summer 17 examination model answer 17301 · 2017-07-14 · relevant answer based on candidate’s...

MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION (Autonomous)

(ISO/IEC - 27001 - 2005 Certified)

______________________________________________________________________________________________

SUMMER – 17 EXAMINATION Model Answer Subject Code:

Important Instructions to examiners: 1) The answers should be examined by key words and not as word-to-word as given in

the model answer scheme. 2) The model answer and the answer written by candidate may vary but the examiner

may try to assess the understanding level of the candidate. 3) The language errors such as grammatical, spelling errors should not be given more

Importance (Not applicable for subject English and Communication Skills. 4) While assessing figures, examiner may give credit for principal components indicated

in the figure. The figures drawn by candidate and model answer may vary. The examiner may give credit for any equivalent figure drawn.

5) Credits may be given step wise for numerical problems. In some cases, the assumed constant values may vary and there may be some difference in the candidate’s answers and model answer.

6) In case of some questions credit may be given by judgement on part of examiner of relevant answer based on candidate’s understanding.

7) For programming language papers, credit may be given to any other program based on equivalent concept.

Q. No.

Sub Q. N.

Answer Marking Scheme

1

a)

Ans

b)

Ans

Attempt any TEN of the following:

Find the point on the curve 2

6 8y x x where the tangent is parallel to X-axis

26 8

2 6

y x x

d yx

d x

2

ta n g e n t is p a ra lle l to X -a x is

0

2 6 0

3

3 6 3 8 1

P o in t is 3 , 1

d y

d x

x

x

y

--------------------------------------------------------------------------------------------------------------

Find the radius of curvature of the curve xy c at point ,c c

0

x y c

d yx y

d x

20

02

½

½

½

½

02

Page No.01/29

17301



______________________________________________________________________________________________

SUMMER – 17 EXAMINATION

Model Answer Subject Code:

Q. No.

Sub Q. N.


1

b)

c)

Ans

d)

Ans

2

2 2

2

2 2

32 2

32 2

2

2

,

1

1 2

11 1

R a d iu s o f c u rv a tu re 2

2 2 2 2

d y y

d x x

d yx y

d y d x

d x x

a t c c

d y c

d x c

c cd y

d x c c

d y

d x

d y

cd x

cc

---------------------------------------------------------------------------------------------------------

1 2

1 2

1

2

1

1E v a lu a te

s in 1

1

s in 1

p u t s in

1

1

1

lo g

lo g s in

d x

x x

d x

x x

x t

d x d t

x

d tt

t c

x c

----------------------------------------------------------------------------------------------------

34

34

1E v a lu a te

2 3

1

2 3

d x

x

d x

x

½

½

½

½

02

½

½

½

½

02

Page No.02/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


1

d)

e)

Ans

f)

Ans

3 1

4

3

4

3

4

1

4

1

4

1

2 3

2 3

2 3 2 31 1. .

3 13 31

4 4

42 3

3

d x

x

x d x

x xc

x c

---------------------------------------------------------------------------------------------------

1

1

1

1

1

2

1

2

1 2

E v a lu a te ta n

ta n .1

ta nta n 1 1 .

ta n .1

1 2ta n

2 1

1ta n lo g 1

2

x d x

x d x

d xx d x d x d x

d x

xx x d x

x

xx x d x

x

x x x c

------------------------------------------------------------------------------------------------------------

2

3

0

2

3

0

2

2

0

2

2

0

E v a lu a te s in

s in

s in . s in

1 c o s s in

P u t c o s w h e n 0 , 1

s in w h e n , 02

xd x

xd x

x xd x

x xd x

x t x t

x d x d t x t

½

1

½

02

½

½

½

½

02

½

½

Page No.03/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


1

f)

g)

Ans

0

2

1

0

2

1

03

1

1

1

3

10 1

3

2

3

t d t

t d t

tt

OR

2

3

0

2

0

2

0

s in

3 s in s in 3

4

1 c o s 33 c o s

4 3

1 10 0 3

4 3

1 8.

