Mathematics-Trigonometry-MCQ.pdf

15 - TRIGONOMETRY

Page 1

( Answers at the end of all questions )

2

(b) c = a + b

1

(d) c + a

[ AIEEE 2005 ]

y 2 2 is equal to = α, then 4x - 4xy cos α + y 2 ( c ) 4 sin

2

α

(

ra

(b) 4

) - 4 sin

2

α

( c ) Arithmetic-Geometric Progression

[ AIEEE 2005 ]

( d ) H.P.

xa

( b ) A.P.

.e

α - β

If

3

(b)

130

= sin

3

(c)

130

6 65

a 2 cos 2 θ + b 2 sin 2 θ +

the maximum and minimum values of u

(a) 2(a

2

[ AIEEE 2005 ]

21 , then the value of 65

is

2

w w w

1

(c) a + b+c

Let α, β be such that π < α - β < 3π. If sin α + sin β = -

(a) -

(7)

[ AIEEE 2005 ]

If in triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in

cos

(6

(d) b = a + c

ce .c

(b) a + b

If cos - x - cos -

( a ) G.P.

(5)

(c) b = c

π . If r is the inradius and R is the circumradius of the 2 triangle ABC, then 2 ( r + R ) equals

( a ) 2 sin 2α

(4)

are the roots of the equation

In triangle ABC, let ∠ C =

(a) b + c

(3)

Q and tan   2

a ≠ 0, then

+ b x + c = 0,

(a) a = b + c

P If tan   2

om

ax

(2)

π . 2

In triangle PQR, ∠ R =

m

(1)

2

+ b )

(b) 2

The sides of a triangle are

a2 + b2

sin α,

(d) -

6 65

[ AIEEE 2004 ]

a 2 sin 2 θ + b 2 sin 2 θ , then difference between 2

is given by

(c) (a + b)

cos α and

2

(d) (a - b)

1 + sin α cos α

2

[ AIEEE 2004 ]

for some 0 < α <

π . 2

Then the greatest angle of the triangle is ( a ) 60°

( b ) 90°

( c ) 120°

( d ) 150°

[ AIEEE 2004 ]

15 - TRIGONOMETRY

Page 2

( Answers at the end of all questions )

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of a river is 60° and when he retires 40 m away from the tree, the angle of elevation becomes 30°. The breadth of the river is

(9)

( b ) 30 m

( c ) 40 m

2  C  If in a triangle a cos    2 

( a ) in A. P.

( d ) 60 m

2  A  + c cos    2 

( b ) in G. P.

3b , then the sides a, b and c are 2

=

( c ) in H. P.

[ AIEEE 2004 ]

om

( a ) 20 m

( d ) satisfy a + b = c

ce .c

(8)

[ AIEEE 2003 ]

( 10 ) The sum of the radii of inscribed and circumscribed ci cles, for an n sided regular polygon of side a, is  π ( b ) b cot    n

(

)

ra

 π  ( a ) a cot    2n 

2

 π  cot    2n 

(d)

a  π  cot   4  2n  [ AIEEE 2003 ]

m

3  3  th portion of a vertic l pole subtends an angle tan - 1   at a point in 4  5  the horizontal plane through it fo t and at a distance 40 m from the foot. The height of the vertical pole is

xa

( 11 ) The upper

( b ) 40 m

.e

( a ) 20 m

2

( c ) 60 m

( d ) 80 m

2

[ AIEEE 2003 ]

2

( 12 ) The value of cos α + cos ( α + 120° ) + cos ( α - 120° ) is 3 2

w w

(a)

b)

1 2

(c) 1

(d) 0

[ AIEEE 2003 ]

( 13 ) The trigonometric equation sin - x = 2 sin - a has a solution for 1 1 1 1 (a) lal< (b) lal ≥ (c) ( d ) all real values of a < lal < 2 2 2 2 [ AIEEE 2003 ] 1

w

1

 θ - φ  ( 14 ) If sin θ + sin φ = a and cos θ + cos φ = b, then the value of tan    2  (a)

(c)

a2 + b2 4 - a2 - b2 a2 + b2 4 + a2 + b2

(b)

(d)

is

4 - a2 - b2 a2 + b2 4 + a2 + b2 a2 + b2

[ AIEEE 2002 ]

