Untitled Document

Question Bank No: 1

1. If tan $\frac{π}{9}$ , x and tan $\frac{5 π}{18}$ are in A.P and tan $\frac{π}{9}$ , y and tan $\frac{7 π}{18}$ are also in A.P., then

a)		2x = y
b)		x > y
c)		x = y
d)		None of these

2. If sin $θ$ + cosec $θ$ = 2, then value of $\sin^{10}$ $θ$ + ${cosec}^{10}$ $θ$ is

a)		2
b)		$2^{10}$
c)		$2^{5}$
d)		None of these

3. The value of log tan $1^{0}$ + tan $2^{0}$ + ...... + log tan $89^{0}$ is

a)		0
b)		-1
c)		1
d)		$\propto$

4. The value of cos $\frac{π}{15}$ cos $\frac{2 π}{15}$ cos $\frac{3 π}{15}$ cos $\frac{4 π}{15}$ cos $\frac{5 π}{15}$ cos $\frac{6 π}{15}$ cos $\frac{7 π}{15}$ is

a)		$\frac{1}{2^{6}}$
b)		$\frac{1}{2^{7}}$
c)		$\frac{1}{2^{8}}$
d)		None of these

5. The value of the expression 3 (sin $θ$ - cos $θ$ $)^{4}$ + 6(sin $θ$ + cos $θ)^{2}$ + 4( $\sin^{6}$ $θ$ + $\cos^{6}$ $θ$ ) is

a)		1
b)		-1
c)		13
d)		0

6. The maximum value of 3+sin $(\frac{π}{2} + θ)$ +2cos $(\frac{π}{2} - θ)$ for real $θ$ is

a)		5
b)		7
c)		6
d)		None of these

7. If cos A = $\frac{3}{4}$ , then the value of cos $\frac{A}{2}$ cos $\frac{5 A}{2}$ is

a)		$\frac{- 3}{32}$
b)		$\frac{- 5}{32}$
c)		$\frac{- 7}{32}$
d)		$\frac{- 7}{16}$

8. $\frac{\cos^{3} θ - \cos 3 θ}{\cos θ}$ $\frac{\sin^{3} θ + \sin 3 θ}{\sin θ}$ is equal to

a)		1
b)		3
c)		5
d)		0

9. If tan $θ$ = $\frac{1}{2}$ and tan $φ$ = $\frac{1}{3}$ , then the value of $θ$ + $φ$ is equal to

a)		$\frac{π}{6}$
b)		$π$
c)		zero
d)		$\frac{π}{4}$

10. If A + B + C = $\frac{3 π}{2}$ then cos 2A + cos 2B + cos 2C is equal to

a)		1 – 4 cos A cos B cos C
b)		4 sin A sin B sin C
c)		1 + 2 cos A cos B cos C
d)		1 – 4 sin A sin B sin C

11. If sec $α$ and cosec $α$ are the roots of $x^{2}$ – px + q = 0, then

a)		$p^{2}$ = q (q – 2)
b)		$p^{2}$ = q (q + 2)
c)		$p^{2}$ + $q^{2}$ = 2q
d)		None of these

12. If tan $α$ equals to the integral solution of the inequality 4 $x^{2}$ - 16x + 15 < 0 and cos $β$ equals to the slope of the bisector of the first quadrant, then sin ( $α$ + $β$ ) sin ( $α$ - $β$ ) is equal to

a)		$\frac{3}{5}$
b)		- $\frac{3}{5}$
c)		$\frac{2}{\sqrt{5}}$
d)		$\frac{4}{5}$

13. $\frac{\sin^{2} 3 A}{\sin^{2} A} -$ $\frac{\cos^{2} 3 A}{\cos^{2} A}$ is equal to

a)		cos 2A
b)		8 cos 2A
c)		$\frac{1}{8}$ cos 2A
d)		None of these

14. If (sec A – tan A) (sec B – tan B) (sec C – tan C) = (sec A + tan A) (sec B + tan B) (sec C + tan C), then each side is equal to

a)		0
b)		1
c)		-1
d)		$\pm$ 1

15. The vlaue of sin $10^{0}$ + sin $20^{0}$ + sin $30^{0}$ + .... + sin $360^{0}$ is

a)		1
b)		0
c)		-1
d)		None of these

16. $\sqrt{3}$ cosec $20^{0}$ - sec $20^{0}$ =

a)		2
b)		4
c)		2 $\frac{\sin 20^{0}}{\sin 40^{0}}$
d)		4 $\frac{\sin 20^{0}}{\sin 40^{0}}$

