Solns of Triangles

1

ST LEVEL-I 1.

If the bisector of angle A of ABC makes an angle  with BC, then sin is equal to B C B C (A) cos  (B) sin     2   2 

 

(C) sin  B 

A  2

 

(D)sin  C 

A  2

2.

If the radius of the circumcircle of an isosceles triangle ABC is equal to AB = AC then the angle A is (A) /6 (B) /3 (C) /2 (D) 2/3

3.

In a triangle ABC, if angle A is (A) 300 (C)600

4.

(B)450 (D) 900

If A = 450, B =750 then a + c 2 is equal to (A) 2b (B) 3b (C)

5.

2b

(D) b

The sides of a triangle inscribed in a given circle subtend angle , and  at the centre. The minimum value of the arithmetic mean of cos( + /2), cos( + /2) and cos( +/2) is equal to (A) 0 (B) 1/ 2 (D) - 3 /2

(C) –1 6.

2 cos A cos B 2 cos C a b     , then the value of the a b c bc ca

A regular polygon of nine sides, each of length 2, is inscribed in a circle. The radius of the circle is (A)sec

 9

(C) cosec

 9  (D) tan 9 (B)sin

 9

7.

In an acute angled triangle ABC, the least value of secA + secB + secC is (A) 6 (B)3 (B) 9 (D) 4

8.

A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is (A) a2/4 (B) a2/6

2

(C) a2/9

(D) 2a2/3

9.

If 3 sin2A + 2sin2B =1 and 3 sin2A – 2 sin2B = 0, where A and B are acute angles, then A + 2B is equal to (A) /3 (B) /4 (C) /2 (D) none of these.

10.

If in a ABC, cos(A - C)cosB + cos2B = 0, then a2, b2, c2 are in (A) A.P. (B) G.P. (C) H. P. (D) none of these

11.

If tan(A+B), tanB, tan(B+C) are in A.P., then tanA, cotB, tanC are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

12.

If twice the square of the diameter of a circle is equal to the sum of the squares of the sides of the inscribed triangle ABC, then sin2A + sin2B + sin2C is equal to (A) 2 (B) 3 (C) 4 (D) 1

13.

Consider a triangle ABC, with given A and side ‘a’. If bc = x2, then such a triangle would exist if, ( x is a given positive real number) . A A (A) a < x sin (B) a >2x sin 2 2 A (C) a < 2 x sin (D) None of these . 2

14.

If in ABC a, b, c are in geometric progression then, (A) cot2A, cot2B, cot2C are in G.P. (B) cosec2A, cosec2B, cosec2C are in A.P. (C) cosec2A, cosec2B, cosec2C are in G.P. (D) none of these.

15.

If in a  ABC, 8R2 = a2 + b2 + c2, then the triangle is (A) Equilateral (B) Right angled (C) Isosceles (D) None of these

16.

In a triangle ABC, angle B is greater than angle A, B –A <

2 . If the values of A 3 and B satisfy the equation 3sinx – 4sin3x - k = 0 (0 < k < 1), then angle C is equal to   (A) (B) 3 6 2 (C) (D) None of these 3

3

17.

If in a triangle ABC, b + c = 4a. Then cot

5 3 5 (C) 8 (A)

B C cot is equal to 2 2 3 (B) 5 (D) None of these

18.

The ex-radai of a triangle r1, r2, r3 are in Harmonic progression, then the sides a, b, c are in (A) A.P (B) G.P (C) H.P (D) none of these

19.

In a ABC A = 300, B = 600, then a : b : c is (A) 1 : 2 : 3 (B) 1 :

3 :2

(C) 1 : 2 :

2 :3

(D) 1 :

3

20.

In a ABC, the value of a (cos B + cos C) + b (cos A + cos C) + c (cos A + cos B) is (A) a + b (B) a + b + c (C) b + c (D) b + c –a

21.

In a triangle a = 13, b = 14, c = 15, r = (A) 4 (C) 2

22.

In a triangle ABC, If b + c = 3a, then the value of cot (A) 1 (C) 3

23.

(B) 8 (D) 6

B C cot is 2 2

(B) 2 (D) 3

In a triangle ABC, then 2ac sin (A) a2 + b2 –c2 (D) b2 –c2 –a2

1 (A –B + C) is 2 (B) c2+ a2 –b2 (D) c2 –a2 –b2

24.

The angle A of the triangle ABC, in which (a + b + c) (b + c –a) = 3bc is (A) 300 (B) 450 0 (C) 60 (D) 1200

25.

In a triangle ABC, Let C = triangle, then (A) a + b (C) c + a

26.

In a triangle ABC,

 , if r is the inradius and R is the circumradius of the 2 2 (r + R) is equal to (B) b + c (D) a + b + c

cb A . tan is equal to c b 2

4

A  (A) tan  B  2   B  (C) tan A   2  27.

A  (B) cot  B  2   (D) none of these

In a ABC, a = 2b and |A –B| =

 , the measure of angle C 3

…………………………………….. 28.

In a ABC, the sides a, b and c are such that they are the roots of x3 –11x2 + 38x cos A cos B cos C –40 = 0 then the value of   = a b c ………………………………………

29.

