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L.K. Gupta (Mathematic Classes) wwwwww..ppiioonneeeerrmmaatthheemmaattiiccss..ccoomm MOBILE: 9815527721, 4617721

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1

PAPER -A

IIT-JEE (2012)

(Vector+3D+Probability)

“Solutions”

“TOWARDS IIT- JEE IS NOT A JOURNEY, IT’S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE” TIME: 60 MINS MAX. MARKS: 80

MARKING SCHEME

In Section I (Total Marks: 21), for each question you will be awarded 3 marks if you darken

ONLY the bubble corresponding to the correct answer and zero marks if no bubble is

darkened. In all other cases, minus one ( 1) mark will be awarded.

In Section II (Total Marks: 16), for each question you will be awarded 4 marks if you darken

ALL the bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise.

There are no negative marks in the section.

In Section III (Total Marks : 15), for each question you will be awarded 3 marks if you

darken ONLY the bubble corresponding to the correct answer and zero marks if no bubble is

darkened. In all other cases, minus one (1) mark will be awarded.

In Section IV (Total Marks: 28), for each question you will be awarded 4 marks if you darken

ONLY the bubble corresponding to the correct answer and zero marks otherwise, There are

no negative marks in this section

NAME OF THE CANDIDATE CONTACT NUMBER

L.K. Gupta (Mathematics Classes)



2

Section-I (Total Marks: 21)

(Single Correct Answer Type)

This section contains 7 multiple choice questions. Each question has 4 choices.(a), (b), (c) and

(d), out of which ONLY ONE is correct.

1. The line x 2 y 1 z 1

3 2 1

intersects the curve 2xy c ,z 0 if c is equal to.

(a) 1 (b) 1

3 (c) 5 (d) none of these

Sol. (c) we have z =0 for the point where the line intersects the curve

2 1 0 1

3 2 1

x y

2 11 1

3 2

x yand

5 1x and y

Put these values in 2xy c

we get 25 c

5c

2. The line joining the points (1, 1, 2) and (3, –2, 1) meets the plane 3x 2y z 6 at

the point. (a) (1, 1, 2) (b) (3, –2, 1) (c) (2, –3, 1) (d) (3 , 2, 1) Sol. (b)

The straight line joining the points 1,1,2 3, 2,1and is

1 1 2

2 3 1

x y zr

(say)

Point is 2 1,1 3 ,2r r r

which lies on 3 2 6x y z

3 2 1 2 1 3 2 6r r r

1r

Required point is 3, 2,1 .

3. If a 2

and b 3

and a.b

= 0 then (a (a (a (a b)))

is equal to

(a) 48b

(b) 48b

(c) 48a

(d) 48a



3

Sol. (a)

. 0a b

a and b

are mutually perpendicular

Also | | 2 | |a and b

=3

Let ˆ ˆ2 2a i then b i

such that ˆ ˆ,a i b j

ˆ ˆ ˆ ˆ ˆ2 2 2 2 3a a a a b i i i i j

ˆˆ48 48j b

4. The value of the x for which the angle between the vectors 22x i 4xj k and

7i 2j xk are obtuse and the angle between the z –axis and 7i 2j xk is acute and

Less than π

6is given by

(a) 1

0 x2

(b) 1

x or x 02

(c) 1

x 152 (d) there is no such value for x

Sol. (d)

Let 2 ˆ ˆˆ ˆ ˆ ˆ2 4 7 2a x i xj k and b i j xk

the angle between a and b

is obtuse.

. 0a b

214 8 0x x x

2 1 0x x

1

0, ..................2

x i

Also it is given .b k x

and . 3

cos6 2| |

b k

b

22 3 53x x

2 159....................x ii

there is one common value for Eqs. (i) and (ii)

5. Let v 2i j k and w i 3k If u

is a unit vector, then maximum value of the

Scalar triple product [uvw]

is.

