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PROPERTIES AND SOLUTIONS OF TRIANGLES 

1. Σa3 cos (B −  C) =
A: 3abc

 

2. In ∆ABC, if sin A, sin B are the roots of c2x2 − c(a + b) x + ab = 0, then sin C =
A: 1

 

3. The sides of a triangle are in A.P. If the angles A and C are the greatest and smallest angles respectively, then 4(1 − cos A)(1 − cos C) =
A: cos A + cos C

 

4. If b = 3, c = 4,   then number of triangles that can be constructed is
A: 0

 

5. In a triangle ABC, if cos A cos B + sin A sin B sin C = 1, then a : b : c =
A: 1 : 1 : 

 

6. If sides of a triangle are sin α, cos α,   for some 0 < α < , then greatest angle of triangle is
A: 120o

 

7. In a triangle ABC, if  the sides a, b and c are in
A: A.P.

 

8.  
A: 60o

9.  in terms of k, where k is the perimeter of the ∆ABC is
A:  

 

10. Two straight roads intersect at an angle of 60°. A bus on one road is 2 km. away from the intersection and a car on the other road is 3 km. away from the intersection. The direct distance between the vehicles is
A:  km

 

11. The distance of the middle point of the side BC from the foot of the altitude from A to BC is (assuming b > c)
A:  

12. In ∆ABC, if the sides a, b, c are the roots of x3 − 11x2 + 38x − 40 = 0, then the value of 

A:  
 

13. If the area of ∆ABC is a2 - (b - c)2, then the value of tan A is
A:   

 

A: −xyz
 

15. ABC is a triangle, D is the midpoint of BC. If AD is perpendicular to AC, then the value of cos A cos C =
A:  

 

16. In ∆ ABC, if cos A + 2cos B + cos C = 2 then a, b, c are in
A: A.P.

 

17. In the given figure, 'P' is any interior point of the equilateral triangle ABC of side length 2 units. If Xa, Xb and Xc represents the distance of P from the sides BC, CA and AB respectively, then Xa + Xb + Xc is equal to       
A: 

 

18. If a = (b − c) sec θ, then tan θ =     
A: 

 

19. If the bisector of the angle A of a triangle ABC meets the opposite side in D, then AD =
A: 

 

20. In ∆ABC, if 5 cos C + 6 cos B = 4 and 6 cos A + 4 cos C = 5, then tan 
A: 

 

Ans:  

 

22. In triangle ABC, 
Ans: 

 

23. There exists a triangle ABC satisfying
Ans: (a + b)2 = c2 + ab

 

Ans: A.P.
 

25. In a ∆ABC, the value of   is equal to
Ans: 

 

26. In ∆ABC, AD is the altitude from A. If b > c, 
Ans: 113°

 

27. If in a triangle ABC, we define

Ans: −xyz

 

28. In triangle ABC, medians AD and BE are drawn. If AD = 4,  then the area of the ∆ABC is
Ans:  

 

29. In a triangle ABC, the median to the side BC is of length  and it divides angle A into angles of 30° and 45°. BC is of length
Ans: 2 Units

 

30. If α, β, γ are the lengths of the altitudes of ∆ABC, then 
Ans: 0

 

31. In a triangle, ABC a = 5, b = 4 and c = 3. 'G' is the centroid of the triangle. Circumradius of triangle GAB is equal to
Ans: 

 

32. In triangle ABC, If   then the triangle is
Ans: Right angled

 

33. In triangle ABC, let DC =  If 'r' is the inradius and 'R' is the circumradius of the  triangle, then 2(r + R) is equal to
Ans: a + b − c

 

34. In any ∆ABC,  is always greater than
Ans: 9

 

35. In a triangle ABC, if r1 = 2r2 = 3r3, then a : b is equal to
Ans:  5/4

 

36. If b2 + c2 = 3a2 then cot B + cot C =
Ans: cot A

 

37. Let AD be a median of the ∆ABC. If AE and AF are medians of the triangles ABD and ADC respectively and AD = m1, AE = m2, AF = m3, then  is equal to
Ans: m22 + m32 − 2m12

 

