# Solved papers for JEE Main & Advanced JEE Main Paper (Held On 9 April 2014)

### done JEE Main Paper (Held On 9 April 2014) Total Questions - 30

• question_answer1) Let P be the relation defined on the set of all real numbers such that $P=\{(a,b):se{{c}^{2}}a-ta{{n}^{2}}b=1\}.$Then P is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
reflexive and symmetric but not transitive.

B)
reflexive and transitive but not symmetric.

C)
symmetric and transitive but not reflexive.

D)
an equivalence relation.

• question_answer2) Let $w(Imw\ne 0)$be a complex number. Then the set of all complex number z satisfying the equation$w-\overline{w}z=k(1-z),$for some real number k, is   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$\left\{ z:\left| z \right|=1 \right\}$

B)
$\left\{ z:z=\overline{z} \right\}$

C)
$\left\{ z:z\ne 1 \right\}$

D)
$\left\{ z:\left| z \right|=1,z\ne 1 \right\}$

• question_answer3) If equations $a{{x}^{2}}+bx+c=0(a,b,c\in R,a\ne 0)$and $2{{x}^{2}}+3x+4=0$have a common root, then a : b : c equals:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
: 2 : 3

B)
2 : 3 : 4

C)
4 : 3 : 2

D)
3 : 2 : 1

• question_answer4) If$\frac{1}{\sqrt{\alpha }}$and$\frac{1}{\sqrt{\beta }}$ are the roots of the equation,$a{{x}^{2}}+bx+1=0$$(a\ne 0,a,b,\in R),$then the equation,$x\left( x+{{b}^{3}} \right)+\left( {{a}^{3}}-3abx \right)=0$has roots:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
${{\alpha }^{{}^{3}/{}_{2}}}$and${{\beta }^{{}^{3}/{}_{2}}}$

B)
$\alpha {{\beta }^{{}^{1}/{}_{2}}}$and${{\alpha }^{{}^{1}/{}_{2}}}\beta$

C)
$\sqrt{\alpha \beta }$and$\alpha \beta$

D)
${{\alpha }^{-\frac{3}{2}}}$and${{\beta }^{-\frac{3}{2}}}$

• question_answer5) If a, b, c are non-zero real numbers and if the system of equations  (a - 1) x = y + z (b - 1) y = z + x (c - 1) z = x + y
has a non-trivial solution, then ab + bc + ca equals:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
a + b + c

B)
abc

C)
1

D)
- 1

• question_answer6) If B is a $3\times 3$matrix such that ${{B}^{2}}=0,$then det. $[{{(I+B)}^{50}}-50B]$is equal to:     [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
1

B)
2

C)
3

D)
50

• question_answer7) The number of terms in the expansion of${{(1+x)}^{101}}{{(1+{{x}^{2}}-x)}^{100}}$in powers of x is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
302

B)
301

C)
202

D)
101

• question_answer8) The sum of the digits in the unit's place of all the 4-digitnumbers formed by using the numbers 3, 4, 5 and 6, without repetition, is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
432

B)
108

C)
36

D)
18

• question_answer9) Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its 4th term is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
8

B)
16

C)
20

D)
24

• question_answer10) If the sum$\frac{3}{{{1}^{2}}}+\frac{5}{{{1}^{2}}+{{2}^{2}}}+\frac{7}{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}+.......+$up to 20 terms is equalto$\frac{k}{21},$then k is equal to:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
120

B)
180

C)
240

D)
60

• question_answer11) If f(x) is continuous and$f\left( \frac{9}{2} \right)=\frac{2}{9},$then $\underset{x\to 0}{\mathop{\lim }}\,f\left( \frac{1-\cos 3x}{{{x}^{2}}} \right)$is equal to:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$\frac{9}{2}$

B)
$\frac{2}{9}$

C)
0

D)
$\frac{8}{9}$

• question_answer12) If$y={{e}^{nx}},$then$\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)\left( \frac{{{d}^{2}}x}{d{{y}^{2}}} \right)$is equal to:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$n{{e}^{nx}}$

B)
$n{{e}^{-nx}}$

C)
1

D)
$-n{{e}^{-nx}}$

• question_answer13) If the Rolle's theorem holds for the function$f(x)=2{{x}^{3}}+a{{x}^{2}}+bx$ in the interval [-1, 1] for the point$c=\frac{1}{2},$then the value of 2a + b is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$1$

B)
$-1$

C)
$2$

D)
$- 2$

• question_answer14) If $f(x)={{\left( \frac{3}{5} \right)}^{x}}+{{\left( \frac{4}{5} \right)}^{x}}=-1,x\in R,$then the equation f(x) =0 has :   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
no solution

B)
one solution

C)
two solutions

D)
more than two solutions

• question_answer15) $\int_{{}}^{{}}{\frac{{{\sin }^{8}}x-{{\cos }^{8}}x}{\left( 1-2\sin x{{\cos }^{2}}x \right)}}dx$is equal to:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$\frac{1}{2}\sin 2x+c$

B)
$-\frac{1}{2}\sin 2x+c$

C)
$-\frac{1}{2}\sin x+c$

D)
$-{{\sin }^{2}}x+c$

• question_answer16) The integral $\int\limits_{0}^{\frac{1}{2}}{\frac{\ln \left( 1+2x \right)}{1+4{{x}^{2}}}}dx,$equals:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$\frac{\pi }{4}\ln 2$