(a) –1 (b) 10 6 (c) 59 (d) 60 Sol. (c)



5

22 2 0

4

Lx Lx

2

2 2 04

Lx L L

2 3

02

Lx L

3 30

2 2

L Lx L x L

2 3 2 30

2 2x L x L

2 3 2 30, ,2

2 2x L L L

Now, x takes value in the interval [0, 2L] Required Probability

2 3

2 2

0 2 3

2

2

0

LL

L

L

dx dx

dx

2 3 2 32

2 2

2 0

L L L

L

2 3

2

L L

L

2 3

2



6

Section – II (Total Marks: 16) (Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c) and

(d) out of which ONE or MORE may be correct.

8. The lines x 2 y 3 z 4 x 1 y 4 z 5

and1 1 k k 2 1

are coplanar if

(a) k = 0 (b) k = 1 (c) k = 3 (d) k = 3 Sol. (a), (c) The given lines are coplanar, if

2 1 3 4 4 5

0 1 1 k

k 2 1

1 1 1

1 1 k

k 2 1

2 2 1 3 3 1Applying C C C and C C C

1 .......0 .......0

1 2 1 k

k k 2 1 k

or 2 1 k k 2 1 k 0 2or k 3k 0

k 0, 3

9. The probabilities that a student in Mathematics, Physics and Chemistry are , and

respectively. Of these subjects, a student has a 75% chance of passing in atleast one, a 50% chance of passing in atleast two and a 40% chance of passing n exactly two subjects. Which of the following relations are true ? (a) 19/20 (b) 27 /20

(c) 1 /10 (d) 1 /14

Sol. (b,c)

Here,P M , P P and P C

The probability of passing in atleast one subject = 0.75 (given)

1 P MPC 0.75

1 P M P P P C 0.75



7

1 1 1 1 0.75

3or

4 ……………(i)

The probability of passing in atleast two subjects = 0.50 (given)

or P M P C P M P C P M P C P M P C 0.50

P M P P P C P M P P P C P M P P P C P M P P P C 0.50

1

1 1 12

12

2 ……………..(ii)

and the probability of passing in exactly two subjects = 0.40 (given)

2P M PC P M P C P M P C

5

2

P M P P P C P M P P P C P M P P P C5

2

1 1 15

23

5 ………………(iii)

Eqs. (ii) and (iii) 1 2

2 32 5

1 2 1(c)

2 5 10

1 1From Eq. (ii),

5 2 ……………(iv)

7

10

Substituting the values of and in Eq. (i), then

∴ 7 1 3

10 10 4

6 3 12 15 27(b)

10 4 20 20

10. The letters of the word PROBABILITY are written down at random in a row, Let E1 denotes the event that two I’s are together and E2 denotes the event that two B’s are together, then

(a) 1 2

2P E P E

11 (b) 1 2

2P E E

55



8

(c) 1 2

18P E E

55 (d) 1 2

1P E / E

5

Sol. (a, b, c, d)

Option (a) : 1 2

10 !

22 !P E P E

11 ! 112 !2 !

Option (b) : 1 2P E E = probability of two I’s are together and two B’s are together

9 !

11 !

2 ! 2 !

4 2

110 55

Option (c) : 1 2 1 2 1 2P E E P E P E P E E

2 2 2

11 11 55

4 2

11 55

18

55

Option (d) :

1 2

1 2

2

2P E E 155P E / E

2P E 511

11. The vector c

directed along the bisectors of the angle between the vectors a 7i 4 j 4k and b 2i j 2 k if c 3 6

is given by

(a) i 7 j 2 k (b) i 7 j 2 k

(c) i 7 j 2 k (d) i 7 j 2 k

Sol. (a, c) 7i 4 j 4k 2i j 2k

c9 3

c i 7j 2k9

…………(i)

2 2

c 1 49 4 5481 81

2 5454 9

81

Putting 9 in Eq. (i)

Then, c i 7 j 2k



9

Section – III (Total Marks: 15)

(Paragraph Type)

This section contains two paragraphs. Based upon one of the paragraph, 2 multiple choice

questions and based on the other paragraph 3 multiple choice questions have to be answered.

