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PATHANIA INSTITUTE OF MATHEMATICS
S.C.F 13 PHASE: 2 MOHALI PH: 98145-06093
TRIGONOMETRIC RATIOS AND IDENTITIES
Instructions
1. Questions 1 to 25 MCQ with one correct answer type
2. Questions 26 to 32 MCQ with more than one correct answer type
3. Questions 33 to 38 Assertion and Reasoning
4. Questions 39 to 43 Numerical based answer. Answers is from 0 to 9.
5. Question 33 to 38 carrying 1 marks
6. Question 39 - 43 (carrying 1 marks)
7. In OMR sheet only use Blue Pen otherwise you will awarded with zero.
M.M.: __________
Date: _________
1
The difference between two acute angles of a
3
right angled triangle is
radians. The angles
10
in degrees are:
(a) 60o, 30o
(b) 70o, 20o
o
o
(c) 72 , 18
(d) none of these
2.
Angle between the hour-hand and the minute
hand in circular measure of half past 4 is


(a)
(b)
3
4

(c)
(d) none of these
2
4.
If the arcs of same length in two circles
subtend angles of 60o and 75o at their centres,
then the ratio of their radii is
(a) 2 : 3
(b) 3 : 4
(c) 4 : 5
(d) 5 : 4
5.
3.
The angles of a triangle are in A.P. The
number of degrees in the least is to be number
of radians in the greatest as 60 : . Then the
greatest angle is
(a) 120o
(b) 90o
o
(c) 135
(d) 105o
p
psin   q cos 
If tan   , then

q
psin   q cos 
p2  q 2
p2  q 2
(c) (p2 + q2) (p2 – q2)
(a)
p2  q 2
p2  q 2
(d) none of these
(b)
9.
6.
7.
8.
If A, B, C, D are the angles of a cyclic
quadrilateral
taken
in
order,
then
o
cos (180o + A) + cos (180 – B) + cos (180o –
C) – sin (90o – D) =
(a) 0
(b) 1
(c) – 1
(d) none of these
3
1

3
If sin A  , tan B 
and
A ,
5
2
2
2
then the value of
7
5
(a)
(b)
2
2
5
7
(c) 
(d) 
2
2
If A, B, C are acute +ve angles, then
(sin A  sin B)(sin B  sin C)(sin C  sin A)
is
sin Asin Bsin C
(a) > 1
(b) < 1
(c) = 2
(d) none of these
If tan 2   1  e 2 , then sec   tan3  cosec  is
equal to
(a) 2  e2
(c) 2 – e2
10.
11.
(b) (2  e2 )3/2
(d) none of these
If sin 6   cos6   K cos 2   1, then K is equal
to
1
1
(a) tan 2 2
(b) tan 2 2
2
4
3
2
(c) 4cot 2
(d) tan 2 2
4
sin(660o ) tan(1050o )sec(420o )
=
cos(225o ) cosec(315o )cos(510o )
(a)
(c)
3
4
2
3
(b)
(d)
3
2
4
3
(c) 1/2
16.
12.
The value of
(a) 0
(c) 3
cot 54o tan 20o


tan 36o cot 70o
(b) 2
(d) 1
17.
18.
13.
If 2sin  cos  sin   sin  . sin(  ). Then
tan , tan  and tan  are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) 5
19.
20.
21.
For m  n, of tan m = tan n, then the
different values of  are in
(a) A.P.
(b) H.P.
(c) G.P.
(d) no particular seq.
8
2
If
and
31tan   2cos   3cos 
3cos2  1, then the general value of  is

(a) 2n 
(b) 2n  cos1 2
3
2
(c) 2n 
(d) none of these
3
The number of solution of the form
x3 + x2 + 4x + 2 sin x = 0 in 0  x  2 is
(a) 0
(b) 1
(c) 2
(d) 4
The number of solutions of the equation
5sec   13  12 tan  in [0, 2] is
(a) 2
(b) 1
(c) 4
(d) 0
The expression (1 + tan x + tan2 x) (1 – cot x +
cot2 x) has the positive value for x, given by

