trigonometric ratios and identity Model Questions & Answers, Practice Test for ssc mts paper 1 2023
ssc mts paper 1 2023 SYLLABUS WISE SUBJECTS MCQs
Number System
LCM & HCF
Ratio Proportion & Partnership
Percentages
Profit & Loss
Time & Work
Time & Distance
Simple Interest & Compound Interest
Mensuration: Area & Volumes
Algebraic Expressions
Trigonometric Ratios & Identity
What is the value of ${5 sin 75° sin 77° + 2 cos 13° cos 15°}/{cos 15° sin 77°} - {7 sin 81°}/{cos 9°}$ ?
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
What is the value of $sin^6 θ + cos^6 θ + 3 sin^2 θ cos^2$ θ – 1?
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
What is ${(sin θ + cos θ) (tan θ + cot θ)}/{sec θ + cosec θ}$ equal to?
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
The angles A, B, C and D of a quadrilateral ABCD are in the ratio 1 : 2 : 4 : 5.
A = 30°, B = 60°, C = 120°, D = 150°
Consider the following statements :
- ABCD is a cyclic quadrilateral.
- sin(B – A) = cos(D – C)
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.
Consider the following statements for 0 ≤ q ≤ 90°.
- The value of sin θ + cos θ is always greater than1.
- The value of tan θ + cot θ is always greater than1.
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.
ssc mts paper 1 2023 IMPORTANT QUESTION AND ANSWERS
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