trigonometric ratios and identity Model Questions & Answers, Practice Test for ibps clerk prelims 2023

Answer: (b)

We know that, if value of cos θ increases, then the value of θ decreases.

∵ cos θ ≥ $1/2$

∴ cos θ ≥ cos ${π}/3 ⇒ θ ≤ {π}/3$

Answer: (b)

∵ In 24 h, Earth rotate about its own axis = 360°

In 1 h Earth rotate about its own axis = ${360°}/{24}$ = 15°

In 4 h Earth rotate about its own axis = 15° × 4 = 60°

Since, in 60 min Earth rotate about its own axis = 15°

In 12 min Earth rotate about its own axis

= ${15° × 12}/{60}$ = 3°

∴ In 4 h 12 min Earth rotate about its own axis

= 60° + 3° = 63°

Answer: (c)

1. If possible, let cos θ + sec θ = $3/2$

∴ cos θ + $1/{cos θ} = 3/2$

⇒ $2 cos^2 θ$ + 2 = 3 cos θ

⇒ $2cos^2$ θ - 3 cos θ + 2 = 0

Solving it as a quadratic in cos θ, we get

cos θ = ${3 ± √{9 - 16}}/4 = {3 ± √{-7}}/4$

Both of these values of cos θ are imaginary and hence both these values are impossible as cos θ is always real.

∴ cos θ + sec θ can never be equal to $3/2.$

2. $sec^2 θ + cosec^2 θ$

$1/{sin^2 θ cos^2 θ}$

$4/{4 sin^2 θ cos^2 θ}$

$4/{(sin 2 θ)^2}$

This expression is less than 4 if sin 2 θ > 1 but it not possible as. sin 2 θ lies in [0, 1]

Answer: (d)

Since we know that 5, 13 and 12 forms a Pythagrean triplet, the side with 13 units is the longest side and the angle between the other two sides is 90°.

Therefore, sin θ = $P/H = {12}/{13}$ and cos θ = $B/H = 5/{13}$

Thus, sin θ + cos θ = ${12}/{13} = 5/{13} = {12 + 5}/{13} = {17}/{13}$

Answer: (c)

${\text"sin x"}/{1 + \text"cos x"} = {1 - \text"cos x"}/{\text"sin x"}$

⇒ $sin^2$x = (1 - cos x) (1 + cos x)

⇒ $(1 - cos^2 x) = (1 - cos^2 x)$

which is possible for all values of x except multiples of 180°

For x = 180°, sin x = 0 and 1 + cos x = 0.