# Solved papers for JEE Main & Advanced JEE Main Paper (Held On 11 April 2014)

### done JEE Main Paper (Held On 11 April 2014) Total Questions - 30

• question_answer1) Let f be an odd function defined on the set of real numbers such that for $x\ge 0,$$f(x)=3sinx+4cosx.$ Then f(x) at $x=-\frac{11\pi }{6}$is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{3}{2}+2\sqrt{3}$

B)
$-\frac{3}{2}+2\sqrt{3}$

C)
$\frac{3}{2}-2\sqrt{3}$

D)
$-\frac{3}{2}-2\sqrt{3}$

• question_answer2) If ${{z}_{1}},{{z}_{2}}$and ${{z}_{3}},{{z}_{4}}$are 2 pairs of complex conjugate numbers, then $\arg \left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+\arg \left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)$equals:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
0

B)
$\frac{\pi }{2}$

C)
$\frac{3\pi }{2}$

D)
$\pi$

• question_answer3) If $\alpha$ and $\beta$ are roots of the equation, ${{x}^{2}}-4\sqrt{2}kx+2{{e}^{4\ln k}}-1=0$ for some k, and ${{\alpha }^{2}}+{{\beta }^{2}}=66,$then ${{\alpha }^{3}}+{{\beta }^{2}}$is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$248\sqrt{2}$

B)
$280\sqrt{2}$

C)
$-32\sqrt{2}$

D)
$-280\sqrt{2}$

• question_answer4) Let A be a $3\times 3$matrix such that$A\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \\ \end{matrix} \right]=\left[ \begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{matrix} \right]$Then ${{A}^{-1}}$is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\left[ \begin{matrix} 3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1 \\ \end{matrix} \right]$

B)
$\left[ \begin{matrix} 3 & 2 & 1 \\ 3 & 2 & 0 \\ 1 & 1 & 0 \\ \end{matrix} \right]$

C)
$\left[ \begin{matrix} 0 & 1 & 3 \\ 0 & 2 & 3 \\ 1 & 1 & 0 \\ \end{matrix} \right]$

D)
$\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & 2 & 3 \\ \end{matrix} \right]$

• question_answer5) Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in $x,p{{'}_{i}}(x)$and $p'{{'}_{i}}(x)$be the first and second order derivatives of ${{p}_{i}}(x)$ respectively. Let,$A(x)=\left[ \begin{matrix} {{p}_{1}}(x) & {{p}_{1}}'(x) & {{p}_{1}}''(x) \\ {{p}_{2}}(x) & {{p}_{2}}'(x) & {{p}_{2}}''(x) \\ {{p}_{3}}(x) & {{p}_{3}}'(x) & {{p}_{3}}''(x) \\ \end{matrix} \right]$and $B(x)={{[A(x)]}^{T}}A(x).$Then determinant of B(x):   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
is a polynomial of degree 6 in x.

B)
is a polynomial of degree 3 in x.

C)
is a polynomial of degree 2 in x.

D)
does not depend on x.

• question_answer6) An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
72 (7!)

B)
18 (7!)

C)
40 (7!)

D)
36 (7!)

• question_answer7) The coefficient of ${{x}^{50}}$in the binomial expansion of${{(1+x)}^{1000}}+x{{(1+x)}^{999}}+{{x}^{2}}$${{(1+x)}^{998}}+....+{{x}^{1000}}$ is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{\left( 1000 \right)!}{\left( 50 \right)!\left( 950 \right)!}$

B)
$\frac{\left( 1000 \right)!}{\left( 49 \right)!\left( 951 \right)!}$

C)
$\frac{\left( 1001 \right)!}{\left( 51 \right)!\left( 950 \right)!}$

D)
$\frac{\left( 1001 \right)!}{\left( 50 \right)!\left( 951 \right)!}$

• question_answer8) In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is:

