CBSE Class 10 Mathematics Introduction to Trigonometry MCQs Set 05

Question. The value of tan³θ/1+ tan²θ + cot³θ/1+ cot²θ
(a) 1−sin²θ cos²θ/2sin θ cos θ
(b) 1+2sin²θ cos²θ/sin θ cos θ
(c) 1−2sin² θ cos² θ/sin θ cos θ
(d) 2sin²θ cos²θ/1− sinθ cosθ

Answer: C

Question. A river flows due North, and a tower stands on its left bank. From a point A upstream and on the same bank as the tower, the elevation of the tower is 60° and from a point B just opposite A on the other bank the elevation is 45°. If the tower is 360 m high, the breadth of the river is :-
(a) 120 √6 m
(b) 240/√ 3 m
(c) 240 √3 m
(d) 240 √6 m

Answer: A

Question. A tower subtends an angle of 30° at a point on the same level as the foot of tower. At a second point h m high above the first, the depression of the foot of tower is 60°. The horizontal distance of the tower from the point is:-
(a) h/√3
(b) h cot60/√3 °
(c) hcot60/3 °
(d) h cot 30°

Answer: A

Question. If tan q = p/q, then psinθ −qcosθ/ psinθ + qcosθ =
(a) (p² + q²)/(p² − q²)
(b) (p² − q²)/(p2 + q²)
(c) (p² + q²)/(p² − q²)
(d) None of these

Answer: B

Question. The angle of elevation of the top of a tower from a point A due south of the tower is a and from a point B due east of the tower is b. If AB = d, then the height of the tower is :-
(a) d/√tan² α −tan²β
(b) d/√tan² α +tan²β 
(c) d/√cot² α +cot² β
(d) d/√cot² α −cot²β

Answer: C

Question. Choose the correct option and justify your choice: 1-tan² 45° /1+ tan² 45°
(a) tan 90°
(b) 1
(c) sin 45°
(d) 0

Answer: D

Question. The value of cosec⁴ A-2 cosec² A+1 is
(a) tan⁴ A
(b) sec⁴ A
(c) cosec⁴ A
(d) cot⁴ A

Answer: D

Question. If A, B and C are interior angles of a triangle ABC, then the value of tan(B+C)/2 is
(a) cot A/2
(b) sin A/2
(c) tan A/2
(d) None of the options

Answer: A

Question. If ΔPQR is right angled at Q, then the value of sin (P + R) is
(a) 1/2
(b) 1
(c) √2
(d) 0

Answer: B

Question. The value of (1 + tan² θ)(1 – sin θ)(1 + sin θ) =
(a) 0
(b) 1
(c) 2
(d) None of the options

Answer: B

Question. If 15 cot A = 8, then the value of cosec A is
(a) 15/ 12
(b) 13/ 15
(c) 4/ 15
(d) 17/ 15

Answer: D

Question. The value of 5 cos²60 ° + 4sec² 30° − tan² 45°/sin2 30° + cos² 30° is
(a) 32/ 35
(b) 14/ 55
(c) 67 /12
(d) 19/ 33

Answer: C

Question. The value of 3sin 30° + 4 cos ²45° − cot² 30°/cos² 30° + sin²30° is
(a) 1/ 2
(b) 1/ 3
(c) 2/ 5
(d) 3/ 8

Answer: A

Question. If x = r cos α, cos β, y = r cosa sinb and z = r sin a then x² + y² + z² is equal to
(a) r²
(b) r⁴
(c) 1
(d) None of the options

Answer: A

Question. A person standing on the bank of a river observes that the angles subtended by a tree on the oppo- site bank is 60°. When he retires 40 m from the bank, he finds the angle to be 30°. The breadth of the river is
(a) 40 m
(b) 60 m
(c) 20 m
(d) 30 m

Answer: C

Question. Find the value of 1/(1+tan²θ) + 1/(1 +cot²θ)
(a) 1/2
(b) 2
(c) 1
(d) 1/4

Answer: C

Question. AB is vertical tower. The point A is on the ground and C is the middle point of AB. The part CB subtend an angle a at a point P on the ground. If AP = nAB, then tan a =
(a) n(n² + 1)
(b) n/2n²−1 
(c) n²/2n² +1 
(d) n/2n² +1 

Answer: D

Question. Given that sinα =1/√2 and cos β=1/√2 , then the value of (α +β) is
(a) 90°
(b) 45°
(c) 60°
(d) 30°

Answer: A

Question. cot A tan A
(a) tan A
(b) sec A
(c) 1
(d) cot A

Answer: C

Question. The value of sin 60° ⋅ cos 30° + sin 30° ⋅ cos 60° is
(a) 0
(b) 1
(c) 2
(d) 8

Answer: B

Question. Value of cos 0° ⋅ cos 30° ⋅ cos 45° ⋅ cos 60° ⋅ cos 90° is
(a) 0
(b) 1
(c) 2
(d) 9

Answer: A

Question. ABC is right-angled triangle, right-angled at B. If BC = 7 cm and AC – AB = 1 cm, then cos A + sin A equals
(a) 31/ 25
(b) 51/ 61
(c) 17 /39
(d) 51/ 53

Answer: A

Question. The value of cos sin cos sin30° + sin 60°/1 + cos 60° +sin 30° is
(a) √3 /2
(b) 2/ √3
(c) 1/ √2
(d) 0

