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For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. An incorrect statement among the following is

1) there is a regular polygon with r/R = 1/2 2) there is a regular polygon with r/R = 1/√2 3) there is a regular polygon with r/R = 2/3 4)... View Article

In a ∆ABC, let ∠C = π/2, if r is the inradius and R is the circumradius of the ∆ABC, then 2 (r + R) equals

1) c + a 2) a + b + c 3) a + b 4) b + c Solution: Given ∠C = π/2 Let r is the inradius and R is the circumradius of the ∆ABC. The triangle... View Article

If p₁, p₂, p₃ are respectively the perpendiculars from the vertices of a triangle to the opposite side, then p₁p₂p₃ is equal to

1) a2b2c2 2) 2 a2b2c2 3) 4 a2b2c3/R2 4) a2b2c2/8R3 Solution: Let p1 be the altitude AD, p2 be the altitude BE and p3 be the altitude CF.... View Article

Inradius of a circle which is inscribed in an isosceles triangle one of whose angle is 2π/ 3, is √3, then the area of the triangle is

1) 4√3 2) 12 - 7√3 3) 12 + 7√3 4) None of these Solution: Given the triangle is isosceles with one angle 1200. Let A = 1200 Then b = c B =... View Article

In a ∆ABC, a : b : c = 4 : 5 : 6. The ratio of the radius of the circumcircle to that of the incircle is

1) 15/4 2) 11/5 3) 16/7 4) 16/3 Solution: Given a : b : c = 4 : 5 : 6 Let R be the radius of the circumcircle and r be the radius of the... View Article

In a triangle, if r₁ = 2r₂ = 3r₃, then a/b + b/c + c/a is equal to

1) 75/60 2) 155/60 3) 176/60 4) 191/60 Solution: Given r1 = 2r2 = 3r3 We know r1 = ∆/(s - a) r2 = ∆/(s - b) r3 = ∆/(s - c) So ∆/(s - a) =... View Article

The value of 1/r₁² + 1/r₂² + 1/r₃² + 1/r² is

(1) (a2 + b2 + c2)/∆2 (2) 0 (3) ∆2/(a2 + b2 + c2) (4) 1 Solution: We know r = ∆/s r1 = ∆/(s - a) r2 = ∆/(s - b) r3 = ∆/(s - c) Where r is... View Article

If A, A₁, A₂ and A₃ and the areas of the incircle and excircles, then 1/√A₁+ 1/√A₂ + 1/√A₃ is equal to

1) 1/√A 2) 2/√A 3) 3/√A 4) 4/√A Solution: The area of excircle, A1 = πr12 A2 = πr22 A3 = πr32 1/√A1 + 1/√A2 + 1/√A3 = 1/√πr1 + 1/√πr2 +... View Article

The area of a circle and the area of a regular polygon of n sides of the perimeter equal to that of that circle are in the ratio

1) tan π/n : π/n 2) cos π/n : π/n 3) sin π/n : π/n 4) cot π/n : π/n Solution: Let r be the radius of the circle and a be the side of... View Article

In a ∆PQR, P is the largest angle and cos P = 1/3, Further in the circle of the triangle touches the sides PQ, QR, and RP at N, L, and M respectively, such that the lengths of PN, QL, and RM are consecutive even integers, Then, possible length(s) of the side (s) of the triangle is (are)

1) 16 2) 18 3) 24 4) 22 Solution: Let s - a = 2k - 2 s - b = 2k s - c = 2k + 2, k ∈ I, k > 1 Adding we get, s = 6k So, a = 4k + 2 b =... View Article

If the sides of the triangles are p, q, √(p² + q² + pq), then the greatest angle is

1) π/2 2) 5π/4 3) 2π/3 4) 7π/4 Solution: Given p, q, √(p2 + q2 + pq) are the sides. cos θ = [p2 + q2 - √(p2 + q2 + pq)]/2pq = (p2 + q2 - p2... View Article

If in a ∆ABC, the altitudes from the vertices A, B, C on the opposite side are in HP, then sin A, sin B, and sin C are in

1) HP 2) Arithmetico - Geometric Progression 3) AP 4) GP Solution: Consider ∆BDA, cos (90 - B) = AD/c AD = c sin B Similarly, BE = a sin... View Article

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Archives

For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. An incorrect statement among the following is

In a ∆ABC, let ∠C = π/2, if r is the inradius and R is the circumradius of the ∆ABC, then 2 (r + R) equals

If p₁, p₂, p₃ are respectively the perpendiculars from the vertices of a triangle to the opposite side, then p₁p₂p₃ is equal to

Inradius of a circle which is inscribed in an isosceles triangle one of whose angle is 2π/ 3, is √3, then the area of the triangle is

In a ∆ABC, a : b : c = 4 : 5 : 6. The ratio of the radius of the circumcircle to that of the incircle is

In a triangle, if r₁ = 2r₂ = 3r₃, then a/b + b/c + c/a is equal to

The value of 1/r₁² + 1/r₂² + 1/r₃² + 1/r² is

If A, A₁, A₂ and A₃ and the areas of the incircle and excircles, then 1/√A₁+ 1/√A₂ + 1/√A₃ is equal to

The area of a circle and the area of a regular polygon of n sides of the perimeter equal to that of that circle are in the ratio

In a ∆PQR, P is the largest angle and cos P = 1/3, Further in the circle of the triangle touches the sides PQ, QR, and RP at N, L, and M respectively, such that the lengths of PN, QL, and RM are consecutive even integers, Then, possible length(s) of the side (s) of the triangle is (are)

If the sides of the triangles are p, q, √(p² + q² + pq), then the greatest angle is

If in a ∆ABC, the altitudes from the vertices A, B, C on the opposite side are in HP, then sin A, sin B, and sin C are in

If in the ∆ABC, ∠B = 45^o, then a⁴ + b⁴ + c⁴ is equal to

In a ∆ABC, if the sides are a = 3, b = 5 and c = 4, then sin B/2 + cos B/2 is equal to

In a ∆ABC, a, c, A are given and b₁, b₂ are two values of the third side b such that b₂ = 2b₁, then sin A is equal to

The value of log tan [(π / 4) + (ix / 2)] is

2 sin h^-1 θ =

If tan^-1 (ɑ + iβ) = x + iy, then x =

The imaginary part of tan^-1 (cos θ + i sin θ)

The value of tan h^-1 (2^-1) is