CBSE Class 12 Mathematics Inverse Trigonometric Functions MCQs Set E

Question. A man from the top of a 100 metre high tower looks a car moving towards the tower at an angle of depression of 30º. After some time, the angle of depression becomes 60º. The distance (in metre) travelled by the car during this time is :
(a) 100√3
(b) 200√3/3
(c) 100√3/3
(d) 200√3
Answer: b

Question. If cos(u +iv) = x +iy, then x2 + y2 +1is equal to:
(a) cos² u + sinh² v
(b) sin² u + cosh² v
(c) cos² u + cosh² v
(d) sin² u + sinh² v
Answer: c

Question. If x = log(y+√y²+1) then y = ?
(a) tanh x
(b) coth x
(c) sinh x
(d) cosh x
Answer: c

Question. If tanh2 x = tan2θ , then cosh 2x is equal to:
(a) − sin 2θ
(b) sec 2θ
(c) cos 3θ
(d) cos 2θ
Answer: b

Question. If 1 cosh⁻¹x log(2+√3), then x = ?
(a) 2
(b) 1
(c) 3
(d) 5
Answer: a

Question. Find imaginary part of sin⁻¹ (cosecθ )
(a) log (cotθ/2)
(b) π/2
(c) 1/2log (cotθ/2)
(d) None of these
Answer: a

Question. The angle of elevation of a tower at a point distant d metres from its base is 30º. If the tower is 20 meters high, then the value of d is:
(a) 10√3m
(b) 20/√3m
(c) 20√3 m
(d) 10m
Answer: c

Question. The number of positive integral solutions of the equation (Image 19)
(a) One
(b) Two
(c) Zero
(d) None of these
Answer: b

Question. α ,β and γ are three angles given by α 2tan–1 (√2–1). β = 3sin^–1 1/√2 + sin^–1 (–1/2) and γ = cos^–1(1/3) Then:
(a) α > β
(b) β >γ
(c) α <γ
(d) None of these
Answer: b,c

Question. If the angle of elevation of the top of a tower at a distance 500 m from its foot is 30º, then height of the tower is:
(a) 1/√3
(b) 500/√3
(c) 3
(d) 1/500
Answer: b

 Question. A vertical pole consists of two parts, the lower part being one third of the whole. At a point in the horizontal plane through the base of the pole and distance 20 metres from it, the upper part of the pole subtends an angle whose tangent is 1/2. The possible heights of the pole are:
(a) 20m and 20√3 m
(b) 20 m and 60 m
(c) 16 m and 48 m
(d) None of these
Answer: b

Question. cos ix + i sin ix equals: 
(a) e^ix 
(b) e^−ix
(c) e^x
(d) e^−x
Answer: d

Question. A person standing on the bank of a river finds that the angle of elevation of the top of a tower on the opposite bank is 45º. Then which of the following statements is correct
(a) Breadth of the river is twice the height of the tower
(b) Breadth of the river and the height of the tower are the same
(c) Breadth of the river is half of the height of the tower
(d) None of these
Answer: b

 Question. The angle of elevation of the top of a tower from a point 20 meters away from its base is 45º. The height of the tower is:
(a) 10 m
(b) 20 m
(c) 40 m
(d) 20√3 m
Answer: b

Question. A person observes the angle of deviation of a building as 30º. The person proceeds towards the building with a speed of 25( 3 −1)m/ hour. After 2 hours, he observes the angle of elevation as 45º. The height of the building (in metres) is:
(a) 100
(b) 50
(c) 50( √3 +1)
(d) 50( √3 −1)
Answer: b

Question. The value of sinh⁻¹ (1) is:
(a) 0
(b) log(√2 + 1)
(c) log(1 − √2)
(d) None of these
Answer: b

Question. The angular depressions of the top and foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first are θ and φ respectively, then the distance between their tops when tanθ 4/3 = and tanφ = 5/2 is:
(a) 150/√3 metres
(b) 100√3metres
(c) 150metres
(d) 100metres
Answer: d

Question. If cos( x + iy) = A + i B , then A equals:
(a) cos x cosh y
(b) sin x sinh y
(c) −sin x sinhy
(d) cos x sinh y
Answer: a

