SADVIDYA COMPOSITE PRE-UNIVERSITY COLLEGE

NO 7, NARAYANASHASTRY ROAD MYSORE – 24

II PUC PREPARATORY EXAMINATION – 2017-18
Time: 3Hrs.15 Min MATHEMATICS [35] Max.Marks:100

General instructions:
1. Candidates are required to give their answers in their own words as far as practicable.
2. Figures in the right hand margin indicate full marks.
3. While answering the candidate should adhere to the word limit as far as practicable.
4. 15 minutes of extra time has been allotted for the candidates to read the questions.
Instructions: i) The question paper has five parts A, B, C, D & E. Answer all the parts.
ii) Use the graph sheet for the questions on Linear programming problems.

Part – A
I. Answer ALL the ten questions: 10 x 1 = 10
1. Let ‘ ¿’ be a binary operation defined on set of all non-zero rational numbers by
ab
a∗b= . Find the identity element.
4
2x
−1
2. Write the set of values of ‘ x ’ for which 2 tan ( x )=tan
−1
( 1−x )
2 holds.

3. Construct a 23 matrix, whose elements are given by a ij=|i− j|.

|3 x| |3 2|
4. Find the values of x for which x 1 = 4 1

5. Differentiate y=cosec ¿ ) w.r.t. x .

dx
6. Evaluate ∫ sin x ∙ cos x .
7. Define collinear vectors.
8. Find the equation of the plane having intercept 3 on y-axis and parallel to ZOX plane.
9. Define ‘optimal solution’ of L.P.P.
7 9 4
10. If P ( A )= 13 , P ( B )= 13 and P ( A ∩B )= 13 , then evaluate P ( A∨B ) .

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 Part – B
II. Answer any TEN questions: 10 x 2 = 20

11.A relation R is defined on the set A={1,2,3,4,5,6 } by R={(x , y)/ y is divisible by x } . Verify
whether R is symmetric and reflexive or not. Give reason.
−1 cos x−sin x π
( )
12.Write the simplest form of tan cos x +sin x , 0< x< 2 .

π −1 −1
(
13.Evaluate sin 3 −sin 2 . ( ))
14.Find the area of the triangle formed by the vertices (3,8) ,(−4,2) and (5,1) using
determinants.
dy y
15.If √ x+ √ y =√ a , prove that
dx √
=− .
x
dy
16.Find dx , if sin2 x+ cos2 y=k , where k is a constant.

x 1+ sinx
(
17.Evaluate ∫ e 1+cosx dx . )
−1
x sin x
18.Evaluate ∫ dx .
√1−x 2
19. If a⃗ =5 i−
^ ^j−3 k^ and b= ^ 3 ^j−5 k^ , then show that the vectors (a⃗ + ⃗b ¿ and (a⃗ −b⃗ ¿ are
⃗ i+

perpendicular.
20. Find λ , if the vectors i+3
^ ^j+ k^ , 2 i−
^ ^j−k^ and λ i+7
^ ^j+3 k^ are coplanar.

21.Find the local maximum value of the function g ( x )=x 3−3 x .

22. Find the order and degree of the differential equation
2 3 2
d y dy dy
( )( ) ( )
dx
2
+
dx
+sin
dx
+ 1=0.

23. Find the angle between the pair of lines

^ ^j+ ^k) and
^ ^j−k^ ) +( i+
r⃗ =( 3 i+5
^ 4 k^ ) + μ ¿).
r⃗ =( 7 i+

24.Two cards are drawn at random without replacement from a pack of 52 playing cards.
Find the probability that both the cards are black.

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 Part – C
III. Answer any TEN questions: 10 x 3 = 30
25.Verify whether the function f : A → B , where A=R−{3} and B=R−{1 }
x−2
defined by f ( x)= x−3 is one-one and onto or not. Give reason.

−1 x−1 −1 x+1 π
( ) ( )
26. If tan x−2 + tan x+2 = 4 , find x .

