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M. Prakash Academy: X 2014 Note that answer key might contain answers of some questions which are not in the set. Please ignore those answers. Trigonometric Ratios, Trigonometric Equations One Correct Option 1 Let α, β be such that π < α − β < 3π. α−β 21 In sin α + sin β = − 65 , cos α + cos β = − 27 , then cos is 65 2 3 3 6 6 (A) − √130 (B) √130 (C) √65 (D) − √65 . √ √ 2 If u = a2 cos2 θ + b2 sin2 θ + a2 sin2 θ + b2 cos2 θ, then the difference between maximum and minimum values of u2 is (A) 2(a2 + b2 ) (B) 2(a2 − b2 ) (C) (a + b)2 (D) (a − b)2 . 3 If cos θ + cos2 θ + cos3 θ = 1 and sin6 θ = a + b sin2 θ + c sin4 θ, then a + b + c = (A) 0 (B) 1 (C) −1 (D) 2. 3pi 5 If tan(π cos θ) = cot(π sin θ), 0 < θ < 4 , then sin θ + π4 = (A) √12 (B) − √12 (C) 2√1 2 (D) − 2√1 2 . 8 The least value of 13 (A) 24 (B) 61 48 n−1 rπ X cos2 10 = n r=1 (A) n 2 (B) 1 2 n−1 2 11 If 0 < A < π4 , then n−2 (D) n+1 . 2 2 √ √ √1+sin 2A+√1−sin 2A = 1+sin 2A− 1−sin 2A (C) (B) − tan A (A) tan A = cos x b (C) cot A (D) − cot A. 1 ak = k, then bc + ck + 1+bk = (A) ka (B) k12 (C) k a − a1 (D) k1 a + a1 . 1 1 16 sec2 α−cos2 α + csc2 α−sin2 α · cos2 α sin2 α = 12 If sin x a + 32 csc2 θ + 38 sec2 θ is (C) 61 (D) 61 . 25 24 = tan x c 1+cos2 α sin2 α 2−cos2 α sin2 α 2 α sin2 α (C) 1−cos 2−cos2 α sin2 α 17 If x = y cos 2π = 3 4 (A) 3 (B) 1 (A) 1+cos2 α sin2 α 2+cos2 α sin2 α 2 α sin2 α (D) 1−cos 2+cos2 α sin2 α 4π , then xy + yz + 3 (B) z cos zx = (C) −1 (D) 0. √ 18 If sin x + sin y = 3(cos y − cos x) and sin 3y + sin 3x = k, then k will have (A) a single value (C) three values (B) two values (D) infinity of values. 19 If cos x + cos 2x + cos 3x = 3, then sin x + sin 2x + sin 3x = (A) 3 (B) 1 (C) 0 (D) none. 23 If A and B are two angles satisfying 0 < A, B < π2 and A + B = π3 , then the minimum value of sec A + sec B is (A) √23 (B) √43 (C) √13 (D) N.O.T. 24 If x + y + z = π, tan x tan z = 2 and tan y tan z = 18, then tan2 z is equal to (A) 15 (B) 16 (C) 19 (D) 20. 26 If A, B, C are angles of an acute angled triangle, then the minimum value of tan4 A + tan4 B + tan4 C will be √ (A) 729 (B) 27 (C) 81 3 (D) N.O.T. 27 If A is an obtuse angle, then sin3 A−cos3 A sin A + √1+tan 2 A − 2 tan A cot A is always equal to sin A−cos A (A) 1 (B) −1 (C) 2 (D) N.O.T. 28 The number of roots of the equation 3 sin2 x = 8 cos x in − π2 , π2 is (A) 1 (B) 2 (C) 3 (D) 4. √ 29 The solution set of |4 sin x − 1| < 5, |x| < π is π π (A) −π, − 4π ∪ − π5 , 10 ∪ 9π ,π (B) − 9π , − 10 ∪ 3π , 7π 5 10 10 10 10 π 3π (C) −π, − 9π ∪ − 10 (D) − 7π , 10 ∪ 7π ,π , 9π . 10 10 10 10 30 The number of solutions of 2 2 3sin 2x+2 cos x + 31−sin 2x+2 sin x = 28 in [0, 2π] is (A) 3 (B) 4 (C) 5 (D) 6. 34 If α, β, γ, δ satisfy the equation tan x + tan α tan β tan γ tan δ = (A) 12 (B) − 12 (C) 1 (D) − 31 . π 4 = 3 tan 3x, then 35 The number of roots of the equation sec2 θ + csc2 θ = 8 in [0, π] is (A) 1 (B) 2 (C) 0 (D) 4. 36 of values of x in [0, 2π] such that √ The number √ 3 sin x + 2 · cos x = 0 is (A) 0 (B) 1 (C) 2 (D) 4. 37 The number of solutions of | cot x| = cot x + csc x, 0 ≤ x ≤ 2π is (A) 0 (B) 1 (C) 2 (D) 3. 38 The number of values of x in [0, 3π] such that 2 sin2 x + 5 sin x − 3 = 0 is (A) 1 (B) 2 (C) 4 (D) 6. 