38. In a ∆ABC, the length of the altitude from A to BC is
Ans: 

 

39. In ∆ABC, a = 5, b = 4 and cos (A − B) =  then side c is
Ans: 4

 

40. If (r2 − r1)(r3 − r1) = 2r2r3 then  =
Ans: 90°

 

41. In traingle ABC,  AA1 and AA2 are the median and altitude respectively. If  AA1 = 1AA2 = 2AC, then   is equal to
Ans:  

 

42.  
Ans:  

 

43. Three circles touch one-another externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4. The ratio of the product of the radii to the sum of the radii of circles is
Ans: 16 : 1

 

44. If 2(r + R) = b + c, then A =
Ans: 90°

 

45. In a right angled triangle ABC, the bisector of the right angle C divides AB into segment x and y and tan  = t, then x : y =
Ans: (1 − t) : (1 + t)

 

46. The roots of the equation x3 − (r + 4R) x2 + s2x − s2 r = 0 are
Ans: r1, r2, r3

 

47. BC is a side of a square. On the perpendicular bisector of BC, two points P, Q are taken, equidistant from the center of the square. BP, CQ are joined and cut in A, then in the triangle ABC, tan A (tan B − tan C)2 =
Ans: −8

 

48. If the sides of a triangle are xy + yz, yz + zx, zx + xy then inradius of the triangle is
Ans: 

 

49. A variable triangle ABC is cirumscribed about a fixed circle of unit radius. Side BC always touches the circle at D and has fixed direction. If B and C vary in such a way that (BD) . (CD) = 2, then locus of vertex A will be a straight line
Ans: parallel to side BC

 

50. The sum of the radii of inscribed and circumscribed circles for an 'n' sided regular polygon of side a, is
Ans: 

 

ONE OR MORE THAN ONE ANSWER TYPE

51. If cos(θ − α), cos θ, cos (θ + α) are in H.P., then cos θ sec   is equal to
    A) −1    B) −     C)       D) 2
Ans: B, C

 

52. In a triangle ABC, 2a2 + 4b2 + c2 = 4ab + 2ac, then the numerical value of cos B is equal to
    A) 0      B)         C)        D)  
Ans: D

 

53. Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is less than or equal to 
A) 3      B) 2        C)  3/2    D) 1
Ans: B, C, D

 

54. If a, b, A be given in a triangle and c1 and c2 two possible values of third side such that c12 + c1c2 + c22 = a2, then A is equal to
    A) 30°      B) 60°      C) 90°      D) 120°
Ans: B, D

 

55. Let A, B, C be angles of a triangle and let 


      A) cot D cot E + cot E cot F + cot F cot D = 1
      B) cot D + cot E + cot F = cot D cot E cot F
      C) tan D tan E + tan E tan F + tan F tan D = 1
      D) tan D + tan E + tan F = tan D tan E tan F
Ans: B, C

 

56. Given an isosceles triangle with equal sides of length b, base angle α < π/4, R, r the radii and O, I the centres of the circumcircle and incircle respectively, then
 A)  B) ∆ = 2b2 sin 2α     C)     D) 
Ans: A, C, D

 

57. If in a triangle ABC, a, b, c are in A.P. and p1,  p2,  p3 are the altitudes from the vertices A, B, C respectively, then
      A) sin A, sin B, sin C are in A.P.       B) sin A, sin B, sin C are in H.P.
      C)                     D) 
Ans: A, D

 

58. If in a triangle, circumcentre and incentre are equidistant from side BC, then
       A) R sin A = 2r                    B) R cos A = 2r
       C) R cos A = r                      D) cos B + cos C = 1
Ans: C, D

 

59. In a triangle ABC, If tan   then we must have (∆ denotes area of triangle)
       A) ∆ = 2c2 sin A      B)       C)         D) 
Ans: A, B, C 

 

60. If A1, A2, A3, ......, An be a regular polygon of n sides and  then
       A) n = 5        B) n = 6      C) n = 7       D) None of these
Ans: C

 

61. In ∆ABC, AD is the bisector of , DE ⊥ AD intersect C at E and meets AB produced at F, then
     A)    B)    C) AE = AF  D) 
Ans: A, B, C, D

