JEE Advance - Mathematics (2014 - Paper 2 Offline)

1
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
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(A)
$${1 \over 2}$$
2
For $$x \in \left( {0,\pi } \right),$$ the equation $$\sin x + 2\sin 2x - \sin 3x = 3$$ has
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(D)
no solution
3
Match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists:
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List-$$I$$
(P.)$$\,\,\,\,$$ Let $$y\left( x \right) = \cos \left( {3{{\cos }^{ - 1}}x} \right),x \in \left[ { - 1,1} \right],x \ne \pm {{\sqrt 3 } \over 2}.$$ Then $${1 \over {y\left( x \right)}}\left\{ {\left( {{x^2} - 1} \right){{{d^2}y\left( x \right)} \over {d{x^2}}} + x{{dy\left( x \right)} \over {dx}}} \right\}$$ equals
(Q.)$$\,\,\,\,$$ Let $${A_1},{A_2},....,{A_n}\left( {n > 2} \right)$$ be the vertices of a regular polygon of $$n$$ sides with its centre at the origin. Let $${\overrightarrow {{a_k}} }$$ be the position vector of the point $${A_k},k = 1,2,......,n.$$ $$$f\left| {\sum\nolimits_{k = 1}^{n - 1} {\left( {\overrightarrow {{a_k}} \times \overrightarrow {{a_{k + 1}}} } \right)} } \right| = \left| {\sum\limits_{k = 1}^{n - 1} {\left( {\overrightarrow {{a_k}} .\,\overrightarrow {{a_{k + 1}}} } \right)} } \right|,$$$ then the minimum value of $$n$$ is
(R.)$$\,\,\,\,$$ If the normal from the point $$P(h, 1)$$ on the ellipse $${{{x^2}} \over 6} + {{{y^2}} \over 3} = 1$$ is perpendicular to the line $$x+y=8,$$ then the value of $$h$$ is
(S.)$$\,\,\,\,$$ Number of positive solutions satisfying the equation $${\tan ^{ - 1}}\left( {{1 \over {2x + 1}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {4x + 1}}} \right) = {\tan ^{ - 1}}\left( {{2 \over {{x^2}}}} \right)$$ is
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$List-$$II$$
(1.)$$\,\,\,\,$$ $$1$$
(2.)$$\,\,\,\,$$ $$2$$
(3.)$$\,\,\,\,$$ $$8$$
(4.)$$\,\,\,\,$$ $$9$$
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(A)
$$P = 4,Q = 3,R = 2,S = 1$$
4
The probability that $${x_1},$$, $${x_2},$$ $${x_3}$$ are in an arithmetic progression, is
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(C)
$${{11} \over {105}}$$
5
The probability that $${x_1} + {x_2} + {x_3}$$ is odd, is
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(B)
$${{53} \over {105}}$$
6
If $$st=1$$, then the tangent at $$P$$ and the normal at $$S$$ to the parabola meet at a point whose ordinate is
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(B)
$${{a{{\left( {{t^2} + 1} \right)}^2}} \over {2{t^3}}}$$
7
Let $${z_k}$$ = $$\cos \left( {{{2k\pi } \over {10}}} \right) + i\,\,\sin \left( {{{2k\pi } \over {10}}} \right);\,k = 1,2....,9$$
List-I