40 The number of solutions of the equation sin 2x + cos 2x + sin x + cos x + 1 = 0 in [0, 2π] is (A) 2 (B) 3 (C) 4 (D) 6. 41 If tan 5θ = cot 2θ, then θ = π π π (A) nπ + 14 (B) 2nπ ± 14 (C) (2n + 1) 14 π (D) (2n − 1) 14 . 51 number of solutions of sin 5θ cos 3θ = sin 9θ · cos 7θ in πThe 0, 2 is (A) 5 (B) 10 (C) 4 (D) 9. √n π π 52 The positive integer n such that sin 2n + cos 2n = 2 is (A) 4 (B) 5 (C) 6 (D) 8. 59 If cos4 x + 4 sin4 x = 1; then general value of x equals to q −1 1 (A) nπ (B) nπ ± sin , nπ (C) 2nπ (D) 2nπ ± π4 . 2 5 3 Asnswer Keys One Option Correct 1A 2D 3A 9C 10 C 11 C 17 D 18 A 19 C 25 A 26 B 27 B 33 A 34 D 35 D 41 C 42 A 43 A 45 C 46 C 47 B 53 A 54 C 55 C 4B 12 D 20 D 28 B 36 B 44 C 48 D 56 A 5C 13 B 21 B 29 C 37 B 6B 7A 14 D 15 C 22 A 23 B 30 B 31 A 38 C 39 B 8D 16 D 24 B 32 C 40 C 49 C 57 D 50 C 58 A 52 C 60 C 51 D 59 B Integer Answer Type 1 If A + B = 225◦ , then value of (1 + tan A)(1 + tan B) is 2 (sin θ + csc θ)2 + (cos θ + sec θ)2 is greater than or equal to 3 The greatest value fo ‘a0 such that the equation cos 2x + a sin x = 2a − 7 possesses a solution is 4 The angle θ whose cosine equal to its tangent is given by sin θ = n sin 18◦ . Then n is equal to 5 The maximum value of 4 sin2 x + 3 cos2 x + sin x2 + cos x2 is √ a + 2, then a is Asnswer Keys Integer Answer Type 12 29 36 42 54 Properties of Triangles and Radii of Circles One Correct Option 1 In a ∆ABC if s−a = s−b = s−c then 11 12 13 (A) a, b, c are in A.P. (B) a, b, c are in G.P. 1 (C) Ar(∆ABC) = 216 (D) tan2 A2 = 20 . 33 4 If three sides a, b, c of triangle ABC are in A.P., then the value of cot A2 cot C2 is (A) 1 (B) 2 (C) 3 (D) 4. √ 9 If the sides of a triangle are in α, cos α, 1 + sin α cos α, 0 < α < π2 , the largest angle is (A) 60◦ (B) 90◦ (C) 120◦ (D) 150◦ . 12 The internal bisectors of the angles of triangle ABC meet = the circumcricle at D, E, F respectively. Then BC EF (A) sin A2 (B) cos A2 (C) 2 sin A2 (D) 2 cos A2 . 14 If r is inradius and R is the circumradius of a right angled triangle, its area is (A) r(r + 2R) (B) 2(2r + R) (C) rR (D) R(2r + R). 15 In a triangle ABC if r = 2, R = 5, r3 = 12, then C = (A) 30◦ (B) 45◦ (C) 60◦ (D) 90◦ . 16 In a triangle ABC, D and E are the midpoints of BC, CA respectively. If AD = 5, BC = BE = 4, then CA = √ √ √ (C) 2 7 (D) 5 5. (A) 5 (B) 7 19 In a triangle ABC if a = 2b, A − B = 90◦ , then tan C = (A) − 43 (B) 34 (C) 23 (D) 23 . 21 In a triangle ABC if r = 3, R = 6, r1 = 9, then A = (A) 30◦ (B) 45◦ (C) 60◦ (D) 90◦ . 22 In a triangle ABC, if a, b, c are in A.P. Then a possible value of B is (A) 45◦ (B) 75◦ (C) 90◦ (D) 120◦ . 23 In a triangle ABC, D is the midpoint of BC and AD is perpendicular to AC. Then (A) a2 = 3b2 + c2 (B) b2 = 3(a2 + c2 ) (C) a2 = b2 + c2 (D) b2 = 5c2 − a2 . 25 In a triangle ABC, if 4 = a2 − (b − c)2 , then tan A = (A) 15 16 (B) 1 2 (C) 8 17 2 (D) 8 . 15 28 In a triangle ABC if b + c2 = 3a2 , then cot B + cot C = (A) 0 (B) cot A (C) 2 cot A (D) 1. 30 In a triangle ABC, if 3s2 = r12 , then A = (A) 30◦ (B) 60◦ (C) 120◦ (D) 150◦ . 41 In a ∆ABC, if a = 3, b = 4, c = 5, then r1 + r2 + r3 = (A) 9 (B) 10 (C) 11 (D) 12. 