Each of these questions has four choices (a), (b), (c) and (d), out of which ONLY ONE is correct.

Passage for questions Nos. 12 and 13

Let a,b,c

be three vector such that a b c 4

and angle between aandb

is π /3, angle

Between b and c

is π /3 and angle between c and a

is π /3 .

On the basis of above information, answer the following questions:

12. The height of the parallelepiped whose adjacent edges are represented by vectors

a,bandc

is

(a) 2

43

(b) 2

33

(c) 3

42

(d) 3

32

Sol. (a)

Volume of the parallelepiped = (base area ) (height )

132 2 2 4 4 sin

2 3h

8 3 h

24

3h

13. The volume of the tetrahedron whose adjacent edges are represented by the vectors

a,bandc

is

(a) 4 3

2 (b)

8 2

3 (c)

16

3 (d)

16 2

3

Sol.(d)

Volume of the tetrahedron1

6abc



10

132 2

6

16 2

3

Paragraph for question 14 to 16

Let two planes 1 2P :2x y z 2 and P : x 2y z 3 are given.

On the basis of above information, answer the following questions 14. The equation of the lane through the intersection of 1 2P andP and the point (3,2 ,1) is

(a) 3x y 2z 9 0 (b) x 3y 2z 1 0

(c) 2x 3y z 1 0 (d) 4x 3y 2z 8 0

Sol. (b) The equation of any plane through the intersection of 1P and 2P is

1 2 0P P

2 2 2 3 0.........x y z x y z i

Since it passes through (3, 2,1) then 6 2 1 2 3 4 1 3 0

1

From Eq. (1) 3 2 1 0x y z

which is the required plane . 15. Equation of the plane which passes through the point (–1, 3, 2) and is perpendicular to each of the planes 1 2P andP is

(a) x 3y 5z 2 0 (b) x 3y 5z 18 0

(c) x 3y 5z 20 0 (d) x 3y + 5z = 0

Sol. (c)

The equation of any plane through 1,3,2 is

1 3 2 0a x b y c z …………………………(ii)

if this plane (ii) is perpendicular to 1P then 2 0......................a b c ii

and if the plane (ii) is perpendicular to 2P then 2 0.....................a b c iii

From Eqs. (ii) and (iii) we get

1 3 5

a b c

Substituting these Proportionate values of a, b, c in Eq. (ii) we get the required

equation as 1 3 3 5 2 0x y z

or 3 5 20 0x y z



11

16. The equation of the acute angle bisector of planes 1 2P andP is

(a) x 3y 2z 1 0 (b) 3x y 5 0

(c) x 3y 2z 1 0 (d) 3x z 7 0

Sol. (a) The given plane can be written as 2 2 0x y z and 2 3 0x y z

Here 2 1 1 2 1 1 1 0 Equation of bisectors

2 2 2 3

1 4 14 1 1

x y z x y z

Acute angle bisector is 2 2 2 3x y z x y z

3 2 1 0x y z

Section IV (Total Marks: 28) (Integer Answer Type)

This section contains 7 questions. The answer to each of the questions is a single-digit integer,

ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the

Answer sheet.

17. ABCD is a parallelogram. A1 and B1 are the mid points of sides BC and CD respectively. If

1 1AA AB AC

, then the value of 2626 3935 must be

Ans. 4 Sol.

Let position vectors of A, B, D be O, b and d

respectively.

AC AB BC

AB AD

b d

1

AB ACAA

2



12

db

2

1

AD AC band AB d

2 2

1 1

3 3AA AB b d AC

2 2

3

2

3Then, 2626 2626

2 3939

18. A, B, C, D are any four points in the space. If

AB CD BC AD CA BD (area of ABC)

then the value of must be

Ans. 4 Sol.

