trigonometric ratios and identity Model Questions & Answers, Practice Test for ibps clerk prelims 2023
ibps clerk prelims 2023 SYLLABUS WISE SUBJECTS MCQs
Ratio & Proportion
Percentages
Profit & Loss
Time & Work
Time & Distance
Simple Interest & Compound Interest
Mensuration: Area & Volumes
Algebraic Expressions
Trigonometric Ratios & Identity
Linear Equations
Quadratic Equations
Logarithm
If cos ≥ $1/2$ in the first quadrant, then which one of the following is correct?
Answer: (b)
We know that, if value of cos θ increases, then the value of θ decreases.
∵ cos θ ≥ $1/2$
∴ cos θ ≥ cos ${π}/3 ⇒ θ ≤ {π}/3$
The Earth takes 24 h to rotate about its own axis. Through what angle will it turn in 4 h and 12 min?
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°
Consider the following statements :
1. cos θ + sec θ can never be equal to 1.5.
2. $sec^2 θ + cosec^2$ θ can never be less than 4.
Which of the statements given above is/are correct?
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]
If a triangle has sides 5, 13 and 12 units and θ is the acute angle of the triangle, then what is the value of (sin θ + cos θ)?
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 = ${12}/{13} = 5/{13} = {12 + 5}/{13} = {17}/{13}$
Under which one of the following conditions is the trigonometrical identify ${sin x}/{(1 + cos x)} = {(1 - cos x)}/{sin x}$ true?
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.
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