A)
7

B)
21

C)
28

D)
42

• question_answer9) The sum of the first 20 terms common between the series 3 + + 11 + 15 + ......... and 1 + 6 + 11 + 16 + ......, is   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
4000

B)
4020

C)
4200

D)
4220

• question_answer10) If$\underset{x\to 2}{\mathop{\lim }}\,\frac{\tan \left( x-2 \right)\left\{ {{x}^{2}}+\left( k-2 \right)x-2k \right\}}{{{x}^{2}}-4x+4}=5,$then k is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
0

B)
1

C)
2

D)
3

• question_answer11) Let f(x) = x|x|, g(x) = sin x and h(x) = (gof) (x). Then

A)
h(x) is not differentiable at x = 0.

B)
h(x) is differentiable at x = 0, but h¢(x) is not continuous at x = 0

C)
h?(x) is continuous at x = 0 but it is not differentiable at x = 0

D)
h?(x) is differentiable at x = 0

• question_answer12) For the curve $y=3\sin \theta \cos \theta ,x={{e}^{\theta }}\sin \theta ,$$0\le \theta \le \pi ,$the tangent is parallel to x-axis when $\theta$is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{3\pi }{4}$

B)
$\frac{\pi }{2}$

C)
$\frac{\pi }{4}$

D)
$\frac{\pi }{6}$

• question_answer13) Two ships A and B are sailing straight away from a fixed point O along routes such that $\angle AOB$ is always $120{}^\circ$. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/ hr):   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{260}{\sqrt{37}}$

B)
$\frac{260}{37}$

C)
$\frac{80}{\sqrt{37}}$

D)
$\frac{80}{37}$

• question_answer14) The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $=\sqrt{3}$ is:

A)
$\frac{4}{3}\sqrt{3}\pi$

B)
$\frac{8}{3}\sqrt{3}\pi$

C)
$4\pi$

D)
$2\pi$

• question_answer15) The integral$\int_{{}}^{{}}{x{{\cos }^{-1}}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)dx(x>0)$is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$-x+1(1+{{x}^{2}})ta{{n}^{-1}}x+c$

B)
$x-(1+{{x}^{2}})co{{t}^{-1}}x+c$

C)
$-x+(1+{{x}^{2}})co{{t}^{-1}}x+c$

D)
$x-(1+{{x}^{2}})ta{{n}^{-1}}x+c$

• question_answer16) If for $n\ge 1,{{P}_{n}}=\int\limits_{1}^{e}{{{\left( \log x \right)}^{n}}dx,}$then${{P}_{10}}-90{{P}_{8}}$ then is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
-9

B)
10e

C)
-9 e

D)
10

• question_answer17) If the general solution of the differential equation$y'=\frac{y}{x}+\Phi \left( \frac{x}{y} \right),$for some function $\Phi ,$ is given by $\ln |cx|=x,$where c is an arbitrary constant, then $\Phi (2)$is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
4

B)
$\frac{1}{4}$

C)
-4

D)
$-\frac{1}{4}$

• question_answer18) A stair-case of length l rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
an ellipse of eccentricity$\frac{1}{2}$

B)
an ellipse of eccentricity$\frac{\sqrt{3}}{2}$

C)
a circle of radius$\frac{1}{2}$

D)
a circle of radius$\frac{\sqrt{3}}{2}l$

• question_answer19) The base of an equilateral triangle is along the line given by 3x + 4y = 9. If a vertex of the triangle is (1, 2), then the length of a side of the triangle is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{2\sqrt{3}}{15}$

B)
$\frac{4\sqrt{3}}{15}$

C)
$\frac{4\sqrt{3}}{5}$

D)
$\frac{2\sqrt{3}}{5}$

• question_answer20) The set of all real values of $\lambda$for which exactly two common tangents can be drawn to the circles ${{x}^{2}}+{{y}^{2}}4x4y+6=0$and ${{x}^{2}}+{{y}^{2}}10x10y+\lambda =0$is the interval:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
(12, 32)