(a) 0  x 
(b) 0  x  
2
(c) for all x  R
(d) x = 0
Let n be an odd integer. If sin n  =
n
b
r 0
14.
15.
If cos5  a cos5   bcos3  c cos , then c =
(a) 1
(b) 2
(c) 3
(d) none of these
Value of
(a) 1
sin 2 20o  cos 4 20o
is
sin 4 20o  cos 2 20o
(b) 2
(a)
(b)
(c)
(d)
Sol.
(d) none of these
r
sin r  for any value of , then
b0 = 1, b1= 3
b0 = 0, b1 = n
b0 = -1, b1 = n
b0 = 0, b1 = n2 – n + 3
22.
If sin A = sin B and cos A = cos B, then A =
(a) 2n + B
(b) 2n - B
(c) n + B
(d) n + (1)n B
Sol.
sin 6 + sin 4 + sin 2 = 0 then  =
n

n

(a)
(b)
or n 
or n 
4
3
4
6
n

(c)
(d) none of these
or 2n 
4
6
Ans. a
24. The general value of  satisfying sin2  + sin 
= 2 is


(a) n  (1)n
(b) 2n 
6
4


(c) n  (1)n
(d) n  (1)n
2
3
Ans. c
x
25. The number of real roots of the equation

2
sin x is
(a) 0
(b) 3
(c) 7
(d) infinite
Ans. a
26. For a positive integer n, let:


f n ()   tan  (1  sec )(1  sec2)
2

(1  sec4)...(1  sec2n ), then
23.

(a) f 2    1
 16 
 
(c) f 4    1
 64 
Sol.
 
(b) f36    1
 32 
  
(d) f5 
 1
 128 
27.
Which of the following is true?
 
 2 
 3  1
(a) cos    cos    cos   
7
 7 
 7  8
 
 4 
 5  1
(b) cos    cos    cos   
7
 7 
 7  8
 3 
 2 
 4  1
(c) sin    sin    sin   
 7 
 14 
 14  8
(d) none of these
Ans. a, b, c, d
28. The
value(s)
of
the
expression
3
3
sin x
cos x

is / are :
1  cos x 1  sin x




(a) 2 cos   x 
(b) 2 cos   x 
4

4





(c) 2 sin   x 
(d) 2 sin   x 
4

4

Sol.
29.
If cos (A  B) 
3
and tan A and tan B = 2,
5
then
1
5
2
(b) sin A sin B = 5
1
(c) cos(A  B)  
5
4
(d) sin Acos B 
5
Ans. a, c
6 
x
30. If sin  x   0 and cos    0, then
5 
5
(a) x  (n  5)
(b) x  6(n  1)
(a) cos A cos B =
1

(c) x  5  n   
2

1

(d) x  5  n   
2

32.
If in a triangle ABC, 3 sin A = 6 sin B = 2 3
sin C, then the angle A is:
(a) 30°
(b) 60°
(c) 90°
(d) 120°
Ans.
Sol.
31.
Ans.
In a triangle ABC, let D be the mid point of
BC. If AB =- 2, BC = 4 and CA = 3, then
(a) AD = 1.85
(b) AD = 1.58
16
11
(c) cos B = 8
(d) cos B =
11
16
If sin  + cosec  = 2 then the value of sin8
+ cosec8 is equal to
(a) 2
(b) 28
4
(c) 2
(d) none of these
2
Ans. sin  + cosec = 2 => sin  - 2 sin  + 1 = 0
=> sin  = 1
39.
40.
The
maximum
value
of






1  sin      2cos     for real values
4
4




of  is
(a) 3
(b) 5
(c) 4
(d)
Ans.
41.
Let a = cos A + cos B – cos (A + B) and
A
B
AB
b  4sin sin cos
. Then a – b is
2
2
2
equal to
(a) 1
(b) 0
(c) -1
(d) none of these
Ans.
42.




tan   tan      tan      k tan 3
3
3


then k is equal to
(a) 1
(b) 3
(c) 1/3
(d) none of these
If
Ans.
If cos 5 = a cos5 + bcos3 then c is equal to
(a) -5
(b) 1
(c) 5
(d) none of these

Ans. Differentiate w.r.t.  and put   .
2
43.
Answers
1.
c
2.
b
3.
d
4.
b
5.
b
6.
a
7.
d
8.
a
9.
b
10.
d
11.
c
12.
b
13.
c
14.
d
15.
a
16.
17.
18.
19.
20.
21.
b
22.
a
23.
a
24.
c
25.
a
26.
a, b, c, d
27.
all correct
28.
a, d
29.
a, c
30.
c, d
31.
b, d
32.
c