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

1. True
2. False
3. 1
4. 2
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(C)
P = 1, Q = 2, R = 3, S = 4
8
The quadratic equation $$p(x)$$ $$ = 0$$ with real coefficients has purely imaginary roots. Then the equation $$p(p(x))=0$$ has
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(D)
neither real nor purely imaginary roots
9
Coefficient of $${x^{11}}$$ in the expansion of $${\left( {1 + {x^2}} \right)^4}{\left( {1 + {x^3}} \right)^7}{\left( {1 + {x^4}} \right)^{12}}$$ is
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(C)
1113
10
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
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(C)
53
11
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
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(D)
$$15$$
12
The value of $$r$$ is
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(D)
$${{{t^2} - 1} \over t}$$
13
The function $$y=f(x)$$ is the solution of the differential equation
$${{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - {x^2}} }}\,$$ in $$(-1,1)$$ satisfying $$f(0)=0$$.
Then $$\int\limits_{ - {{\sqrt 3 } \over 2}}^{{{\sqrt 3 } \over 2}} {f\left( x \right)} \,d\left( x \right)$$ is
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(B)
$${\pi \over 3} - {{\sqrt 3 } \over 4}$$
14
Let $$f:\left[ {0,2} \right] \to R$$ be a function which is continuous on $$\left[ {0,2} \right]$$ and is differentiable on $$(0,2)$$ with $$f(0)=1$$. Let
$$F\left( x \right) = \int\limits_0^{{x^2}} {f\left( {\sqrt t } \right)dt} $$ for $$x \in \left[ {0,2} \right]$$. If $$F'\left( x \right) = f'\left( x \right)$$ for all $$x \in \left[ {0,2} \right]$$, then $$F(2)$$ equals
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(B)
$${e^4} - 1$$
15
In a triangle the sum of two sides is $$x$$ and the product of the same 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
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(B)
$${{3y} \over {2c\left( {x + c} \right)}}$$
16
The following integral $$\int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\left( {2\cos ec\,\,x} \right)}^{17}}dx} $$ is equal to
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(A)
$$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {2{{\left( {{e^u} + {e^{ - u}}} \right)}^{16}}\,du} $$
17
List - $$I$$
P.$$\,\,\,\,$$ The number of polynomials $$f(x)$$ with non-negative integer coefficients of degree $$ \le 2$$, satisfying $$f(0)=0$$ and $$\int_0^1 {f\left( x \right)dx = 1,} $$ is
Q.$$\,\,\,\,$$ The number of points in the interval $$\left[ { - \sqrt {13} ,\sqrt {13} } \right]$$
at which $$f\left( x \right) = \sin \left( {{x^2}} \right) + \cos \left( {{x^2}} \right)$$ attains its maximum value, is
R.$$\,\,\,\,$$ $$\int\limits_{ - 2}^2 {{{3{x^2}} \over {\left( {1 + {e^x}} \right)}}dx} $$ equals
S.$$\,\,\,\,$$ $${{\left( {\int\limits_{ - {1 \over 2}}^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)} \over {\left( {\int\limits_0^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)}}$$

List $$II$$
1.$$\,\,\,\,$$ $$8$$
2.$$\,\,\,\,$$ $$2$$
3.$$\,\,\,\,$$ $$4$$
4.$$\,\,\,\,$$ $$0$$
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(D)
$$P = 2,Q = 3,R = 1,S = 4$$
18
The value of $$g'\left( {{1 \over 2}} \right)$$ is
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(D)
$$0$$
19
The value of $$g\left( {{1 \over 2}} \right)$$ is
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(A)
$$\pi $$
20
Let f₁ : R $$ \to $$ R, f₂ : [0, $$\infty $$) $$ \to $$ R, f₃ : R $$ \to $$ R, and f₄ : R $$ \to $$ [0, $$\infty $$) be defined by
$${f_1}\left( x \right) = \left\{ {\matrix{ {\left| x \right|} & {if\,x < 0,} \cr {{e^x}} & {if\,x \ge 0;} \cr } } \right.$$

f₂(x) = x² ;
$${f_3}\left( x \right) = \left\{ {\matrix{ {\sin x} & {if\,x < 0,} \cr x & {if\,x \ge 0;} \cr } } \right.$$
and

$${f_4}\left( x \right) = \left\{ {\matrix{ {{f_2}\left( {{f_1}\left( x \right)} \right)} & {if\,x < 0,} \cr {{f_2}\left( {{f_1}\left( x \right)} \right) - 1} & {if\,x \ge 0;} \cr } } \right.$$

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(D)
P - 1, Q - 3, R - 2, S - 4