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

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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^-1x + cos^-1y 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^-1x = 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 x²⁰¹⁷ + y²⁰¹⁸ + z²⁰¹⁹ – \(\frac{9}{x^{101}+y^{101}+z^{101}}\) 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(x² – 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(x² – 3)
Domain of sin^-1 (x) is [-1, 1]
⇒ -1 ≤ x² – 3 ≤ 1 ⇒ 2 ≤ x² ≤ 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) x² – x – 6 = 0
(b) x² – x – 12 = 0
(c) x² + x – 12 = 0
(d) x² + x – 6 = 0
Solution:
(b) x² – x – 12 = 0
Hint:

Question 14.
sin^-1(2 cos² x – 1) + cos^-1(1 – 2 sin² x) = ……………

Solution:
(a) \(\frac{\pi}{2}\)
Hint:
2 cos²x – 1 = cos 2x
1 – 2 sin² 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) tan² 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: