8
1. Find the value of Tan(Cos −1 𝑥) and hence evaluate Tan (Cos −1 17)
2. Consider 𝑓: 𝑅+ → [4, ∞]given by 𝑓(𝑥) = 𝑥 2 + 4. Show that 𝑓 is invertible with inverse
𝑓 −1 of 𝑓 given by f −1(𝑦) = √𝑦 − 4, where 𝑅+ is the set of all non-negative real numbers.
3. Let 𝑓: 𝑅 → 𝑅 be defined as 𝑓(𝑥) = 10𝑥 + 7. Find the function 𝑔: 𝑅 → 𝑅such that 𝑓𝑜𝑔 =
𝑔𝑜𝑓 = 𝐼𝑅
𝑎 + 𝑏, 𝑖𝑓 𝑎 + 𝑏 < 6
4. A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as 𝑎 ∗ 𝑏 = {
𝑎 + 𝑏 − 6, 𝑖𝑓 𝑎 + 𝑏 ≥ 6
show that zero is the identity for this operation and each element ‘a’ of the set is invertible with
6-a, being the inverse of ‘a’.
5. Consider the binary operation * on the set {0, 1, 2, 3, 4, 5} defined as 𝑎 ∗ 𝑏 = 𝑚𝑖𝑛{𝑎, 𝑏} write
the operation table of the operation *.
6. Let ∗ be a binary operation on Q defined by 𝑎 ∗ 𝑏 =
3𝑎𝑏
.
5
Show that ∗ is
commutative as well as associative. Also find its identity element, if exists.
7. Consider 𝑓: 𝑅 → 𝑅 given by 𝑓(𝑥) = 4𝑥 + 3. Show that ‘f’ is invertible and also
find the inverse of ‘f’.
8. If 𝑓(𝑥) =
4𝑥+3
6𝑥−4
2
3
2
3
, 𝑥 ≠ . Show that 𝑓𝑜𝑓(𝑥) = 𝑥 for all 𝑥 ≠ . What is inverse of ‘f’
1
4
9. Evaluate cos( sin−1 4 + sec −1 3)
3
17
𝜋
10. Prove that 2 𝑠𝑖𝑛−1 5 − tan−1 31 = 4 .
11. Prove that 𝑐𝑜𝑡 −1 7 + 𝑐𝑜𝑡 −1 8 + 𝑐𝑜𝑡 −1 18 = 𝑐𝑜𝑡 −1 3
1−𝑥
1
tan−1 𝑥 , 𝑥
2
−1
−1
12. Solve for x: 𝑡𝑎𝑛−1 (1+𝑥) =
13. Solve for the equation sin
6𝑥 + sin
>𝑜
𝜋
2
6√3𝑥 = − .
𝜋
2
14. Find the real solution of the equation : tan−1 √𝑥(𝑥 + 1) + sin−1 √𝑥 2 + 𝑥 + 1 = .
1
2
3
15. Show that tan−1 2 + tan−1 11 = tan−1 4
3𝑥−𝑥 3
2𝑥
16. Prove that tan−1 𝑥 + tan−1 (1−𝑥2 ) = tan−1 (1−3𝑥 2 ) , |𝑥| <
1
1
1
.
√3
31
17. Prove that 2 tan−1 2 + tan−1 7 = tan−1 17
1
18. Find the value of tan−1 (2 cos (2 sin−1 2))
1
1−𝑦 2
2𝑥
19. Find the value of 𝑇𝑎𝑛 2 [sin−1 (1−𝑥2 ) + cos −1 (1+𝑦2 ) ], |𝑥| < 1, |𝑦| > 0and 𝑥𝑦 < 1.
1
20. If 𝑠𝑖𝑛 (sin−1 5 + cos −1 𝑥) = 1, then find the value of ‘𝑥’.
𝑥−1
𝑥+1
𝜋
21. If tan−1 𝑥−2 + tan−1 𝑥+2 = 4 , then find the value of x.
3
5
−1 12
sin 13 +
8
=
17
4
cos −1 5 +
22. Show that sin−1 − sin−1
23. Show that
26.
27.
28.
𝜋.
𝑎 cos 𝑥−𝑏 sin 𝑥
𝑎
], if 𝑏 tan 𝑥 > −1.
𝑏 cos 𝑥+𝑎 sin 𝑥
𝜋
Solve tan−1 2𝑥 + tan−1 3𝑥 = 4 .
12
3
56
Show that cos −1 13 + sin−1 5 = sin−1 65.
1
1
1
1
𝜋
Show that tan−1 5 + tan−1 7 + tan−1 3 + tan−1 8 = 4
𝑥
𝜋
√1+sin 𝑥+√1−sin 𝑥
Prove that cot −1 [
] = 2 , 𝑥 ∈ (0, 2 )
𝑥−√1−sin
𝑥
√1+sin
24. Simplify tan−1 [
25.
84
.
85
63
tan−1 16 =
cos−1