4 3

2

3

d x

x xd x

xx

------------------------------------------------------------------------------------------------------------

Find the area of the region bounded by 21 6 , 1, 4x y y y and Y-axis in first

quadrant

4

1

4

3

2

1

4

43

2

b

a

A xd y

y d y

y

½

½

½

½

½

½

02

½

½

Page No.04/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


1

g)

h)

Ans

i)

Ans

j)

Ans

3 3

2 28

4 13

87

3

5 6

3

---------------------------------------------------------------------------------------------------------------------

Determine the order and degree of 2

2

2

d y d yx x m y

d x d x

Order=2

Degree=1

-----------------------------------------------------------------------------------------------------------------------

2

2

2

2

2

.W h e re is a rb i t ra ry c o n s ta n t

8

F o rm th e d if fe re n t ia l e q u a t io n

14

8

14

i

6

f

4

4

6

4

1

x A A

y x A

d yx A

d x

d yy

d x

d yy

d x

d yy

d x

y

-----------------------------------------------------------------------------------------------------------------------

A fair die is rolled. What is the probability that the number on the die is a prime number

1, 2 , 3 , 4 , 5 , 6

n 6

2 , 3 , 5

n 3

3

6

1 o r 0 .5

2

S

s

A

A

n Ap A

n s

-----------------------------------------------------------------------------------------------------------------------

½

½

02

1

1

02

½

½

½

½

02

½

½ [

1

Page No.05/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


1

k)

Ans

l)

Ans

m)

Ans.

From 4 men and 2 women, 3 persons are chosen at random to form a committee. Find the probability that the committee consists of at least one person of either sex.

6

3

4 2 4 2

1 2 2 1

2 0

1 6

1 6 4 0 .8

2 0 5

n S c

n A c c c c

p A o r

---------------------------------------------------------------------------------------------------------------------

A person fires 10 shorts at target. The probability that any shot will hit the target is

3

5.Find the probability that the target is hit exactly 5 times.

5 5

1 0

5

3 3 2G iv e n : 1 0 , , 1 1

5 5 5

3 2 5

5 5

0 .2 0 0 7

n r n r

r

n p q p

p r c p q

p c

-----------------------------------------------------------------------------------------------------------------------

2

22

2 22

2

1 1 1

1E v a lu a te

9 4

1 1 O R

99 42

4

1 1 1 =

23 2 3

2

2 1 1 1 2s in . = s in s in

3 2 2 3 / 2 2 3

d x

x

d x d x

xx

d x d x

xx

x x xc c

c

-----------------------------------------------------------------------------------------------------------------------

02

½

1

½

02

1

1

02

1

1

Page No.06/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


1

2

n)

Ans

a)

1

1

1

0

12

0

1E v a lu a te . lo g

1 . lo g

P u t lo g w h e n 1 lo g 1 0

1 w h e n lo g 1

2

1

2

e

e

xd xx

xd xx

x t x t

d x d t x e t ex

td t

t

-----------------------------------------------------------------------------------------------------------Attempt any FOUR of the following:

3

2

2

2

2

2

2

2

E v a lu a te ta n

C o n s id e r in g in d e x a s 3

ta n

ta n ta n

s e c 1 ta n

ta n s e c ta n

ta n s e c ta n

In f irs t in te g ra l , p u t ta n

s e c

lo g s e c

lo g s e c2

ta nl

2

x d x

x d x

x x d x

x x d x

x x x d x

x x d x x d x

x t

x d x d t

td t x c

tx c

x

o g s e c x c

Note: If student attempted to solve the problem assuming any

index value then consider it and reward appropriate marks to it.

----------------------------------------------------------------------------------------------

02

½

½

½

½

16

04

½

1

1

1

½

Page No.07/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


2

b)

Ans

c)

Ans

lo gE v a lu a te

2 lo g 3 lo g

lo g

2 lo g 3 lo g

P u t lo g

1

2 3

c o n s id e r 2 3 2 3

3 2

P u t 2

2

P u t 3

3

2 3

2 3 2 3

2 3

2 3 2 3

2 lo g 2 3 lo g 3

xd x

x x x

xd x

x x x

x t

d x d tx

td t

t t

t A B

t t t t

t A t B t

t

A

t

B

t

t t t t

td t d t

t t t t

t t c

2 lo g 2 lo g 3 lo g 3 lo gx x c

------------------------------------------------------------------------------------------------------------