27.If A and B are square matrices of same order, then show that ( AB )−1=B−1 A−1.
28. Verify mean value theorem for the function f ( x )=x 2−4 x−3 in [1,4]
29. Prove that the curves x= y 2 and xy=k cut at right angles, if 8 k 2 =1.
2
x
30. Express ∫ e dx as limit of a sum.
0

dx
31.Find ∫ 1+tanx .

32.Find the area of the region bounded by the curve y=x 2 and the line y=4 .
33. Form the differential equation of the family of circles touching the y-axis at the origin.
34. If a⃗ =2 i+2
^ ^j +3 k^ , b=−
⃗ ^ ^j+ k^ and c⃗ =3 i^ + ^j are such that (a⃗ + ⃗b ¿ is perpendicular to c⃗ , then
i+2
find the value of .
35.If three vectors a⃗ , b⃗ , c⃗ satisfies the condition a⃗ + ⃗b+ c⃗ = ⃗0. Evaluate the quantity
μ=⃗a ∙ b⃗ + ⃗b ∙ ⃗c + ⃗c ∙ ⃗a, if |⃗a|=1 ,|b⃗|=4 and |c⃗|=2.

36. Find the vector equation of the plane passing through the intersection of the planes
^ ^j+3 k^ )=9 and through the point (2,1,3).
^ ^j−3 ^k )=7 , r⃗ ∙ ( 2 i+5
r⃗ ∙ ( 2 i+2

√ 1+ x 2−1 , then prove that dy 1

37. If y=tan −1 ( x ) dx 2(1+ x 2 ) .
=

38.Find the probability distribution of number of heads in two tosses of a coin.

Part – D
IV. Answer any SIX questions: 6 x 5 = 30
39.Let f : N → R be defined by f ( x )=4 x 2 +12 x +15. Show that f : N → S , where ‘ S’ is range of f is
invertible. Also find inverse of f .
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 0 6 7 0 1 1 2

[ ] [ ] []
40. If A= −6 0 8 , B= 1 0 2 and C = −2 . Calculate AC , BC and ( A+ B)C . Also, verify
7 −8 0 1 2 0 3

that ( A+ B)C= AC+ BC .

41. Solve the following system of equations by matrix method:
3 x−2 y+ 3 z =8 , 2 x + y−z=1 , 4 x−3 y+ 2 z=4.

42. Find the integral of √ x 2+ a2 w.r.t. x and hence evaluate ∫ √ 4 x 2+ 9 dx.

2
43. If y=( tan−1 x ) , then show that ( x 2+ 1)2 y ' ' +2 x ( x 2+1 ) y ' ¿ 2 .
44. A particle moves along the curve 6 y=x 3 +2. Find the points on the curve at which the y-
co-ordinates is changing 8 times as fast as the x-co-ordinate.
dy
45.Find the particular solution of the differential equation + y cot x=4 x cosec x , x ≠ 0 given
dx

π
that y=0, when x= 2 .

46.Find the area enclosed between the two parabolas y 2=4 ax and x 2=4 ay by the method of
integration.
47.Derive the equation of the line in space passing through a point and parallel to a vector
both in vector and cartesian forms.
48. If a fair coin is tossed 10 times, find the probability of
(a) exactly six heads. (b) at least six heads.
PART- E
V. Answer any ONE of the following questions: 1x10 =10
π
b b 3

49. a) Prove that ∫ f ( x ) dx=∫ f ( a+b−x ) dx and hence evaluate ∫ 1+√dxtanx . (6)
a a π
6

x x 2 yz
b) Prove that

of determinants.
| y
z
2

2
z xy
|
y zx = ( x − y )( y −z )( z−x ) (xy + yz + zx ) using properties

(4)
50. a) Maximize and minimize Z=3 x+ 9 y subjected to the constraints
x +3 y ≤ 60 , x + y ≥ 10 , x ≤ y , x ≥0 , y ≥0. Solve this LPP graphically. (6)

b) Find the values of a and b , such that the function defined by

5 ,if x ≤ 2
{
f ( x )= ax +b ,if 2< x <10 , is a continuous function.
21 ,if x ≥10
(4)

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**************

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SADVIDYA COMPOSITE PRE UNIVERSITY COLLEGE
#7, NARAYANASHASTRY ROAD, MYSORE-570024
II PUC SECOND PREPARATORY -2017
 SADVIDYA COMPOSITE PRE-UNIVERSITY COLLEGE
NO 7, NARAYANASHASTRY ROAD, MYSORE-570024
II PUC – FIRST PREPARATORY EXAMINATION –2019-20
MATHEMATICS (35)
Time: 3 Hours 15 Minutes] [Total No. of questions: 50] [Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Use the Graph sheet for the question on Linear programming problem in part-E.

