Download JEE Advanced 2014 Mathematics Question Paper - 2

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Marking Scheme

• For each question in Section 1, 2 and 3 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.

SECTION - 1 (Only One Option Correct Type)

This section contains 10 multiple choice questions. Each question has four choices (A), B), (C) and (D) out of which only one option is correct.

1. The function $𝑦$ = $𝑓(𝑥)$ is the solution of the differential equation
   $\frac{dy}{dx}$+$\frac{xy}{x^2-1}$=$\frac{x^4+2x}{\sqrt{1-x^2}}$
   in (-1, 1) satisfying $f(0)$=0. Then
   $\int \limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}f(x)dx$
   is
   1. $\frac{\pi}{3}-\frac{\sqrt{3}}{2}$
   2. $\frac{\pi}{3}-\frac{\sqrt{3}}{4}$
   3. $\frac{\pi}{6}-\frac{\sqrt{3}}{4}$
   4. $\frac{\pi}{6}-\frac{\sqrt{3}}{2}$
2. The following integral
   $\int \limits_{\frac{\pi}{4}}^{\frac{\pi}{2}}(2cosecx)^{17}dx $ is equal to
   1. $\int \limits_{0}^{log(1+\sqrt{2})}2(e^u+e^{-u})^{16}du$
   2. $\int \limits_{0}^{log(1+\sqrt{2})}(e^u+e^{-u})^{17}du$
   3. $\int \limits_{0}^{log(1+\sqrt{2})}(e^u-e^{-u})^{17}du$
   4. $\int \limits_{0}^{log(1+\sqrt{2})}2(e^u-e^{-u})^{16}du$
3. Coefficient of $𝑥^{11}$ in the expansion of $(1 + 𝑥^2)^4$$ (1 + 𝑥^3)^7$$(1 + 𝑥^4)^{12}$ is
   1. 1051
   2. 1106
   3. 1113
   4. 1120
4. Let $𝑓:[0, 2]$ → $ℝ$ be a function which is continuous on [0, 2] and is differentiable on (0, 2) with $𝑓(0)$ =1. Let
   $F(x)$ = $\int \limits_{0}^{x^2}f(\sqrt{t})dt$
   for $x \in [0, 2]$. If $F'(x)$=$f'(x)$ for all $x \in (0, 2)$, then $F(2)$ equals
   1. $e^2-1$
   2. $e^4-1$
   3. e-1
   4. $e^4$
5. The common tangents to the circle $𝑥^2$ + $𝑦^2$ = 2 and the parabola $𝑦^2$ = $8𝑥$ touch the circle at the points $𝑃$, $𝑄$ and the parabola at the points $𝑅$, $𝑆$. Then the area of the quadrilateral $𝑃QRS$ is
   1. 3
   2. 6
   3. 9
   4. 15
6. For $𝑥 ∈ (0, 𝜋)$, the equation $sin 𝑥$ + 2 $sin 2𝑥$ − $sin 3𝑥$ = 3 has
   1. infinitely many solutions
   2. three solutions
   3. one solution
   4. no solution
7. In a triangle the sum of two sides is $𝑥$ and the product of the same two sides is $𝑦$. If $𝑥^2$ − $𝑐^2$ = $𝑦$, where $𝑐$ is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is
   1. $\frac{3y}{2x(x+c)}$
   2. $\frac{3y}{2c(x+c)}$
   3. $\frac{3y}{4x(x+c)}$
   4. $\frac{3y}{4c(x+c)}$
8. Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is
   1. 264
   2. 265
   3. 53
   4. 67
9. Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is
   1. $\frac{1}{2}$
   2. $\frac{1}{3}$
   3. $\frac{2}{3}$
   4. $\frac{3}{4}$
10. The quadratic equation $𝑝(𝑥)$ = 0 with real coefficients has purely imaginary roots. Then the equation $𝑝(𝑝(𝑥))$ = 0 has
    1. only purely imaginary roots
    2. all real roots
    3. two real and two purely imaginary roots
    4. neither real nor purely imaginary roots

SECTION - 2 (Comprehension Type)

This section contains 3 paragraphs each describing theory, experiments, data etc. Six questions relate to the three paragraphs with two questions on each paragraph. Each question has only one correct answer among the four given options (A), (B), (C) and (D).

