# Solved papers for JEE Main & Advanced JEE Main Paper (Held On 12 April 2014)

### done JEE Main Paper (Held On 12 April 2014) Total Questions - 30

• question_answer1) A relation on the set $a=\{x:|x|<3,x\in Z\},$where Z is the set of integers is defined by $R=\{x,y):y=|x|,x\ne -1\}.$Then the number of elements in the power set of R is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
32

B)
16

C)
8

D)
64

• question_answer2) Let $z\ne -i$ be any complex number such that $\frac{z-i}{z+i}$is a purely imaginary number. Then$z+\frac{1}{z}$is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
0

B)
any non-zero real number other than 1.

C)
any non-zero real number.

D)
a purely imaginary number.

• question_answer3) The sum of the roots of the equation, ${{x}^{2}}+|2x-3|-4=0,$ is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$2$

B)
$-2$

C)
$\sqrt{2}$

D)
$-\sqrt{2}$

• question_answer4) If $\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ {{(a+\lambda )}^{2}} & {{(b+\lambda )}^{2}} & {{(a+\lambda )}^{2}} \\ {{(a-\lambda )}^{2}} & {{(b-\lambda )}^{2}} & {{(c+\lambda )}^{2}} \\ \end{matrix} \right|=$ $k\lambda \left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix} \right|,\lambda \ne 0$then k is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$4\lambda abc$

B)
$-4\lambda abc$

C)
$4{{\lambda }^{2}}$

D)
$-4{{\lambda }^{2}}$

• question_answer5) If$A=\left[ \begin{matrix} 1 & 2 & x \\ 3 & -1 & 2 \\ \end{matrix} \right]$andB=\left[ \begin{align} & y \\ & x \\ & 1 \\ \end{align} \right]be such thatAB=\left[ \begin{align} & 6 \\ & 8 \\ \end{align} \right],then:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$y=2x$

B)
$y=2x$

C)
$y=x$

D)
$y=-x$

• question_answer6) 8-digit numbers are formed using the digits 1, 1, 2, 2, 2, 3, 4, 4. The number of such numbers in which the odd digits do no occupy odd places, is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
160

B)
120

C)
60

D)
48

• question_answer7) If${{\left( 2+\frac{x}{3} \right)}^{55}}$is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
7th and 8th

B)
8th and 9th

C)
28th  and 29th

D)
27th  and 28th

• question_answer8) Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $\frac{1}{a}$and$\frac{1}{b}.$If$\frac{1}{M}:G$is 4 : 5 then a : b can be:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
1 : 4

B)
1 : 2

C)
2 : 3

D)
3 : 4

• question_answer9) The least positive integer n such that $1-\frac{2}{3}-\frac{2}{{{3}^{2}}}-....-\frac{2}{{{3}^{n-1}}}<\frac{1}{100},$is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
4

B)
5

C)
7

D)
7

• question_answer10) Let $f,g:R\to R$be two functions defined by f(x)=\left\{ \begin{align} & x\sin \left( \frac{1}{x} \right),x\ne 0 \\ & 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.,and $g(x)=xf(x)$ Statement I: f is a continuous function at x = 0. Statement II: g is a differentiable function at x = 0.   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
Both statement I and II are false.

B)
Both statement I and II are true.

C)
Statement I is true, statement II is false.

D)
Statement I is false, statement II is true.

• question_answer11) If $f(x)={{x}^{2}}-x+5,x>\frac{1}{2},$and g(x) is its inverse function, then g'(7) equals:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$-\frac{1}{3}$

B)
$\frac{1}{13}$

C)
$\frac{1}{3}$

D)
$-\frac{1}{13}$

• question_answer12) Let f and g be two differentiable functions on R such that f'(x) > 0 and g'(x) < 0 for all $x\in R$. Then for all x:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
f (g (x)) > f (g (x - 1))

B)
f (g (x)) > f (g (x + 1))

C)
g(f (x)) > g (f (x - 1))

D)
g(f (x)) < g (f (x + 1))

• question_answer13) If $1+{{x}^{4}}+{{x}^{5}}=T\sum\limits_{i=0}^{5}{{{a}_{i}}}{{\left( 1+x \right)}^{i}},$for all x in R, then ${{a}_{2}}$is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
-4

B)
6

C)
-8

D)
10

• question_answer14) The integral $\int_{{}}^{{}}{\frac{\sin x{{\cos }^{2}}x}{{{\left( {{\sin }^{3}}x+{{\cos }^{3}}x \right)}^{2}}}dx}$is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$\frac{1}{\left( 1+{{\cot }^{3}}x \right)}+c$

B)
$-\frac{1}{3\left( 1+{{\cot }^{3}}x \right)}+c$

C)
$-\frac{{{\sin }^{3}}x}{\left( 1+{{\cot }^{3}}x \right)}+c$

D)
$-\frac{{{\cos }^{3}}x}{3\left( 1+{{\cot }^{3}}x \right)}+c$

• question_answer15) If [ ] denotes the greatest integer function, then the integral $\int\limits_{0}^{\pi }{\left[ \cos x \right]}dx$is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

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

B)
0

C)
-1

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

• question_answer16) If for a continuous function $f(x),\int\limits_{-\pi }^{t}{\left( f\left( x \right)+x \right)dx}={{\pi }^{2}}-t2,$for all$t\ge -\pi ,$then$\left( -\frac{\pi }{3} \right)$is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$\pi$

