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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 4 Inverse Trigonometric Functions Ex 4.6

Choose the correct or the most suitable answer from the given four alternatives.

Question 1.

The value of sin^{-1}(cos x), 0 ≤ x ≤ π is …………

(a) π – x

(b) x – \(\frac{\pi}{2}\)

(c) \(\frac{\pi}{2}\) – x

(d) π – x

Solution:

(c) \(\frac{\pi}{2}\) – x

Hint:

Question 2.

If sin^{-1} + sin^{-1} y = \(\frac{2 \pi}{3}\); then cos^{-1}x + cos^{-1}y is equal to ………….

(a) \(\frac{2 \pi}{3}\)

(b) \(\frac{\pi}{3}\)

(c) \(\frac{\pi}{6}\)

(d) π

Solution:

(b) \(\frac{\pi}{3}\)

Hint:

Question 3.

……………

(a) 2π

(b) π

(c) 0

(d) tan^{-1}\(\frac{12}{65}\)

Solution:

(c) 0

Hint:

Question 4.

If sin^{-1}x = 2 sin^{-1} α has a solution, then ……………

Solution:

(a) |α| ≤ \(\frac{1}{\sqrt{2}}\)

Hint:

Question 5.

sin^{-1} (cos x) = \(\frac{\pi}{2}\) – x is valid for ……………..

(a) -π ≤ x ≤ 0

(b) 0 ≤ x ≤ π

(c) \(-\frac{\pi}{2}\) ≤ x ≤ \(\frac{\pi}{2}\)

(d) \(-\frac{\pi}{4}\) ≤ x ≤ \(\frac{3 \pi}{4}\)

Solution:

(b) 0 ≤ x ≤ π

Question 6.

If sin^{-1} x + sin^{-1} y + sin^{-1} z = \(\frac{3 \pi}{2}\), the value of x^{2017} + y^{2018} + z^{2019} – \(\frac{9}{x^{101}+y^{101}+z^{101}}\) is ……………….

(a) 0

(b) 1

(c) 2

(d) 3

Solution:

(a) 0

Hint:

The maximum value of sin^{-1} x is \(\frac{\pi}{2}\) and sin^{-1} 1 = \(\frac{\pi}{2}\)

Here it is given that

sin^{-1} x + sin^{-1} y + sin^{-1} z = \(\frac{3 \pi}{2}\)

⇒ x = y = z = 1

and so 1 + 1 + 1 – \(\frac{9}{1+1+1}\) = 3 – 3 = 0

Question 7.

If cot^{-1} x = \(\frac{2 \pi}{5}\) for some x ∈ R, the value of tan^{-1} x is …………

(a) \(-\frac{\pi}{10}\)

(b) \(\frac{\pi}{5}\)

(c) \(\frac{\pi}{10}\)

(d) \(-\frac{\pi}{5}\)

Solution:

(c) \(\frac{\pi}{10}\)

Hint:

Question 8.

The domain of the function defined by f(x) = sin^{-1} \(\sqrt{x-1}\) is …………….

(a) [1, 2]

(b) [-1, 1]

(c) [0, 1]

(d) [-1, 0]

Solution:

(a) [1, 2]

Hint:

The domain for sin^{-1} x is [0, 1]

So \(\sqrt{x-1}\) = 0 ⇒ x – 1 = 0 ⇒ x = 1

\(\sqrt{x-1}\) = 1 ⇒ x – 1 = 0 ⇒ x = 2

∴ The domain is [1, 2]

Question 9.

If x = \(\frac{1}{5}\), the value of cos(cos^{-1} x + 2 sin^{-1} x) is …………….

Solution:

(d) \(-\frac{1}{5}\)

Hint:

Question 10.

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to …………

Solution:

(d) tan^{-1}\(\frac{1}{2}\)

Hint:

Question 11.

If the function f(x) = sin^{-1}(x^{2} – 3), then x belongs to …………..

(a) [1, -1]

(b) [\(\sqrt{2}\), 2]

(c) \([-2,-\sqrt{2}] \cup[\sqrt{2}, 2]\)

(d) \([-2,-\sqrt{2}] \cap[\sqrt{2}, 2]\)

Solution:

(c) \([-2,-\sqrt{2}] \cup[\sqrt{2}, 2]\)

Hint:

f(x) = sin^{-1}(x^{2} – 3)

Domain of sin^{-1} (x) is [-1, 1]

⇒ -1 ≤ x^{2} – 3 ≤ 1 ⇒ 2 ≤ x^{2} ≤ 4

⇒ \(\sqrt{2}\) ≤ x ≤ 2 ⇒ \(\sqrt{2}\) ≤ |x| ≤ 2

x ∈ \([-2,-\sqrt{2}] \cup[\sqrt{2}, 2]\)

Question 12.

