Direction: Study the following questions carefully and choose the right answer:
Important for :
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
 y y2 – x2
 y y2 – x2

Hence, option A is correct.

2
The degree measure of 1 radian (taking π = 22/7) is
» Explain it
B

 ∴    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 = ( 3π ) radian 4
...(i)

 ⇒    A – B = ( π ) radian 12
...(ii)

 2A = 3π + π 4 12

 = 9π + π = 10π = 5π 12 12 6

 ∴     A = 5π 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.