Find the equations of the tangent and normal to the ellipse 2 22 3 5x y which is

perpendicular to the line 3 2 7 0x y

1

2 2

S lo p e o f lin e 3 2 7 0 is

3

2

2 3 5

4 6 0

4 2

6 3

x y

m

x y

d yx y

d x

d y x x

d x y y

04

½

½

½

½

½

½

½

½

04

Page No.08/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


2

c)

d)

Ans

2

1 2

22

2 2

2

2s lo p e o f ta n g e n t

3

L in e a n d ta n g e n t a re p e rp e n d ic u la r

. 1

3 2. 1

2 3

2 3 5

2 3 5

1

1

if 1 1 p o in t is 1, 1

if 1 1 p o in t is 1,1

E q u a t io n

xm

y

m m

x

y

x y y x

x x

x x

x

x

x y

x y

o f ta n g e n t a t (1, 1) is

21 1

3

2 3 5 0

E q u a tio n o f ta n g e n t a t ( 1,1) is

21 1

3

2 3 5 0

E q u a tio n o f n o rm a l a t (1, 1)

31 1

2

3 2 1 0

E q u a tio n o f n o rm a l a t ( 1,1) is

31 1

2

3 2 1

y x

x y

y x

x y

y x

x y

y x

x y

0

-----------------------------------------------------------------------------------------------------------------------

Find the radius of curvature for the curve 3 3co s , s in

4x a y a a t

3 3

2 2

c o s s i n

3 c o s s in 3 s in c o s

x a y a

d x d ya a

d d

½

½

½

½

½

½

½

½

04

½+½

Page No.09/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


2

d)

2

2

2

2

2

2

2

2

22

2

2

2

2

3 s in c o s

3 c o s s in

ta n

s e c

1s e c

1s e c

3 c o s s in

a t 4

ta n 14

1s e c

43 c o s s in

4 4

12 .

1 13

2 2

d y

d y ad

d xd x a

d

d y

d x

d y d

d x d x

d x

d

a

d y

d x

d y

d xa

a

32 2

2

2

32 2

2

13

2 2

4 2

3

1

R a d iu s o f c u rv a tu re

1 1

4 2

3

3 o r 1 .5

2

a

a

d y

d x

d y

d x

a

aa

-----------------------------------------------------------------------------------------------------------------------

½

½

½

½

½

½

Page No.10/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


2

e)

Ans

f)

Ans

A bullet is fired into a mud tank and penetrates 2

1 2 0 3 6 0 0t t meters in ‘t’

seconds after impact. Calculate maximum depth of penetration.

2

2

2

2

1 2 0 3 6 0 0

1 2 0 7 2 0 0

7 2 0 0 0 D e p th is m a x im u m .

L e t 0

1 2 0 7 2 0 0 0

1

6 0

1 1M a x im u m d e p th y 1 2 0 3 6 0 0

6 0 6 0

1 m e te r

y t t

d yt

d t

d y

d t

d y

d t

t

t

-----------------------------------------------------------------------------------------

2

2

2

2

2

5 4E v a lu a te

8 1 2

5 4

8 1 2

5 4 5 4C o n s id e r

8 1 2 6 2

5 4

6 2 6 2

5 4 2 6

P u t 6

2 6 1 3

4 2

P u t 2

6 3

4 2

1 3 3

5 4 2 2

8 1 2 6 2

1 3 3

5 4 2 2

8 1 2 6 2

xd x

x x

xd x

x x

x x

x x x x

x A B

x x x x

x A x B x

x

A

x

B

x

x x x x

xd x

x x x x

d x

04

½

½

½

1

½

1

04

½

½

½

½

½

½

Page No.11/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


2

3

f)

a)

Ans

b)

Ans

1 3 3

lo g 6 lo g 22 2

I x x c

-----------------------------------------------------------------------------------------------------------------------

Attempt any FOUR of the following:

12

2

32

12

2

32

12

232

2

2

12

232

12

2

3 22

12

1

32

1

1E v a lu a te

4 1 2 1 3

1

4 1 2 1 3

1 1

1 343

4

3 9T h ird te rm

44 .