If AD, BE and CF are the medians of a ABC, then (AD2 + BE2 + CF2) : (BC2 + CA2 + AB2) = ………………………………………………..

30.

sin A, sin B, sin C are in A.P for the ABC then (A) altitudes are in A.P (B) sides are in A.P (C) altitudes are in H.P (D) medians are in A.P

31.

In a triangle ABC, tan C< 0, then (A) tan A . tan B < 1 (C) tan A + tan B + tan C < 0

32.

If in a triangle ABC, b + c = 4a. Then cot

5 3 these (A)

33.

(D) None of

sin B sin C sin2 A , then the triangle is   sin C sin B sin B sin C (B) isosceles (C) scalene (D) None of

In a triangle, the lengths of the two larger sides are 10 and 9 respectively. If the angles are in A.P., then the length of third side can be (A) 5 – 6 (B) 3 (C) 5

35.

3 5

B C cot is equal to 2 2 5 (C) 8

If in a triangle ABC, cosA = (A) right angled these

34.

(B)

(B) tan A . tan B > 1 (D) tan A + tan B + tan C > 0

(D) 3 3

In a ABC, maximum value of c cos (A - ) + a cos(C + ), equals (A) a (B) b (C) c

(D)

a2  c 2

5

36.

In a triangle ABC, a2 ( cos2B - cos2C) + b2 ( cos2C – cos2A) + c2 ( cos2Acos2B) equals (A) 0 (B) 1 (C) -1 (D) none of these

37.

In a ABC, the angles A and B are two values of  satisfying 3 sin+ cos = , || < 2. Then C equals (A) 60 (B) 90 (C) 120 (D) none of these

38.

If the ex-radii of a triangle ABC are in H.P., then the sides a, b, c are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

6

LEVEL-II 1.

The expression

(a  b  c )(b  c  a )(c  a  b)(a  b  c) is equal to 4b 2 c 2

(A) cos2A (C) cosA cosB cosC

(B) sin2A (D) None of these

2.

The perimeter of a triangle ABC is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is (A) /6 (B) /3 (C) /2 (D) 

3.

If a2, b2,c2 are in A.P , then cotA, cotB, cotC are in (A) A.P (B) G.P (C) H.P (D) None of these

4.

The area of the circle and the regular polygon of n sides and of equal perimeter are in the ratio of (A) tan(/n) : /n (B) cos (/n) : /n (C) sin(/n) : /n (D) cot(/n) : /n

5.

In a triangle ABC, (a+b+c)(b+c-a) = bc if (A)  < 0 (C) 0 < < 4

6.

(B)  > 0 (D)  > 4

In a triangle ABC, AD is the altitude from A. Given b > c, C =230 and AD =

abc then B is equal to b  c2 2

(A) 230 (C) 670

(B) 1130 (D) 900

7.

In any triangle ABC, a3cos(B-C) + b3 cos(C-A) + c3cos(A-B) is equal to (A) 6abc (B) 9abc (C) 3abc (D) None

8.

In a triangle ABC, a  b  c is (A) always positive (B) always negative (C) positive only when c is smallest (D) none of these .

9.

In a triangle with sides a,b, and c, a semicircle touching the sides AC and CB is inscribed whose diameter lies on AB. Then , the radius of the semicircle is (A) a/ 2 (B) / s 2abc 2 A B C (C) (D) cos cos cos s a  b ab 2 2 2

10.

A triangle is inscribed in a circle. The vertices of the triangle divide the circle in to three arcs of length 3, 4 and 5 units, then area of the triangle is equal to,

7









9 3 1 3 2 9 3 1 3 (C) 2 2 (A)

11.









9 3 3 1 2 9 3 3 1 (D) 22 (B)

If a sinx + bcos(C + x) + bcos (C –x) = , then the minimum value of |cosC| is (A) (C)

 2  a2 b2  2  a2 16b 2

(B)

 2  a2 4b 2

(D) none of these

12.

In a ABC, the point D divides BC in the ratio 1:2 . Also AD is perpendicular to AB. Then the value of the expression tanB(1+2tanA tanC) – 2tanC is (A) 0 (B) 1 (C) –1 (D) none of these

13.

If in ABC , secA , secB, secC are in Harmonic progression, then (A) a, b, c, are in harmonic progression. A B C (B) cot , cot , cot are in harmonic progression 2 2 2 (C) r1, r2, r3 are in arithmetic progression A B C (D) cot , cot , cot are in arithmetic progression . 2 2 2

14.

In a triangle ABC a = 7, b = 8 and c= 9. Then the length of median from B to AC is given by (A) 9 (B) 8 (C) 7 (D) 6

15.

If sinA and sinB of a triangle ABC satisfy c2x2 – c(a+b)x + ab = 0, then the triangle is (A) Equilateral (B) Isosceles (C) Right angled (D) Acute angled

16.

The number of triangles that can be made with the given data: b = 2cm, c = 6 cm and B = 30°, is (A) 1 (B) 2 (C) zero (D) None of these

17.