Part – A
I. Answer ALL the ten questions: 10 x 1 = 10
1. Define a diagonal matrix.
2. Find the value of tan−1 ( √3 )−cot −1 (−√ 3 ) .
3. Find the number of possible matrices of order 3 ×3 with entries 0 or 1.
4. If A is an invertible matrix of order 2, find det( A−1 ).
dy
5. Find dx if y=cos−1 (sinx)
2
se c x
6. Evaluate ∫ dx .
cosec 2 x
7. Find the value of i∙^ ( ^j× k^ ) + ^j ∙ ( i^ × k^ ) + k^ ∙ ( i^ × ^j ).
8. Find a unit vector in the direction of a⃗ =2 i+3
^ ^j+ k^

x+3 y −5 z +6
9. The Cartesian equation of a line is 2 = 4 = 2 . Find the vector equation of the line.

10.Define Feasible region of a linear programming problem.

Part – B
II. Answer any TEN questions: 10 x 2 = 20
11.Show that the signum function f : R → R , given by
1 , x> 0
{
f ( x )= 0 , x =0 is neither 1-1 nor onto
−1 , x< 0
−1 −1 π
12.Solve tan (2 x ) +tan ( 3 x )= 4 .
2
13.Simplify tan−1 √
1+ x −1
[
, x ≠ 0.
x ]
14.Find the value of k if the area of a triangle is 35 sq units and vertices are
( 2 ,−6 ) , ( 5 , 4 )∧( k , 4 ) .
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 2 3 2 −2
[ ]
15.Find X ∧Y if 2 X +3 Y = 4 0 ∧3 X +2 Y = −1 5 [ ]
dy xlog e a− y
16.If x y =a x prove that dx = x log x
e

17.Differentiate sin2 x with respect to e cosx .

18.Find points at which the tangent to the curve y=x 3−3 x 2−9 x +7 is parallel to the x−axis .
19.Evaluate ∫ cotx∙ log ( sinx ) dx .
2
3
20.Find dx .
∫ 4 +9 x2
0

21.If a⃗ =5 i−
^ ^j−3 k^ and b= ^ 3 ^j−5 k^ then show that the vectors a⃗ + ⃗b∧⃗a−b⃗ are perpendicular.
⃗ i+

22.If the position vectors of the points A & B are respectively i+2
^ ^j−3 k^ ∧ ^j− k^ , than find the

direction cosines of ⃗
AB .

23.Find the distance of the point (3 ,−2, 1) from the plane 2 x− y +2 z +3=0.
24.Find the vector equation of a plane that passes through the point (1, 0, -2) and the normal
to the plane is i+
^ ^j− k^ .

Part – C
III. Answer any TEN questions: 10 x 3 = 30
25.If R1∧R 2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence
relation.
−1 √ 1+ x− √ 1−x π 1
26.Prove that tan
√ 1+ [
x + √ 1−x 2 ]
= − cos−1 x ,
4
−1
√2
≤ x ≤ ,1

27.By using elementary row operations find the inverse of 5 7 [2 3 ]

1 2 0 0
[ ][ ]
28.For what values of x ,[ 12 1 ] 2 0 1 2 =O
1 0 2 x

dy 3 y
29.If x=acos 3 θ , y =a sin3 θ , prove that
dx
=−
√
x
30.Verify Rolle’s theorem for the function f ( x )=x 2 +2 x−8, x ∈ [−4 , 2]
dy −1
31.If x √1+ y + y √1+ x=0 ,−1< x <1 prove that dx = 2
(1+ x)

32.Find the approximate value of f ( 3.02 ) , f ( x )=3 x 2+ 5 x +3

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 x dx
33.Evaluate ∫
( x−1 )2 ( x +2)
x
( x ¿¿ 2+1) e
34. Evaluate ∫ dx ¿
( x +1 )2

35. Show that the vectors a⃗ , b∧⃗

⃗ c are coplanar if a⃗ + ⃗b , ⃗b+ c⃗ ∧⃗c + ⃗a are coplanar.

