Class XII Mathematics Set-3 Time: 3 hrs M.M: 100 Marks ...icsk-kw.com/pdf/pqp/12/mat/mat12-1.pdf ·...

Material downloaded from http://myCBSEguide.com and http://onlineteachers.co.in

Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

Class XII

Mathematics

Set-3

Time: 3 hrs M.M: 100 Marks

General Instructions:

(i) All questions are compulsory.

(ii) Please check that this Question Paper contains 26 Questions.

(iii) Marks for each question are indicated against it.

(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions

carrying one mark each.

(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4

marks each.

(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying

6 marks each.

(vii) Please write down the serial number of the Question before attempting it.

Section A

Q1 The two vectors ˆj k+ and $3 4i j k− +$ represent the two sides AB and AC,

respectively of a. Find the length of the median through A.

Q2 Find the vector equation of a plane which is at a distance of 5 units from

the origin and its normal vector is $ $2 3 6i j k− +$

Q.3 Find the maximum value of 1 1 1

1 1 sin 1

1 1 1 cos

θθ

++

Q4 If A is a square matrix such that A2 = I, then find the simplified value of (A

– I)3 + (A + I)3 – 7A.

Q5 Matrix

0 2 2

3 1 3

3 3 1

b

A

a

−=

− is given to be symmetric, find values of a and b.



Q.6 Find the position vector of a point which divides the join of points with

position vectors 2 2a b and a b− +r r rv

externally in the ratio 2 : 1.

Section B

Q7 Find the general solution of the following differential equation :

( ) ( )2 tan 11 0 y dyy x e

dx−+ + − =

Q8 Show that the vectors ,a b and cur r r

are coplanar if , ,a b b c+ur r r r

and c a+r r

are coplanar.

Q9 Find the vector and Cartesian equations of the line through the point (1, 2,

−4) and perpendicular to the two lines.

( ) ( )ˆ ˆˆ ˆ ˆ ˆ 8 19 10 3 16 7r i j k i j kλ= − + + − +r and

( ) ( )ˆ ˆˆ ˆ ˆ ˆ 15 29 5 3 8 5 . r i j k i j kµ= + + + + −r

Q10 Three persons A, B and C apply for a job of Manager in a Private Company.

Chances of their selection (A, B and C) are in the ratio 1 : 2 :4. The

probabilities that A, B and C can introduce changes to improve profits of

the company are 0.8, 0.5 and 0.3, respectively. If the change does not take

place, find the probability that it is due to the appointment of C.

OR

A and B throw a pair of dice alternately. A wins the game if he gets a total

of 7 and B wins the game if he gets a total of 10. If A starts the game, then

find the probability that B wins.

Q11 Prove that: 1 7 3 11 1 1 1

5 7 3 8 4tan tan tan tan

π− − − −+ + + =

OR

Solve for x: ( ) ( )1 12 2 tan cos x tan cosec x− −=

Q12 The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their

monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per



month, find their monthly incomes using matrix method. This problem

reflects which value?

Q13 If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), find the values of dy

dx

at 4

tπ= and

3t

π= .

OR

If 22

2

1, 0.x d y dy y

y X prove thatdx y dx x

= − − =

Q14 Find the values of p and q for which 3

2

2

1 sin,

3cos 2

( ) ,2

(1 sin ),

( 2 ) 2

x xif x

x

f x P if x

q xif x

x

π

ππ

− <= =

− > −

is continuous at 2

xπ=

Q15. Show that the equation of normal at any point t on the curve 3 3 – x cos t cos t= and 3 3 – y sin t sin t= is

( )3 34 – 3 4 . y cos t sin t sin t=

Q16. Find 2

(3sin 2)cos.

5 cos 4sin

θ θ θθ θ−

∫− −

OR

Evaluate 2

0. sin

4xe x dx

π π +

∫

Q17. Find 3 3

. x

dxa x

∫−

Q18 Evaluate 2 31 . x x dx−∫ −∣ ∣

Q19 Find the particular solution of the differential equation

( ) ( )21 – 1 2 0y logx dx xydy+ + = given that y = 0 when x = 1.



SECTION C

Q20 Find the coordinate of the point P where the line through A(3, –4, –5) and

B(2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3,

0, 1) and N(4, –1, 0).

Also, find the ratio in which P divides the line segment AB.

Q.21 An urn contains 3 white and 6 red balls. Four balls are drawn one by one

with replacement from the urn. Find the probability distribution of the

number of red balls drawn. Also find mean and variance of the

distribution.

Q22 A manufacturer produces two products A and B. Both the products are

processed on two different machines. The available capacity of first

machine is 12 hours and that of second machine is 9 hours per day. Each

unit of product A requires 3 hours on both machines and each unit of

product B requires 2 hours on first machine and 1 hour on second

machine. Each unit of product A is sold at Rs 7 profit and B at a profit of

Rs 4. Find the production level per day for maximum profit graphically.

Q23 Let : f N N→ be a function defined as ( ) 29 6 5f x x x= + − . Show that

: f N S→ , where S is the range of f, is invertible. Find the inverse of f and

hence find ( ) ( )1 143 163 . f and f− −

Q24 Prove that

2 2 2

2 2 2

2 2 2

yz x zx y xy z

zx y xy z yz x

xy z yz x zx y

− − −− − −− − −

is divisible by (x + y + z) and hence find

the quotient.

OR

Using elementary transformations, find the inverse of the matrix

8 4 3

2 1 1

1 2 2

A

=

and use it to solve the following system of linear equations :

8 4 3 19

2 5

2 2 7

x y z

x y z

x y z

+ + =+ + =

+ + =



Q.25 Show that the altitude of the right circular cone of maximum volume that

can be inscribed in a sphere of radius4

3

rr is . Also find maximum volume

in terms of volume of the sphere.

OR

Find the intervals in which ( ) 3 – 3 , 0 ,f x sin x cos x x π= < < is strictly

increasing or strictly decreasing.

