains SEVEN 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. The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes $2x$ + $y −$ $2z$ = 5 and $3x −$ $6y −$ $2z$ = 7, is
   1. $14x$ + $2y −$ $15z$ = 1
   2. $14x −$ $2y$ + $15z$ = 27
   3. $14x$ + $2y$ + $15z$ = 31
   4. $ −14x$ + $2y$ + $15z$ = 3
2. Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point (1, 1, 1) is such that $\vec{OP} ⋅ \vec{OQ}$ + $\vec{OR} ⋅ \vec{OS}$ = $\vec{OR} ⋅ \vec{OP}$ + $\vec{OQ} . \vec{OS}$ = $\vec{OQ} ⋅ \vec{OR}$ + $\vec{OP} ⋅ \vec{OS}$ Then the triangle $PQR$ has $S$ as its
   1. centroid
   2. circumcentre
   3. incentre
   4. orthocenter
3. If $y$= $y(x)$ satisfies the differential equation $8\sqrt{x}\left(\sqrt{9+\sqrt{x}} \right)dy$=$\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)^{-1}dx$, $x$>0 and $y(0)=\sqrt{7}$, then $y(256)$=
   1. 3
   2. 9
   3. 16
   4. 80
4. If $f: ℝ → ℝ$ is a twice differentiable function such that $f"(x)$ > 0 for all $x ∈ ℝ$, and $f\left(\frac{1}{2}\right)$=$\frac{1}{2}$, $f(1)$ = 1, then
   1. $f'(1) \leq 0$
   2. $0 < f'(1) \leq \frac{1}{2}$
   3. $\frac{1}{2} < f'(1) \leq 1$
   4. $f'(1)$ > 1
5. How many 3×3 matrices $M$ with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of $M^TM$ is 5?
   1. 126
   2. 198
   3. 162
   4. 135
6. Let $S$ = {1, 2, 3, … , 9} . For $k$ = 1, 2, … ,5, let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1$ + $N_2$ + $N_3$ + $N_4$ + $N_5$ =
   1. 210
   2. 252
   3. 125
   4. 126
7. Three randomly chosen nonnegative integers $x$, $y$ and $z$ are found to satisfy the equation $x$ + $y$ + $z$ = 10. Then the probability that $z$ is even, is
   1. $\frac{36}{55}$
   2. $\frac{6}{11}$
   3. $\frac{1}{2}$
   4. $\frac{5}{11}$

SECTION 2 (Maximum Marks:28)

• This section contains SEVEN 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.