36. If a⃗ , b⃗ ,∧⃗c are three vectors such that |⃗a|=3 ,∨⃗b∨¿=4, |c⃗ ∨¿=5 and each one of them being
perpendicular to the sum of the other two, find ¿ a⃗ + ⃗b+ c⃗ ∨.
37.Show that the points A (−2 i+
^ 3 ^j+5 k^ ) , B ( i+2 ^ k^ ) are collinear
^ ^j+3 k^ ) , C ( 7 i−

38.Find the angle between the planes 3 x−6 y +2 z=7∧2 x +2 y−2 z=5.

Part – D
IV. Answer any SIX questions: 6 x 5 = 30
0 6 7 0 1 1 2
39.If A= −6
[ 7 −8 0 ] [ ] []
0 8 , B= 1 0 2 , C= −2 verify that ( A+ B ) C=AC +BC .
1 2 0 3

1 0 2
40.If A=
[ ]
0 2 1 , Prove that A3 −6 A 2 +7 A +2 I =0.
2 0 3

2 −3 5
41.If A= 3
[ ] 2 −4 find A−1 . Using A−1solve the system of equations
1 1 −2
2 x−3 y+ 5 z =11, 3 x+ 2 y −4 z=−5 , x + y−2 z=−3
42.If y=3 cos ( logx ) +4 sin ( logx ) ,show that x 2 y 2 + x y1 + y =0.
2
2 d y dy 2
43.If y=ea cos x ,−1 ≤ x ≤1 , show that ( 1−x ) 2 −x dx −a y =0.
−1

dx
44.A man of height 2mts walks at a uniform speed of 5km/hour away from a lamp post of
6mt high. Find the rate at which the length of his shadow increases.
45.Sand is pouring from a pipe at the rate of 12cm3/s. The falling sand forms a cone on the
ground in such a way that the height of the cone is always one-sixth of the radius of the
base. How fast is the height of the sand cone increasing when the height is 4cm.
dx dx
46.Find ∫ 2 2
∧¿ ¿hence evaluate ∫ .
√ a −x √ 7−6 x −x2
47.Find ∫ √a 2−x 2 dx and hence evaluate ∫ √1−4 x 2 dx .

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48.Derive the equation of a line passing through a point A( x 1 , y1 , z 1) and parallel to a vector b⃗
both in vector and Cartesian form
Part - E
V. Answer any ONE of the following questions: 1x10 =10
49.a) Minimize and Maximize Z ¿ 5 x+10 y subject to the constraints
x +2 y ≤ 120 ,
x + y ≥ 60 ,
x−2 y ≥ 0 ,
x , y ≥0 [6]
b) Using properties of determinants prove that
2 2
1+a −b 2 ab

| |
−2b
2 2 2 2 3
2 ab 1−a +b 2a =( 1+a + b ) [4]
2 2
2b −2 a 1−a −b
a

{
a
2∫ f ( x ) dx , f ( x ) is even
50. a) Prove that ∫ f ( x ) dx= 0
−a
0 , f ( x ) is odd
π
4

and hence evaluate ∫ sin 2 x dx . [6]

−π
4

5 , x≤2
b) Find the values of a & b if the function defined by f ( x ) = ax +b , 2< x <10
21 , x ≥ 10 {
is a continuous function. [4]

************

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 SADVIDYA COMPOSITE PRE-UNIVERSITY COLLEGE
No-7, Narayana Shastry Road, Mysore
II PUC – SECOND PREPARATORY EXAMINATION – 2020
MATHEMATICS (35)
Time: 3 Hours 15 Minutes] [Total No. of questions: 50] [Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Use the Graph sheet for the question on Linear programming problem in part-E.