17. If cos A = $\frac{3}{4}$ , then 32 sin $(\frac{A}{2})$ sin $(\frac{5 A}{2})$ is equal to

a)		7
b)		8
c)		11
d)		None of these

18. The value of cos $1^{0}$ cos $2^{0}$ cos $3^{0}$ ........ cos $179^{0}$ is equal to

a)		$\frac{1}{\sqrt{2}}$
b)		-1
c)		0
d)		2

19. The maximum value of 5 cos $θ$ + 3 cos $(θ + \frac{π}{3})$ +3 is

a)		5
b)		10
c)		11
d)		-11

20. If tan $θ$ = a – $\frac{1}{4 a}$ a then sec $θ$ - tan $θ$ =

a)		2a
b)		$\frac{- 1}{2 a}$ , 2a
c)		-2a, $\frac{1}{2 a}$
d)		$\frac{1}{2 a}$ , 2a

21. If cosec $θ - \sin θ = a^{3}$ and sec $θ - \cos θ = b^{3}$ then ab( $a^{2} + b^{2}) is$

a)		1
b)		-1
c)		2
d)		none of these

22. If $x^{2} - 4 x + 5 - siny = 0, y ε (0, 2 π) then$

a)		x=1, y=0
b)		x=1, y= $\frac{π}{2}$
c)		x=2,y=0
d)		x=2,y= $π / 2$

23. If sin $\propto$ and cos $\propto$ are the roots of equation P $x^{2} + qx + r = 0 then$

a)		$p^{2} + q^{2} - 2 pr = 0$
b)		${(p - r)}^{2} = q^{2} + r^{2}$
c)		$p^{2} - q^{2} + 2 pr = 0$
d)		${(p + r)}^{2} = q^{2} - r^{2}$

24. If sin $θ = \frac{24}{25}$ and $θ$ lies in the second quartarnt then sec $θ + \tan θ is$

a)		-7
b)		6
c)		4
d)		-5

25. If for real value of x, cos $θ = x + \frac{1}{x}$ then

a)		$θ$ is an acute angle
b)		$θ is a$ right angle
c)		$θ$ is an obtuse angle
d)		No value of $θ$ is possible

26. The maximum value of 4 $\sin^{2} x + 3 \cos^{2} x + \sin \frac{x}{2} + \cos \frac{x}{2} is$

a)		4+ $\sqrt{2}$
b)		3+ $\sqrt{2}$
c)		9
d)		4

27. If tan $θ + \sin θ = m$ and tan $θ - \sin θ = n,$ then $m^{2} - n^{2} is$

a)		4mn
b)		2 $\sqrt{mn}$
c)		4 $\sqrt{mn}$
d)		2 $\sqrt{\frac{m}{n}}$

28. If x=psec $θ$ and y=qtan $θ then$

a)		$x^{2} - y^{2}$ = $p^{2} q^{2}$
b)		$x^{2} q^{2} - y^{2} p^{2} = pq$
c)		$x^{2} q^{2} - y^{2} p^{2} = \frac{1}{p^{2} q^{2}}$
d)		$x^{2} q^{2} - y^{2} p^{2} = p^{2} q^{2}$

29. If sin $θ + cosec θ = 2,$ the value of $\sin^{10} θ + {cosec}^{10} θ is$

a)		2
b)		1
c)		0
d)		4

30. If tan $θ = a (\neq 0),$ tan 2 $θ = b (\neq 0)$ and tan $θ + \tan 2 θ = \tan 3 θ$ then

a)		a=b
b)		ab=1
c)		a+b =0
d)		b=2a

31. If tan $θ + \sec θ = \sqrt{3}$ , 0 $< θ < π$ then $θ$ is equal to

a)		$\frac{5 π}{6}$
b)		$\frac{2 π}{3}$
c)		$\frac{π}{6}$
d)		$\frac{π}{3}$

32. If tan $θ$ = $\sqrt{n}$ for some non -square natural number n, then sec 2 $θ is$

a)		a rational number
b)		an irrational number
c)		a positive number
d)		none of these

