JEE Main 2013 Maths Question Paper Set 2 With Solution

Be Paitent While your exam is being loaded

View Solution

Participate to compete with other aspirants at an all India level.

Attempt Mock Test

Participate to compete with other aspirants at an all India level.

General Instructions

Choose Your Default Language

Please read the instructions carefully

General Instructions:

- Total duration of is 60 min.
The clock will be set at the server. The countdown timer in the top right corner of screen will display the remaining time available for you to complete the examination. When the timer reaches zero, the examination will end by itself. You will not be required to end or submit your examination.
The Questions Palette displayed on the right side of screen will show the status of each question using one of the following symbols:
1. You have not visited the question yet.
2. You have not answered the question.
3. You have answered the question.
4. You have NOT answered the question, but have marked the question for review.
5. The question(s) "Answered and Marked for Review" will be considered for evalution.
You can click on the ">" arrow which apperes to the left of question palette to collapse the question palette thereby maximizing the question window. To view the question palette again, you can click on "<" which appears on the right side of question window.
You can click on your "Profile" image on top right corner of your screen to change the language during the exam for entire question paper. On clicking of Profile image you will get a drop-down to change the question content to the desired language.
You can click on to navigate to the bottom and to navigate to top of the question are, without scrolling.

Navigating to a Question:

To answer a question, do the following:
1. Click on the question number in the Question Palette at the right of your screen to go to that numbered question directly. Note that using this option does NOT save your answer to the current question.
2. Click on Save & Next to save your answer for the current question and then go to the next question.
3. Click on Mark for Review & Next to save your answer for the current question, mark it for review, and then go to the next question.

Answering a Question:

Procedure for answering a multiple choice type question:
1. To select you answer, click on the button of one of the options.
2. To deselect your chosen answer, click on the button of the chosen option again or click on the Clear Response button
3. To change your chosen answer, click on the button of another option
4. To save your answer, you MUST click on the Save & Next button.
5. To mark the question for review, click on the Mark for Review & Next button.
To change your answer to a question that has already been answered, first select that question for answering and then follow the procedure for answering that type of question.

Navigating through sections:

Sections in this question paper are displayed on the top bar of the screen. Questions in a section can be viewed by click on the section name. The section you are currently viewing is highlighted.
After click the Save & Next button on the last question for a section, you will automatically be taken to the first question of the next section.
You can shuffle between sections and questions anything during the examination as per your convenience only during the time stipulated.
Candidate can view the corresponding section summery as part of the legend that appears in every section above the question palette.

Please note all questions will appear in your default language. This language can be changed for a particular question later on.

I have read and understood the instructions. All computer hardware allotted to me are in proper working condition. I declare that I am not in possession of / not wearing / not carrying any prohibited gadget like mobile phone, bluetooth devices etc. /any prohibited material with me into the Examination Hall.I agree that in case of not adhering to the instructions, I shall be liable to be debarred from this Test and/or to disciplinary action, which may include ban from future Tests / Examinations

Proceed

Back
Total Question		Total Attempted
Correct Answers		Incorrect Answers
Score

Category: IIT-JEE 2013 Math Paper 2

Question 1:

Let

w = \frac{\sqrt{3} + i}{2}

and

P = \{w^{n} : n = 1, 2, 3, . . . .\} .

Further

H_{1}

\{z \in ℂ : R e z > \frac{1}{2}\}

and

H_{2} = \{z \in ℂ : R e z < \frac{- 1}{2}\},

where

ℂ

is the set of all complex numbers. If

z_{1} \in p \cap H_{1}, z_{2} \in p \cap H_{2} a n d O

represents the origin, then

∠ z_{1} O_{z_{2}} =

A	$\frac{π}{2}$	B	$\frac{π}{6}$
C	$\frac{2 π}{3}$	D	$\frac{5 π}{6}$

Correct Answer : C and D

Category: IIT-JEE 2013 Math Paper 2

Question 2:

3^{x} = 4^{x - 1}, t h e n x =

A	$\frac{2 \log_{3} 2}{2 \log_{3} 2 - 1}$	B	$\frac{2}{2 - \log_{2} 3}$
C	$\frac{1}{1 - \log_{4} 3}$	D	$\frac{2 \log_{2} 3}{2 \log_{2} 3 - 1}$