Answer: A

Question. A tower of height h standing at the centre of a square with sides of length a makes the same angle a at each of the four corners. Then a²/h² cot² α is :-
(a) 1
(b) 3/2
(c) 2
(d) 4

Answer: C

Question. If sec A + tan A = x, then sec A =
(a) x² -1/x
(b) x² - 1/2x
(c) x² + 1/x
(d) x² +1/ 2x

Answer: D

Question. If cosec θ − sinθ = m and sec θ − cosθ = n then (m²n)^2/3 + (mn²)^2/3 =
(a) −1
(b) 1
(c) 0
(d) None of the options

Answer: B

Question. At the foot of a mountain, the elevation of its summit is 45°. After ascending one kilometer the mountain upon and incline of 30°, the elevation changes to 60°. The height of the mountain is
(a) 1.366 km
(b) 1.266 km
(c) 1.166 km
(d) 1.466 km

Answer: A

Question. Choose the correct option and justify your choice: 2tan 30°/1-tan²30°
(a) cos60°
(b) sin 30°
(c) sin 60°
(d) tan 60°

Answer: D

Question. If 5 tan θ = 3, then what is the value of (5sin θ -3 cos θ/4sin θ +3cos θ) ?
(a) 0
(b) 1
(c) 2
(d) 3

Answer: A

Question. The value of sin ⁶ θ + cos6 θ + 3sin² θ cos² θ is
(a) 0
(b) 1
(c) 2
(d) 1/ 4

Answer: B

Question. The angle of depression of a car standing on the ground, from the top of a 75 m high tower is 30°. The distance of the car from the base of the tower (in m) is:
(a) 25 √3
(b) 50 √3
(c) 75 √3
(d) 150

Answer: C

Question. One side of a parallelogram is 12 cm and its area is 60 cm². If the angle between the adjacent sides is 30°, then its other side is
(a) 10 cm
(b) 8 cm
(c) 6 cm
(d) 4 cm

Answer: A

Question. If sin θ=1/2 and cosΦ=1/2, then the value of (θ +Φ ) is
(a) 0°
(b) 30°
(c) 90°
(d) 60°

Answer: C

Question. If sin θ = x and sec θ = y, then the value of cot θ is
(a) xy
(b) 2xy
(c) 1/ xy
(d) x + y

Answer: C

Question. If (1 + cos A)(1 – cos A) = 3/ 4 , the value of sec A is
(a) 2
(b) –2
(c) ±2
(d) 0

Answer: C

Question. If x tan 45° sin 30° = cos 30° tan 30°, then x is equal to
(a) √3 
(b) 1/2
(c) 1/√2
(d) 1

Answer: D

Question. If A, B and C are interior angles of a ΔABC, then cos (B+C/2) is equal to
(a) sin A/ 2
(b) − sin A/2
(c) cos A/ 2
(d) − cos A/2

Answer: A

Question. ABC is a triangle right angled at C and AC = √3 BC. Then ∠ABC =
(a) 30°
(b) 60°
(c) 90°
(d) 0°

Answer: B

Question. Given sin (A – B) = √3/ 2 and cos (A + B) = √3/ 2 . Then A and B respectively are
(a) 30°, 45°
(b) 45°, –15°
(c) 60°, 45°
(d) None of the options

Answer: B

Question. (sec θ +cosθ ) (secθ - cos θ) =
(a) tan²θ + cos²
(b) tan²θ - cos²θ
(c) tan²θ +sin²θ
(d) tan²θ -sin² θ

Answer: C

Question. The value of sin 60° cos 30° + sin 30° cos 60° is
(a) 1
(b) 2
(c) 11
(d) 0

Answer: A

Question. If tan(A + B) = √3 and tan(A – B) = 1/√3 , A > B, then the value of A is
(a) A = 30°
(b) A = 60°
(c) A = 90°
(d) A = 45°

Answer: D

Question. If tan θ + cot θ = 2, the value of √tan² θ + cot² θ is
(a) 2
(b) 2
(c) 3
(d) 3

Answer: B

One Word Questions :

Question. If cos³θ =1 , then find the value of θ .
Answer: 0°

Question. If, A, B and C are the angles of a triangle, then find the value of tan (A + B)/2 in terms of angle C.
Answer: cot C/₂

Question. If tan α = 1/√3 and sin β = 1/√2 , find the value of α +β .
Answer: 75°

Question. If cosec² and cotθ = √3k , then find the value of k.
Answer: 1

Question. If sec²θ (1+ sinθ )(1− sinθ ) = k , then find the value of k.
Answer: 1

Question. If CosA 3/5 ,then find the value of tan2 A− sec² A .
Answer: - 1

Question. If sin 3θ = cos 4θ , then find the value of 7θ .
Answer: 90°

Question. Find the value of sin 20⁰ sin 70⁰ − cos 20⁰ cos 70⁰
Answer: 0

Question. Find the value of cos ecAsec(900 − A) − cot Atan(900 − A).
Answer: 1

Question. Find the value of sin θ - sin3 θ/cos θ - cos3 θ .
Answer: cot θ

Question. In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion (A): If the angle of elevation of Sun, above a perpendicular line (tower) decreases, then the shadow of tower increases.
Reason (R): It is due to decrease in slope of the line of sight.

Answer: A

Question. Assertion (A): When we move towards the object, angle of elevation decreases.
Reason (R): As we move towards the object, it subtends large angle at our eye than before.

Answer: D