1 1

9 9 1 343

4 4 4

1 1

4 31

2

1 3ta n

4 2

1 1 3 1ta n ta

4 2 2 4

d xx x

d xx x

d x

x x

x

x

d x

x x

d x

x

x

1

1

3 3n

2 2

1ta n 2

4

---------------------------------------------------------------------------------------------------------------------

3

3

6

3

3

6

1E v a lu a te

1 c o t

1

1 c o t

d x

x

d x

x

1

16

04

½

½

½

½

1

½

½

04

Page No.12/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


3

b)

3

3

6

3

36

3

3

63

3 3

3 3

6

33

3 36

3

1

1 c o t

1

c o s1

s in

1

c o s1

s in

s in -------------------- 1

s in c o s

s in3 6

s in c o s3 6 3 6

s in2

s in2

I d x

x

d x

x

x

d x

x

x

xI d x

x x

x

I d x

x x

x

I

x

3

3 36

3 3

3 3

6

3 33 3

3 3 3 3

6 6

3 3 3

3 3

6

3

6

3

6

c o s2

c o s -------------------- 2

c o s s in

a d d (1 ) a n d (2 )

s in c o s +

s in c o s c o s s in

s in c o s2

s in c o s

2 1

2

23

d x

x

xI d x

x x

x xI I d x

x x x x

x xI d x

x x

I d x

I x

I

6

1 2I

½

1

½

½

½

½

½

Page No.13/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


3 c)

[

Ans

d)

Ans

Find the area of the region enclosed between parabola 21y x and the line

2 1y x

2

2

2

1 2

2

2

0

2

2

0

2

2

0

22 3

0

3

2

1 a n d 2 1

1 2 1

2 0

2 0

0 & 2

A re a

2 1 1

2 1 1

2

2

2 3

22 0

3

4

3

b

a

y x y x

x x

x x

x x

x x

y y d x

x x d x

x x d x

x x d x

x x

-----------------------------------------------------------------------------------------------------------------------

2 2 2

2 2 2

2 2

2

22

2

2 2 2 2

2

S o lv e .

.

P u t

d yx x x y y

d x

d yx x x y y

d x

d y x x y y

d x x

y v x

d y d vv x

d x d x

x x v x v xd vv x

d x x

d v x v x v xv x

d x x

04

1

½

1

1

½

04

½

½

Page No.14/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


3 d)

e)

Ans

f)

Ans

2 2

2

2

2

2

2

2

1

1

1

1

1

1

1 1

1

1 1

1

ta n lo g

ta n lo g

x v vd vv x

d x x

d vv x v v

d x

d vx v

d x

x d v v d x

d v d xv x

d v d xv x

v x c

yx c

x

-----------------------------------------------------------------------------------------------------------------------

2

2

2

2

2

S o lv e c o s 2 1 2

c o s 2 1 2

P u t 2

1 2

c o s

1

c o s

s o lu t io n is

s e c

ta n

ta n 2

d yx y

d x

d yx y

d x

x y v

d y d v

d x d x

d vv

d x

d x d vv

d x v d v

x v c

x x y c

----------------------------------------------------------------------------------------------------------------------

1

1

2 ta n

ta n

2 2

S o lv e 1

1

1 1

C o m p a rin g w ith

x

x

d yx y e

d x

d y ey

d x x x

d yP y Q

d x

1

½

1

½

04

1

1

½

1

½

04

Page No.15/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


3

4

f)

a)

Ans

1

-12

1

1 1

1

1

ta n

2 2

1

ta n1

ta n

ta n ta n

2

2ta n

ta n

2

1

1,

1 1

In te g ra t in g fa c to r

. .

1

1

P u t ta n O R P u t

x

d xP d xxx

x

x x

x

x

eP Q

x x

e e e

y IF Q IF d x

ey e e d x

x

e

y e d xx

x t e

-1

1

1 1

1

ta n

ta n

2 2

2ta n 2 ta n

2

ta n ta

1 1 .