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

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

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

• question_answer17) The general solution of the differential equation, $\sin 2x\left( \frac{dy}{dx}-\sqrt{\tan x} \right)-y=0,$is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$y\sqrt{\tan x}=x+c$

B)
$y\sqrt{\cot x}=\tan x+c$

C)
$y\sqrt{\tan x}=\cot x+c$

D)
$y\sqrt{\cot x}=x+c$

• question_answer18) If a line intercepted between the coordinate axes is trisected at a point A(4, 3), which is nearer to x-axis, then its equation is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$4x\text{ }-\text{ }3y\text{ }=\text{ }7\text{ }~$

B)
$3x\text{ }+\text{ }2y\text{ }=\text{ }18$

C)
$3x\text{ }+\text{ }8y\text{ }=\text{ }36$

D)
$x\text{ }+\text{ }3y\text{ }=\text{ }13$

• question_answer19) If the three distinct lines x + 2ay + a = 0, x + 3by + b = 0 and x + 4ay + a = 0 are concurrent, then the point (a, b) lies on a:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
circle

B)
hyperbola

C)
straight line

D)
parabola

• question_answer20) For the two circles ${{x}^{2}}+{{y}^{2}}=16$ and ${{x}^{2}}+{{y}^{2}}-2y=0,$there is/are   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
one pair of common tangents

B)
two pair of common tangents

C)
three pair of common tangents

D)
no common tangent

• question_answer21) Two tangents are drawn from a point (- 2, - 1) to the curve, ${{y}^{2}}=4x.$ If $\alpha$ is the angle between them, then $|\tan \alpha |$is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$\frac{1}{3}$

B)
$\frac{1}{\sqrt{3}}$

C)
$\sqrt{3}$

D)
$3$

• question_answer22) The minimum area of a triangle formed by any tangent to the ellipse$\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{81}=1$and the co-ordinate axes is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
12

B)
18

C)
26

D)
36

• question_answer23) A symmetrical form of the line of intersection of the planes $x\text{ }=\text{ }ay\text{ }+\text{ }b\text{ }and\text{ }z\text{ }=\text{ }cy\text{ }+\text{ }d$ is   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$\frac{x-b}{a}+\frac{y-1}{1}=\frac{z-d}{c}$

B)
$\frac{x-b-a}{a}=\frac{y-1}{1}=\frac{z-d-c}{c}$

C)
$\frac{x-a}{b}=\frac{y-0}{1}=\frac{z-c}{d}$

D)
$\frac{x-b-a}{b}=\frac{y-1}{0}=\frac{z-d-c}{d}$

• question_answer24) If the distance between planes, $4x-2y-4z+1=0$and $2y-4z+d=0$ is 7, then d is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$41 or - 42$

B)
$42 or - 43$

C)
$- 41 or 43$

D)
$- 42 or 44$

• question_answer25) If $\hat{x},\hat{y}$and $\hat{z}$ are three unit vectors in three-dimensional space, then the minimum value of$|\hat{x}+\hat{y}{{|}^{2}}+|\hat{y}+\hat{z}{{|}^{2}}+|\hat{z}+\hat{x}{{|}^{2}}$   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

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

B)
3

C)
$3\sqrt{3}$

D)
6

• question_answer26) Let$\overline{X}$and M.D. be the mean and the mean deviation about X of n observations ${{x}_{i}},$ i = 1, 2, ........, n. If each of the observations is increased by 5, then the new mean and the mean deviation about the new mean, respectively, are :   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$\overline{X},M.D.$

B)
$\overline{X}+5,M.D.$

C)
$\overline{X},M.D.+5$

D)
$\overline{X}+M.D.+5$

• question_answer27) A number x is chosen at random from the set {1, 2, 3, 4, ...., 100}. Define the event: A = the chosen number x satisfies$\frac{\left( x-10 \right)\left( x-50 \right)}{\left( x-30 \right)}\ge 0$ Then P (A) is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
0.71

B)
0.70

C)
0.51

D)
0.20

• question_answer28) Statement I : The equation${{(si{{n}^{-1}}x)}^{3}}+$${{(co{{s}^{-1}}x)}^{3}}+a{{\pi }^{3}}=0$has a solution for all$a\ge \frac{1}{32}.$ Statement II: For any $x\in R,$ ${{\sin }^{-1}}x{{\cos }^{-1}}x=\frac{\pi }{C}$and$0\le {{\left( {{\sin }^{-1}}x-\frac{\pi }{4} \right)}^{2}}\le \frac{9{{\pi }^{2}}}{16}$   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
Both statements I and II are true.

B)
Both statements I and II are false.

C)
Statement I is true and statement II is false.

D)
Statement I is false and statement II is true.

• question_answer29) If $f(\theta )=\left| \begin{matrix} 1 & \cos \theta & 1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \\ \end{matrix} \right|$and A and B are respectively the maximum and the minimum values of f(q), then (A, B) is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$(3, - 1)$

B)
$\left( 4,2-\sqrt{2} \right)$

C)
$\left( 2+\sqrt{2},2-\sqrt{2} \right)$

D)
$\left( 2+\sqrt{2},-1 \right)$

• question_answer30) Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement$p\Rightarrow \left( q\vee r \right)$ is:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

A)
$\left( p\vee q \right)\Rightarrow r$

B)
$\left( p\Rightarrow q \right)\vee \left( p\Rightarrow r \right)$

C)
$\left( p\Rightarrow \tilde{\ }q \right)\wedge \left( p\Rightarrow r \right)$

D)
$\left( p\Rightarrow q \right)\wedge \left( p\Rightarrow \tilde{\ }r \right)$