8. If $g(x)$ = $\int \limits_{sinx}^{sin2x}sin^{-1}t dt$, then
   1. $g'\left(\frac{\pi}{2}\right)=-2 \pi$
   2. $g'\left(-\frac{\pi}{2}\right)=2 \pi$
   3. $g'\left(\frac{\pi}{2}\right)=2 \pi$
   4. $g'\left(-\frac{\pi}{2}\right)=-2 \pi$
9. Let $\alpha$ and $\beta$ be nonzero real numbers such that 2 cos $\beta$ − cos $\alpha$ + cos $\alpha$ cos $\beta$ = 1. Then which of the following is/are true?
   1. $tan\left(\frac{\alpha}{2}\right)$ + $\sqrt{3} tan\left(\frac{\beta}{2}\right)=0$
   2. $\sqrt{3} tan\left(\frac{\alpha}{2}\right)$ + $ tan\left(\frac{\beta}{2}\right)=0$
   3. $tan\left(\frac{\alpha}{2}\right)$ - $\sqrt{3} tan\left(\frac{\beta}{2}\right)=0$
   4. $ \sqrt{3} tan\left(\frac{\alpha}{2}\right)$ - $ tan\left(\frac{\beta}{2}\right)=0$
10. If $f: ℝ → ℝ$ is a differentiable function such that $f′(x)$ > 2$f(x)$ for all $x ∈ ℝ$, and $f(0)$ = 1, then
    1. $f(x)$ is increasing in (0, ∞)
    2. $f(x)$ is decreasing in (0, ∞)
    3. $f(x)$ > $e^{2x}$ in (0, ∞)
    4. $f'(x)$ < $e^{2x}$ in (0, ∞)
11. Let $f(x)$ =$\frac{1-x(1+|1-x|)}{|1-x|}$ $cos \left(\frac{1}{1-x}\right)$ for $x \neq 1$. Then
    1. $\lim \limits_{x \to 1^{-}} f(x)$=0
    2. $\lim \limits_{x \to 1^{-}} f(x)$ does not exist
    3. $\lim \limits_{x \to 1^{+}} f(x)$=0
    4. $\lim \limits_{x \to 1^{+}} f(x)$ does not exist
12. Let $f(x)$ =$\begin{equation*} \left| \begin{array} {ccc}\cos(2x) & \cos(2x) & \sin(2x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x \end{array} \right| \end{equation*}$, then
    1. $f'(x)$ = 0 at exactly three points in $(−\pi, \pi)$
    2. $f'(x)$ = 0 at more than three points in $(−\pi, \pi)$
    3. $f(x)$ attains its maximum at $x$ = 0
    4. $f(x)$ attains its minimum at $x$ = 0
13. If the line $x= \alpha$ divides the area of region $R$ = { $(x, y)$ ∈ $ℝ^2$: $x^3$ ≤ $y$ ≤ $x$, 0 ≤ $x$ ≤ 1} into two equal parts, then
    1. $0 < \alpha \leq \frac{1}{2}$
    2. $\frac{1}{2} < \alpha \leq 1$
    3. $2 \alpha ^4$ - $ 4 \alpha^2$+1=0
    4. $\alpha ^4$ + $ 4 \alpha^2$-1=0
14. If $I$=$\sum \limits_{k=1}^{98}\int \limits_{k}^{k+1} \frac{k+1}{x(x+1)}dx$, then
    1. $I>log_e99$
    2. $I < log_e99$
    3. $I < \frac{49}{50}$
    4. $I > \frac{49}{50}$

SECTION 3 (Maximum Marks:12)

• This section contains TWO paragraphs.
• Based on each paragraph, there will be TWO questions.
• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.
• 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 In all other cases.

Paragraph 1

Let $O$ be the origin, and $\vec{OX}$, $\vec{OY}$, $\vec{OZ}$ be three unit vectors in the directions of the sides $\vec{QR}$, $\vec{RP}$, $\vec{PQ}$, respectively, of a triangle $PQR$.

15. ∣ $\vec{OX}$×$\vec{OY}$ ∣=
    1. $sin(P + Q)$
    2. $sin2R$
    3. $sin(P + R)$
    4. $sin(Q + R)$
16. If the triangle $PQR$ varies, then the minimum value of cos $(P + Q)$ +cos $(Q + R)$ +cos $(R + P)$
    1. $-\frac{5}{3}$
    2. $-\frac{3}{2}$
    3. $\frac{3}{2}$
    4. $\frac{5}{3}$

Paragraph 2

Let $p$, $q$ be integers and let $\alpha$, $\beta$ be the roots of the equation, $x^2 − x − 1$ = 0, where $\alpha \neq \beta$. For $n$ = 0, 1, 2, … , let $\alpha_n$ = $p \alpha^n+ q \beta^n$. FACT: If $a$ and $b$ are rational numbers and $a$ + $b\sqrt{5}$ = 0, then $a$ = 0 = $b$.

17. $a_{12}$=
    1. $a_{11}-a_{10}$
    2. $a_{11}+a_{10}$
    3. $2a_{11}+a_{10}$
    4. $a_{11}+2a_{10}$
18. If $a_4$=28, then $p$+$2q$=
    1. 21
    2. 14
    3. 7
    4. 12

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