50 In a ∆ABC, a = 90◦ , B = 30◦ and D is the foot of perpendicular from A on BC. If AC = 2, then BD = (A) 1 (B) 2 (C) 3 (D) 4. 51 In a ∆ABC, the length of the altitude from A to BC is 2rr1 2 r3 1 2 r3 (B) r2r2 +r (C) r2rr (D) r2r2 −r . (A) r+r 1 3 1 −r 3 52 In a ∆ABC, if the median and altitude from A trisect angle A, then |B − C| = ◦ (C) 30◦ (D) 45◦ . (A) 15◦ (B) 22 21 3 2 55 In ∆ABC, P cos Aif the sides a, b, c are roots of x −11x +38x−40 = = 0, then a 3 9 (A) 4 (B) 43 (C) 16 (D) 16 . 9 Asnswer Keys One Option Correct 1A 2A 3A 9C 10 A 11 B 17 C 18 D 19 B 25 D 26 C 27 B 33 C 34 D 35 A 41 C 42 D 43 D 44 D 45 C 51 C 52 C 53 C 4C 5D 12 C 13 C 20 C 21 C 28 B 29 B 36 D 37 C 6A 7B 8A 14 A 15 D 16 C 22 A 23 A 24 C 30 C 31 A 32 B 38 C 39 C 40 D 46 B 54 C 48 C 56 B 47 A 55 D 49 D 57 C 50 C 58 C Integer Answer Type 1 The sides of a triangle are 3 consecutive numbers and the largest angle is double the least, then the greatest side is 2 The number of triangles can be constructed with the data a = 5, b = 7, sin A = 43 , is 5 In a right angled ∆ABC, sin2 A + sin2 B + sin2 C is equal to 6 Points D, E are taken on the side BC of a triangle ABC, such that BD = DE = EC. If ∠BAD = x, ∠DAE = y, ∠EAC = sin(y+z) z, then the value of sin(x+y) is equal to sin x sin z 7 If in a triangle ABC, right angled at B, s − a = 3, s − c = 2, then the value of a is 8 If length of the sides AB and AC of a ∆ABC are 3 cm and 6 cm respectively and if cosine of angle BAC is 81 , then length of the angle bisector of angle BAC is 9 If I is the incentre of ∆ABC and AD is the angle bisector of angle BAC so that AI : ID = 2 : 1, then value of tan B2 tan C2 is equal to n1 , where n is 10 If the sum of the squares of the sides of a triangle is equal to twice the square of its circumdiameter, then sin2 A+ sin2 B + sin2 C is equal to 11 In a triangle ABC, the median to the side BC is of length √ 1 √ and it divides the angle A into angles of 30◦ and 45◦ . 11−6 3 Find the length of the side BC. 12 In a ∆ABC, find side c, when a = 5, b = 4 and cos(A − B) = 31/32. 13 In a ∆ABC, the tangent of half the difference of two angles is one-third the tangent of half the sum of the angles. The ratio of the sides opposite to the angles is k : 2, then find k. 14 Two sides of√a triangle are a, b given by the roots of the equation x2 − 2 3x + 2 = 0. The angle between the sides is π/3. Find the value of c2 . 15 Find the value of of ∆ABC. b−c r1 + c−a + a−b , if a, b, c are length of sides r2 r3 16 If A0 , A1 , A2 , A3 , A4 and A5 be the consecutive vertices of a regular hexagon inscribed in a unit circle. Then find the product of length of A0 A1 , A0 A2 and A0 A4 . 17 If the area of cyclic quadrilateral ABCD is √ 3 3 . 4 The ra√ dius of the circle circumscribing it is 1. If AB = 1, BD = 3 then evaluate BC · CD. 18 The two adjacent sides of a cyclic quadrilateral are 2 ◦ and 5 and the angle √ between them is 60 . If the area of the quadrilateral is 4 3, Find the sum of remaining two sides. 20 If x, y, z are perpendicular from the circumcentre of the abc . Find sides of the ∆ABC respectively, then xa + yb + zc = kxyz the value of k Asnswer Keys Integer Answer Type 16 20 33 41 11 2 12 6 13 1 14 6 52 64 15 0 16 3 74 83 17 2 18 5 93 10 2 19 3 20 4