B)
$\frac{\pi }{8}\ln 2$

C)
$\frac{\pi }{16}\ln 2$

D)
$\frac{\pi }{32}\ln 2$

• question_answer17) Let $A=\{(x,y):{{y}^{2}}\le 4x,y-2x\ge -4\}.$The area (in square units) of the region A is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
8

B)
9

C)
10

D)
11

• question_answer18) If the differential equation representing the family of all circles touching x-axis at the origin is $\left( {{x}^{2}}-{{y}^{2}} \right)\frac{dy}{dx}=g\left( x \right)y,$then g(x) equals:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

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

B)
$2{{x}^{2}}$

C)
$2x$

D)
$\frac{1}{2}{{x}^{2}}$

• question_answer19) Let a and b be any two numbers satisfying$\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}=\frac{1}{4}.$ Then, the foot of perpendicular from the origin on the variable line,$\frac{x}{a}+\frac{y}{b}=1,$lies on:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
a hyperbola with each semi-axis $=\sqrt{2}$

B)
a hyperbola with each semi-axis = 2

C)
a circle of radius = 2

D)
a circle of radius $=\sqrt{2}$

• question_answer20) Given three points $P, Q, R$ with $P(5, 3)$ and R lies on the x-axis. If equation of RQ is $x - 2y = 2$ and PQ is parallel to the x-axis, then the centroid of $DPQR$ lies on the line:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$2x+y-9=0\text{ }~$

B)
$x-2y+1=0$

C)
$5x - 2y = 0$

D)
$2x - 5y = 0$

• question_answer21) If the point (1, 4) lies inside the circle ${{x}^{2}}+{{y}^{2}}-6x-10y+p=0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of p is the interval:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$(0, 25)$

B)
$(25, 39)$

C)
$(9, 25)$

D)
$(25, 29)$

• question_answer22) If OB is the semi-minor axis of an ellipse, ${{F}_{1}}$ and ${{F}_{2}}$ are its foci and the angle between ${{F}_{1}}B$and ${{F}_{2}}B$ is a right angle, then the square of the eccentricity of the ellipse is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

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

B)
$\frac{1}{\sqrt{2}}$

C)
$\frac{1}{2\sqrt{2}}$

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

• question_answer23) Equation of the plane which passes through the point of inter section of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$and$\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$and has the largest distance from the origin is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$7x\text{ }+\text{ }2y\text{ }+\text{ }4z\text{ }=\text{ }54~~$

B)
$3x\text{ }+\text{ }4y\text{ }+\text{ }5z\text{ }=\text{ }49$

C)
$4x\text{ }+\text{ }3y\text{ }+\text{ }5z\text{ }=\text{ }50$

D)
$5x\text{ }+\text{ }4y\text{ }+\text{ }3z\text{ }=\text{ }57$

• question_answer24) A line in the 3-dimensional space makes an angle$\theta \left( 0<\theta \le \frac{\pi }{2} \right)$with both the x and y axes. Then the set of all values of $\theta$ is the interval:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$\left( 0,\frac{\pi }{4} \right]$

B)
$\left[ \frac{\pi }{6},\frac{\pi }{3} \right]$

C)
$\left[ \frac{\pi }{4},\frac{\pi }{2} \right]$

D)
$\left( \frac{\pi }{3},\frac{\pi }{2} \right]$

• question_answer25) If$\left| \overset{\to }{\mathop{a}}\, \right|=2,\left| \overset{\to }{\mathop{b}}\, \right|=3$and$\left| 2\overset{\to }{\mathop{a}}\,-\overset{\to }{\mathop{b}}\, \right|=5,$then$\left| 2\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\, \right|$ equals:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
17

B)
7

C)
5

D)
1

• question_answer26) In a set of 2n distinct observations, each of the observations below the median of all the observations is increased by 5and each of the remaining observations is decreased by 3. Then the mean of the new set of observations:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
increases by 1

B)
decreases by 1

C)
decreases by 2

D)
increases by 2

• question_answer27) If A and B are two events such that $P\left( A\cup B \right)=P\left( A\cap B \right),$then the incorrect statement amongst the following statements is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
A and B are equally likely

B)
$P\left( A\cap B' \right)=0$

C)
$P\left( A'\cap B \right)=0$

D)
$P(A)+P(B)=1$

• question_answer28) The number of values of $\alpha$ in $[0,2\pi ]$for which $2{{\sin }^{3}}\alpha -7{{\sin }^{2}}\alpha +7\sin \alpha =2,$is:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
6

B)
4

C)
3

D)
1

• question_answer29) If$\cos ec\theta =\frac{p+q}{p-w}(p\ne q\ne 0),$then$\left| \cot \left( \frac{\pi }{4}+\frac{\theta }{2} \right) \right|$then cotto:   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
$\sqrt{\frac{p}{q}}$

B)
$\sqrt{\frac{q}{p}}$

C)
$\sqrt{pq}$

D)
$pq$

• question_answer30) The contrapositive of the statement "I go to school if it does not rain" is   [JEE Main Online Paper ( Held On 09 Apirl  2014  )

A)
If it rains, I do not go to school.

B)
If I do not go to school, it rains.

C)
If it rains, I go to school.

D)
If I go to school, it rains.