Let PV of A, B, C and D be a , b , c and O

AB CD b a c b c a c

BC AD c b a c a b a

and CA BD a c b a b c b

Adding all we get

AB CD BC AD CA BD

2 a b b c c a

AB CD BC AD CA BD

2 a b b c c a

2 c a b a

2 AC AB

14. AC AB

2

4 (Area of ABC)

4

Then, 125 500



13

19. Let a and b

be unit vectors such that a b 3

, then the value of

200 2a 5b

. 3a b a b

3894 must be

Ans. 6 Sol.

2

a b 3 a b 3

2 2a b 2a. b 3

1a . b

2

200 2a 5b . 3a b a b

2 2200 6a 2a.b 0 15 a. b 5b 0

15200 6 1 5

2

39200 3900

2

20. If angle between i and line of intersection of the planes

r . i 2 j 3k 0 and r . 3i 3j k 0 is

1 21cos 1090 then the value of must be

Ans. 8 Sol.

Line of intersection of r. i 2j 3k 0 and r. 3i 3j k 0

will be parallel to

3i 3j k i 2j 3k , ie, 7 i 8j 3k , then required angle is

1 1 . 7cos

1 49 64 9

1 7cos

122

1 21or cos

1098

1098



14

21. If the distance of the point B i 2j 3k from the line which is passing through

A 4i 2j 2k and which is parallel to the vector c 2i 3j 6k is , then the value of

6 4 2 1 1110 must be

Ans. 1 Sol.

AB OB OA

i 2j 3k 4i 2j 2k

3i 0j k

AB 9 0 1 10

Now, AM = Projection of AB along c

AB . c

c

3i 0j k . 2i 3j 6k

4 9 36

6 0 60 BA AM

7

Perpendicular distance of B from AM = AB 10

(given) 6 4 2 1 1000 100 10 1

= 1111 22. If the area of the triangle whose vertices are A (1, 2, 3), B (2, 1, 1) and C(1, 2, 4) is

sq unit then 20 10 700

Ans. 0 Sol.



15

The coordinates of the projections of A, B, C on the yz-plane are (0, 2, 3), (0, 1, 1) and (0, 2, 4) respectively

x = area of projection of ABC on yz-plane

2 3 11

1 1 12

2 4 1

1 1 2 3 3 2Applying R R 2R and R R 2R

x

0 5 3

1 21then 1 1 1

2 2

0 2 3

sq unit

Similarly, the projection of A, B and C on zx and xy-planes are (1, 0, 3), (2, 0, 1), (1, 0, 4) and (1, 2, 0), (2, 1, 0), (1, 2, 0) respectively. Also let y zand be the areas of the projection of

the ABC on zx and xy-planes respectively.

Then,

y

1 3 11

2 1 12

1 4 1

2 2 1 3 3 1Applying R R R and R R R

Then, y

1 3 11

1 2 02

0 7 0

7

2 sq unit

and z

1 2 11

2 1 12

1 2 1

1 1 3Applying C C C

z

0 2 1

:1

Then, 1 1 1 02

:

0 2 1



16

2 2 2x y zThe required area

2 2

221 70

2 2

710 sq unit

2

710 sq unit

2

720 10 20 10 10 700

2

23. If the distance of the point P (4, 3, 5) from the axis of y is unit, then the value of

25 204 must be

Ans. 1 Sol.

The equations of y-axis are x y z

0 1 0 ,

Any point N on y-axis is (0, r, 0) ……….(i) The direction cosines of the line PN are 0 4, r 3, 0 5 ie, 4, r 3, 5 ……..(ii) Let N be the foot of the perpendicular from P to y-axis, then PN is to the y-axis whose direction cosines are 0, 1, 0 and so from Eq. (ii), we have 0. ( 4) + 1 . (r 3) + 0. ( 5) = 0 r = 3 From Eq. (i) the coordinates of N are (0, 3, 0) Required distance = PN

2 2 2

4 0 3 3 5 0

41 unit

41 25 5 41 205

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