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PREPARED BT TEAM GURUKUL, www.thegurukulinstitute.in Properties of triangles, Class - XII

THEGURUKULINSTITUTEPLOT 5C, 2ND FLOOR,GANAPATI COMPLEX,SEC-13,OPP. JAIPURIA SCHOOL,VASUNDHARA,GHAZIABAD (U.P)

PROPERTIESOFTRIANGLE

CLASSXII

1.

If the angles of ABC are in the ratio 1: 2 : 3 then the corresponding sides are in the ratioa) 2:3:1 b)

3 : 2 : 1 c) 2 :

3 : 1 d) 1 :

3 : 2

2.

In triangle ABC + + is equal toa)

++ b)

+ c)

+ d) None of these

3. In a triangle ABC, if A = 60, then

++

+equalsa) 0 b)1 c) d)None of these

4. In a triangle ABC, 2ac sin+

2is equal to

a) a2+ b

2c2 b) c2+ a2b2 c) b2c2a2 d) c2a2b2

5. In a triangle ABC, if c42(a2+ b2)c2+ a4+ a2b2+b4= 0 then angle C=

a) 60or 120 b) 45or 135 c) 30or 150 d) 456. In a triangle ABC, a(b cos C c cos B) =

a) a2 b) b2c2 c) 0 d) None of these

7. The expression +++ + (+ )422 is equal to

a) Cos2A b) sin

2A c) cos A cos B cos C d) None of these

8. In a triangle ABC, if2

+cos

+2 cos

=

+

is equal toa) 30 b) 45 c) 60 d) 90

9. If the radius of the circumcircle of an isosceles triangle ABC is equal to AB= AC, then the angleA is

a)

6

b)3 c)

2 d)

23

10.

If the sides of a triangle are in ratio 2 : 6 : (3 + 1) then the largest angle of the triangle will be a) 60 b) 75 c) 90 d) 120

11.

IfA=45,B=75,then a + c2 is equal toa)

2b b) 3b c) 2b d) b12.With usual notation, if in a triangle ABC, +11

=+12

=+13

, then cos A equals

a) 1/5 b) 19/35 c) 5/7 d) Data inadequate

13.In an isosceles triangle ABC, AB = AC. If vertical angle A is 20 , then a3+b3is equal toa) 3ac2 b) 3a2b c) abc d) None of these

14.

In a triangle ABC of 1 + + 1 + = k , then k lies in the intervala) [-1, 1] b) [0, 1] c) [- 4, 4] d) [0, 4]

15.Which of the following pieces of data does not uniquely determine acute angled ABC ( R =circumradius)

a) a , sin A, sin B b) a, b, c c) a, sin B, R d) a, sin A, R

16.in a triangle ABC a3cos (B

C)+ b

3cos (CA ) + c3cos (A + B) =

a)

3abc b) 3(a + b+ c) c) abc (a + b + c) d) 0

17.In a triangle ABC,2 sin()sin +sin +

2sin ()sin +sin +

2 sin()sin +sin =

a) abc b) a + b+ c c) 0 d) 1

18.In triangle ABC, a sin (B C) + b sin (CA )+ c sin (A B) isa) abc b) a + b+ c c) 1 d) None of these

19.In a triangle ABC, (b2c2)cot A + (c2a2) cot B + (a2b2)cot C is

a)

0 b) 1 c)1

+1

+1

d)++

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PREPARED BT TEAM GURUKUL, www.thegurukulinstitute.in Properties of triangles, Class - XII

20.If semi perimeter of a triangle is 15 then value of (b+c)cos (B + C)+( c + a)cos (C+ A)+ (a +b)cos (A+B) is equal toa) 15 b) 30 c) cannot be determined d) None of these

21.In a triangle ABC, (sin C(a sin A + b sin B) cos C(a cos A + b cos B)) is equal toa) a + b + c b) a c) b d) c

22.In a triangle ABC, a = 5, b = 4, and cos (A B) = 31/32 then the third side is equal toa) 36 b) 66 c) 6 d) (216)1/3

23.

In a triangle if 3a = b+ c, then cot

2cot

2=

a) 1 b) 2 c) cot A/2 d) 2 cot A/2

24.If cos2= +

2 , thena) a2+ b2= c2 b) b2+ c2= a2 c) c2+ a2= b2 d) None of these

25.

In a ABC, bc cos2A2

+ca cos22+ ab cos2

2

=

a) (sa )2 b) (sb )2 c) (sc )2 d) s2

26.

In a triangle ABC, 1tan2tan

2=

a)2 b)

2 c)

2 d) 0

27.In a triangle ABC, if a = 8, b = 10 and c = 12 then bc cos 2(B/2) is equals toa) 49 b) 25 c) 9 d) 180

28.