Correct Answer : A,B and C

Category: IIT-JEE 2013 Math Paper 2

Question 3:

Let

ω

be a complex cube root of unity with

ω \neq 1 a n d P = [P_{i j}]

be a

n \times n

matrix with

P_{i j} = ω^{i + j} . T h e n P^{2} \neq 0, w h e n n =

A	57	B	55
C	58	D	56

Correct Answer : B, C and D

Category: IIT-JEE 2013 Math Paper 2

Question 4:

The function

f (x) = 2 |x| + |x + 2| - |x + 2| |- 2| |x|

has a local minimum or a local maximum at x=

A	-2	B	$\frac{- 2}{3}$
C	2	D	$\frac{2}{3}$

Correct Answer : B and C

Category: IIT-JEE 2013 Math Paper 2

Question 5:

For

α \in ℝ

(the set of all real numbers),

a \neq - 1,

\lim_{n \to \infty} \frac{(1^{a} + 2^{a} + . . . + n^{a})}{{(n + 1)}^{a - 1} [(n a + 1) + (n a + 2) + . . . + (n a + n)]} = \frac{1}{60}

Then a=

A	5	B	7
C	$\frac{- 15}{2}$	D	$\frac{- 17}{2}$

Correct Answer : B

Category: IIT-JEE 2013 Math Paper 2

Question 6:

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length

2 \sqrt{7}

on the y-axis is (are)

A	$x^{2} + y^{2} - 6 x + 8 y + 9 = 0$	B	$x^{2} + y^{2} - 6 x + 7 y + 9 = 0$
C	$x^{2} + y^{2} - 6 x - 8 y + 9 = 0$	D	$x^{2} + y^{2} - 6 x + 7 y + 9 = 0$

Correct Answer : A and C

Category: IIT-JEE 2013 Math Paper 2

Question 7:

Two lines

L_{1} : x = 5, \frac{y}{3 - α} = \frac{z}{- 2} a n d L_{2} : x = α, \frac{y}{- 1} = \frac{z}{2 - α} .

are coplanar, Then

α

can take value(s)

A	1	B	2
C	3	D	4

Correct Answer : A and D

Category: IIT-JEE 2013 Math Paper 2

Question 8:

In a triangle PQR, P is the largest angle and

\cos P = \frac{1}{3} .

Further, the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length (s) of the side(s) of the triangle Is (are)

A	16	B	18
C	24	D	22

Correct Answer : B and D

Category: IIT-JEE 2013 Math Paper 2

Question 9:

Let

S = S_{1} \cap S_{2} \cap S_{3}, w h e r e S_{1} = \{z \in ℂ : |z| < 4\}, S_{2} = \{z \in ℂ : I m [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0\} a n d S_{3} = \{z \in ℂ : R e z > 0\} .

min

|1 - 3 i - z| = z \in S

A	$\frac{2 - \sqrt{3}}{2}$	B	$\frac{2 + \sqrt{3}}{2}$
C	$\frac{3 - \sqrt{3}}{2}$	D	$\frac{3 + \sqrt{3}}{2}$

Correct Answer : C

Category: IIT-JEE 2013 Math Paper 2

Question 10:

A box

B_{1}

contains 1 white ball, 3 red balls and 2 black balls. Another box

B_{2}

contains 2 white balls, 3 red balls and 4 black balls. A third box

B_{3}

contains 3 white balls, 4 red balls and 5 black balls. If 1 ball is drawn from each of the boxes

B_{1}, B_{2} a n d B_{3}

, the probability that all 3 drawn balls are of the same colour is

A	$\frac{82}{648}$	B	$\frac{90}{648}$
C	$\frac{558}{648}$	D	$\frac{566}{648}$

Correct Answer : A

Category: IIT-JEE 2013 Math Paper 2

Question 11:

A box

B_{1}

contains 1 white ball, 3 red balls and 2 black balls. Another box

B_{2}

contains 2 white balls, 3 red balls and 4 black balls. A third box

B_{3}

contains 3 white balls, 4 red balls and 5 black balls. If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from the box

B_{2}

A	$\frac{116}{181}$	B	$\frac{126}{181}$
C	$\frac{65}{181}$	D	$\frac{55}{181}$

Correct Answer : D

Category: IIT-JEE 2013 Math Paper 2

Question 12:

Let

f [0, 1] \to ℝ

(the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies

f'' (x) - 2 f' (x) + f (x) \geq e^{x}, x \in [0, 1] .