1 1

2

x

x

x t t x

t

x

t

d x d t e d x d tx x

y e e d t e d t y e td t

ey e c y e

-1

-1

1

1 1

2

n

2ta n

2 ta n

ta n ta n

2

2 2

x

xx

x x

tc

ee

y e c y e c

-----------------------------------------------------------------------------------------------------------------------Attempt any FOUR of the following:

27

22

3

27

22

3

27

22

3

27

2 2

3

27 2

2 22

3

1 0E v a lu a te

1 0

1 0L e t --------------------(1 )

1 0

1 0 7 3

7 3 1 0 7 3

----------------------(2 )

1 0

A d d in g (1 ) a n d (2 )

1 0

1 0 1 0

xd x

x x

xI d x

x x

xI d x

x x

xI d x

x x

x xI I

x x x

7

2

3

d x

x

½

1

1

½

½

½

16

04

1

1

Page No.16/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


4

a)

b)

Ans

2 27

22

3

7

3

7

3

1 02

1 0

2 1

2

2 7 3

4

2

2

x xI d x

x x

I d x

I x

I

I

I

----------------------------------------------------------------------------------------------------------------------

1

1

0

1

1

0

11 1 1

1

0 0 0

12 2

1

2

0

12 1 2

2

0

12 1 2

2 2

0

2 1

2

E v a lu a te . s in

. s in

s ins in . .

1s in . .

2 2 1

. s in 1 1 1

2 2 1

. s in 1 1 1

2 2 1 1

. s in 1 11

2 2

x xd x

x xd x

d xx x d x x d x d x

d x

x xx d x

x

x x xd x

x

x x xd x

x x

x xx

1

2

0

12 1 2

2 1 1

0

12 1

2 1

0

2 1

1

1

. s in 1 11 s in s in

2 2 2 2

. s in 11 s in

2 4 4

1 . s in 1 10 s in 1 0

2 4

12.

2 4 2

d x

x

x x xx x x

x x xx x

½

½

½

½ [

04

½

½

½

1

½

Page No.17/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


4 b)

c)

Ans

d)

Ans

4 8

8

-----------------------------------------------------------------------------------------------------------------------

Find the area of the region in the first quadrant enclosed by the X-axis the line

y x and the circle 2 28x y

2 2

2 2

2

2

, 8

8

2 8

4

y x x y

x x

x

x

2 1

22

2

0

22

22

2 1

0

22

22 1

2 In f i rs t q u d ra n t

0 to 2

8

8

8 s in2 2 28

8 22 28 2 s in 0

2 2 28

2 4 24

b

a

x

x x

A y y d x

A x x d x

x x xA x

A

A

A

------------------------------------------------------------------------------------------------------------

3 2 2 2

3 2 2 2

2 2 2 2

S o lv e . s e c 3 ta n s e c 0

C o m p a rin g w ith

. s e c N = 3 ta n s e c

3 s e c 3 s e c

y x d x y x y d y

M d x N d y c

M y x y x y

M Ny x y x

y x

M N

y x

1

04

1

½

1

½

1

04

1

1

Page No.18/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


4

d)

e)

Ans

f)

3 2 2

3

D .E . is e x a c t

S o lu tio n is

. s e c s e c

ta n ta n

y xd x yd y c

y x y c

----------------------------------------------------------------------------------------------------------------------

2

2

2 2

2 2

2

2

2

2

2

2

2

1

S o lv e 1 1

P u t 1

1

1

1 1

1

1

1

s o lu tio n is

1

1 1

1

11

1

ta

n

d yx y

d x

x y v

d y d v

d x d x

d y d v

d x d x

d vv

d x

d vv v

d x

d vv v

d x

vd v d x

v

vd v d x

v

vd v d x

v

d v d xv

v v x c

1

1

1 ta n 1

1 ta n 1

x y x y x c

y x y c

----------------------------------------------------------------------------------------------------------------------

Verify that 1

sinm xy e

is a solution of differential equation

2

2 2

21 0

d y d yx x m y

d x d x

1

1

04

1

1

½

1

½

04

Page No.19/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


4

Ans

[[[[[

1sin

C onsider m x

y e

1s in

2

2

2

2

2 2 2

22

2 2

2

2

2 2

2

2

2 2

2

2

2 2

2

1.

1

1

1

S q u a r in g ,

1

1 2 2 2

2 1 2 2

1

1 0

m xd ye m

d x x

d y m y

d x x

d yx m y

d x

d yx m y

d x

d y d y d y d yx x m y

d x d x d x d x

d y d y d y d yx x m y

d x d x d x d x

d y d yx x m y

d x d x

d y d yx x m y

d x d x

OR

1

1

2

2 2

2

s in

s in

2

2

2

2

2

22

22

2 2

2

2

2

2

1 0

C o n s id e r

1.