Q26 Using integration find the area of the region

( ) 2 2 2, : 2 , { } , , 0 . x y x y ax y ax x y+ � � �



Class XII

Mathematics

Set-1

Time Allowed: 3 hrs. M.M: 100 Marks

General Instructions:



(iii) Marks for each question are indicated against it.

(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark

each.

(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each.

(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying 6 marks

each.

(vii) Please write down the serial number of the Question before attempting it.

Section A

Q1 Find the maximum value of

1 1 1

1 1 sin 1

1 1 1 cos

θθ

++

Q2 If A is a square matrix such that A2 = I, then find the simplified value of

( ) ( )3 37A I A I A− + + −

Q3 Matrix

0 2 2

3 1 3

3 3 1

b

A

a

−=

− is given to be symmetric, find values of a and b.

Q4 Find the position vector of a point which divides the join of points with position

vectors 2 2a b and a b− +r r rv

externally in the ratio 2 : 1.

Q5 The two vectors ˆj k+ and $3 4i j k− +$ represent the two sides AB and AC, respectively

of a. Find the length of the median through A.

Q6 Find the vector equation of a plane which is at a distance of 5 units from the origin and

its normal vector is $ $2 3 6i j k− +$

Section B

Q7 Prove that: 1 7 3 11 1 1 1

5 7 3 8 4tan tan tan tan

π− − − −+ + + =



OR

Solve for x:

( ) ( )1 12 2 tan cos x tan cosec x− −=

Q8 The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly

expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their

monthly incomes using matrix method. This problem reflects which value?

Q9 If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), find the values of dy

dx at

4t

π= and

3t

π= .

OR

If

22

2

1, 0.x d y dy y

y X prove thatdx y dx x

= − − =

Q10 Find the values of p and q for which 3

2

2

1 sin,

3cos 2

( ) ,2

(1 sin ),

( 2 ) 2

x xif x

x

f x P if x

q xif x

x

π

ππ

− <= =

− > −

is continuous at 2

xπ=

Q11 Show that the equation of normal at any point t on the curve 3 3 – x cos t cos t= and 3 3 – y sin t sin t= is

( )3 34 – 3 4 . y cos t sin t sin t=

Q12 Find 2

(3sin 2)cos.

5 cos 4sin

θ θ θθ θ−

∫− −

OR

Evaluate 2

0. sin

4xe x dx

π π +

∫



Q13 Find 3 3

. x

dxa x

∫−

Q14 Evaluate 2 31 . x x dx−∫ −∣ ∣

Q15 Find the particular solution of the differential equation

( ) ( )21 – 1 2 0y logx dx xydy+ + =

given that y = 0 when x = 1.

Q16 Find the general solution of the following differential equation:

( ) ( )2 tan 11 0 y dyy x e

dx−+ + − =

Q17 Show that the vectors ,a b and cur r r

are coplanar if , ,a b b c+ur r r r

and c a+r r

are

coplanar.

Q18 Find the vector and Cartesian equations of the line through the point (1, 2, −4) and

perpendicular to the two lines.

( ) ( )ˆ ˆˆ ˆ ˆ ˆ 8 19 10 3 16 7r i j k i j kλ= − + + − +r and

( ) ( )ˆ ˆˆ ˆ ˆ ˆ 15 29 5 3 8 5 . r i j k i j kµ= + + + + −r

Q19 Three persons A, B and C apply for a job of Manager in a Private Company. Chances of

their selection (A, B and C) are in the ratio 1 : 2 :4. The probabilities that A, B and C can

introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively.

If the change does not take place, find the probability that it is due to the appointment

of C.

OR

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B

wins the game if he gets a total of 10. If A starts the game, then find the probability that

B wins.

SECTION C

Q20 Let : f N N→ be a function defined as ( ) 29 6 5f x x x= + − . Show that : f N S→ ,

where S is the range of f, is invertible. Find the inverse of f and hence find

( ) ( )1 143 163 . f and f− −



Q21 Prove that

2 2 2

2 2 2

2 2 2

yz x zx y xy z

zx y xy z yz x

xy z yz x zx y

− − −− − −− − −

is divisible by (x + y + z) and hence find the

quotient.

OR

Using elementary transformations, find the inverse of the matrix

8 4 3

2 1 1

1 2 2

A

=

and

use it to solve the following system of linear equations :

8 4 3 19

2 5

2 2 7

x y z

x y z

x y z

+ + =+ + =

+ + =

Q.22 Show that the altitude of the right circular cone of maximum volume that can be

inscribed in a sphere of radius4

3

rr is . Also find maximum volume in terms of volume

of the sphere.

OR

Find the intervals in which ( ) 3 – 3 , 0 ,f x sin x cos x x π= < < is strictly increasing or

strictly decreasing.

Q23 Using integration find the area of the region ( ) 2 2 2, : 2 , { } , , 0 . x y x y ax y ax x y+ � � �

Q24 Find the coordinate of the point P where the line through A (3, –4, –5) and B(2, –3, 1)

crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0).

Also, find the ratio in which P divides the line segment AB.

Q25 An urn contains 3 white and 6 red balls. Four balls are drawn one by one with

replacement from the urn. Find the probability distribution of the number of red balls

drawn. Also find mean and variance of the distribution.

Q26 A manufacturer produces two products A and B. Both the products are processed on

two different machines. The available capacity of first machine is 12 hours and that of

second machine is 9 hours per day. Each unit of product A requires 3 hours on both

machines and each unit of product B requires 2 hours on first machine and 1 hour on

second machine. Each unit of product A is sold at Rs 7 profit and B at a profit of Rs 4.

Find the production level per day for maximum profit graphically.

65/2/1/F 1 [P.T.O.

¸üÖê»Ö ®ÖÓ.

Roll No.