In a triangle ABC, a = 4, b = 3 , angle A = 60 . Then c is a root of the equationa) c23c7 = 0 c) c23c + 7 = 0

b) c2+ 3c + 7 = 0 d) c2+ 3c7 =029.In a triangle ABC, angle A is greater than angle B. if the measure of angles A and B satisfy the

equation 3 sin x4 sin3xk = 0, 0< k< 1, then the measure of angle C is

a)3 b)

2 c)

23

d)56

30.If a cos A = b cos B, then the triangle is

a) Equilateral b) right angled c) isosceles d) right angled or isosceles

31.In a triangle ABC, sin A + sin B + sin C = 1 + 2, cos A + cos B + cos C = 2, if the triangle isa)

Equilateral b) isosceles c) right angled d)right angled isosceles

32.In a triangle ABC, if cot A/2 = (b + c)/a , then the triangle ABC is

a)

Isosceles b) equilateral c) right angled d) None of these33.If

a

cos A=

b

cos B=

c

cos C, then the triangle is

a) right angled b) isosceles c) obtuse angle d) equilateral

34.Ifsin A

sin C=

sin (AB)sin(BC) , then the sides of ABC are in

a)

A.P b) G.P c) H.P d) None of these35.

ABC is a triangle such that sin (2A + B)= sin (C - A) = - sin (B + 2C) = . If A, B, C are in A.P

thena) A = 45, B = 60, C = 75 c) angle A = B = C = 60

b) A = 75, B = 60, C = 45 d) A = 30, B = 60 , C = 90

36.

The perimeter of a triangle ABC is 6 times the arithmetic mean of the sines of its angles. If theside a is 1, then the angle A is

a)6 b)

3 c)

2 d)

37.If a2, b2, c2are in A.P, then cot A, cot B, cot C are in

a) A.P b) G.P c) H.P d) None of these38.In a triangle ABC, tan A/2 = 5/6 , tan C/2 = 2/5, then

a) a, c, b are in A.P. c) b, a, c are in A.Pb) a, b, c are in A.P. d) a, b, c are in G.P.

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PREPARED BT TEAM GURUKUL, www.thegurukulinstitute.in Properties of triangles, Class - XII

39.If the angles A, B, C of ABC are in A.P. thena) c2 = a2+ b2ab c) c2= a2+ b2

b) b2= a2+ c2ac d) None of these

40. If a2, b

2, c

2are in A.P. then which of the following are also in A.P.

a)

Sin A, sin B, sin C c) cot A , cot B , cot Cb) Tan A, tan B, tan C d) None of these

41.If the sides of a triangle ABC are in A.P. then tan A/2, tan C/2 equals

a)

1/ 3 b) 3 c) 1 d) None of these42.In a triangle PQR, sin P, sin Q, sin R are in A.P. then

a)

The altitudes are in A.P. c) The medians are in A.P.b) The altitudes are in H.P. d) The medians are in H.P.

43.If stands for the area of a triangle ABC then a 2sin 2B + b2sin 2A =a)

3 b) 2 c) 4 d) 44.In a ABC, angle C= 90, a = 3 and b = 4 and D is a point on AB, so that angle BCD = 30 , then

CD =

a)24

4 +43 b)24

3 +43 c)24

4 +34 d)24

3 +3345.In a triangle ABC, median AD and CE are drawn. If AD = 5, angle DAC = /8 and angle ACE =

/4 , then area of isa)

25/3 b) 25/6 c) 25/9 d) 25/1246.In a triangle ABC, angle C = 60, angle A = 75. If D is a point on AC such that area of BAD

is 3 times that of BCD, then angle ABD isa) 30 b) 15 c) 22

1

2

0 d) None of these

47.

If the angles of a triangle are 30 and 45and the included side is (3 + 1) cms, then the area ofthe triangle is

a)1

2(31 ) b) (31 ) c) 1

2(3 + 1 ) d) (3 +1 )

48.If b2 + c2= 3a2, then cot B + cot C cot A =

a) 1 b)ab

4 c) 0 d)ac

449.

If c2= a2+ b2, 2s = a + b+ c, then 4s(s a )( sb )( sc )is equal toa) s4 b) b2c2 c) c2a2 d) a2b2

50.

two medians AD and BE of a triangle ABC are of length 42 and 6 and angle AGB betweenthem is 45, where G is centroid of the triangle, then area of the triangle ABC is

a) 16 b) 162 c) 32 d) Data inadequate51.If sides of a triangle are 6, 8 and 10. Then area of the triangle ABC is

a)

4 b) 8 c) 16 d 652.In a triangle ABC if

cos =

cos =

cos and a = 2 then area of the triangle ABC is

a) 1 b) 2 c) 32 d) 353.The sides of a triangle inscribed in a given circle subtend angle , and at centre. The

minimum value of the arithmetic mean cos (+ /2), cos (+ /2) and cos (+ /2) is equal toa) 0 b) 1/2 c)1 d) - 3/ 2

54.In an acute angled triangle ABC, the least value of sec A + sec B + sec C is

a)

6 b) 3 c) 9 d) 455.The medians AD of a triangle ABC is r to AB, then tan A + 2 tan B =

a) 1 b)1 c) 2 d) 056.In a triangle ABC, angle B = / 3 and angle c = / 4. Let D divide BC internally in the ratio1: 3.