Question. The angle of elevation of a cliff at a point A on the ground and a point B, 100 m vertically at A are α and β respectively. The height of the cliff is:
(a) 100cot α/cot α − cotβ
(b) 100cot β/cot α − cotβ
(c) 100cot β/cot β − cotα
(d) 100cot β/cot β + cotα
Answer: c

Question. A tower is situated on horizontal plane. From two points, the line joining these points passes through the base and which are a and b distance from the base. The angle of elevation of the top are α and 90º −α and θ is that angle which two points joining the line makes at the top, the height of tower will be:
(a) a+b/a–b
(b) a–b/a+b
(c) √ab
(d) (ab)^1/3
Answer: c

Question. The greatest and least values of (sin⁻¹ x)³ (cos⁻¹ x)³ are:
(a) π³/32
(b)−π³/8
(c) 7π³/8
(d) π/2
Answer: a,c

Question. A tower of height b subtends an angle at a point O on the level of the foot of the tower and at a distance a from the foot of the tower. If a pole mounted on the tower also subtends an equal angle at O, the height of the pole is:
(a) b(a²–b²/a²+b²)
(b) b(a²+b²/a²–b²)
(c) a(a²–b²/a²+b²)
(d) a(a²+b²/a²–b²)
Answer: b

Question. α ,β and γ are the angles given by α = 2tan⁻¹(√2−1). β = 3 sin−1(−1/√2) and γ = cos⁻¹(1/3) then:
(a) α > β
(b) β >γ
(c) γ >α
(d) none of these
Answer: b,c

Question. For a man, the angle of elevation of the highest point of the temple situated east of him is 60º. On walking 240 metres to north, the angle of elevation is reduced to 30º, then the height of the temple is:
(a) 60√6m
(b) 60m
(c) 50√3m
(d) 30√6m
Answer: a

Question. A flag-staff of 5m high stands on a building of 25 m high.
At an observer at a height of 30m. The flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag-staff is:
(a) 5√3/2
(b) 5√(3/2)
(c) 5√(2/3)
(d) None of these
Answer: b

Assertion and Reason
Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.
(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.
(c) If assertion is true but reason is false.
(d) If the assertion and reason both are false.
(e) If assertion is false but reason is true.

Question. Assertion: A pole standing in the centre of a rectangular field of area 2500 sq. units subtends angle α and β respectively at the mid-points of two adjacent sides of the field such that , α +β = π/2 the height of the pole is 25 sq.units.
Reason: Area of a rectangle is equal to the product of the length of the adjacent sides. 
Answer: a

Question. Assertion: Apoorv, standing on the ground wants to observe the angle α of elevation of the top of a tower in front of him. He walks half the distance towards the foot of the tower and finds the angle of elevation is π / 4. He observes α = tan−¹ (1/ 2)
Reason: If the angles of elevation of the top of a tower at three distinct points on the ground is α , the points lie on a circle with centre at the foot of the tower.
Answer: b

Question. Mansi observes that the angle of elevation of a vertical pole of height h at two points A and B on the horizontal plane through the foot O of the pole is π / 3. AB = (a)
Assertion: If AB subtends an angle π / 2 at the foot of the tower, then 2h = 3a
Reason: If AB subtends and angle π / 3 at the foot of the tower than h = a√3
Answer: d

Question. ABC in an equilateral triangle on the horizontal ground with length of each side equal to (a)
Assertion: If a tower standing at the centre O of the triangle makes an angle α at each corner such that α = tan⁻¹ 9, the height of the tower is 3√3
Reason: If a tower of height 2a standing at one corner of the triangle makes an α at any other corner, then 1 a tan 2. =
Answer: b

Question. Assertion: Three poles of height a, b, c stand at the points A, B, C respectively and subtend to same angle α at a point O on the horizontal line through the feet of the poles. If a, b, c are in (a)P., then AB = B(c)
Reason: O is the centre of a circular field and A is any point on its boundary. Two poles standing at A and O subtend the same angle α
at a point B on the other end of the diameter through (a) Height of the pole at A is twice the height of the plot at O. 
Answer: b