ÝÖ×ÞÖŸÖ MATHEMATICS

×®Ö¬ÖÖÔ×¸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü †×¬ÖÛúŸÖ´Ö †ÓÛú : 100

Time allowed : 3 hours Maximum Marks : 100

ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :

(i) ÃÖ³Öß ¯ÖÏ¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü … (ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …

(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æîü …

(iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æïü …

(v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æïü …

(vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …

Series : ONS/2 ÛúÖê›ü ®ÖÓ. Code No.

65/2/1/F

• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´Öã×¦üŸÖ ¯ÖéÂšü 8 Æïü …

• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê ü ×¤ü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü ¯Ö¸ü ×»ÖÜÖë …

• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …

• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …

• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê ×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî üÖ®Ö ¾Öê ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …

• Please check that this question paper contains 8 printed pages.

• Code number given on the right hand side of the question paper should be written on the

title page of the answer-book by the candidate.

• Please check that this question paper contains 26 questions.

• Please write down the Serial Number of the question before attempting it.

• 15 minute time has been allotted to read this question paper. The question paper will be

distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the

question paper only and will not write any answer on the answer-book during this period.

¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü ¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë … Candidates must write the Code on

the title page of the answer-book.

SET – 1

65/2/1/F 2

General Instructions :



(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one

mark each.

(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks

each.

(v) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying

6 marks each

(vi) Please write down the serial number of the Question before attempting it.

ÜÖÞ›ü – †

SECTION – A

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü … Question numbers 1 to 6 carry 1 mark each.

1. µÖ×¤ü (2 1 3)

–1 0 –1

–1 1 0

0 1 1

1

0

–1

= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …

If (2 1 3)

–1 0 –1

–1 1 0

0 1 1

1

0

–1

= A, then write the order of matrix A.

2. µÖ×¤ü

x sin θ cos θ

–sin θ –x 1

cos θ 1 x

= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …

If

x sin θ cos θ

–sin θ –x 1

cos θ 1 x

= 8, write the value of x.

3. µÖ×¤ü A =

3 5

7 9 ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú

×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …

If A =

3 5

7 9 is written as A = P + Q, where P is a symmetric matrix and Q is skew

symmetric matrix, then write the matrix P.

65/2/1/F 3 [P.T.O.

4. µÖ×¤ →a , →

b ŸÖ£ÖÖ →c ‹êÃÖê ´ÖÖ¡ÖÛú ÃÖ×¤ü¿Ö Æïü ×Ûú →a + →

b + →c =

→

0 Æîü, ŸÖÖê →a ⋅ →

b + →

b ⋅ →c +

→c ⋅

→a ÛúÖ ´ÖÖ®Ö

×»Ö×ÜÖ‹ …

If →a ,

→

b , →c are unit vectors such that

→a +

→

b + →c =

→

0 , then write the value of

→a ⋅

→

b + →

b ⋅ →c +

→c ⋅

→a .

5. µÖ×¤ü →a ×

→b

2

+ →a ⋅

→b

2

= 400 Æîü ŸÖ£ÖÖ →a = 5 Æîü, ŸÖÖê

→

b ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …

If →a ×

→b

2

+ →a ⋅

→b

2

= 400 and →a = 5, then write the value of

→

b .

6. ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ´Öæ»Ö Ø²Ö¤ãü ÃÖê 5 3 Ûúß ¤æü¸üß ¯Ö¸ü Æîü ŸÖ£ÖÖ ×•ÖÃÖÛúÖ †×³Ö»ÖÓ²Ö †õÖÖë ¯Ö¸ü ÃÖ´ÖÖ®Ö ºþ¯Ö ÃÖê —ÖãÛúÖ Æîü …

Write the equation of a plane which is at a distance of 5 3 units from origin and the

normal to which is equally inclined to coordinate axes.

ÜÖÞ›ü – ²Ö

SECTION – B

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 7 ÃÖê 19 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 4 †ÓÛú ÛúÖ Æîü …

Question numbers 7 to 19 carry 4 marks each.

7. ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú

cot–1 1 + sin x + 1 – sin x

1 + sin x – 1 – sin x =

x

2, 0 < x <

π

2

†£Ö¾ÖÖ

x Ûêú ×»Ö‹ Æü»Ö Ûúß×•Ö‹ :

tan–1

x – 2

x – 1 + tan–1

x + 2

x + 1 =

π

4

Prove that :

cot–1 1 + sin x + 1 – sin x

1 + sin x – 1 – sin x =

x

2, 0 < x <

π

2

OR

Solve for x :

tan–1

x – 2

x – 1 + tan–1

x + 2

x + 1 =

π

4

65/2/1/F 4

8. †ÓÝÖÏê•Öß ×¾ÖÂÖµÖ ¯ÖœÌüÖ®Öê ¾ÖÖ»Öß ÛúÖêØ“ÖÝÖ ÃÖÓÃ£ÖÖ ¤üÖê ²Öî“Ö I †Öî ü II ´Öë ÛúõÖÖ »ÖêŸÖß Æîü ×•Ö®Ö´Öë †´Öß¸ü ¾Ö ÝÖ¸üß²Ö ²Ö““ÖÖë Ûêú ×»Ö‹ ±úßÃÖ †»ÖÝÖ-†»ÖÝÖ Æîü … ²Öî“Ö I ´Öë 20 ÝÖ¸üß²Ö ŸÖ£ÖÖ 5 †´Öß¸ü ²Ö““Öê Æïü †Öî ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö ` 9,000 Æîü, •Ö²Ö×Ûú ²Öî“Ö II ´Öë 5 ÝÖ¸üß²Ö †Öî ü 25 †´Öß¸ü ²Ö““Öê Æïü †Öî ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö ` 26,000 Æî … †Ö¾µÖæÆü ×¾Ö×¬Ö ÃÖê ¤üÖê®ÖÖë ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏŸµÖêÛú ²Ö““Öê «üÖ¸üÖ ¤üß ÝÖ‡Ô ´ÖÖ×ÃÖÛú ±úßÃÖ –ÖÖŸÖ Ûúß×•Ö‹ … ‹êÃÖÖ Ûú¸üÛêú ÛúÖêØ“ÖÝÖ ÃÖÓÃ£ÖÖ ÃÖ´ÖÖ•Ö ´Öë ŒµÖÖ ´Öæ»µÖ ¸üÜÖ ¸üÆüß Æîü ?