If cot^{-1} 2 and cot^{-1} 3 are two angles of a triangle, then the third angle is …………..

Solution:

(b) \(\frac{3 \pi}{4}\)

Hint:

Question 13.

\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}(\sqrt{\frac{3}{x}})=\frac{\pi}{6}\). Then x is a root of the equation …………..

(a) x^{2} – x – 6 = 0

(b) x^{2} – x – 12 = 0

(c) x^{2} + x – 12 = 0

(d) x^{2} + x – 6 = 0

Solution:

(b) x^{2} – x – 12 = 0

Hint:

Question 14.

sin^{-1}(2 cos^{2} x – 1) + cos^{-1}(1 – 2 sin^{2} x) = ……………

Solution:

(a) \(\frac{\pi}{2}\)

Hint:

2 cos^{2}x – 1 = cos 2x

1 – 2 sin^{2} x = cos 2x

∴ sin^{-1} x(cos 2x) + cos^{-1}(cos 2x) = \(\frac{\pi}{2}\) (∵ sin^{-1} x + cos^{-1} x = \(\frac{\pi}{2}\))

Question 15.

If \(\cot ^{-1}(\sqrt{\sin \alpha})+\tan ^{-1}(\sqrt{\sin \alpha})\) = u, then cos 2u is equal to …………..

(a) tan^{2} α

(b) 0

(c) -1

(d) tan 2α

Solution:

(c) -1

Hint:

cot^{-1} x + tan^{-1} x = \(\frac{\pi}{2}\) ⇒ u = \(\frac{\pi}{2}\) so 2u = π

∴ cos 2u = cos π = -1

Question 16.

If |x| ≤ 1, then 2tan^{-1} x – sin^{-1} \(\frac{2 x}{1+x^{2}}\) is equal to ………….

(a) tan^{-1} x

(b) sin^{-1} x

(c) 0

(d) π

Solution:

(c) 0

Hint:

Let x = tan θ so \(\frac{2 x}{1+x^{2}}\) = sin 2θ.

Now 2 tan^{-1}(tanθ) – sin^{-1}(sin 2θ) = 2θ – 2θ = 0

Question 17.

The equation tan^{-1} x – cot^{-1} x = tan^{-1} \(\left(\frac{1}{\sqrt{3}}\right)\) has …………..

(a) no solution

(b) unique solution

(c) two solutions

(d) infinite number of solutions

Solution:

(b) unique solution

Hint:

Question 18.

If sin^{-1} x + cot^{-1} \(\left(\frac{1}{2}\right)=\frac{\pi}{2}\), then x is equal to …………

Solution:

(b) \(\frac{1}{\sqrt{5}}\)

Hint:

sin^{-1} x + cot^{-1} \(\left(\frac{1}{2}\right)=\frac{\pi}{2}\)

Question 19.

If sin^{-1}\(\frac{x}{5}\) + cosec^{-1}\(\frac{5}{4}\) = \(\frac{\pi}{2}\), then the value of x is ………

(a) 4

(b) 5

(c) 2

(d) 3

Solution:

(d) 3

Hint:

Question 20.

sin(tan^{-1}), |x| < 1 is equal to ………….

Solution:

(d) \(\frac{x}{\sqrt{1+x^{2}}}\)

Hint:

### Samacheer Kalvi 12th Maths Solutions Chapter 4 Inverse Trigonometric Functions Ex 4.6 Additional Questions

Question 1.

Find the principal value of

Solution:

Question 2.

Find the principal value of

Solution:

Question 3.

Solution:

Question 4.

Evaluate

Solution:

Question 5.

Evaluate

Solution:

Question 6.

Question 7.

Solution:

Question 8.

Solution:

x = \(\frac{1}{6}\)

Question 9.

Find the values of each of the following:

Solution:

Question 10.

Solve for x:

Solution:

Question 11.

Prove:

Question 12.

Evaluate: sin(tan^{-1} x + cot^{-1} x)

Question 13.

The value of sin^{-1}(1) + sin^{-1}(0) is …….

Solution:

(a) \(\frac{\pi}{2}\)

Hint:

Question 14.

Solution:

(d) \(\frac{\pi}{2}\)

Hint:

Question 15.

tan^{-1}x + cot^{-1}x = ……..

(a) 1

(b) – π

(c) \(\frac{\pi}{2}\)

(d) π

Solution:

(c) \(\frac{\pi}{2}\)

Hint:

Question 16.

Solution:

(a) \(\frac{-\pi}{2}\)

Hint:

Question 17.

Solution:

(b) \(\frac{\pi}{2}\)

Hint:

Question 18.

Solution:

(a) \(\sin ^{-1} \frac{1}{\sqrt{2}}\)

Hint:

Question 19.

Solution:

(d) 2π

Hint:

Question 20.

Solution:

(c) 2π

Hint: