TS Inter 1st Year Model Paper 2021 Maths IA

0166
Total No. of Questions - 30 Regd.
Total No. of Printed Pages - 3 No.

Part - III
MATHEMATICS, Paper - IA
(English Version)
MODEL QUESTION PAPER (FOR IPE 2020-21 ONLY)
Time : 3 Hours Max. Marks : 75
Note: This question paper consists of three section A, B and C.
Section - A
Very short answer type questions.
(i) Answer all questions.
(ii) Each question carries 2 marks. 10×2=20

    
1. If A  0, , , ,  and F : AB is a surjection defined by f(x) = cos x, then find
 6 4 3 2
B.

1
2. Find the domain of the real valued function f  x   .
log  2  x 

 2 3 1 1 0 1
3. If A    and B    then find A+B.
7 8 5  2 4 1

 i 0
4. If A    , find A2.
0 i 

5. if a  2i  5 j  k and b  4i  m j  nk are collinear, then find m and n.
6. Find the vector equation of the line passing through the point 2i  3 j  k and parallel
to the vector 4i  2 j  3k .

1

7. If a  i  2 j  3k and b  3i  j  2k then show that a  b and a  b are perpendicular
to each other.
cos 9º  sin 9º
8. Prove that  cot 36º .
cos 9º  sin 9º

9. Find the period of the function defined by f  x   tan  x  4 x  9 x  ......  n2 x  .

10. 
If sinhx = 3, then show that x  log e 3  10 . 
Section - B
Short answer type questions. 5×4=20
(ii) Each question carries four marks.

1 0  0 1 
11. If I    and E    , then show that (aI + bE)3 = a3I + 3a2bE where ‘I’ is unit
0 1  0 0 
matrix of order 2.

1 2 1 
12. Show that A  3 2 3  is non-singular and find A–1.
1 1 2 

13. Let ABCDEF be regular hexagone with centre O, show that
AB  AC  AD  AE  AF  3AD  6AO .
14. Find the equation of the plane passing through the point a  2i  3 j  k and
perpendicular to the vector 3 i  2 j  2k and the distance of this plane from the origin.

15. If the vectors a  2i  j  k , b  i  2 j  3k and c  3 i  p j  5k are coplanar, then find
‘P’.

16. If A is not an integral multiple of , then prove that
2
(i) tanA + cotA = 2 cosec2A
(ii) cotA – tanA = 2 cot2A
17. Find the range of 7 cos x  24sin x  5 .
cosh x sinh x
18. Prove that   sinh x  coth x for x  0.
1  tanh x 1  coth x

2

A B C s2
19. Prove that cot  cot  cot  .
2 2 2 

a 2 bc A
20. If sin   then show that cos   cos .
b+c b+c 2
Section - C
Long Answer type questions. 5×7=35
(ii) Each question carries seven marks.

21. If f  1, 2  ,  2,  3 ,  3,  1 then find (i) 2f (ii) 2+f (iii) f2 (iv) f

1 2 1 
22. If A  0 1 1 , then find A3 – 3A2 – A – 3I, where I is unit matrix of order 3.
 3 1 1 

23. Solve the following system of equations by Cramer’s rule
x  y  z  1, 2 x  2 y  3 z  6 , x  4 y  9 z  3 .
24. Solve the following system of equations by Matrix Inversion method
2 x  y  3 z  9, x  y  z  6 , x  y  z  2 .

25. Find the vector equation of the plane passing through points 4 i  3 j  k , 3 i  7 j  10k
and 2 i  5 j  7 k and show that the point i  2 j  3k lies in the plane.

26. If a  7 i  2 j  3k , b  2i  8k and c  i  j  k , then compute a  b, a  c and

 
a  b  c . Verify whether the cross product is distributive over vector addition.

27. If [ b c d ] + [ c a d ] + [ a b d] = [ a b c ]. Then show that the points with position
vectors a, b, c and d are coplanar.
28. If A, B, C are angles in a triangle, then prove that
A B C
sinA + sinB – sinC = 4sin sin cos .
2 2 2

A B C
29. If cot : cot : cot  3 : 5 : 7 , then show that a:b:c = 6:5:4.
2 2 2

65 21
30. If a = 13, b = 14, c = 15, show that R  , r  4, r1  , r2  12 and r3 = 14.
8 2

3