Part-A
I Answer ‘ALL’ the questions (One mark each): 10x1=10
𝑎𝑏
1) Let ‘∗’ be a binary operation defined on Q the set of all rational numbers by a ∗ b = .
4

Find the identity element.

2𝜋
2) Find the principal value of sin−1 (𝑠𝑖𝑛 ).
3

3) If A is a square matrix of order 3 and |A| = 4, then find |adjA|.

4) Differentiate : y = log(cos(ex)) with respect to x.
5) Evaluate:∫ 𝑠𝑒𝑐𝑥(𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥)𝑑𝑥.
6) Show that the vectors 𝑎⃗ = 2𝑖̂ − 3𝑗̂ + 4𝑘̂ and 𝑏⃗⃗ = −4𝑖̂ + 6𝑗̂ − 8𝑘̂ are collinear.
𝑥+2 𝑦−3
7) If ( ) is a scalar matrix, find x and y.
0 4
8) Find the equation of the plane having intercept 4 on the z-axis and parallel to xoy plane.
9) Define the term ‘Feasible solution’ of L.P.P.
10) If P(A) = 0.6, P(B) = 0.3 and P(A∩B) = 0.2, then find P(A/B).

Part - B
II Answer any ‘TEN’ of the following questions: 10X2=20
1
11) Find fog and gof, if f(x) = 8x3 and g(x) = 𝑥 3 .
𝜋 1
12) Evaluate: sin ( − sin−1 (− )).
3 2
1−𝑥 1
13) Solve: tan−1 ( )= tan−1 (𝑥).
1+𝑥 2

14) Using determinants, show that the points A(a,b+c), B(b,c+a) and C(c,a+b) are collinear.

1
 𝑑𝑦
15) Find , if xy = yx.
𝑑𝑥
𝑑𝑦
16) If x2+xy+y2 = a2, find .
𝑑𝑥
𝑥−1
17) Find the slope of the tangent to the curve y = , x ≠ 2 at x = 10.
𝑥−2
𝑑𝑥
18) Evaluate: ∫ .
𝑥+𝑙𝑜𝑔𝑥

tan4 (√𝑥).sec2 (√𝑥).𝑑𝑥

19) Evaluate: ∫ .
√𝑥

20) If two vectors 𝑎⃗ and 𝑏⃗⃗ are such that |𝑎⃗| = 2 , |𝑏⃗⃗| = 3 and 𝑎⃗. 𝑏⃗⃗ = 4. Find |𝑎
⃗⃗⃗⃗ − ⃗⃗⃗⃗
𝑏|
21) Find the projection of the vector 𝑖̂ + 3𝑗̂ + 7𝑘̂ on the vector 7𝑖̂ − 𝑗̂ + 8𝑘̂ .
𝑥−5 𝑦+2 𝑧 𝑥 𝑦 𝑧
22) Show that the lines = = and = = are perpendicular.
7 −5 1 1 2 3

23) A die is thrown. If E is the event “the number appearing is a multiple of 3” and F be
event “the number of appearing is even”, then prove that E and F are independent
events.
3
𝑑2𝑦 𝑑𝑦 4
24) Find the order and degree of the differential equation ( 2
) + ( ) + sin2 𝑦 = 0.
𝑑𝑥 𝑑𝑥

Part - C
III Answer any ‘TEN’ of the following questions: 10X3=30
25) Show that relation R on the set A = {1,2,3,4,5} given by R = {(a,b) / |a – b| is even} is an
equivalence relation.
𝑥−1 𝑥+1 𝜋
26) Find the value of x, if tan−1 ( ) + tan−1 ( )= .
𝑥−2 𝑥+2 4

1 −1
27) If A = [ ], find A-1 using elementary operations.
2 3
𝑑𝑦 𝑡
28) Find , if x = a[cost + log(𝑡𝑎𝑛 ( ))] , y = asint.
𝑑𝑥 2

29) Verify mean value theorem for the function f(x) = x3 – 5x2 – 3x in [a,b], where a = 1 and
b = 3.
30) Find two positive numbers, whose sum is 15 and sum of whose squares is minimum.
1+𝑠𝑖𝑛𝑥
31) Evaluate : ∫ 𝑒 𝑥 ( ) 𝑑𝑥.
1+𝑐𝑜𝑠𝑥
2
32) Evaluate : ∫0 (𝑥 2 + 1)𝑑𝑥 as the limit of a sum.