A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is ` 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is ` 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society ?

9. µÖ×¤ü ±ú»Ö®Ö f(x) = x2 + 3x + a , x ≤ 1

bx + 2 , x > 1

x = 1 ¯Ö¸ü †¾ÖÛú»Ö®ÖßµÖ Æîü, ŸÖÖê a ŸÖ£ÖÖ b Ûêú ´ÖÖ®Ö –ÖÖŸÖ Ûúß×•Ö‹ …

Find the values of a and b, if the function f defined by

f(x) = x2 + 3x + a , x ≤ 1

bx + 2 , x > 1

is differentiable at x = 1.

10. tan–1

1 + x2 – 1

x ÛúÖ sin–1

2x

1 + x2 Ûêú ÃÖÖ¯ÖêõÖ †¾ÖÛú»Ö®Ö Ûúß×•Ö‹, •Ö²Ö×Ûú x ∈ (–1, 1) Æîü …

†£Ö¾ÖÖ

µÖ×¤ü x = sin t Æîü ŸÖ£ÖÖ y = sin pt Æîü, ŸÖÖê ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú (1 – x2) d2y

dx2 – xdy

dx + p2y = 0

Differentiate tan–1

1 + x2 – 1

x w.r.t. sin–1

2x

1 + x2 , if x ∈ (–1, 1)

OR

If x = sin t and y = sin pt, prove that (1 – x2) d2y

dx2 – xdy

dx + p2y = 0.

11. ¾ÖÛÎúÖë y2 = 4ax ŸÖ£ÖÖ x2 = 4by Ûêú ²Öß“Ö ÛúÖ ¯ÖÏ×ŸÖ“”êû¤üß ÛúÖêÞÖ –ÖÖŸÖ Ûúß×•Ö‹ …

Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.

12. ´ÖÖ®Ö –ÖÖŸÖ Ûúß×•Ö‹ : ⌡⌠

0

π

x

1 + sin α sin x dx

Evaluate : ⌡⌠

0

π

x

1 + sin α sin x dx

65/2/1/F 5 [P.T.O.

13. –ÖÖŸÖ Ûúß×•Ö‹ : ⌡⌠.

.(2x + 5) 10 – 4x – 3x2 dx

†£Ö¾ÖÖ

–ÖÖŸÖ Ûúß×•Ö‹ : ⌡⌠

(x2 + 1) (x2 + 4)

(x2 + 3) (x2 – 5) dx

Find : ⌡⌠.

.(2x + 5) 10 – 4x – 3x2 dx

OR

Find : ⌡⌠

(x2 + 1) (x2 + 4)

(x2 + 3) (x2 – 5) dx

14. –ÖÖŸÖ Ûúß×•Ö‹ : ⌡⌠

x sin–1x

1 – x2 dx

Find : ⌡⌠

x sin–1x

1 – x2 dx

15. ×®Ö´®Ö †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ÛúÖê Æü»Ö Ûúß×•Ö‹ : y2dx + (x2 – xy + y2)dy = 0

Solve the following differential equation :

y2dx + (x2 – xy + y2)dy = 0

16. ×®Ö´®Ö †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ÛúÖê Æü»Ö Ûúß×•Ö‹ : (cot–1y + x) dy = (1 + y2) dx

Solve the following differential equation :

(cot–1y + x) dy = (1 + y2) dx

17. µÖ×¤ü →a × →

b = →c ×

→

d †Öî ü →a × →c =

→

b × →

d Æîü, ŸÖÖê ¤ü¿ÖÖÔ‡‹ ×Ûú →a – →

d, →

b – →c Ûêú ÃÖ´ÖÖÓŸÖ¸ü Æîü, •Ö²Ö×Ûú →a ≠

→

d

†Öî ü →

b ≠ →c Æîü …

If →a ×

→

b = →c ×

→

d and →a ×

→c =

→

b × →

d, show that →a –

→

d is parallel to →

b – →c , where

→a ≠

→

d

and →

b ≠ →c .

18. ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú Ø²Ö¤ãü†Öë A(0, –1, –1) ŸÖ£ÖÖ B(4, 5, 1) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß êüÜÖÖ Ø²Ö¤ãü†Öë C(3, 9, 4) ŸÖ£ÖÖ D(–4, 4, 4) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß êüÜÖÖ ÛúÖê ¯ÖÏ×ŸÖ“”êû¤ü Ûú¸üŸÖß Æîü …

Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through

C(3, 9, 4) and D(–4, 4, 4).

65/2/1/F 6

19. ‹Ûú ×›ü²²Öê ´Öë 20 ¯Öê®Ö Æîü ×•ÖÃÖ´Öë ÃÖê 2 ÜÖ¸üÖ²Ö Æïü … µÖ×¤ü ˆ¢Ö¸üÖê¢Ö¸ü ¯ÖÏ×ŸÖÃ£ÖÖ¯Ö®ÖÖ Ûêú ÃÖÖ£Ö 5 ¯Öê®Ö ×®ÖÛúÖ»Öê •ÖÖ‹Ñ, ŸÖÖê ¯ÖÏÖ×µÖÛúŸÖÖ –ÖÖŸÖ Ûúß×•Ö‹ ×Ûú †×¬ÖÛúŸÖ´Ö 2 ¯Öê®Ö ÜÖ¸üÖ²Ö ÆüÖëÝÖë …