Thensin sin equals

a) 1/6 b) 1/ 3 c) 1/ 3 d) 23

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PREPARED BT TEAM GURUKUL, www.thegurukulinstitute.in Properties of triangles, Class - XII

57.In a triangle ABC, AD is the altitude from A. Given b> c. C = 23 and AD =

22 then angle Bis equal to

a) 23 b) 113 c) 67 d) 9058.In a triangle ABC the line joining the circumcentre to the incentre is parallel to BC then cos B +

cos C is equal toa)

3/2 b) 1 c) d) 0

59.If the hypotenuse of right angled triangle is four times the length of perpendicular drawn from

opposite vertex to it, then the difference of two acute angles will bea)

75 b) 15 c) 60 d) 3060.If in a triangle ABC cos A cos B + sin A sin B. sin C = 1 then the sides are proportio nal to

a) 1 : 1 : 2 b) 1 : 2 : 1 c) 2 : 1 : 1 d) None of these61.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 c) a2/9 d) 2a2/3

62.If r1, r2, r3are ex-radii of triangle AB then

1 +

2 +

3 is equal toa)

0 b) a+ b+ c c) (a + b + c) /2 d) 3r

63.

In a triangle ABC,1

+1

+1

is equal toa) 2/Rr b) 1/Rr c) 1/4Rr d) 1/2Rr

64.

If sides of a triangle are 4,5,6 then length of its smallest median is

a)46

2 b) 46

2 c)

232

d) None of these

65.In a triangle ABC; b = 2 , c = 3 and angle A = 30, then in radius of the triangle isa)

3+12

b) 3+1 c) 312

d) 3166.If ex radii of a ABC are 3, 4 and 6 then its in- radius is

a) 3 /4 b) 4/3 c) 5/6 d) None of these

67.In a ABC, let angle C= /2. If r and R are in - radius and circum radius respectively of thetriangle then 2(r + R) is equal to

a) a + b b) b + c c) c + a d) a + b+ c68.if R = circum radius and r = in radius of a triangle ABC, then which of the following CANNOT

follow the ordered pair (R, r)a) (14, 7) b) (13, 7) c) (15, 7) d) None of these69.

A circle of radius 4 unit is inscribed in an equilateral triangle ABC. Sides AB and AC of a

triangle are produced and a circle is drawn touching the sides BC and produced part of sides ABand AC . The area of the circle drawn is

a) 16 sq. unit b) 64 sq. unit c) 144 sq. unit d) 256 sq. unit70.If r1, r2and r3are exradii and r1= 2r2= 3r3then a : b : c equal

a)

3 : 4 : 5 b) 5 : 4 : 3 c) 1 : 2 : 3 d) 1 : 4 : 9

71.If notations have their usual meaning in a triangle and 1 12 1 13 = 2 then the triangle is

a) Equilateral b) isosceles c) right angled d) none of these

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

a)

16 : 5 b) 7 : 16 c) 5 : 16 d) 16 : 773.Two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60 . If the

third side is 3, then the fourth side isa)

2 b) 3 c) 4 d) 574.

A regular polygon of nine sides, each of length 2 is inscribed in a circle , he radius of the circle

is

a) Sec / 9 b) sin / 9 c) cosec / 9 d) tan /9

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PREPARED BT TEAM GURUKUL, www.thegurukulinstitute.in Properties of triangles, Class - XII

75.Let A0, A1, A2, A3, A4, A5 be a regular hexagon inscribed in a circle of unit radius. Then theproduct of length of segment A 0A1, A0A2and A0A4is

a) 3/4 b) 33 c) 3 d) 33 / 276.The area of a circle and the area of a regular polygon of n sides of perimeter equal to that of the

circle are in the ratio

a)

Tan:

b) cos

:

c) sin

:

d) cot

:

In the following (77 to 82 ) one or more than one alternatives may be correct

77.There exists a triangle ABC satisfying the condition

a) b sin A = a , A 2 d) b sin A < a, A a

78.If a, b, A be given in a triangle and c 1and c2two possible values of third side such that12 + 12 + 22 = 2, then A is equal toa) 30 b) 60 c) 90 d) 120

79.

In a triangle ABC, points D and E are taken on sides BC such that DB = DE = EC. If angle ADE

= angle AED = thena)

Tan = 3 tan B b) tan = 3 tan C c) tan A =6 tan

tan 29 d) 9 cot2

2= tan2

2 80.For a triangle ABC, which of the following is true? a)

cos =

cos =

cos c)

cos +

cos +

cos =

3

2b)

cos +

cos +

cos =

2+2+22 d)

sin 22 =

sin 2 2 =

sin 2 2

81.If cos (), cos , cos ( + ) one in H.P then cos sec2/2 is equal toa) 1 b) 2 c) 2 d) 2

82.

In a triangle the length of two larger sides are 10 and 9 respectively. If the angles are in A.P. thelength of third side can bea) 5 6 b) 5 +6 c) 65 d) 6 + 5