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MARCH – 2020MATHEMATICS PAPER- IA

Time :3 Hrs Total Marks : 75 M

SECTION-A

I. Answer all the questions. Each question carries ‘2’ marks. 10 x 2 = 20M

1) If :f Q Q is defined by 5 4f x x for all x Q , find 1f x .

2) Find the domain of the real valued function 2 2x xf x

x

.

3) Give examples of two square matrices A and B of the same order for which

0AB but 0BA .

4) If

1 0 0

2 3 4

5 6

A

x

and det 45A , then Find x .

5) Show that the points whose position vectors are 2 3 5a b c , 2 3a b c ,

7a c are collinear, when , ,a b c are non - coplanar vectors.

6) Let 2 4 5a i j k , b i j k and 2c j k . Find unit vector in the

opposite direction of a b c .

7) If 2 3 5a i j k , 4 2b i j k then find a b a b and unit vector

perpendicular to both a b and a b .

8) Prove that 0 0

00 0

cos90 sin 90cot 36

cos90 sin 90

.

9) Sketch the region enclosed by siny x , cosy x and X-axis in the interval

0, .

10)If 3

sinh4

x , find cosh 2x and sinh 2x .

Disclaimer: This Question paper is purely for preparation purpose only. Manabadi.com is no where claiming this to be the Main Examination Paper.

SECTION-B

II. Answer ANY FIVE questions. Each question carries ‘4’ marks. 5 x 4 = 20M

11)If 1 0

0 1I

and 0 1

0 0E

then, show that 3 3 23aI bE a I a bE .

12) , ,a b c are non – coplanar vectors. Prove that the four points 6 2a b c ,

2 3a b c , 2 4a b c , 12 3a b c are coplanar.

13)Find the vector having magnitude 6 units and perpendicular to both 2i k

and 3 j i k .

14)Prove that 4 4 4 43 5 7 3sin sin sin sin

8 8 8 8 2

.

15)Find the general solution of the equation 21 sin 3sin cos .

16)Prove that 2

1 12

1cos Tan sin cot

2

xx

x

.

17)In triangle ABC, show that 2 2 2cot cot cotabc

a A b B c CR

.

SECTION-C

III. Answer ANY FIVE questions. Each question carries ‘7’ marks. 5 x 7 = 35M

18)If : ,f A B :g B C are two bijective functions then prove that

1 1 1gof f og and if 1 1 1 2, , 4, , 1, , 3,gof f og a b c d

and then

show that 1 1 1gof f og .

19)By using mathematical induction, show that n N ,

3 3 3 3 3 31 1 2 1 2 3.......

1 1 3 1 3 5

up to ‘n’ terms 22 9 13

24

nn n .

20)Show that 2 2 2 2

22 2 2 3 3 3

2 2 2

2

2 3

2

a b c bc a c b

b c a c ac b a a b c abc

c a b b a ab c

.

Disclaimer: This Question paper is purely for preparation purpose only. Manabadi.com is no where claiming this to be the Main Examination Paper.

21)Solve the equations by using Gauss – Jordan Method 1x y z ,

2 2 3 6x y z , 4 9 3x y z .

22)Find the shortest distance between the skew lines

6 2 2 2 2r i j k t i j k and 4 3 2 2r i k s i j k .

23)In triangle ABC, prove that

sin sin sin 1 4cos cos sin2 2 2 4 4 4

A B C A B C

24)In triangle ABC, show that 31 2 1 1

2

rr r

bc ca ab r R .