†£Ö¾ÖÖ

´ÖÖ®ÖÖ ×Ûú X, ÛúÖò»Öê•ÖÖë Ûúß ÃÖÓÜµÖÖ ÃÖæ×“ÖŸÖ Ûú üŸÖÖ Æîü •ÖÆüÖÑ ¯Ö ü †Ö¯Ö †¯Ö®ÖÖ ¯Ö× üÞÖÖ´Ö †Ö®Öê Ûêú ²ÖÖ¤ü †Ö¾Öê¤ü®Ö Ûú ëüÝÖë †Öî ü P (X = x) ¯ÖÏÖ×µÖÛúŸÖÖ ÃÖæ×“ÖŸÖ Ûú¸üŸÖÖ Æîü •Ö²Ö×Ûú †Ö¯ÖÛúÖê x ÛúÖò»Öê•Ö ´Öë ¯ÖÏ¾Öê¿Ö ×´Ö»Ö ÃÖÛúŸÖÖ Æîü … ×¤üµÖÖ ÝÖµÖÖ Æîü ×Ûú

P(X = x) =

kx , µÖ×¤ü x = 0 µÖÖ 12 kx , µÖ×¤ü x = 2

k(5 – x) , µÖ×¤ü x = 3 µÖÖ 40 , µÖ×¤ü x > 4

,

•Ö²Ö×Ûú k ‹Ûú ¬Ö®ÖÖŸ´ÖÛú †“Ö¸ü Æîü …

k ÛúÖ ´ÖÖ®Ö –ÖÖŸÖ Ûúß×•Ö‹ … µÖÆü ¯ÖÏÖ×µÖÛúŸÖÖ ³Öß –ÖÖŸÖ Ûúß×•Ö‹ ×Ûú †Ö¯ÖÛúÖê (i) ‹Ûú †Öî ü Ûêú¾Ö»Ö ‹Ûú ÛúÖò»Öê•Ö ´Öë ¯ÖÏ¾Öê¿Ö ×´Ö»ÖêÝÖÖ (ii) †×¬ÖÛú ÃÖê †×¬ÖÛú ¤üÖê ÛúÖò»Öê•ÖÖë ´Öë ¯ÖÏ¾Öê¿Ö ×´Ö»ÖêÝÖÖ (iii) Ûú´Ö ÃÖê Ûú´Ö ¤üÖê ÛúÖò»Öê•ÖÖë ´Öë ¯ÖÏ¾Öê¿Ö ×´Ö»ÖêÝÖÖ …

A box has 20 pens of which 2 are defective. Calculate the probability that out of

5 pens drawn one by one with replacement, at most 2 are defective.

OR

Let, X denote the number of colleges where you will apply after your results and

P (X = x) denotes your probability of getting admission in x number of colleges. It is

given that

P(X = x) =

kx , if x = 0 or 1

2 kx , if x = 2

k(5 – x) , if x = 3 or 4

0 , if x > 4

,

where k is a positive constant. Find the value of k. Also find the probability that you

will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2

colleges.

ÜÖÞ›ü – ÃÖ

SECTION – C

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 20 ÃÖê 26 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 6 †ÓÛú ÛúÖ Æîü …

Question numbers 20 to 26 carry 6 marks each.

20. ´ÖÖ®ÖÖ ×Ûú f, g : R → R ¤üÖê ±ú»Ö®Ö ‡ÃÖ ¯ÖÏÛúÖ¸ü ¯Ö×¸ü³ÖÖ×ÂÖŸÖ Æïü

f(x) = | x | + x †Öî ü g(x) = | x | – x, ∀x ∈ R

fog †Öî ü gof –ÖÖŸÖ Ûúß×•Ö‹ …

†ŸÖ: fog (–3), fog(5) †Öî ü gof (–2) –ÖÖŸÖ Ûúß×•Ö‹ …

If f, g : R → R be two functions defined as f(x) = | x | + x and g(x) = | x | – x, ∀x ∈ R.

Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).

65/2/1/F 7 [P.T.O.

21. µÖ×¤ü a, b †Öî ü c ÃÖ³Öß ¿Öæ®µÖêŸÖ¸ü Æïü,

†Öî

1 + a 1 1

1 1 + b 1

1 1 1 + c

= 0 Æîü, ŸÖÖê ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú 1

a +

1

b +

1

c + 1 = 0 Æîü …

†£Ö¾ÖÖ

µÖ×¤ü A =

cos α – sin α 0

sin α cos α 0

0 0 1

Æîü, ŸÖÖê adj. A –ÖÖŸÖ Ûúß×•Ö‹ ŸÖ£ÖÖ ÃÖŸµÖÖ×¯ÖŸÖ Ûúß×•Ö‹ ×Ûú

A(adj⋅A) = (adj⋅A)A = | A | I3 ü

If a, b and c are all non-zero and

1 + a 1 1

1 1 + b 1

1 1 1 + c

= 0, then prove that 1

a +

1

b +

1

c + 1= 0

OR

If A =

cos α – sin α 0

sin α cos α 0

0 0 1

, find adj⋅A and verify that A(adj⋅A) = (adj⋅A)A = | A | I3.

22. ‹Ûú ‘Ö®ÖÖ³Ö ×•ÖÃÖÛúß ³Öã•ÖÖ‹Ñ x, 2x ŸÖ£ÖÖ x

3 Æïü, ŸÖ£ÖÖ ‹Ûú ÝÖÖê»Öê Ûêú ¯ÖéÂšüßµÖ õÖê¡Ö±ú»ÖÖë ÛúÖ µÖÖêÝÖ †“Ö¸ü Æîü … ×ÃÖ¨ü Ûúß×•Ö‹

×Ûú ˆ®ÖÛêú †ÖµÖŸÖ®ÖÖë ÛúÖ µÖÖêÝÖ ®µÖæ®ÖŸÖ´Ö ÆüÖêÝÖÖ µÖ×¤ü x, ÝÖÖê»Öê Ûúß ×¡Ö•µÖÖ ÛúÖ ŸÖß®Ö ÝÖã®ÖÖ Æîü … ˆ®ÖÛêú †ÖµÖŸÖ®ÖÖë Ûêú µÖÖêÝÖ ÛúÖ ®µÖæ®ÖŸÖ´Ö ´ÖÖ®Ö ³Öß –ÖÖŸÖ Ûúß×•Ö‹ …

†£Ö¾ÖÖ

¾ÖÛÎú y = cos(x + y), –2π ≤ x ≤ 2π Æîü Ûúß ˆ®Ö ÃÖ³Öß Ã¯Ö¿ÖÔ êüÜÖÖ†Öë Ûêú ÃÖ´ÖßÛú¸üÞÖ –ÖÖŸÖ Ûúß×•Ö‹ •ÖÖê ¸êüÜÖÖ x + 2y = 0 Ûêú ÃÖ´ÖÖÓŸÖ¸ü Æïü …

The sum of the surface areas of a cuboid with sides x, 2x and x

3 and a sphere is given to

be constant. Prove that the sum of their volumes is minimum, if x is equal to three

times the radius of sphere. Also find the minimum value of the sum of their volumes.