2
 33) Prove that [𝑎 ⃗⃗⃗⃗, 𝑏
⃗⃗⃗⃗ + 𝑏 ⃗⃗⃗⃗ + 𝑐⃗⃗⃗, 𝑐⃗⃗⃗ + 𝑎 ⃗⃗⃗⃗, ⃗⃗⃗⃗
⃗⃗⃗⃗ ] = 2[𝑎 𝑐 ].
𝑏 , ⃗⃗⃗

34) Three vectors 𝑎 ⃗⃗⃗⃗ 𝑎𝑛𝑑 𝑐⃗⃗⃗, satisfy the condition ⃗⃗⃗⃗
⃗⃗⃗⃗, 𝑏 𝑎 + 𝑏⃗⃗⃗⃗ + 𝑐⃗⃗⃗ = 0. Find the value of the

quantify 𝜇 = 𝑎 ⃗⃗⃗⃗ + 𝑏
⃗⃗⃗⃗ . 𝑏 ⃗⃗⃗⃗. 𝑐⃗⃗⃗ + 𝑐⃗⃗⃗. ⃗⃗⃗⃗,
𝑎 𝑖𝑓 |⃗⃗⃗⃗ ⃗⃗⃗⃗| = 4 𝑎𝑛𝑑 |𝑐⃗⃗⃗| = 2.
𝑎 | = 1, |𝑏
35) Find the Cartesian and vector equation of the line that passes through the points ( -1,0,2)
and (3,4,6)
36) A die is tossed thrice. Find the probability of getting an odd number at least once.
37) Form the differential equation of family of circles touching the y-axis at the origin.
𝑥2 𝑦2
38) Find the area of the region bounded by the ellipse 2
+ = 1, by the method of
𝑎 𝑏2

integration.

Part - D
Answer any ‘SIX’ of the following questions: 6x5=30
39) Consider f: R→[ - 5,∞) given by f(x) = 9x2 + 6x – 5. Show that f is invertible with
√𝑦+6−1
f -| = { }.
5

0 6 7 0 1 1 2
40) If A = [−6 0 8], B = [1 0 2] and C = [−2]. Calculate AC, BC and (A+B)C.
7 −8 0 1 2 0 3
Verify that (A+B)C = AC + BC.
41) Solve the following system of equations by using matrix method. 3x – 2y + 3z = 8,
2x + y – z = 1 and 4x – 3y + 2z = 4.
42) If y = Aemx + Benx, then prove that y|| – (m+n)y| + mny = 0.
43) The volume of a cube is increasing at the rate of 9 cubic cm per second. How fast is the
surface area increasing when the length of an edge is 10 cm?
𝑑𝑥 𝑑𝑥
44) Find the integral of ∫ 𝑎𝑛𝑑 ℎ𝑒𝑛𝑐𝑒 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∫ 2 .
𝑎2 +𝑥 2 𝑥 −6𝑥+13

45) Using integration, find the area of the region bounded by the triangle whose vertices are
( -1, 0), (1,3) and (3,2).
46) Derive the equation of a plane passing through the given points and perpendicular to the
given vector.

3
 47) A die is thrown 6 times. ‘If getting an odd number is success’. What is the probability
of (i) 5 success?
(ii) atleast 5 success?
(iii) atmost 5 success?
𝑑𝑦 1
48) Find the particular solution of the differential equation (1+x2) + 2𝑥𝑦 = , when
𝑑𝑥 1+𝑥 2

y = 0 and x = 1.
Part - E
Answer any ‘ONE’ completely from the following questions: 1x10=10
𝜋
𝑏 𝑏 1
49) a) Prove that ∫𝑎 𝑓 (𝑥)𝑑𝑥 = ∫𝑎 𝑓 (𝑎 + 𝑏 − 𝑥)𝑑𝑥 and hence evaluate ∫ 𝜋
3
1+
𝑑𝑥. (6m)
6 √𝑡𝑎𝑛𝑥