1

1

1

11 2

2 1

1 1

1

m x

m x

d y d yx x m y

d x d x

y e

d ye m

d x x

d y m y

d x x

d yx m y

d x

d y d y d yx x m

d x d x d xx

d y d y d yx x m x

d x d x d x

d y d yx x m m y

d x d x

½

½

1

1

1

½

1

1

½

Page No.20/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


4

5

a)

Ans

b)

Ans

2

2 2

2

2

2 2

2

1

1 0

d y d yx x m y

d x d x

d y d yx x m y

d x d x

----------------------------------------------------------------------------------------------------------------------


A Card is drawn at random from a well shuffled pack of 52 playing cards.

A=event that the card drawn is not a spade. B=event that the card drawn is king. V erify th a t even ts A an d B a re in d ep en d e n t.

5 2

1

3 9

1

4

1

3

1

5 2

3 9

4

3 9 3

5 2 4

4 1

5 2 1 3

3 1 3 C o n s id e r =

4 1 3 5 2

E v e n t th a t th e c a rd d ra w n is a k in g o f h e a r t o ro f d ia m o n d o r o f c lu b

n 3

n 3

5 2

A

n s C

n A C

n B C

n Ap A

n s

n Bp B

n s

p A p B

A B

A B C

A Bp A B

n s

p A B p A p B

a n d B In d e p e n d e n t e v e n ts .

0 .2 5

1A s s u m in g th a t th e p ro b a b ili ty o f a fa ta l a c c id e n t d u r in g th e ye a r is .

1 2 0 0

C a lc u la te th e p ro b a b ili ty th a t in a fa c t o ry e m p lo y in g 3 0 0 w o rk e rs th e re

w ill b e a t le a s t tw o fa ta l a c c id e n ts in a ye a r . e

0 .7 7 8 8

1G iv e n 3 0 0 ,

1 2 0 0

3 0 00 .2 5

1 2 0 0

a t le a s t tw o fa ta l a c c id e n ts 1 0 1

n p

m n p

p p p

1

16

04

½

½

½

½

1

½

½

04

1

1

Page No.21/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


5 b)

c)

Ans

d)

Ans

0 10 .2 5 0 .2 5.0 2 5 .0 2 5

a t le a s t tw o fa ta l a c c id e n ts 10 ! 1!

= 1 0 .7 7 8 8 0 .7 7 8 8 0 .2 5

= 0 .0 2 6 5

e ep

-------------------------------------------------------------------------------------------------------------------

In c e rta in e x a m in a tio n 5 0 0 s tu d e n ts a p p e re rd .M e a n sc o re is 6 8 w ith S .D 8 .F in d th e

n u m b e r o f s tu d e n ts s c o rin g

i) le s s th a n 5 0

ii) m o re th a n 6 0

G iv e n 6 8 , 8x

i ) w h e n 5 0

5 0 6 8

8

2 .2 5

ii) w h e n 6 0

6 0 6 8

8

1

x

x xz

x

x xz

NOTE : As the areas for the above problem are not given, the students cannot solve the problem completely. If students attempted to solve the problem and calculated upto value z. Full marks to be rewarded.

----------------------------------------------------------------------------------------------------------------------

2 2

2

2

1E v a lu a te

4 5 s in 2

1

4 5 s in 2

2P u t ta n , , s in 2

1 1

1

24 5

1

d xx

d xx

d t tx t d x x

t t

d t

t

t

t

1

1

04

2

2

04

1

Page No.22/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


5

d)

2

2

2

2 2

4 4 1 0

4 1 0 4

1 0 0 1 0 04 1 0 4

1 6 1 6

5 32

2 2

5 32

1 12 2lo g

3 5 3 22 2

2 2 2

1 2 1lo g

6 2 4

1 2 ta n 1lo g

6 2 ta n 4

O R

1

4 5 s in 2

P u t t

d t

t t

d t

t t

d t

t t

d t

t

t

c

t

tc

t

xc

x

d xx

2 2

2

2

2

2

2

2

2a n , , s in 2

1 1

1

24 5

1

4 4 1 0

4 1 0 4

1

541

2

1

5 2 5 2 541

2 1 6 1 6

d t tx t d x x

t t

d t

t

t

t

d t

t t

d t

t t

d t

t t

d t

t t

½

½

1

½

½

1

½

Page No.23/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


5

d)

e)