OR

Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel

to the line x + 2y = 0.

23. ÃÖ´ÖÖÛú»Ö®ÖÖë Ûêú ¯ÖÏµÖÖêÝÖ ÃÖê ¾ÖÛÎúÖë y = 4 – x2 ŸÖ£ÖÖ x2 + y2 – 4x = 0 ŸÖ£ÖÖ x-†õÖ ÃÖê ×‘Ö êü õÖê¡Ö ÛúÖ õÖê¡Ö±ú»Ö –ÖÖŸÖ Ûúß×•Ö‹ …

Using integration find the area of the region bounded by the curves y = 4 – x2,

x2 + y2 – 4x = 0 and the x-axis.

65/2/1/F 8

24. ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ –ÖÖŸÖ Ûúß×•Ö‹ •ÖÖê ÃÖ´ÖŸÖ»ÖÖë x + 2y + 3z – 4 = 0 ŸÖ£ÖÖ 2x + y – z + 5 = 0 Ûúß ¯ÖÏ×ŸÖ“”êû¤üß ¸êüÜÖÖ ÛúÖê †ÓŸÖÙ¾ÖÂ™ü Ûú¸üŸÖÖ Æîü ŸÖ£ÖÖ ×•ÖÃÖ «üÖ¸üÖ x-†õÖ ¯Ö¸ü ÛúÖ™üÖ ÝÖµÖÖ †ÓŸÖ:ÜÖÓ›ü, z-†õÖ ¯Ö¸ü ÛúÖ™êü ÝÖ‹ †ÓŸÖ:ÜÖÓ›ü ÛúÖ ¤ãüÝÖã®ÖÖ Æîü …

†ŸÖ: ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ×¤ü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê Ø²Ö¤ãü (2, 3, –1) ÃÖê ÆüÖêÛú¸ü •ÖÖŸÖÖ Æîü ŸÖ£ÖÖ ˆ¯Ö¸üÖêŒŸÖ ¯ÖÏÖ¯ŸÖ ÃÖ´ÖŸÖ»Ö Ûêú ÃÖ´ÖÖÓŸÖ¸ü Æîü …

Find the equation of the plane which contains the line of intersection of the planes

x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 and whose x-intercept is twice its

z-intercept.

Hence write the vector equation of a plane passing through the point (2, 3, –1) and

parallel to the plane obtained above.

25. ‹Ûú £Öî»Öê A ´Öë 3 »ÖÖ»Ö ŸÖ£ÖÖ 5 ÛúÖ»Öß ÝÖë¤ëü Æïü, •Ö²Ö×Ûú £Öî»Öê B ´Öë 4 »ÖÖ»Ö ŸÖ£ÖÖ 4 ÛúÖ»Öß ÝÖë¤ëü Æïü … £Öî»Öê A ´Öë ÃÖê 2 ÝÖë¤êü £Öî»Öê B ´Öë µÖÖ¥ü“”ûµÖÖ Ã£ÖÖ®ÖÖ®ŸÖ×¸üŸÖ Ûúß ÝÖ‡Ô ŸÖ£ÖÖ ×±ú¸ü £Öî»Öê B ´Öë ÃÖê ‹Ûú ÝÖë¤ü µÖÖ¥ü“”ûµÖÖ ×®ÖÛúÖ»Öß ÝÖ‡Ô ŸÖ£ÖÖ »ÖÖ»Ö ¸ÓüÝÖ Ûúß ¯ÖÖ‡Ô ÝÖ‡Ô … ¯ÖÏÖ×µÖÛúŸÖÖ –ÖÖŸÖ Ûúß×•Ö‹ ×Ûú A ÃÖê B ´Öë Ã£ÖÖ®ÖÖÓŸÖ×¸üŸÖ Ûúß ÝÖ‡Ô ¤üÖê®ÖÖë ÝÖë¤ëü »ÖÖ»Ö ÓüÝÖ Ûúß £Öà …

Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls.

Two balls are transferred at random from bag A to bag B and then a ball is drawn from

bag B at random. If the ball drawn from bag B is found to be red, find the probability

that two red balls were transferred from A to B.

26. †¯Ö®Öê ¤îü×®ÖÛú ÃÖÓŸÖã×»ÖŸÖ †ÖÆüÖ¸ü ÛúÖê ¯Öæ üÛú Ûú¸ü®Öê Ûêú ×»Ö‹ ‹Ûú ¾µÖ×ŒŸÖ X †Öî ü Y ÝÖÖê×»ÖµÖÖÑ »Öê®ÖÖ “ÖÖÆüŸÖÖ Æîü … X †Öî ü Y ¯ÖÏÛúÖ¸ü Ûúß ‹Ûú-‹Ûú ÝÖÖê»Öß ´Öë »ÖÖêÆêü, Ûîú×»¿ÖµÖ´Ö ¾Ö ×¾Ö™üÖ×´Ö®Ö Ûúß ´ÖÖ¡ÖÖ (×´Ö»ÖßÝÖÏÖ´Ö ´Öë) ®Öß“Öê ¤üß ÝÖ‡Ô Æïü :