15 - TRIGONOMETRY

Page 3

( Answers at the end of all questions )

1

2

(a)

(b) 3

( 16 ) The value of tan -

(a)

π 2

(b)

1

2π , then the value of x is 3

(c)

(d)

3

3 - 1 3 + 1

[ AIEEE 2002 ]

 1  -1  1  -1  1  -1   + tan   + tan   + … + tan  3   7   13 

π 4

(c)

2π 3

ce .c

1

om

( 15 ) If tan - ( x ) + 2 cot - ( x ) =

(d) 0

    n

   + n + 1 1

is

[ AIEEE 2002 ]

(b)

3

a

3

3 - 1

(c)

a(3 + 2

m

(a) a

ra

( 17 ) The angles of elevation of the top of a tower ( A ) from the top ( B ) and bottom ( D ) at a building of height a are 30° and 45° resp ctively. If the tower and the building stand at the same level, then the height f the tower is

( 18 ) If cos ( α - β ) = 1 and co ( α + β ) =

xa

ordered pairs ( α, β ) = (b) 1

c) 2

1 , e

(d) a(

3

- 1)

[ AIEEE 2002 ]

- π ≤ α, β ≤ π, then the number of

(d) 4

[ IIT 2005 ]

.e

(a) 0

3)

w w

( 19 ) Which of the following is correct for triangle ABC having sides a, b, c opposite to the angles A, B, C respectively ( a ) a sin

A  B - C  = ( b - c ) cos 2 2  

A  B - C ( d ) sin   = a cos 2 2  

w

A  B + C ( c ) ( b + c ) sin   = a cos 2 2  

A  B + C ( b ) a sin   = ( b + c ) cos 2 2  

( 20 ) Three circles of unit radii are inscribed in an equilateral triangle touching the sides of the triangle as shown in the figure. Then, the area of the triangle is (a) 6 + 4

3

(c) 7 + 4

3

( b ) 12 + 8 3 7 (d) 4 + 3 2 [ IIT 2005 ]

[ IIT 2005 ]

15 - TRIGONOMETRY

Page 4

( Answers at the end of all questions ) If θ and φ are acute angles such that sin θ = lies in  π π  (a)  ,   3 2 

 π 2π  (b)  , 3   2

1 2

and cos θ =

 2π 5π  (c)  , 3   3

1 , then θ and φ 3

 5π  (d)  , π  6 

[ IIT 2004 ]

om

( 21 )

( 22 ) For which value of x, sin [ cot - ( x + 1 ) ] = cos ( tan - x ) ? 1

(b) 0

(d) -

(c) 1

[ IIT 2004 ]

If a, b, c are the sides of a triangle such that a : b : then A : B : C is (a) 3 : 2 : 1

( 24 ) Value of

(b) 3 : 1 : 2

tan 2 α

x2 + x +

(c) 1 : 3 : 2

,

x2 + x

(b)

5 2

( c ) 2 tan α

:

3 : 2,

 

π   2 

[ IIT 2004 ]

is always greater than or

( d ) sec α

[ IIT 2003 ]

xa

(a) 2

=

(d) 1 : 2 : 3

α ∈  0,

x >

m

equal to

If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the largest side to the perimeter is equal to

.e

( 25 )

1 2

ra

( 23 )

1 2

ce .c

(a)

1

(a) 1:1 +

(b) 2:3

w w

3

( 26 ) The natural domain of

sin - 1 ( 2x ) +

 1 1 (b) - ,   4 4 

3 :2 +

π 6

3

(d) 1:2 +

3

[ IIT 2003 ]

for all x ∈ R, is

 1 1 (c) - ,   2 2 

w

 1 1 (a) - ,   4 2 

(c)

 1 1 (d) - ,   2 4 

[ IIT 2003 ]

3

( 7 ) The length of a longest interval in which the function 3 sin x - 4 sin x is increasing is (a)

( 28 )

π 3

(b)

π 2

(c)

3π 2

(d)

π

[ IIT 2002 ]

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle ABC ( R being the radius of the circumcircle ) ? ( a ) a sin A, sin B

