Direction: Study the following questions carefully and choose the right answer:
Important for :
1
The angles of a triangle are (x + 5)°, (2x – 3)° and (3x + 4)°. The value of x is
» Explain it
C
Sum of angles of a triangle = 180°

∴ x + 5 + 2x – 3 + 3x + 4 = 180°

⇒ 6x + 6 = 180°

⇒ 6x = 174°

⇒ x = 29

Hence, option C is correct.

2
The value of cot 10° . cot 20° . cot 60° . cot 70° . cot 80° is
» Explain it
D
cot 10° . cot 80° . cot 20° . cot 70° . cot 60°

= cot 10° . tan 10° . cot 20° . tan 20° . cot 60°

[∵ cot (90° – Θ) = tan Θ & cot Θ . tan Θ = 1]
 = 1 × 1 × 1 [ ∵  cot 60° = 1 ] 3 3
 = 1 3
Hence, option D is correct.

3
If Θ be an acute angle and 7 sin2 Θ + 3 cos2 Θ = 4, then the value of tan Θ is
» Explain it
B
7 sin2 Θ + 3 cos2 Θ = 4

 ⇒  7 sin2 Θ + 3 = 4 = 4 sec2 Θ cos2 Θ cos2 Θ

⇒  7 tan2 Θ + 3 = 4(1 + tan2 Θ)

⇒  7 tan2 Θ – 4 tan2 Θ = 4 – 3

⇒  3 tan2 Θ = 1

 ⇒  tan2 Θ = 1 3

 ⇒  tan Θ = 1 3

Hence, option B is correct.

4
The value of sin2 1° + sin2 5° + sin2 9° + ..... + sin2 89° is
» Explain it
A
No. of terms in 1 + 5 + 9 + ... + 89 = n

∴  a + (n – 1) d = Tn

⇒ 1 + (n – 1) × 4 = 89

⇒ (n – 1) × 4 = 89 – 1 = 88

⇒ n – 1 = 22

⇒ n = 23

Now,

sin2 1° + sin2 89° + sin2 5° + sin2 85° + ..... to 22 terms + sin2 45°

= (sin2 1° + cos2 1°) + (sin2 5° + cos2 5°) + ..... to 11 terms + sin2 45°

[ sin (90° – Θ) = cos Θ]
 = 1 + 1 + ..... + to 11 terms + ( 1 ) 2 2
[∵  sin2 Θ + cos2 Θ = 1]
 = 11 + 1 2
 = 11 1 2
Hence, option A is correct.

5
The numerical value of
 cot 18° ( cot 72° cos2 22° + 1 ) is tan 72° sec2 68°
» Explain it
A
 cot 18° ( cot 72° cos2 22° + 1 ) tan 72° sec2 68°

 = cot 18° cot 72° cos2 22° + cot 18° tan 72° sec2 68°

 = cot 18° tan 18° cos2 22° + tan 72° tan 72° sec2 68°

[∵  cot (90° – Θ) = tan Θ]

 = cos2 22° + 1 sec2 68°

[∵  cot Θ tan Θ = 1]

= cos2 22° + cos2 68°

= sin2 68° + cos2 68° = 1.

[∵  cos (90° – Θ) = sin Θ]

Hence, option A is correct.