ÝÖÖê»Öß »ÖÖêÆüÖ Ûîú×»¿ÖµÖ´Ö ×¾Ö™üÖ×´Ö®Ö

X 6 3 2

Y 2 3 4

¾µÖ×ŒŸÖ ÛúÖê Ûú´Ö ÃÖê Ûú´Ö 18 ×´Ö»ÖßÝÖÏÖ´Ö »ÖÖêÆêü, 21 ×´Ö»ÖßÝÖÏÖ´Ö Ûîú×»¿ÖµÖ´Ö †Öî ü 16 ×´Ö»ÖßÝÖÏÖ´Ö ×¾Ö™üÖ×´Ö®Ö Ûúß †Öî ü †Ö¾Ö¿µÖÛúŸÖÖ Æîü … X †Öî ü Y Ûúß 1 ÝÖÖê»Öß Ûúß Ûúß´ÖŸÖ ÛÎú´Ö¿Ö: ` 2 †Öî ü ` 1 Æîü … ¯ÖÏŸµÖêÛú ÝÖÖê»Öß Ûúß ×ÛúŸÖ®Öß ÃÖÓÜµÖÖ ¾µÖ×ŒŸÖ ÛúÖê »Öê®Öß “ÖÖ×Æü‹ ×Ûú ‰ú¯Ö¸ü ¤üß ÝÖ‡Ô †Ö¾Ö¿µÖÛúŸÖÖ Ûú´Ö ÃÖê Ûú´Ö Ûúß´ÖŸÖ ´Öë ¯Öæ üß ÆüÖê •ÖÖ‹ … ‹Ûú ¸îü×ÜÖÛú ¯ÖÏÖêÝÖÏÖ´Ö®Ö ÃÖ´ÖÃµÖÖ ²Ö®ÖÖÛú¸ü ÝÖÏÖ±ú «üÖ¸üÖ Æü»Ö Ûúß×•Ö‹ …

In order to supplement daily diet, a person wishes to take X and Y tablets. The

contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given

as below :

Tablets Iron Calcium Vitamin

X 6 3 2

Y 2 3 4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of

calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is ` 2 and

` 1 respectively. How many tablets of each type should the person take in order to satisfy

the above requirement at the minimum cost ? Make an LPP and solve graphically.

��

��

��

��

��

� �� !��"�#��$�%$��&��'(��)��*+��,��-�� -�� ./�� $��0��%�� 1��2��31�45��

��(��$%� ��(��(��)��1 ��./��%��#��#1�6��1��*+��,��4�*+��"�� -��%�

� ��

��

�� ! ��

��"��#��

��

��

�� !�

��

��

��

�� !��"�#��

�� $�%�� &"'(��)� ��*�� +��

�� $�%� , �� -.��"'(�� *�� +��

�� $�%��-.��"'(��*�� +��

�� '(�� $��/0�� )�� $+�

��

��

��

�� !��"��#��$��#��#��%��

�� !��&��%$��#��#��%��

�� !��'��%$��#��#��%��

�� (��)��(�)��%��

"�� !�

�� #� $!��%�&

��!��"�� #�� $�'�� (��)��*��(��

�� *��"��7�� -$�� i j k∧∧ ∧→

� � �a 2 2 ��!�� j k→ ∧∧� �b ��

�(�� $��

i j k∧∧ ∧→

� � �a 2 2 �� j k→ ∧∧� �b �

�� 0�� %�%�� 8"� �8�1� #�9�3�� "� ��7��:� �� ;�� #1�1� �#� %�#!1�� :� �-$ $�� %�%�� #�%��

"� ��&�#��

x

x x

� ��

�

3 2 8

3 2��:��<��"��$

'��&��

x

x x

� ��

�

3 2 8

3 2#��

+� *��(��,!�� -$��(��#��:��,!�� 23 a b→→

� ��!��2 a 3 b→→

�

��:�� 1� �� -�-�&��%�(��#��(�0�� $�� )��

�� 23 a b→→

� ��2 a 3 b→→

� �� %�(��

+��

�� *�� "��2��<��"��$:��#�#�#��*�#=��>��#��*�#��%�#��>-�&��0��

"�� #��*��%��#�#��*�+��

� ��'8�� ,��8��%�� '��#�9�3��"��2�� %��$� >

2 1 3 1 1 0

2 0 2 0 1 1

��

,�� %�� +��(

2 1 3 1 1 0

2 0 2 0 1 1

��

�� %

�#� $!��%�,

��!��-��.�"�� #�� +��$��

'�� -��.�(��)�+��* ��(��

-� ��1��#�� (��-%#��.��,��4� ��-��"��2��#�*��:��!�� <��"��$

! ��-%#��.�� #��#�*��"��

�� !�

/� *��(��4��"��$��(��#��/-�#�*#��.�#��0-�#��#�1.��1� "��-��23�� ?)��"�� 1��2��8"�<��"��$��-��23�� !�(��"��

"�� /-�#�*#��.��0-�#��#�1.�� 23��/�� 23��

.� <��"��$� >� x x x x∫ � � � 2(3 1) 4 3 2 d

"��(� x x x x∫ � � � 2(3 1) 4 3 2 d

�0� $��8�4��"��#��@��8��$� i j k∧∧ ∧

� �2 4 5 �#�� i j k∧∧ ∧

� �2 2 3 �� 1��2��A��<��"��$ ��1��2��A��%��8�4�� =��B� � <��"��$

! ��)�� i j k∧∧ ∧

� �2 4 5 �� i j k∧∧ ∧

� �2 2 3 �

"�� ,�� #��

�� C�"��!�A�� $��1�+�� #1�� "��2�� <��"��$�� 4��#=�� ,��4� ��

"�� +��

��

�� $��B��-� ��$��9��D��"��*��*��(&/"��7��E��"��#F�!��1�� 1��9��D��(��B��2�4�� (&/"��7��E��-� �)��&/�� C��"�"4��"��1� "��"��;�� <�� "��$