𝑥 𝑥2 𝑦𝑧
b) Prove that |𝑦 𝑦2 𝑧𝑥 | = (𝑥 − 𝑦)(𝑦 − 𝑧)(𝑧 − 𝑥)(𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥). (4m)
𝑧 𝑧2 𝑥𝑦

50) a) Minimize and maximize z = 3x + 9y subjected to the constraints x + 3y ≤ 60;

x + y ≥ 10; x ≤ y; x,y≥ 0 by graphical method. (6m)
𝑘𝑥 + 1; 𝑖𝑓 𝑥 ≤ 5
b) Find the value of k, so that the function f(x) = { is a continuous
3𝑥 − 5; 𝑖𝑓 𝑥 > 5
function. (4m)

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 SADVIDYA COMPOSITE PRE-UNIVERSITY COLLEGE
No-7, Narayana Shastry Road, Mysore
II PUC – SECOND PREPARATORY EXAMINATION – 2021
MATHEMATICS (35)
Time: 3 Hours 15 Minutes] [Total No. of questions: 50] [Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Use the Graph sheet for the question on Linear programming problem in part-E.

Part-A
I Answer ‘ALL’ the questions (One mark each): 10x1=10
ab
1) Let ¿ be an operation defined on the set of all non zero rational number by a ¿ b = ,
4
find the identity element.
2) Write the principal value branch of sec-1(x).
3) What is the number of possible square matrices of order 3 with each entry 0 or 1?

4) Find the values of x for which x 1 = 4 1 |3 x| |3 2|

dy
5) Find , if y = sin(cos(x2)).
dx

6) Evaluate : ∫ cosec
2
( x2 ) dx
7) If the vector ⃗ ^ ^j+ k^ , ⃗
AB=2 i− ^
OB =3 i−4 ^j+ 4 k^ , find the positive vector ⃗
OA
8) If a line makes angle 900, 600 and 300 with the positive direction of x, y and z axis
respectively, find its direction cosines.
9) Define optional solution in a linear programming problem.
7 9 4
10)If P(A) = , P(B) = and P(A∩B)= . Evaluate P(A / B).
13 13 13
Part - B
II Answer any ‘TEN’ of the following questions: 10X2=20
a+b
11)On R, ¿ is defined by a ¿ b = , verify whether ¿ is associative.
2

12)Evaluate : tan-1(1) + cos-1 ( −12 ) + sin ( −12 ).

-1

1
 13)If the area of the triangle with the vertices (-2, 0), (0,4) and (0,k) is 4 square units, find
the values of k using determinants.
14)Differentiate : y = (sinx)cosx with respect to x.
dy
15)Find , if sin2x + cos2y = 1.
dx
d2 y
16)Find , if y = x3 + tanx.
d x2
x−1
17)Find the slope of the tangent to the curve y = , x≠2 at x = 10.
x−2
4
tan √ x . sec2 √ x . dx
18)Evaluate : ∫
√x
dx
19)Evaluate : ∫ .
sin x cos 2 x
2

20)Find the angle between the vectors a⃗ =i+

^ ^j− k^ , b=
⃗ i+
^ ^j + k^

21)Find the area of the parallelogram, whose adjacent sides are given by the vectors
^ ^j+3 k^ and b=2
a⃗ =i− ⃗ ^
i−7 ^j+ k^ .
dy
d 3 y 2 dx
22)Find the order and degree of the differential equation + y +e =0.
d x3
23)Find the distance of the point (3,-2,1) from the plane 2x – y + 2z = -3.
24)Two cards are drawn at random and without replacement from a pack of 52 playing
cards. Find the probability that both the cards are black.