( b ) a, b, c

( c ) a, sin B, R

( d ) a, sin A, R

[ IIT 2002 ]

15 - TRIGONOMETRY

Page 5

( Answers at the end of all questions )

The number of integral values of k for which the equation 7 cos x + 5 sin x = 2k + 1 has a solution is (a) 4

( 30 )

(b) 8

( c ) 10

( d ) 12

[ IIT 2002 ]

om

( 29 )

π be a fixed angle. If P = ( cos θ, sin θ ) 2 sin (α - θ ) ], then Q is obtained from P by Let

0 < α <

and

Q = [ cos ( α - θ ),

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a po t X on the circumference of the circle, then 2r equals

α + β =

(b)

3

w w w ( 34 )

(c)

2 PQ ⋅ RS PQ + RS

(d)

PQ 2 + RS 2 2

[ IIT 2001 ]

A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depress on of 30°. After some time, the angle of depression becomes 60°. The distance n ( metres ) traveled by the car during this time is ( a ) 100

( 33 )

PQ + RS 2

(b)

.e

( 32 )

PQ ⋅ RS

xa

(a)

m

( 31 )

[ IIT 2002 ]

ra

ce .c

( a ) clockwise rotation around origin through an angle α ( b ) anticlockwise rotation around origin through an angle α ( c ) reflection in the line through origin with slope tan α α ( d ) reflection in the line through origin with slope t 2

π 2

and

( a ) 2 ( tan β + tan γ ) ( c ) tan β + 2tan γ

200 3 3

(c)

100 3 3

( d ) 200

3

[ IIT 2001 ]

β + γ = α , then tan α equals ( b ) tan β + tan γ ( d ) 2tan β + tan γ

[ IIT 2001 ]

    π x2 x3 x4 x6 + + - ...  + cos - 1  x 2 - ...  = sin - 1  x for 0 < l x l <     2 4 2 4 2     then x equals If

(a)

1 2

(b) 1

(c) -

1 2

(d) - 1

2,

[ IIT 2001 ]

15 - TRIGONOMETRY

Page 6

( Answers at the end of all questions )

(b)

n

2

2 2

(c)

n

1 2n

( 36 ) The number of distinct real roots of π π ≤ x ≤ 4 4

(a) 0

is

(b) 2

( cos αn ),

under the restrictions

(c) 1

(d) 1

sin x cos x cos x

cos x sin x cos x

[ IIT 2001 ]

cos x cos x sin x

= 0 in the interval

(d) 3

[ IIT 2001 ]

ra

-

…..

( cos α1 ) ⋅ ( cos α2 ) ….. ( cos αn ) = 1 is

and

1

( cos α2 )

om

1

(a)

⋅

( cos α1 )

The maximum value of π 0 ≤ α1, α2, ….. αn ≤ 2

ce .c

( 35 )

( 37 ) If f ( θ ) = sin θ ( sin θ + sin 3θ ), then f ( θ )

( b ) ≤ 0 for all real θ ( d ) ≤ 0 only when θ ≤ 0

xa

m

( a ) ≥ 0 only when θ ≥ 0 ( c ) ≥ 0 for all real θ

( 38 ) In a triangle ABC, 2ac sin 2

2

2

(b)

w w

.e

(a) a +b -c

2

2

2

w

(a) a + b

( 40 )

(A - B + C) = 2

+ a - b

( 39 ) In a triangle ABC, if ∠ C = (b) b + c

[ IIT 2000 ]

2

2

(c) b - c - a

2

2

2

2

(d) c - a - b

[ IIT 2000 ]

π , r = inradius and R = circum-radius, then 2 ( r + R ) = 2

(c) c + a

(d) a + b + c

[ IIT 2000 ]

A pole stands vertically inside a triangular park ∆ ABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in ∆ ABC, the foot of the pole is at the ( a ) centroid

( b ) circumcentre

( c ) incentre

( d ) orthocentre

[ IIT 2000 ]

15 - TRIGONOMETRY

Page 7

( Answers at the end of all questions )