�&��

$��!� ��*�%�� ?)��%��(��,!�� "�%�4�#��B��4%�4 � � ��"�D�� !� ��8"�%��B��G

'��#�� *�� #�� ! �� *��"�� +�� 4��

!�

/��*��!��-�� .�� $ �� 5

�"� $��0H�,0��(�2&��I#��(�6��J��6��1�� (�2&��6K��#��3��(�2&��%6�K�� 0H�,0��%#7�� K��E��#� ��F��0H�,0��#� ��(� "��(�2&��6�� %��:��*��K��E�� #�9�3�� 1�6� �� <�� "��$� �� 0H�,0�� 6�� 1�� E�K��L�$��&��$%� ��3L��4��%��G

/� �� ! �� 6�� %6��! ��%#7��8��#�� #�� ,��+�� #�� '�� 8��'��$ �� 5

-�� !�

�+� ��-��.��=��#1�� "��$

�&��

��#�%��-��.��-��.��:�� M��"��$�� y yx x y

xx� � �

22

2d d

0dd

��

9��-��.��

!�

'��#�%��-��.��-��.#�� y y

x x yxx

� � �2

22

d d 0

dd�

�� $�� "��$�>��-��.��

�&��

�� yx� �� 1 1cos cos a b

��:�� M��"��$� �� xy yx� ��

222

2 2 2 cos sinaba b

��

:�� (��-��.��

!�

'��yx� �� 1 1cos cos

a b#��

xy yx� ��

222

2 2 2 cos sinaba b

�� %�-��%�.��!��#��%�-��%�.��:�� t 4�

� �� yx

dd

�<��"��$

'��%�-��%�.��#��%�-��%�.#�� yx

dd

�� t 4�

� �

/��

�-� #1�� "��2�� "��$� >

y yy x x y

x x� � �

d d

d d:�� (

y yy x x y

x x� � �

d d

d d

�/� �� <��"��$� >�

xx

x x∫�

�

22

0 sinsin d

cos

�&��

��<��"��$� (� x x x∫ �

32

0 cos d

;��(�

xx

x x∫�

�

22

0 sinsin d

cos

!�

;��(� x x x∫ �

32

0 cos d

�.� <��"��$� >� x xx x∫

� �

2

4 2 d 2

"��(�x x

x x∫� �

2

4 2 d 2

.�� !�

��

�#� $!��%��

��!��0��"�� #�� $��

'�� 0��(��)��* ��(��

�0� ��2��%�2�6��A��%�� M��"��$��/0��$��C(�� 8��

� � � �

� � �2 2 2

1 1 11 cosA 1 cosB 1 cosC 0

cos A cosA cos B cosB cos C cosC

��

�&��

$��"�� 1�8@�� </=#� <0=�#�� <�=�� "�3��;��$��$�� %��-��"�� "1��</=��*��:�<0=��#��<�=��%��1��-��"��(��-�� </=��1��:� <0=��%��#��<�=��>��-��"�� #�9�3��1�6��;��3L��<��"��$

,��#� �� /0��(

� � � �

� � �2 2 2

1 1 11 cosA 1 cosB 1 cosC 0

cos A cosA cos B cosB cos C cosC

!�

/� �� </=#�<0=��<�=��?�� %�� @�� *�� </=�� #��<0=�� %��<�=�� 1��$ ��: �� 1��</=�� #�%��<0=�� <�=�� >��,��+�� #��

�0��

�� -��</=�#��<0=�� </=��%6��0H��#��6�B��,B��$��&��(��<0=��*6��0H��#��6�B��,B��$��&�� N"��,!��"=�2��(�� <�� #�� *�� B�� $� �� %� ��O%��O� ��0H��#�� %� ��O%��OB��,B��$��&��"�#�1�� </=��3L��O%��O�#��<0=��3L��7��O%��O��#� �-�C�� "��$��*��;��"��"�-��%� ��"��$� �� "�� P�� ;1�� "�#�1�� 3��"� ��$

! �� </=��<0=��</=��%6��6�� <0=��*6��6�� /�� #�� %��%�� '��</=��<0=��7��#��

�� M��"��$��C(��8��:��Q��$��#��1��-"��%��:��F�3�� 6 3 r ��

�&��

��$��2��8��2�4��!��$��8��%��%��:��4�$��8��

�� =��B� �#�6��%��(��*��("��2��3� ��%�

��

��6 3 r �

!�

'�� #� �� +��#��

��3� �

�� !�

�"� %��#?)�� -��(� �� #��;�� 2�� %$� �� *+��+��,!�� %$:��-��(�� "��7��(�0��<�� "��$ � � (�0�� R�� !�� 2�� 8"� <�� "��$ "�� +�� %��'�� #� � �� 8��

�+� ��M� ��"��$� �� 1��#��*��#��*##�*��1%4� �� =��B� � �� "�� (��(�� 8�%�� (�0�� -�#��#��*#�#�*�#��#��C��(M�� #��*��*#�� #��*#�#�*��#��

�� 4�$� �� $�� C#�6��"� �� /��A��B� �� 8"� #� �� /� �� $��C�� 8��P�� 1�� !�� 4�� /�� ;��#1�1� <�� "��$��!�� M��"��$� ��/�� ;��#1�1� 9��;��2�"�� : �� /��A��B��#��/��/��/�� /�� /��

�� (��#�� ,!�� i j k∧∧ ∧

� �2 3 4 �� ( )i j k∧∧ ∧→

�� r . 2 3 26 0 ��

-"�� %$� '(� �� ,!�� !�� '(1�S� �3��"� <�� "��$ � � � � ��('(� 8"� <�� "��$ "��

�� i j k∧∧ ∧

� �2 3 4 � ��

( )i j k∧∧ ∧→

�� r . 2 3 26 0 ��/��

Date post:	17-Jul-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

Class XII Mathematics Set-3 Time: 3 hrs M.M: 100 Marks ...icsk-kw.com/pdf/pqp/12/mat/mat12-1.pdf ·...

Documents