Part - C
III Answer any ‘TEN’ of the following questions: 10X3=30
25)Prove that the relation R in the set Z of all integers defined by R = {(x,y) | (x – y) is an
integer} is an equivalence relation.
26)Find the values of x and y from the following equation:

2 x 5 + 3 −4 = 7 6
[
7 ][
y−3 1 2 15 14
. ][ ]
t dy
27)If x = a[cost + log tan ] , y = asint, then prove that =tant .
2 dx

2
 2
dy cos (a+ y)
28)If cosy = xcos(a +y) with cosa ≠ 1, then prove that = .
dx sina
29)Find the intervals in which the function f given by f(x) = 2x3 – 3x2 – 36x + 7 is strictly
increasing.
1+ sinx
30)Evaluate : ∫ e
x
( 1+cosx )dx.
4
31)Evaluate : ∫ tan ( x )dx.
x
32)Find ∫ dx.
( x+ 1 ) ( x+ 2)
33)Find the area of the region bounded between the curve y2 = 4x and the line x = 3.
34)Find the equation of the curve passing through the point (-2,3), given that slope of the

2x
tangent to the curve at any point (x, y) is 2 .
y
35)Show that the position vector of the point P, which divides the line joining the points

⃗ n ⃗a
m b+
A and B having position vectors a⃗ and b⃗ internally in the ratio m:n is .
m+ n
36)Find a unit vector perpendicular to each of the vectors a⃗ + b⃗ and a⃗ - b⃗ , where
^ 2 ^j+ 2 k^ and b=
a⃗ =3 i+ ^ 2 ^j−2 k^ .
⃗ i+

37)Find the shortest distance between the lines r⃗ =i+ ^ ^j+ k^ ),

^ ^j + λ(2 i−
^ ^j−k^ + μ(3 i−5
r⃗ =2 i+ ^ ^
^j+ 2 k)

38)A bag I contains 3 red and 4 black balls, while bag II contains 5 red and 6 black balls.
One ball is drawn at random from one of the bags and it is found to be red. Find the
probability that it was drawn from bag II.
Part - D
Answer any ‘SIX’ of the following questions: 6x5=30
39)Verify whether the function f:R→R defined by f(x) = 1 + x2 is one-one, onto and
bijective.
1 2 −3 3 −1 2 4 1 2

[
40)If A = 5 0
] [ ] [ ]
2 , B = 4 2 5 and C = 0 3 2 , then compute (A+B) and (B –
1 −1 1 2 0 3 1 −2 3

C). Also verify that (A + B) – C = A + (B – C).

3
 41)Solve the system of linear equations by matrix method: 3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
42)If y = Aemx + Benx, then show that y|| – (m + n)y| + mny = 0
43)A ladder 5 mtr long is leaning against a wall. The bottom of the ladder is pulled along
the ground away from the wall at the rate of 2cm/sec. How fast is its height on the wall
decreasing when the foot of the ladder is 4 mtr away from the wall?
1 1
44)Find the integral of 2 2 with respect to x and hence find ∫ dx .
√ a −x √ 7−6 x −x2
45)Find the area of the circle x 2+ y 2=25 by the method of integration.
dy 1
46)Solve the differential equation (1 + x2) dx +2 xy= 2 , given that y = 0 when x = 1.
1+ x
47)Derive the equation of a plane in normal form (both in the vector and Cartesian forms).
48) Two balls are drawn at random with replacement from a box containing 10 black
and 8 red balls. Find the probability that (i) Both balls are red
(ii) First ball is black and second ball is red
(iii) one of them is black and other is red.
Part - E
Answer any ‘ONE’ completely from the following questions: 1x10=10
a

{
a
2∫ f ( x ) dx , if ∧f ( x ) is an even function
49) a) Prove that ∫ f ( x ) dx= 0 and hence
−a
0 ,if f ( x ) is an odd function

1
5 4
evaluate ∫ sin x . cos x dx (6m)
−1

kcosx π
b) Find the value of k, if f ( x )=
{
π −2 x
,∧x ≠

3 ,∧x=
π
2
2
is continuous at x =
π
2
. (4 m)

50) a) Maximize : z = 3x + 2y subjected to the constraints x + 2y ≤ 10; 3x + y ≤ 15, x≥ 0, y

≥ 0 by graphical method. (6m)

4
 [ 2 3]
b) If A = 1 2 satisfying the equation A2 – 4A + I = O, where I = [ 10 01] and
[0 0]
O = 0 0 . Find A-1. (4m)

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