 P  If tan    2  2 equation ax + bx + c = 0 ( a ≠ 0 ), then PQR, ∠ R =

(a) a+b =c

(b) b+c =a

(c) c+a =b

tan - 1

( 42 ) The number of real solutions of ( a ) zero

( b ) one

( c ) two

and

x ( x + 1 ) + sin - 1

(c) 2

[ IIT 1999 ]

x2 + x +

( d ) infinite

= cos x + cos (

( d ) infinit

=

π

2

is

[ IIT 1999 ]

2x ) attains its

[ IIT 1998 ]

ra

(b) 1

are the roots of the

(d) b = c

( 43 ) The number of values of x where the function f ( x maximum is (a) 0

 Q  tan    2 

om

π . 2

In a triangle

ce .c

( 41 )

 π  f2   = 1  16 

(c)

 π  f4   = 1  64 

(b

 π  f3   = 1  32 

d)

 π  f5   = 1  128 

.e

xa

(a)

m

( 44 ) If, for a positive integer n, θ   fn ( θ ) =  tan  ( 1 + sec θ ) ( 1 + sec 2θ ) ... ( 1 + sec 2 n θ ) , then 2  

sin P, sin Q, sin R are in A. P., then

w w

( 45 ) If in a triangle PQR,

( a ) he altitudes are in A. P. ( c ) the medians are in G. P.

( b ) the altitudes are in H. P. ( d ) the medians are in A. P.

w

46 ) The number of values of 2 3 sin x - 7 sin x + 2 = 0 is (a) 0

(b ) 5

[ IIT 1999 ]

(c)

x

6

[ IIT 1998 ]

in the interval [ 0, 5π ] satisfying the equation

( d ) 10

[ IIT 1998 ]

( 47 ) Which of the following number( s ) is / are rational ? ( a ) sin 15°

( b ) cos 15°

( c ) sin 15° cos 15°

( d ) sin 15° cos 75°

[ IIT 1998 ]

15 - TRIGONOMETRY

Page 8

( Answers at the end of all questions )

n

( 48 )

∑ br sinr θ,

Let n be an odd integer. If sin nθ =

for every value of θ, then b0 and

r =0

( a ) 1, 3

( c ) - 1, n

( b ) 0, n

( d ) 0, n

2

1 cos ( p - d ) x sin ( p - d ) x

(b) p

a2 cos ( p + d ) x sin ( p + d ) x (c) d

[ IIT 1998 ]

does not depend upon is

(d) x

ra

(a) a

a cos px sin px

- 3n + 3

ce .c

( 49 ) The parameter, on which the value of the determinant

om

b1 respectively are

[ IIT 1997 ]

2

2 ) - cos ( x + 1 ) is

( 50 ) The graph of the function cos x cos ( x

xa

m

 π  ( a ) a straight line passing through the point  , - sin 2 1  and parallel to the X-axis  2  2 ( b ) a straight line passing th oug ( 0, - sin 1 ) with slope 2 ( c ) a straight line passing through ( 0, 0 ) 2

( d ) a parabola with ve t x ( 1, - sin 1 )

.e

[ IIT 1997 ]

w w

( 51 ) If A0 A1 A A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius, then the product f the lengths of the line segments A0 A1, A0 A2 and A0 A4 is 3 4

(a)

w

( 52

( 53 )

(b) 3

2

sec θ =

3

4 xy

( x + y )2

(a) x + y ≠ 0

(c) 3

(d)

3

3

[ IIT 1998 ]

2

is true if and only if

( b ) x = y, x ≠ 0

(c) x = y

( d ) x ≠ 0, y ≠ 0

[ IIT 1996 ]

The minimum value of the expression sin α + sin β + sin γ, where α, β, γ are the real numbers satisfying α + β + γ = π is ( a ) positive

( b ) zero

( c ) negative

(D) -3

[ IIT 1995 ]

15 - TRIGONOMETRY

Page 9

( Answers at the end of all questions )

sin ∠ BAD sin ∠ CAD

equals

1 3

(c)

1

(a)

(b)

6

( 55 )

If x =

(b) 1

∞ ∑

n=0

(c) 2

cos 2n φ ,

∞ ∑

y =

.e

w w w

2 3

[ IIT 1995 ]

tan x + sec x = 2 cos x

lying in the interval

n=0

sin 2n φ ,

∞ ∑

n=0

[ IIT 1993 ]

cos 2n φ sin 2n φ , for 0 < φ <

m

z =

( b ) xyz xy + z ( d ) xyz = yz + x

If f ( x ) = cos [ π ] function, then  π  (a) f   = -1  2 

( 58 )