33. If 3 sin $θ + 5 \cos θ = 5$ then the value of 5 sin $θ - 3 \cos θ$ is equal to

a)		5
b)		3
c)		4
d)		none of these

34. If sin $θ and$ cos $θ$ are the roots of the equation a $x^{2} - bx + c = 0$ then a,b and c satisfy the relation

a)		$a^{2} + b^{2} + 2 ac = 0$
b)		$a^{2} - b^{2} + 2 ac = 0$
c)		$a^{2} + c^{2} + 2 ab = 0$
d)		$a^{2} - b^{2} - 2 ac = 0$

35. If cosec $θ - \cot θ = \frac{1}{2}$ 0 < $θ < \frac{π}{2}$ then cos $θ$ is equal to

a)		$\frac{- 3}{5}$
b)		$\frac{- 5}{3}$
c)		$\frac{5}{3}$
d)		$\frac{3}{5}$

36. If sin $θ = \frac{2 t}{1 + t^{2}}$ then cos $θ is$ equal to

a)		$\frac{2 t}{1 - t^{2}}$
b)		$\frac{2 t}{1 + t^{2}}$
c)		$\frac{{1 - t}^{2}}{1 + t^{2}}$
d)		$\frac{{1 + t}^{2}}{1 - t^{2}}$

37. If sin A = $\frac{1}{2}$ then A $\cos^{3} A - 3 \cos A is$ equal to ( $0^{o} < A < 90^{o}$ )

a)		1
b)		0
c)		$\frac{\sqrt{3}}{2}$
d)		$\frac{1}{\sqrt{2}}$

38. If $α and β$ between 0 and $\frac{π}{2}$ and if cos ( $α + β) = \frac{12}{13}$ sin ( $α - β) = \frac{3}{5}$ then sin 2 $α$ is

a)		$\frac{16}{15}$
b)		0
c)		$\frac{56}{65}$
d)		$\frac{64}{65}$

39. If sin $θ + \cos θ = a$ then the value of $| \sin θ - \cos θ |$ is

a)		$\sqrt{2 - a^{2}}$
b)		$\sqrt{2 + a^{2}}$
c)		$\sqrt{a^{2} - 2}$
d)		None of these

40. $\frac{\cos 21 - \sin 21}{\cos 21 + \sin 21}$ is equal to

a)		${\tan 21}^{o}$
b)		${\tan 66}^{o}$
c)		${\tan 24}^{o}$
d)		${\tan 69}^{o}$

41. The value of ${\cos 1}^{o} {\cos 2}^{o} {\cos 3}^{o}$ ------------------- ${\cos 179}^{o}$ is equal to

a)		$\frac{1}{\sqrt{2}}$
b)		0
c)		1
d)		2

42. If sin x + $\sin^{2} x$ = 1, then $\cos^{2} x + \cos^{4} x$ is equal to

a)		1
b)		-1
c)		0
d)		2

43. In a triangle ABC $∠$ A = $\frac{π}{2}$ then $\cos^{2} B$ + $\cos^{2} C =$

a)		-2
b)		-1
c)		0
d)		1

44. The value of tan 15 + cot 15 is

a)		$\sqrt{3}$
b)		2 $\sqrt{3}$
c)		4
d)		-4

45. For a cyclic quadrilateral ABCD, cos B+cos D is equal to

a)		1
b)		0
c)		-1
d)		2

46. The value of $\frac{\cot 54}{\tan 36} + \frac{\tan 20}{\cot 70}$ =

a)		3
b)		2
c)		0
d)		1

47. cos $20^{o}$ cos $40^{o}$ cos $60^{o}$ cos $80^{o}$ is equal to

a)		$\frac{3}{8}$
b)		$\frac{1}{8}$
c)		$\frac{1}{16}$
d)		$\frac{3}{16}$

48. sin $200^{o}$ +cos $200^{o}$ is

a)		negative
b)		positive
c)		zero
d)		zero or positive

49. If sin A + cos A = 1, then sin 2A is equal to

a)		1
b)		2
c)		0
d)		$\frac{1}{2}$

50. The value of the expression ${\tan 1}^{o} {\tan 2}^{o} {\tan 3}^{o}$ ---------- ${\tan 88}^{o} {\times \tan 89}^{o}$ is equal to

a)		0
b)		1
c)		2
d)		$\propto$