Question. Assertion: Rajat observes that the angle of elevation of the top P of a tower OP at a point A on the ground is α.
He travels a distance a in the direction AP and reaches the point (b) He then travels a horizontal distance a towards the tower and reaches the point C, where the angle of elevation of the top of the tower is , π/4 the height of the tower is
a(cos α +1–sinα)/cot α–1 (Image 75)
Reason: On the top of building a pole of height equal to 1/3 of the height of the building is placed so that the angles of elevation of the top of the pole and the top of the
building at a point on the ground are α and β respectively then α = (3/ 4)β
Answer: c

Question. Assertion: ABC is a triangular field with AC = b and AB = (c)
A pole standing at a point D on BC subtends angles α and B and β at (c) If BAD = DAC then b cotα = c cotβ .
Reason: Bisector of an angle of a triangle divides the opposite side in the ratio of the side containing the angle.
Answer: a

Question. Assertion: The angle of elevation of the top P of a tower OP at a point A on the ground is α , the angle of elevation of the mid-point Q of the tower at the mid-point B of OA
is also α.
Reason: The line joining the mid-points of two sides of a triangle is parallel to the third side.
Answer: a

Question. Assertion: A tower standing at the centre of a square field subtends an angle α at each corner. If the height of the tower is twice the length of a side of the square, then
α = tan⁻¹ 2.
Reason: A, B, C are three points on the horizontal line through the foot of a tower and the angles of elevation of top of the tower at these points are 30º45º and 60º respectively, 3
AB/BC = √3 .
Answer: d

Comprehension Based

Question. The principal value of sin⁻¹ (sin 5) cos⁻¹ (cos5) is:
a. 0
b. 2π −10
c. −π
d. 3π −10
Answer: c

Question. The value of sin⁻¹ [cos⁻¹{cos (cos x) sin−1 (sin x)}], where x ε (π/2 ,π) is:
a. π/2
b. −π
c. π
d. −π/2
Answer: d

Question. Match Column I with Column II and select the correct answer using the code given below the lists: (Image 90)
a. A→ 4, B→ 3, C→ 2, D→ 1
b. A→ 2, B→ 4, C→ 3, D→ 1
c. A→ 3, B→ 4, C→ 2, D→ 1
d. A→ 3, B→ 4, C→ 1, D→ 2
Answer: a

Question. Let (x, y) be such that sin−1(ax) cos (y) cos⁻¹ (bxy) π/2. (Image 89)
a. A→ 1, B→ 2, C→ 1, D→ 4
b. A→ 2, B→ 4, C→ 3, D→ 1
c. A→ 1, B→ 3, C→ 2, D→ 4
d. A→ 4, B→ 1, C→ 3, D→ 2
Answer: a

Question. If in a triangle s−a/11 = s−b/12 = s−c/13 Then λtan2(A/2) = 455 if λ must be: 
Answer: 1155

Question. In the adjacent figure ‘P’ is any arbitrary interior point of the triangle ABC. , H_a , H_b and , H_c are the length of altitudes drawn from vertices A, B and C respectively.
Answer: 1719

Question. If p₁ , p₂ , p₃ are the altitudes of a triangle which circumscribes a circle of diameter 16/3 unit, then the least value of p₁ + p₂ + p₃ must be:
Answer: 24

Question. If λ = cos4 [tan⁻¹{sin(cot⁻¹ 5)}], then the value of 3645 λ must be
Answer: 3380

Question. In a triangle ABC, right angled at A. The radius of the inscribed circle is 2 cm. Radius of the circle touching the side BC and also sides AB and AC produced is 15 cm. The length of the side BC measured in cm is:
Answer: 13

Question. If θ cot⁻¹ 7 + cot⁻¹8 + cot⁻¹18, = then the value of 4 81cot θ must be:
Answer: 6561

Question. If in a -ABC, BC = 5, CA = 4, AB = 3 and D, E are the points on BC such that BD = DE = EC, then 576tan(∠CAE) must be:
Answer: 216

Question. If sin⁻¹x + sin⁻¹y =π and, if x = λ y, then the value of 39^2λ + 5^λ must be:
Answer: 1526