(d)

(d) 3

xa

( a ) xyz = xz + y ( c ) xyz = x + y + z

π . If D divides BC internally in the ratio 4

3

then

( 57 )

1

Number of solutions of the equation [ 0, 2π ], is (a) 0

( 56 )

and ∠ C =

ce .c

1 : 3, then

π 3

om

In a triangle ABC, ∠ B =

ra

( 54 )

π , 2

[ IIT 1993 ]

cos [ - π ] x , where [ x ] stands for the greatest integer 2

(b) f(π) = 1

(c) f(-π) = 0

 π  (d) f   = 2  4 

[ IIT 1991 ]

2

he equation ( cos p - 1 ) x + ( cos p ) x + sin p = 0 in the variable x has real roots. Then p can take any value in the interval

( a ) ( 0, 2π )

 π π  (c)  - ,   2 2 

( b ) ( - π, 0 )

( d ) ( 0, π )

[ IIT 1990 ]

( 59 ) In a triangle ABC, angle A is greater than angle B. If the measures of angles A and B 3 satisfy the equation 3 sin x - 4 sin x - k = 0, 0 < k < 1, then the measure of angle C is (a)

π 3

(b)

π 2

(c)

2π 3

(d)

5π 6

[ IIT 1990 ]

15 - TRIGONOMETRY

Page 10

( Answers at the end of all questions )

x

( 60 ) The number of real solutions of the equation sin ( e ) = 5 (b) 1

(c) 2

+ 5–

x

is

( d ) infinitely many

[ IIT 1990 ]

om

(a) 0

x

( 61 ) The general solution of sin x - 3 sin 2x + sin 3x = cos x - cos 2x + cos 3x is π nπ + 2 8

(b)

( d ) 2 n π + cos -

( 62 ) The value of the expression

(b) 4

(c)

3 cosec 20°

2 sin 20 o

1 + cos 2 θ

sin 2 θ 7π 24

w w w

(

)

4 s n 4θ

π 2

[ IIT 1988 ]

and satisfying the equation

= 0

are

4 sin 4θ

1

5π 24

nd θ =

sin 40 o

(c)

11 π 24

(d)

π 24

[ IIT 1988 ]

In a tr angle, the lengths of the two larger sides are 10 and 9 respectively. If the angle a e in A. P., then the lengths of the third side can be a) 5 -

65

cos 2 θ

.e

(a)

( 64 )

m

sin 2 θ

4 sin 20 o

4 sin 4θ

xa

cos 2 θ

[ IIT 1989 ]

- se 20° is equal to

(d)

sin 40 o

( 63 ) The values of θ lying between θ = 0 1 + sin 2 θ

3 2

1

ra

(a) 2

ce .c

π 8 π n nπ + (c) (-1) 2 8 (a) nπ +

6

(b) 3

3

(c) 5

(d) 5 +

The smallest positive root of the equation tan x = x lies π  3π    π   (b)  , π  ( c )  π, (d) ( a )  0,   2 2 2      

6

in 3π    π,  2  

[ IIT 1987 ]

[ IIT 1987 ]

( 66 ) The number of all triplets ( a1, a2, a3 ) such that 2 a1 + a2 cos 2x + a3 sin x = 0 for all x is (a) 0

(b) 1

(c) 3

( d ) infinite

( e ) none of these

[ IIT 1987 ]

15 - TRIGONOMETRY

Page 11

( Answers at the end of all questions )

2π 3

(a) -

2π 3

(c)

2π    sin  3   4π 3

is

(d)

5π 3

( e ) none of these

[ IIT 1986 ]

The expression      3π   π  + α  + sin 6 ( 5 π - α )  is equal to - α  + sin 4 ( 3π + α )  - 2  sin 6  3  sin 4   2   2      (a) 0

(b) 1

ce .c

( 68 )

(b)

1

om

( 67 ) The principal value of sin –

( d ) sin 4α + cos 4α

(c)3

( e ) none

f these

[ IIT 1986 ]