 

62. A1B1C1 is the triangle formed by joining the feet of perpendiculars drawn from A, B, C upon the opposite sides; in like manner A2B2C2 is the triangle obtained by joining the feet of perpendiculars from A, B and C, on the opposite sides so on, then
    A)          B) 
     C)            D) 
Ans: A, D

 

63. There exists a triangle ABC satisfying the conditions
     A)     B)   C)   D) 
Ans: A, D

 

64. In inside a big circle exactly n  small circles, each of radius r, can be drawn in such a way that each small circle touches the big circle and also touches the both its adjacent small circles as shown in the figure, then radius of big circle is     
      A)            B) 
      C)        D)  
Ans: A, D

 

65. ABC be a triangle. D, E, F be the feet of the perpendiculars from A, B, C on opposite sides, then perimeter of triangel DEF is
    A) a cos A + b cos B + c cos C       B)    C)       D) 
Ans: A, C

 

COMPREHENSION PASSAGE

PASSAGE - 1:
       A polygon has 'n' sides. If all the sides and all the angles are same, then this polygon is called regular polygon. Let A1A2A3 ..... An be a regular polygon of n sides, R be the radius of circumscribed circle of regular polygon and r be the radius of inscribed circle of regular polygon. If A1 A2 = A2 A3 = A3 A4 = ....  An A1 = a .
Answer the following:

 

66. The value of A1 Aj (j = 1, 2, 3, ..., n) is
Ans: 

 

67. The value of r is
Ans: 

 

68. A regular pentagon and a regular decagon have the same perimeter then their areas are in the ratio
Ans: 2 : 

 

PASSAGE - 2:
      AL, BM and CN are diameter of the circumcircle of a triagle ABC. ∆1, ∆2, ∆3 and ∆ are the areas of the triangles BLC, CMA, ANB and ABC respectively.

 

69. 1 is equal to
Ans: 2R2 sin A cos B cos C

 

70.1 + ∆2 + ∆3 is equal to
Ans:

 

71. If BL2 + CM2 + AN2 = x and CL2 + AM2 + BN2 = y, then
Ans: x − y = 0

 

MATRIX MATCHING

72.

Ans: A-Q; B-S; C-R; D-P.

 

73. If in a trinagle ABC, b + C = 3a


Ans: A-Q; B-R; C-P; D-S.

 

INTEGER TYPE 

74. If in ∆ABC, line joining the circumcentre and orthocentre is parallel to side AC, then value of tan A. tan C is equal to
Ans: 3

 

75. The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. If k is the largest side of the triangle, then  =
Ans: 1

 

76. In a triangle ABC, the median AD is perpendicular to AC. If b = 5 and c = 11, then 'a' is equal to 'pq' (0 < p, q < 9), find  (where [.] represents gratest integer function).
Ans: 4

 

77. In a triangle ABC, the incircle touches the sides BC, CA and AB at D, E, F respectively. If radius of circle is 4 units and BD, CE, and AF be consecutive natural numbers, and if length of side AB is k then  =
Ans: 7

 

78. In a triagle ABC, if  then the value of cos A + cos B + cos   = ... (where (.) denotes least integer function) 
Ans: 2

 

79. If the radius of the circumcircle of a triangle is 12 and that of the incircle is 4, then  of the sum of radii of the escribed circles is .......
Ans: 1

 

80. If b = 3, c = 4, B = 60°, then the number of triangles that can be constructed is
Ans: 0

 

81. The diagonals of parallelogram are inclined to each other at an angle of 45°, while its sides a and b (a > b) are inclined to each other at an angle of 30°. The value of the least interger greater than or equal to  is equal to
Ans: 2

 

82. If in a ∆ABC, sines of angles A and B satisfy 4x2 − 2 + 1 = 0, and cos (A − B) =  (0 < p, q < 9), then  ([.] denotes greatest integer function)
Ans: 5

 

83. In a triangle with one angle  the lengths of two sides form an A.P. If the length of largest side is 7 cm, the radius of circumcentre of triangle is  where are k, m are prime numbers then k − m =
Ans: 4

Posted Date : 19-02-2021

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