Paragraph For Questions 11 and 12

Let $𝑎$, $𝑟$, $𝑠$, $𝑡$ be nonzero real numbers. Let $𝑃(𝑎𝑡^2, 2𝑎t)$, $𝑄$, $𝑅(𝑎𝑟^2, 2𝑎r)$ and $𝑆(𝑎𝑠^2, 2𝑎s)$ be distinct points on the parabola $𝑦^2 = 4𝑎x$. Suppose that $𝑃Q$ is the focal chord and lines $𝑄R$ and $𝑃K$ are parallel, where $𝐾$ is the point $(2𝑎, 0)$.

11. The value of $𝑟$ is
    1. $-\frac{1}{t}$
    2. $\frac{t^2+1}{t}$
    3. $\frac{1}{t}$
    4. $\frac{t^2-1}{t}$
12. If $𝑠t$ = 1, then the tangent at $𝑃$ and the normal at $𝑆$ to the parabola meet at a point whose ordinate is
    1. $\frac{(t^2+1)^2}{2t^3}$
    2. $\frac{a(t^2+1)^2}{2t^3}$
    3. $\frac{a(t^2+1)^2}{t^3}$
    4. $\frac{a(t^2+2)^2}{t^3}$

Paragraph For Questions 13 and 14

Given that for each $𝑎$ ∈ (0, 1),
$\lim \limits_{h \to 0^+} \int \limits_{h}^{1-h} t^{-a}(1-t)^{a-1}dt$
exists. Let this limit be $𝑔(𝑎)$. In addition, it is given that the function $𝑔(𝑎)$ is differentiable on (0, 1).

13. The value of $g\left(\frac{1}{2}\right)$ is
    1. $\pi$
    2. $2\pi$
    3. $\frac{\pi}{2}$
    4. $\frac{\pi}{4}$
14. The value of $g'\left(\frac{1}{2}\right)$ is
    1. $\frac{\pi}{2}$
    2. $\pi$
    3. $-\frac{\pi}{2}$
    4. 0

Paragraph For Questions 15 and 16

Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $𝑥_𝑖$ be the number on the card drawn from the $𝑖^{𝑡ℎ}$ box, 𝑖 = 1,2,3.

15. The probability that $𝑥_1$ + $𝑥_2$ + $𝑥_3$ is odd, is
    1. $\frac{29}{105}$
    2. $\frac{53}{105}$
    3. $\frac{57}{105}$
    4. $\frac{1}{2}$
16. The probability that $𝑥_1$, $𝑥_2$, $𝑥_3$ are in an arithmetic progression, is
    1. $\frac{9}{105}$
    2. $\frac{10}{105}$
    3. $\frac{11}{105}$
    4. $\frac{7}{105}$

SECTION - 3 (Matching List Type)

This section contains four questions, each having two matching lists. Choices for the correct combination of elements from List-I and List-II are given as options (A), (B), (C) and (D), out of which one is correct.

17. Let $𝑧_𝑘$ = cos $\left(\frac{2k \pi}{10}\right)$ + 𝑖 sin $\left(\frac{2k \pi}{10}\right)$ ; $𝑘$ = 1, 2, … ,9.                        

    	List - I		List - II
    P.	For each $𝑧_𝑘$ there exists a $𝑧_𝑗$ such that $𝑧_𝑘$ ∙ $𝑧_𝑗$	1.	True
    (Q)	There exists a $𝑘$ ∈ {1,2, … , 9} such that $𝑧_1$ ∙ 𝑧 = $𝑧_𝑘$ has no solution $𝑧$ in the set of complex numbers.	2.	False
    R.	$\frac{|1-z_1||1-z_2|...|1-z_9|}{10}$ equals	3.	1
    S.	1 - $\sum \limits_{k=1}^{9} cos \left(\frac{2k \pi}{10}\right)$ equals	4.	2

    1. P -> 1; Q -> 2; R -> 4; S -> 3
    2. P -> 2; Q -> 1; R -> 3; S -> 4
    3. P -> 1; Q -> 2; R -> 3; S -> 4
    4. P -> 2; Q -> 1; R -> 4; S -> 3
18.                        