B)
(18, 42)

C)
(12, 24)

D)
(18, 48)

• question_answer21) Let ${{L}_{1}}$be the length of the common chord of the curves ${{x}^{2}}+{{y}^{2}}=9$ and${{y}^{2}}=8x,$ and ${{L}_{2}}$ be the length of the latus rectum of ${{y}^{2}}=8x,$ then:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
${{L}_{1}}>{{L}_{2}}$

B)
${{L}_{1}}={{L}_{2}}$

C)
${{L}_{1}}<{{L}_{2}}$

D)
$\frac{{{L}_{1}}}{{{L}_{2}}}=\sqrt{2}$

• question_answer22) Let $P(3sec\theta ,2tan\theta )$and $Q(3sec\phi ,2tan\phi )$ where $\theta +\phi =\frac{\pi }{2},$be two distinct points on the hyperbola $\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1.$Then the ordinate of the point of intersection of the normals at P and Q is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{11}{3}$

B)
$-\frac{11}{3}$

C)
$\frac{13}{2}$

D)
$-\frac{13}{2}$

• question_answer23) Let A (2, 3, 5), B (? 1, 3, 2) and $C(\lambda ,5,\mu )$be the vertices of a DABC. If the median through A is equally inclined to the coordinate axes, then:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$5\lambda -8\mu =0$

B)
$8\lambda -5\mu =0$

C)
$10\lambda -7\mu =0$

D)
$7\lambda -10\mu =0$

• question_answer24) The plane containing the line $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$and parallel to the line$\frac{x}{1}=\frac{y}{1}=\frac{z}{4}$passes through the point:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$(1, - 2, 5)$

B)
$(1, 0, 5)$

C)
$(0, 3, -5)$

D)
$(-1, -3, 0)$

• question_answer25) If $\overset{\to }{\mathop{{{\left| c \right|}^{2}}}}\,=60$and $\overset{\to }{\mathop{c}}\,\times \left( \hat{i}+2\hat{j}+5\hat{k} \right)=\overset{\to }{\mathop{0}}\,,$then a value of$\overset{\to }{\mathop{c}}\,.\left( -7\hat{i}+2\hat{j}+3\hat{k} \right)$is: [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$4\sqrt{2}$

B)
12

C)
24

D)
122

• question_answer26) A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that $x\in A$is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{1}{2}$

B)
$\frac{64}{127}$

C)
$\frac{63}{128}$

D)
$\frac{31}{128}$

• question_answer27) If X has a binomial distribution, B(n, p) with parameters n and p such that P(X = 2) = P (X = 3), then E(X), the mean of variable X, is   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
2 ? p

B)
3 ? p

C)
$\frac{p}{2}$

D)
$\frac{p}{3}$

• question_answer28) If $2\cos \theta +\sin \theta =1\left( \theta \ne \frac{\pi }{2} \right),$then $7\cos \theta +6\sin \theta$is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{1}{2}$

B)
2

C)
$\frac{11}{2}$

D)
$\frac{46}{5}$

• question_answer29) The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be $\alpha .$ After moving a distance 2 metres from P towards the foot of the tower, the angle of elevation changes to $\beta .$ Then the height (in metres) of the tower is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
$\frac{2\sin \alpha \sin \beta }{\sin \left( \beta -\alpha \right)}$

B)
$\frac{\sin \alpha \sin \beta }{\cos \left( \beta -\alpha \right)}$

C)
$\frac{2\sin \left( \beta -\alpha \right)}{\sin \alpha \sin \beta }$

D)
$\frac{\cos \left( \beta -\alpha \right)}{\sin \alpha \sin \beta }$

• question_answer30) The proposition $\tilde{\ }\left( p\vee \tilde{\ }q \right)\vee \tilde{\ }\left( p\vee q \right)$is logically equivalent to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A)
p

B)
q

C)
$\tilde{\ }p$

D)
$\tilde{\ }q$