In ABC, if AB = c , AC = b, BC = a and A : B : C = 1 : 2 : 5, then (A) b2 = a(c + a) (B) b2 = a( c – a) 2 (C) b = a( a – c) (D) None of these.

18.

In ABC, if

c a ab bc   , then 12 14 18

8

11 r 7 2 (C) r3  r 11

(A) r1 

19.

20.

(B) r2 = 11r (D) None of these

If a cos A = b cos B, the triangle is (A) equilateral (C) isosceles

(B) right angled (D) right angled or isosceles

The sides of a triangle are a, b and  (A) 3 2 (C) 3

a 2  ab  b2 , then the greatest angle is  (B) 2

(D) none of these

21.

Two sides of a  are given by the roots of the equation x2 –2 3 x + 2 = 0. The  angle between the sides is . The perimeter of the triangle is 3

22.

In a triangle ABC, R = circumradius and r = inradius. The value of a cos A  b cos B  c cos C is equal to abc R R (A) (B) r 2r r 2r (C) (D) R R

23.

In a triangle ABC, 2 cos

AC  2

ac a 2  c 2  ac

 3 (D) A, B, C are in A.P

(A) B =

, then

(B) B = C (D) B + C = A

24.

The distance of the circumcentre of the acute angled ABC from the sides BC, CA and AB are in the ratio (A) a sin A : b sin B : c sin C (B) cos A : cos B : cos C (C) a cot A : b cot B : c cot C (D) none of these

25.

If twice the square of the diameter of a circle is equal to the sum of the squares of the sides of the inscribed triangle ABC, then sin2A + sin2B + sin2C is equal to (A) 2 (B) 3 (C) 4 (D) 1

26.

In ABC, if

c a ab bc   , then 12 14 18

9

(A) r1 

11 r 7

(B) r2 = 11r

(C) r3 

2 r 11

(D) None of

these 27.

In a triangle ABC, 2 sinA cosC = 1 and (A) right angled at A (C) right angled at C

28.

In a triangle ABC,

r1  r2 r2  r3 r3  r1 

(A) 4

29.

In a ABC, 8 abc these

(A)

tan A 1  then triangle is tan C 2 (B) right angled at B (D) none of these

Rs 2

(B) 4 abc

is equal to (C)

4abc

(D) 

2

a cos A  b cos B  c cos C is equal to 2 2 83 (B) (C) R abc

(D) None of

30.

If p1, p2 and p3 are respectively the lengths of perpendiculars from the vertices of a triangle ABC to the opposite sides, then the value of p1p2p3 is a 2b2c 2 a 2b2c 2 a 2b2c 2 a 2b2c 2 (A) (B) (C) (D) 8R2 8R3 8R 4 4R 2

31.

If in a triangle cos2A + cos2B – cos2C = 1, then the triangle is (A) Right angled at A (B) Right angled at B (C) Right angled at C (D) not a right triangle

32.

If in a triangle ABC, (A) right angled these

SinB  SinA CosB  CosA   0 then the triangle is SinC CosC (B) equilateral (C) isosceles (D) None of

33.

If sin and - cos are the roots of the equation ax2 – bx – c = 0, where a, b, c are the sides of a triangle ABC then c b c (A) cosB = 1 (B) cosB = 1(C) cosB = 1 + (D) cosB = 1 2a 2a 2a b + 2a

34.

In a right angled triangle ABC, with right angle at B, 8R2 2 these

(A)

(B)

2 R2 2

(C)

4 R2 

1 1 1 1  2  2  2 = 2 r r1 r2 r3 (D) None of

10

35.

If in a triangle ABC, C = 1350, then value of tan A + tan B + tan A tan B equals (A) 0 (B) 1 (C) –1 (D) none of these

36.

Suppose the angles of a triangle ABC are in A.P. and sides b and c satisfy b : c = 3 : 2 then the angle A equals (A) 450 (B) 600 (C) 750 (D) 900

37.

If a2, b2, c2 are the roots of the equation x3 –Px2 + Qx – R = 0 where a, b, c be cos A cos B cos C the sides of a triangle ABC then the value of   equals a b c P P (A) (B) R 2 R P (C) (D) none of these 4 R

38.

In a triangle ABC, (A) R (B) 2R

b2  c 2 c 2  a2  equals a sinB  C b sinC  A  1 (B) 2R (D) none of these

11

ANSWERS LEVEL −I 1. 5.

A D

2. 6.

D C

3. 7.

D A

4. 8.

A B

9. 13. 17. 21.

C D A A

10. 14. 18. 22.

A C A B

11. 15. 19. 23.

C B B B

12. 16. 20. 24.

26.

D

27.

A C

30. 34. 38.

B A A

31. 35.

C B

32. 36.

A C B C 9 16 A A

1. 5.

B C

2. 6.

A B

3. 7.

A C

4. 8.

A A

9. 13. 17. 21.

C B,C A 6 1 2

10. 14. 18. 22.

A C A C

11. 15. 19. 23.

B C D

12. 16. 20. 24.

A C C C

32. 36.

C

25. 29. 33. 37.

28.

LEVEL −II

25. 28. 29. 33.

37.





26.

27.

A C

30. 34.

A

B

38.

D

31. 35.

B