trigonometric ratios and identity Model Questions & Answers, Practice Test for ssc mts paper 1 2023

Answer: (a)

${\text"5 sin 75° sing 77° + 2 cos 13° cos 15°"}/{\text"cos 15° sin 77°"} - {\text"7 sin 81°"}/{\text"cos 9°"}$

= ${\text"5 cos 15° sin 77° + 2 sin 77° cos 15°"}/{\text"cos 15° sin 77°"} - {\text"7 cos 9°"}/{\text"cos 9°"}$

= ${\text"7 cos 15 . sin 77"}/{\text"cos 15 sin 77"} - {\text"7 cos 9"}/{\text"cos 9"}$ = 7 - 7 = 0

Answer: (b)

$sin^6 θ + cos^6 θ + 3 sin^2 θ, cos^2 θ$ - 1

$(sin^2 θ)^3 + (cos^2 θ)^3 + 3 sin^2 θ . cos^2 θ$ - 1

$(sin^2 θ + cos^2 θ)^3$ - 1

1 - 1 = 0

Answer: (b)

${(\text"sin θ + cos θ") (\text"tan θ + cot θ")}/{\text"sec θ + cosec θ"}$

= ${(\text"sin θ + cos θ") ({\text"sin θ"}/{\text"cos θ"} + {\text"cos θ"}/{\text"sin θ"})}/{1/{\text"cos θ"} + 1/{\text"sin θ"}}$

= ${(\text"sin θ + cos θ") ({sin^2 θ + cos^2 θ}/{\text"sin θ cos θ"})}/{{\text"sin θ + cos θ"}/{\text"sin θ cos θ"}}$

[∵ $sin^2 θ + cos^2 θ = 1$]

= ${(\text"sin θ + cos θ") (1/{\text"sin θ cos θ"})}/{{\text"sin θ + cos θ"}/{\text"sin θ cos θ"}}$

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

Answer: (d)

If ABCD is a cyclic quadrilateral, then sum of opposite angles is 180°.

30° + 120° = 150° ≠ 180° and 60° + 150° = 210° ≠ 180°

So, Statement I is not correct.

Statement II:

sin (B - A) = cos (D - C)

⇒ sin (60° - 30°) = cos (150° - 120°)

⇒ sin 30° = cos 30° ⇒ $1/2 ≠ {√3}/2$

So, Statement II is also not correct.

Answer: (a)

Let f(θ) = sin θ + cos θ

Maximum and minimum value of a cos θ + b sin θ is

- $√{a^2 + b^2} \text"≤ a cos θ + b sin θ≤" √{a^2 + b^2}$

∴ - $√{1 + 1} \text" ≤ cos θ + sin θ≤" √{1 + 1}$

⇒ $- √2 ≤ cos θ + sin θ≤√2$

⇒ - 1.414 ≤ cos θ + sin θ ≤ 1.414

∴ f(θ) = (sin θ + cos θ) ∈ [- 1.414, 1.414]

and let g(θ) = tan θ + cot θ = tan θ + $1/{tan θ}$

(∵ AM ≥ GM)

⇒ ${\text"tan θ" + 1/{\text"tan θ"}}/2 ≥ (\text"tan θ" . 1/{\text"tan θ"})^{1/2}$

⇒ (tan θ + cot θ) > 2

So, (tan θ + cot θ) is always greater than 1.

Hence, Statement 1 is false and Statement II is true.