Ans

2 2

1

4 5 3

4 4

5 3

1 1 4 4lo g

3 5 342

4 4 4

1 2 1lo g

6 2 4

1 2 ta n 1lo g

6 2 ta n 4

d t

t

t

t

t

t

x

x

----------------------------------------------------------------------------------------------------------------------

2

2

0

2

2

0

2

2

0

s in 2E v a lu a te

4 s in

s in 2

4 s in

2 s in c o s

4 s in

P u t s in w h e n 0 0

c o s w h e n 12

2

4

xd x

x

xd x

x

x xd x

x

x t x t

x d x d t x t

td t

t

1

2

0

2

2 2c o n s id e r ,

4 2 2

2

2 2 2 2

2 2 2

p u t 2

1

p u t 2

1

t t

t t t

t A B

t t t t

t A t B t

t

A

t

B

½

1

½

½

04

1

½

½

Page No.24/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


5

e)

f)

Ans

1 1

2

0 0

1

0

2

2

0

2

2

0

2 1 1

4 2 2

lo g 2 lo g 2

lo g 3 lo g 1 lo g 2 lo g 2

lo g 3 lo g 4

O R

s in 2

4 s in

2 s in c o s

4 s in

P u t s in

td td t

t t t

t t

xd x

x

x xd x

x

x t

1

2

0

1

2

0

12

0

2 2

w h e n 0 0

c o s w h e n 12

2

4

2

4

lo g 4

lo g 4 1 lo g 4 0

lo g 3 lo g 4

x t

x d x d t x t

td t

t

td t

t

t

-----------------------------------------------------------------------------------------------------------------------

S o lv e 2 lo g 1 0

c o m p a r in g w ith 0

2 lo g N = 1

,

g iv e n D .E .e q u a tio n is e x a c t

x

x

x

x

x x

ex e y d x d y

y

M d x N d y

eM x e y

y

M e N e

y y x y

M N

y x

1

1

1

1

1

1

04

½ +½

1

Page No.25/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


5

6

a)

Ans

b)

Ans

ta n

ta n

2

2

so lu tio n is

2 lo g 1

2lo g

2

lo g

y co n s t te rm s fr e e fro m x

x

y co n s t

x

x

M d x N d y c

x e y d x d y c

xe y y c

x e y y c


A H u s b a n d a n d w ife a p p e a re d in a n in te rv ie w fo r tw o v a c a n c ie s in a n o ff ic e

1 1T h e P ro b a b ili ty o f h u s b a n d 's s e le c t io n i s a n d th a t o f w ife 's s e le c t io n is

7 5

in d th e p ro b a b ili ty th a t

i ) b o th o f th e m a re

F

' '

' ' ' '

s e le c te d .

i i) o n ly o n e o f th e m is s e le c te d .

1 1G iv e n p

7 5

6 41 1

7 5

1 1 1)

7 5 3 5

) p p

1 4 6 1 =

7 5 7 5

p H W

p H p H a n d p W p W

i p H W p H p W

ii p H W p H W p H W p H W

2

7

------------------------------------------------------------------------------------------------------------

A company manufactures electric motors. The probability that an electric

motor is defective is 0.01. What is the probability that a sample of 300

electric motors will contain exactly 5 defective motors? 3

0 .0 4 9 8e

3 5

G iv e n 3 0 0 , 0 .0 1

3 0 0 0 .0 1 3

35

5 !

0 .1 0 0 8

n p

m n p

eP

1

1

16

04

1

1

1

1

04

1

2

1

Page No.26/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


6 c)

Ans

d)

Ans

In a test of 2000 electric bulbs it was found that the life of particular make was normally distributed with average life of 2040 hrs. and S.D.of 60 hrs. Estimate the number of bulbs likely to burn for

(i) between 1920hrs. and 2160 hrs.

(ii) more than 2150 hrs.

