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The angle of elevation of the top of a hill from a point is α. After walking b meters towards the top up a slope inclined at an angle β to the horizon, the angle of elevation of the top becomes γ. Then, the height of the bill is

1) [b sin α sin (γ - β)] / [sin (γ - α)] 2) [b sin α sin (γ - α)] / [sin (γ - β)] 3) [b sin α sin (γ - β)] / [sin (γ - α)] 4) [sin (γ - β)] /... View Article

P is a point on the segment joining the feet of two vertical poles of heights a and b. The angles of elevation of the tops of the poles from P are 45° each. Then, the square of the distance between the tops of the poles is

1) (a2 + b2) / 2 2) (a2 + b2) 3) 2 (a2 + b2) 4) 4 (a2 + b2) Solution: (3) 2 (a2 + b2) OA = a, AB = b, DE = a + b, CE = b - a l = (a +... View Article

ABC is a triangular park with AB = AC = 100 m. A vertical tower is situated at the midpoint of BC. The angle of elevation, if the top of the tower at A and B are cot^-1 3√2 and cosec^-1 2√2, respectively. The height of the tower is

1) 16 m 2) 25 m 3) 50 m 4) 20 m Solution: (4) 20 m cosec β = 2 √2 cot ɑ = 3 √2 x / h = 3 √2 --- (i) h / (√104 - x2) = 1 / √7 --- (ii)... View Article

From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be 30° and 45°. The height of tower is

1) 50 m 2) 50√3 m 3) 50 (√3 - 1) m 4) 50 (1 - √3 / 3) m Solution: (4) 50 (1 - √3 / 3) m From the figure, BD = 50m, AE = h = height of... View Article

From the top of a tower, the angle of depression of a point on the ground is 60°. If the distance of this point from the tower is 1 / (√3 + 1) m, then the height of the tower is

1) (4√3)/2 m 2) (√3 + 3)/2 m 3) (3 - √3)/ 2 m 4) √3/2 m 5) √3 + 1m Solution: (3) (3 - √3) / 2 m From the figure, PQ is the tower and... View Article

Angles of elevation of the top of a tower from three points (collinear) A, B and C on a road leading to the foot of the tower are 30°, 45° and 60°, respectively. The ratio of AB to BC is

1) √3 : 1 2) √3 : 2 3) 1 : 2 4) 2 : √3 Solution: (1) √3 : 1 In triangle EDA, tan 30o = ED / AD 1 / √3 = ED / AD = h / AD AD = h √3 In... View Article

The angles of elevation of the top of a tower at two points, which are at distances a and b from the foot in the same horizontal line and on the same side of the tower, are complementary. The height of the tower is

1) ab 2) √ab 3) √(a/b) 4) √(b/a) Solution: (2) √ab In triangle ABC, tan θ = h / a h = a tan θ --- (i) In triangle ADB, tan (90 - θ) =... View Article

A man from the top of a 100 m high tower sees a car moving towards the tower at an angle of depression of 30°. After some time, the angle of depression becomes 60°. The distance travelled by car during this is

1) 100√3 m 2) 200√3/3 m 3) 100√3 /3 m 4) 200√3 m Solution: (2) 200√3/3 m BC = 100 / tan 60o DB = 100 / tan 30o DC = 100 √3 - (100 /... View Article

A ladder rests against a wall so that it’s top touches the roof of the house. If the ladder makes an angle of 60° with the horizontal and height of the house is 6√3 in, then the length of the ladder is

1) 12√3 m 2) 12 m 3) 12/√3 m 4) None of these Solution: (2) 12 m √3 / 2 = 6 √3 / x x = 12 meters... View Article

A vertical pole PO is standing at the centre 0 of a square ABCD. If AC subtends an ∠90° at the top P of the pole, then the angle subtended by a side of the square at P is

1) 30° 2) 45° 3) 60° 4) None of these Solution: (3) 60° ∠OPA = ∠OPC = π / 4 Let OP = n, OA = n, AC = 2n OP / OA = cos (π / 4) PA = √2 n... View Article

ABCD is a square plot. The angle of elevation of the top of a pole standing at D from A or C is 30° and that from B is θ, then tan θ is equal to

1) √6 2) 1/√6 3) √3/2 4) √2/2 Solution: (2) 1 / √6 In triangle Δ PCD, tan 30° = h / a h / a = 1 / √3 .............. (1) In triangle Δ... View Article

If in a triangle ABC, s – a, s – b, s – c are in GP, then (sin²A + sin²C)/(sin A + sin C) is equal to

(1) sin (A + C) (2) sin (A - C) (3) b/R (4) none of these Solution: Given that s - a, s - b, s - c are in GP. So (s - b)2 = (s - a)(s - c)... View Article

In a triangle ABC, abc/2∆ is equal to

(1) (r1 + r2)/(1 + cos C) (2) (r1 + r2)/(1 + sin C) (3) (r1 + r2)/(1 + tan C) (4) none of these Solution: We know r1 = ∆/(s - a) r2 =... View Article

In a triangle ABC, [latex]frac{a(a+c-b)}{b(b+c-a)}[/latex] is equal to

(1) (1 - cos A)/(1 - cos B) (2) (1 + cos B)/(1 + cos A) (3) (1 - cos A)/(1 + cos B) (4) none of these Solution: a(a + c - b)/b(b + c - a) =... View Article

In a triangle ABC, if A = π/4 and tan B tan C = k, then k must satisfy

(1) k2 + 6k + 1 = 0 (2) k2 - 6k + 1 < 0 (3) k2 - 6k + 1 ≥ 0 (4) none of these Solution: Given tan B tan C = k We know in a triangle ABC,... View Article

If in a triangle ABC, 5 cos C + 6 cos B = 4 and 6 cos A + 4 cos C = 5, then tan(A/2) tan (B/2) is equal to

(1) 3/2 (2) 2/3 (3) 1/5 (4) 5 Solution: Given 5 cos C + 6 cos B = 4 6 cos A + 4 cos C = 5 This is of the form b cos C + c cos B = a and c... View Article

If the sides of a triangle are 3, 5, and 7 and the smallest and greatest angles are α and β respectively, then 4(1 – cos α) (1 – cos β) is equal to

(1) 3/5 (2) 3/7 (3) 3/4 (4) 3/11 Solution: Let a = 3, b = 5 and c = 7 Here a< b<c So A is the smallest angle and C is the greatest... View Article

If in a triangle, a² + b² + c² = ca + ab√3, then the triangle is

(1) equilateral (2) isosceles (3) right angled with A = 900, B = 600 and C = 300 (4) none of these Solution: Given a2 + b2 + c2 = ca +... View Article

The angles of a triangle ABC are in A.P. The largest angle is twice the smallest and the median to the largest side divides the angle at the vertex in the ratio 2 : 3. If the length of the median is 2√3 cm, then the length of the largest side is

1) 2 sin 320 2) 4 sin 320 3) 6 sin 320 4) 8 sin 320 Solution: Let A, B, C are the angles in AP. Then 2B = A + C Also A + B + C = 180 So B... View Article

In a cyclic quadrilateral ABCD; a, b, c, d denote the length of the sides AB, BC, CD, and DA respectively. Then cos A is equal to

(1) (a2 + d2 - b2 - c2)/2(ad + bc) (2) (a2 + d2 - b2 - c2)/2(ab - dc) (3) (a2 - d2 + b2 - 2c2)/2(ad + bc) (4) none of these Solution:... View Article

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