Direction: Study the following questions carefully and choose the right answer:
1
If sin 21° =  x , then sec 21° – sin 69° is equal to
y
» Explain it
A
sin21° =  x
y

cos 21° =  1 – sin2 21°

=  
1 –  x2
y2
 = 
y2 – x2
y

∴    sec 21° =  y
y2 – x2

∴    sec 21° – sin 69°

= sec 21° – sin (90° – 21°)

= sec 21° – cos 21°

=   y  – 
y2 – x2
y2 – x2
y

=   y2 – (y2 – x2)  =  x2
y2 – x2
y2 – x2

Hence, option A is correct.

» Explain it
B
π = radian = 180°

∴    1 radian =  180°
π

=   180 × 7°  =  630  = 57 3 °
22 11 11  

=  57° 3  × 60'  =  57° 180'
11 11

=   57°16' 4'  × 60''  =  57°16'22''
11

Hence, option B is correct.

3
If x sin60° –  sec 60° tan30° +  sin2 45° tan60 = 0 then x is
» Explain it
C
x sin60° –  sec 60° tan30° +  sin2 45° tan60 = 0

⇒ x.
3
2 3  × 2 1 2 4  ×  1 2 × (
3
)= 0
2   2
3
  5
2
 

⇒    3x  –  3  × 2 ×  1  +  4  ×  1  × 3 = 0
4 2 3 5 2

⇒   3x  – 1 +  6 = 0
4 5

⇒   x = –  1  ×  4  = –  4  
5 3 15

Hence, option C is correct.

4
If the sum of two angles is 135° and their difference is (π/12), then the circular measure of the greater angle is
» Explain it
C
Two angles = A and B where A > B     A + B = 135°

=    ( 135 × π )  radian
180

⇒ A + B =  ( )  radian
4
...(i)

⇒    A – B =   ( π ) radian
12
...(ii)

On adding these equations,

2A =    +  π
4 12

=   9π + π  =  10π  = 
12 12 6

∴     A =   radian
12

Hence, option C is correct.

5
The minimum value of 2 sin2 Θ + 3 cos2 Θ is
» Explain it
C
2 sin2 Θ + 3 cos2 Θ

= 2 sin2 Θ + 2 cos2 Θ + cos2 Θ

= 2 (sin2 Θ + cos2 Θ) + cos2 Θ

= 2 + cos2 Θ

[ ∵ sin2 Θ + cos2 Θ = 1 ]

∵  Minimum value of cos Θ = –1

But cos2 Θ ≥ 0, when Θ = 90°

[ ∵ cos 0° = 1, cos 90° = 0 ]

∴  Required minimum value = 2 + 0 = 2.

Hence, option C is correct.