2 0 4 0 , 6 0

i) 1 9 2 0 , 2 1 6 0

1 9 2 0 2 0 4 0 2

6 0

2 1 6 0 2 0 4 0

G iv e n th a t 2 0 .4 7 7 2

1 .8 3 0 .

26 0

( B e tw e e n 1 9 2 0 h rs . a n d 2 1 6 0 h rs .) 2 2 0 .4 7 7 2 0 .4 7 7 2

6 6 4

4

A

A

x

x x

x xz

x xz

P A A

0 .9 5 4 4

N u m b e r o f b u lb s h a v in g life b e tw e e n 1 9 2 0 h rs . a n d 2 1 6 0 h rs .

0 .9 5 4 4 2 0 0 0

1 9 0 8 .8 1 9 0 9

ii) 2 1 5 0

2 1 5 0 2 0 4 0 1 .8 3

. 6 0

M o re th a n 2 1 5 0 h rs. 0 .5 1 .8 3

x

x xz

S D

P A

0 .5 0 .4 6 6 4

0 .0 3 3 6

N u m b e r o f b u lb s h a v in g life m o re th a n 2 1 5 0 h rs .

0 .0 3 3 6 2 0 0 0 6 7 .2 6 7

3 2

3 2

2

2

2

F in d th e m a x im u m a n d m in im u m v a lu e s o f 2 3 3 6 1 0

2 3 3 6 1 0

6 6 3 6

1 2 6

0

x x x

y x x x

d yx x

d x

d yx

d x

d y

d x

04

½

1

½

½

1

½

½

½

Page No.27/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


6

d)

e)

Ans

2 2

2

2

3 2

m in

2

2

3 2

m a x

6 6 3 6 0 o r 6 0

3 o r 2

3

1 2 3 6 3 0

is m in im u m a t 3

2 3 3 3 3 6 3 1 0 7 1

2

1 2 2 6 3 0

y is m a x im u m a t 2

2 2 3 2 3 6 2 1 0 5 4

x x x x

x x

fo r x

d y

d x

y x

y

fo r x

d y

d x

x

y

-----------------------------------------------------------------------------------------------------------------------

2 2

2 2

F in d th e e q u a t io n o f th e ta n g e n t a n d th e n o rm a l to th e c u rv e

2 3 1 8 a t 3 ,1

2 3 1 8 a t 3 ,1

4 6 0

4 6 0

4

6

a t 3 ,1

1 4 3 1 1

6 1 3 3

s lo p e o f ta n g e n t

x x y y

x x y y

d y d yx x y y

d x d x

d y d yx x y y

d x d x

d y y x

d x y x

d y

d x

1 1

3

E q u a tio n o f ta n g e n t is

1 11 3

3

3 3 1 1 3 3

1 1 3 3 6 0

1 3s lo p e o f n o rm a l

1 1

E q u a tio n o f n o rm a l is

y x

y x

x y

d y

d x

1

½

½

½

½

04

1

½

1

½

Page No.28/29

17301



______________________________________________________________________________________________



Q. No.

Sub Q. N.


6 e)

f)

Ans

3

1 31 1

1 1 1 1 3 9

3 1 1 2 0

y x

y x

x y

------------------------------------------------------------------------------------------------------------

2 2

2 2

2 2

2 2

2 2

3

2 2

0

3

2 2

0

32

2 2 1

0

F in d b y in te g ra t io n th e a re a o f th e e lli p s e 4 9 3 6

4 9 3 6

19 4

49

9

23

3

4

24 3

3

83

3

383 s in

3 2 2 3

b

a

x y

x y

x y

y x

y x

A re a y d x

x d x

x d x

x xx

2

2 2 1

1

38 3 33 3 s in 0

3 2 2 3

8 90 s in 1

3 2

8 9

3 2 2

6

Important Note In the solution of the question paper, wherever possible all the possible alternative

methods of solution are given for the sake of convenience. Still student may follow a

method other than the given herein. In such case, first see whether the method falls

within the scope of the curriculum, and then only give appropriate marks in

accordance with the scheme of marking.

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1

04

½

1

1

½

1

Page No.29/29

17301

summer 17 examination model answer 17301 · 2017-07-14 · relevant answer based on candidate’s...

Documents