    	List - I		List - II
    P.	The number of polynomials $𝑓(𝑥)$ with non-negative integer coefficients of degree ≤ 2, satisfying $𝑓(0)$ = 0 and $\int \limits_{0}^{1} 𝑓(𝑥)𝑑𝑥$ = 1, is	1.	8
    Q.	The number of points in the interval $[−\sqrt{13}, \sqrt{13}]$ at which $𝑓(𝑥)$ = sin$(𝑥^2)$ + cos$(𝑥^2)$ attains its maximum value, is	2.	2
    R.	$\int \limits_{-2}^{2} \frac{3 x^2}{1+e^x} dx$ equals	3.	4
    S.	$\frac{\left(\int \limits_{-\frac{1}{2}}^{\frac{1}{2}}cos 2x log \left(\frac{1+x}{1-x}\right)dx \right)}{\left(\int \limits_{0}^{\frac{1}{2}}cos 2x log \left(\frac{1+x}{1-x}\right)dx \right)}$ equals	4.	0

    1. P -> 3; Q -> 2; R -> 4; S -> 1
    2. P -> 2; Q -> 3; R -> 4; S -> 1
    3. P -> 3; Q -> 2; R -> 1; S -> 4
    4. P -> 2; Q -> 3; R -> 1; S -> 4
19.                          

    	List - I		List - II
    P.	Let $𝑦(𝑥)$ = $cos(3cos^{−1}𝑥)$ , $𝑥$ ∈ [−1,1], $𝑥$ ≠ ± $\frac{\sqrt{3}}{2}$. Then $\frac{1}{𝑦(𝑥)}$$\left\{(𝑥^2 −1) \frac{𝑑^2𝑦(𝑥)}{𝑑𝑥^2}\right.$$\left. + 𝑥\frac{𝑑y (𝑥)}{𝑑x}\right\}$ equals	1.	1
    Q.	Let $𝐴_1$, $𝐴_2$, … , $𝐴_𝑛$ ($𝑛$ > 2) be the vertices of a regular polygon of $𝑛$ sides with its centre at the origin. Let $\vec{𝑎_𝑘}$ be the position vector of the point $𝐴_𝑘$, $𝑘$ = 1,2, … , $𝑛$. If $|\sum \limits_{k=1}^{n-1}(\vec{𝑎_𝑘} × \vec{𝑎_{𝑘+1}} )|$ = $|\sum \limits_{k=1}^{n-1}(\vec{𝑎_𝑘}∙ \vec{𝑎_{𝑘+1}})|$, then the minimum value of $𝑛$ is	2.	2
    R.	If the normal from the point $𝑃(ℎ, 1)$ on the ellipse $\frac{𝑥^2}{6}+ \frac{𝑦^2}{3}$ = 1 is perpendicular to the line $𝑥$ + $𝑦$ = 8, then the value of $ℎ$ is	3.	8
    S.	Number of positive solutions satisfying the equation $ tan^{-1}\left(\frac{1}{2x+1}\right)$+$tan^{-1}\left(\frac{1}{4x+1}\right)$=$tan^{-1}\left(\frac{2}{x^2}\right)$ is	4.	9

    1. P -> 4; Q -> 3; R -> 2; S -> 1
    2. P -> 2; Q -> 4; R -> 3; S -> 1
    3. P -> 4; Q -> 3; R -> 1; S -> 2
    4. P -> 2; Q -> 4; R -> 1; S -> 3
20. Let $f_1 : R \to R$, $f_2 : [0, \infty) \to R$, $f_3 : R \to R$ and $f_4 : R \to [0, \infty)$ be defined by
    $$ \begin{array}{l} f_1(x)=\left\{\begin{array}{ll} |x| & \text { if } x<0, \\ e^x & \text { if } x \geq 0 ; \end{array}\right. \\ f_2(x)=x^2 ; \\ f_3(x)=\left\{\begin{aligned} \sin x & \text { if } x<0, \\ x & \text { if } x \geq 0 \end{aligned}\right. \end{array} $$ and $$ f_4(x)=\left\{\begin{array}{ll} f_2\left(f_1(x)\right) & \text { if } x<0 \\ f_2\left(f_1(x)\right)-1 & \text { if } x \geq 0 \end{array}\right. $$
                           

    	List - I		List - II
    P.	$f_4$ is	1.	onto but not one-one
    Q.	$f_3$ is	2.	neither continuous nor one-one
    R.	$𝑓_2$𝑜$𝑓_1$ is	3.	differentiable but not one-one
    S.	$f_2$ is	4.	continuous and one-one

    1. P -> 3; Q -> 1; R -> 4; S -> 2
    2. P -> 1; Q -> 3; R -> 4; S -> 2
    3. P -> 3; Q -> 1; R -> 2; S -> 4
    4. P -> 1; Q -> 3; R -> 2; S -> 4

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