( 69 ) There exists a triangle ABC satisfying the conditi ns

xa

( e ) b sin A < a,

3π   π  ( 70 )  1 + cos   1 + cos  1 8 8   

.e

1 2

(b

cos

π 8

w w

(a)

π 2 π A < , b > a 2

ra

( c ) b sin A > a,

π ( b ) b sin A > a 2 π A < ( d b s n A < a, 2 π A > , b = a 2 A <

A >

m

( a ) b sin A = a,

cos

(c)

[ IIT 1986 ]

5π   7π    1 + cos  8  8  1 8

is equal to

1+

(d)

2

2

[ IIT 1984 ]

2

w

( 71 ) From the top of a light-house 60 m high with its base at the sea-level, the angle of depr ssion of a boat is 15°. The distance of the boat from the foot of the lighthouse is 2

 (a)   

3 - 1  60 metres 3 + 1 

 (b)   

 (c)   

3 + 1  60 metres 3 - 1 

( d ) None of these

3 + 1  3 - 1 

  4  -1  2   ( 72 ) The value of tan  cos - 1   + tan     5   3   (a)

6 17

(b)

7 16

(c)

16 7

metres

[ IIT 1983 ]

is

( d ) None of these

[ IIT 1983 ]

15 - TRIGONOMETRY

Page 12

( Answers at the end of all questions ) ( 73 ) If f ( x ) = cos ( ln x ), then f ( x ) f ( y ) -

(b)

1 2

(c) -2

has the value

( d ) none of these

[ IT 1983 ]

om

(a) -1

 1   x   + f ( xy )  f 2   y  

( 74 ) The general solution of the trigonometric equation sin x + cos x = 1 is giv n by

(c) x + nπ + (-1)

2

n

π π - , n = 0, ± 1, ± 2, … 4 4

4

π , n = 0, ± 1 ± 2, … 2

ce .c

( a ) x = 2 n π, n = 0, ± 1, ± 2, … ( b ) x = 2 n π +

( d ) none of these

[ IIT 1981 ]

( 75 ) If A = sin θ + cos θ, then for all real values of θ 3 ≤ A ≤ 1 4 1 3 ≤ A (d) 16 4

(a) 1 ≤ A ≤ 2

ra

13 ≤ A ≤ 1 16

m

(c)

(b)

xa

π  2  1 2 2 2 ( 76 ) The equation 2 cos  x  sin x = x + x - , 0 < x ≤ 2 2   ( a ) no real solution ( b one real solution ( c ) more than one rea sol ti n

.e u not

w w

-4 5 4 5

(c)

w

( 78

has [ IIT 1980 ]

-4 , then sin θ is 3

( 77 ) If tan θ =

(a)

[ IIT 1980 ]

but not

4 5 -4 5

(b)

-4 5

or

4 5

( d ) none of these

[ IIT 1979 ]

If α + β + γ = 2 π, then

γ β α + tan + tan 2 2 2 β β α ( b ) tan tan + tan tan 2 2 2 γ β α ( c ) tan + tan + tan 2 2 2 ( d ) none of these ( a ) tan

β γ α tan tan 2 2 2 γ γ α + tan tan = 1 2 2 2 β γ α = - tan tan tan 2 2 2 = tan

[ IIT 1979 ]

15 - TRIGONOMETRY

Page 13

( Answers at the end of all questions )

2 b

3 c

4 b

5 a

6 d

7 c

8 a

9 a

10 c

11 b

12 a

13 a

14 b

21 b

22 d

23 d

24 c

25 c

26 a

27 a

28 d

29 b

30 d

31 a

32 b

33 c

34 b

41 a

42 c

43 a

61 b

62 b

63 a,c

44 a,b,c,d 65 a

66 d

46 c

47 c 67 e

48 b 68 b

49 b 69 a,d

50 a 70 c

51 c

71 c

52 b

72

w

w w

.e

xa

m

ra

64 a,c

45 d

15 c

16 b

1 c

18 d

19 a

20 a

35 a

36 c

37 c

38 b

39 a

40 b

ce .c

1 b

om

Answers

53 c

73 d

54 a

74 c

55 d

56 b

57 a,c

58 b

59 c

60 0

75 b

76 a

77 a

78 a

79

80