Which of the following is true for

0 < x < 1 ?

A	$0 < f (x < \infty)$	B	$- \frac{1}{2} < f (x) < \frac{1}{2}$
C	$- \frac{1}{4} < f (x) < 1$	D	$- \infty < f (x) < 0$

Correct Answer : D

Category: IIT-JEE 2013 Math Paper 2

Question 13:

Let

f [0, 1] \to ℝ

(the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies

f'' (x) - 2 f' (x) + f (x) \geq e^{x}, x \in [0, 1] .

If the function

e^{- x} f (x)

assumes its minimum in the interval[0,1] at

x = \frac{1}{4},

which of the following is true?

A	$f' (x) < f (x), \frac{1}{4} < x < \frac{1}{4}$	B	$f' (x) > f (x), 0 < x < \frac{1}{4}$
C	$f' (x) < f (x), 0 < x < \frac{1}{4}$	D	$f' (x) < f (x), \frac{3}{4} < x < 1$

Correct Answer : C

Category: IIT-JEE 2013 Math Paper 2

Question 14:

Let PQ be a focal chord of the parabola

y^{2} = 4 a x .

The tangents to the parabola at P and Q meet at a point lying on the line

y = 2 x + a, a > 0 .

Length of chord PQ is

A	7a	B	5a
C	2a	D	3a

Correct Answer : B

Category: IIT-JEE 2013 Math Paper 2

Question 15:

Let PQ be a focal chord of the parabola

y^{2} = 4 a x .

The tangents to the parabola at P and Q meet at a point lying on the line

y = 2 x + a, a > 0 .

If chord PQ subtends an angle

θ

at the vertex of

y^{2} = 4 a x, t h e n \tan θ =

A	$\frac{2}{3} \sqrt{7}$	B	$\frac{- 2}{3} \sqrt{7}$
C	$\frac{2}{3} \sqrt{5}$	D	$\frac{- 2}{3} \sqrt{5}$

Correct Answer : D

Category: IIT-JEE 2013 Math Paper 2

Question 16:

A line L:y=mx+3 meets the y-axis at E(0,3) and the arc of the parabola

y^{2} = 16 x, 0 \leq y \leq 6

at the point

F (x_{0,} y_{0}) .

The tangent to the parabola at

F (x_{0,} y_{0})

intersects the y-axis at

G (0, y_{1})

. The slope m of the line L is chosen such the area of the triangle EFG has a local maximum. LISTI LISTII p. m= 1.

\frac{1}{2}

Q Maximum area of

∆ E F G

is/ 2 4 R

y_{0} =

3 2 S

y_{1} =

4 1

	P	Q	R	S
A	4	1	2	3
B	3	4	1	2
C	1	3	2	4
D	1	3	4	2

Correct Answer : A

Category: IIT-JEE 2013 Math Paper 2

Question 17:

Match List -I with List-II and select the correct answer using the code given below the lists :-

	List -I		List-II
P	${(\frac{1}{y^{2}} {(\frac{\cos (\tan^{- 1} y) + y \sin (\tan^{- 1} y)}{c o t (\sin^{- 1} y) + \tan (\sin^{- 1} y)})}^{2} + y^{4})}^{1 / 2}$	1.	$\frac{1}{2} \sqrt{\frac{5}{3}}$
Q	If cos x+ cos y + cos z=0= sin x + sin y + sin z then possible value of cos $\frac{x - y}{2} i s$	2.	$\sqrt{2}$
R	If cos $(\frac{π}{4} - x)$ cos 2x+ sin 2x sec x= cos x sin 2 x sec x+ $\cos (\frac{π}{4} + x)$ cos 2x then possible value of sec x is	2.	$\frac{1}{2}$
s.	If cot $(\sin^{- 1} \sqrt{1 - x^{2}}) = \sin (\tan^{- 1} (x \sqrt{6})), x \neq 0,$	4.	1

then possible value of x is 4. 1

	P	Q	R	S
A	4	3	1	2
B	4	3	2	1
C	3	4	2	1
D	3	4	1	2

Correct Answer : B

Category: IIT-JEE 2013 Math Paper 2

Question 18:

Consider the lines

L_{1} : \frac{x - 1}{2} = \frac{y}{- 1} = \frac{z + 3}{1}, \frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2}

and the planes

P_{1} : 7 x + y + 2 z = 3, P_{2}; 3 x + 5 y - 6 z = 4 .

Let ax+ by+cz=d be the equation of the plane passing through the point of intersection of lines

L_{1} a n d L_{2},

and perpendicular to planes

P_{1} a n d p_{2}

. LIST I LIST II P a= 1. 13 Q b= 2.-3 R c= 3. 1 S d= 4. -2

	P	Q	R	S
A	3	2	4	1
B	1	3	4	2
C	3	2	1	4
D	2	4	1	3

Correct Answer : A

Category: IIT-JEE 2013 Math Paper 2

Question 19:

Match List -I with List-II and select the correct answer using the code given below the lists :-

LIST I	LIST II
P. Volume of parallelepiped determined by vectors $\vec{a,} \vec{b} a n d \vec{c} i s 2 .$ Then the volume of the parallelepiped determined by vectors $2 (\vec{a} \times \vec{b}), 3 (\vec{b} \times \vec{c}) a n d (\vec{c} \times \vec{a}) i s$	1 100
Q. The volume of parallelepiped determined by vectors $\vec{a}, \vec{b} a n d \vec{c} i s 2$ Then the volume of the parallelepiped determined by vectors $3 (\vec{a} + \vec{b}), (\vec{b} + \vec{c}) a n d 2 (\vec{c} \times \vec{a}) i s$	2 30
R.Area of a triangle with adjacent sides determined by vectors $\vec{a} a n d \vec{b}$ is 20. Then the area of the triangle with adjacent sides determined by vectors $(2 \vec{a} + 3 \vec{b}) a n d (\vec{a} - \vec{b}) i s$	3 24
S.Area of a parallelogram with adjacent sides determined vectors $\vec{a} a n d \vec{b} i s 30 .$ Then the area of the parallelogram with addicts sides determined by $(\vec{a} + \vec{b}) a n d \vec{a} i s$	4 60

	P	Q	R	S
A	4	2	3	1
B	2	3	1	4
C	3	4	1	2
D	1	4	3	2

Correct Answer : C

Category: IIT-JEE 2013 Math Paper 2

Question 20:

Let

S = S_{1} \cap S_{2} \cap S_{3}, w h e r e S_{1} = \{z \in ℂ : |z| < 4\}, S_{2} = \{z \in ℂ : I m [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0\} a n d S_{3} = \{z \in ℂ : R e z > 0\} .

Area of S

A	$\frac{10 π}{3}$	B	$\frac{20 π}{3}$
C	$\frac{16 π}{3}$	D	$\frac{32 π}{3}$

Correct Answer : B

Not Visited	0	Not Answered
Answered	0	Marked for Review
Answered & Marked for Review (will be considered for evaluation)

JEE Main 2013 Maths Question Paper Set 2 With Solution

View Solution

Attempt Mock Test

General Instructions

Please read the instructions carefully

General Instructions:

Navigating to a Question:

Answering a Question:

Navigating through sections:

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Question 6:

Question 7:

Question 8:

Question 9:

Question 10:

Question 11:

Question 12:

Question 13:

Question 14:

Question 15:

Question 16:

Question 17:

Question 18:

Question 19:

Question 20:

Exam Summary

Are you sure you want to submit for final marking? No changes will be allowed after submission.

Thank you, Submitted Successfully.

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Question 6:

Question 7:

Question 8:

Question 9:

Question 10:

Question 11:

Question 12:

Question 13:

Question 14:

Question 15:

Question 16:

Question 17:

Question 18:

Question 19:

Question 20:

Are you sure you want to submit for final marking?
No changes will be allowed after submission.