Untitled Document

Question Bank No: 2

1. tan $75^{o} - \cot 75^{o}$ is equal to

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

2. If sin $α = \sin β$ and cos $α = \cos β$ then

a)		$α = β$
b)		$α = 2 n π + β,$ n is a natural number
c)		$α = \pm β$
d)		$α = 2 n π + β$ , n is any integer

3. A and B are +ve actue angles satisfying the equations 3 $\cos^{2} A + 2 \cos^{2} B = 4$ and 3 $\frac{\sin A}{sinB} = \frac{2 \cos B}{\cos A}$ then A+2B equals

a)		$\frac{π}{4}$
b)		$\frac{π}{3}$
c)		$\frac{π}{6}$
d)		$\frac{π}{2}$

4. tan 20+ tan 40 + $\sqrt{3}$ tan 20.tan 40 is equal to

a)		$\frac{\sqrt{3}}{2}$
b)		$\frac{\sqrt{3}}{4}$
c)		$\sqrt{3}$
d)		1

5. If sin $θ = \sqrt{3}$ cos $θ,$ then $θ$ is equal to ( $0^{o} < θ < 90^{o}$ )

a)		$45^{o}$
b)		$30^{o}$
c)		$75^{o}$
d)		$60^{o}$

6. The value of sin 28 cos 17+cos 28 sin 17 is

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

7. The value $\sin^{2} 20 + \sin^{2} 70$ is equal to

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

8. sin 50 $- \sin 70 + \sin 10$ is equal to

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

9. Minimum value of sin x+cos x is

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

10. Maximum value of sin x+cos x is

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

11. The maximum value of 12 sin $θ - 9 \sin^{2} θ$ is

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

12. The minimum of sin $θ \cos θ is$

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

13. The maximum value of sin $θ \cos θ is$

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

14. If cos $20^{o}$ =K and cos x = 2 $K^{2} - 1$ then possible values of x between $0^{o}$ and $360^{o}$ are

a)		$140^{o}$
b)		$40^{o}$ and $140^{o}$
c)		$50^{o}$ and $130^{o}$
d)		$40^{o}$ and $320^{o}$

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

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

16. The value of $\cos^{2} θ$ + $\sec^{2} θ$ is always

a)		less than 1
b)		equal to 1
c)		greater than 1 but less than 2
d)		greater than 2

17. If tan $θ = \frac{a}{b}$ , then the value of $\frac{a \sin θ + b \cos θ}{a \sin θ - b \cos θ}$ is

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

18. A circular wire of radius 3cm is cut and bent so as to tie along the circumference at a loop where radius is 48cm. The angle in grades which is subtended at the centre of the loop is

a)		50 grades
b)		20 grades
c)		25 grades
d)		none of these

19. The angles of a triangle are in AP and the least angle is $30^{o}$ . the greatest in radians is

a)		$\frac{π}{2}$ radians
b)		$\frac{π}{3}$ radians
c)		$\frac{π}{4}$ radians
d)		$π$ radians

20. The minute hand of a clock is 10cm long. How far does the tip of the hand move in 20 minute

a)		$\frac{3}{20}$ cm
b)		$\frac{3 π}{20}$ cm
c)		$\frac{20}{3}$ cm
d)		$\frac{20 π}{3} cm$

21. The perimeter of a certain sector of a circle is equal to half of the circle of which, it is a part. The circular of the angle of the sector is

a)		2
b)		$\frac{π}{2}$
c)		$π$ -2
d)		$π$ +2

22. If the length of a chord of a circle is equal to that of the radius of the circle then the angle subtended in radians at the centre of the circle by the chord is

a)		1
b)		$\frac{π}{2}$
c)		$\frac{π}{3}$
d)		$\frac{π}{4}$

23. If $0 < x < π$ and cos $x$ + sin $x$ = $\frac{1}{2}$ , then tan $x$ =

a)		$\frac{- (4 + \sqrt{7})}{3}$
b)		$\frac{1 + \sqrt{7}}{4}$
c)		$\frac{1 - \sqrt{7}}{4}$
d)		$\frac{4 - \sqrt{7}}{3}$

24. If sinA sinB sinC + cosA cosB =1, then sinC

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

25. Let $θ \in (0, \frac{π}{4})$ and $t_{1} = {(\tan θ)}^{\tan θ}$ , $t_{2} = {(\tan θ)}^{\cot θ}$ , $t_{3} = {(\cot θ)}^{\tan θ}$ , $t_{4} = {(\cot θ)}^{\cot θ}$ , then

a)		$t_{1} > t_{2} > t_{3} > t_{4}$
b)		$t_{4} > t_{3} > t_{1} > t_{2}$
c)		$t_{3} > t_{1} > t_{2} > t_{4}$
d)		$t_{2} > t_{3} > t_{1} > t_{4}$

26. A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length x. The maximum area of the park is

a)		$\frac{x^{2}}{2}$
b)		$π x^{2}$
c)		$\frac{3 x^{2}}{2}$
d)		$\sqrt{\frac{x^{3}}{8}}$

27. The sides a, b, c of a triangle ABC are in A.P. and cos $θ_{1}$ = $\frac{a}{b + c}$ , cos $θ_{2}$ = $\frac{b}{c + a}$ , cos $θ_{3}$ = $\frac{c}{a + b}$ . Then $\tan^{2} \frac{θ_{1}}{2} + \tan^{2} \frac{θ_{3}}{2}$ =

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

28. The internal bisector of $∠ A$ of triangle ABC meets side BC at D. A line drawn through D perpendicular to AD meets the side AC at E and the side AB at F. If a, b, c are the sides of the triangle , then

a)		triangle AEF is equilateral
b)		triangle AEF is right angled
c)		$A E^{2} = E F^{2} + A F^{2}$
d)		triangle AEF is isosceles

29. Given an isosceles triangle, whose one angle is $120^{\circ}$ and inradius $\sqrt{3}$ , the area of the triangle is

a)		7 + 12 $\sqrt{3}$
b)		12 - 7 $\sqrt{3}$
c)		12 + 7 $\sqrt{3}$
d)		4 $π$

30. In a $Δ ABC$ , if $r_{1} = \sqrt{3} s$ , then A=

a)		$\frac{π}{3}$
b)		$\frac{π}{2}$
c)		$\frac{2 π}{3}$
d)		$\frac{3 π}{4}$

31. In a $Δ ABC$ , if tan A: tan B: tan C = 2: 3: 4, then $\sec^{2}$ C =

a)		17
b)		7
c)		11
d)		9

32. In a $Δ ABC$ , a =1, c =2 and A is given. If $b_{1}$ , $b_{2}$ are two values of b such that 2 $b_{2}$ = $b_{1}$ , $\sin^{2} A$ =

a)		$\frac{5}{8}$
b)		$\frac{5}{16}$
c)		$\frac{5}{32}$
d)		$\frac{3}{16}$

33. In a $Δ ABC$ if s = $3 + \sqrt{3} + \sqrt{2}$ , $3 B - C = 30^{\circ}$ , $A + 2 B = 120^{\circ}$ , then the largest side is

a)		2
b)		$2 \sqrt{2}$
c)		$2 (\sqrt{3 + 1})$
d)		$\sqrt{3} - 1$

34. In a $Δ ABC$ , if a, b, c are in G.P and the largest angle exceeds the smallest one by $60^{\circ}$ , then cos B =

a)		$\sqrt{2}$ - 1
b)		$\frac{\sqrt{13} - 1}{2}$
c)		$\frac{- (\sqrt{13} + 1)}{4}$
d)		$\frac{\sqrt{13} - 1}{4}$

35. Let $A_{0}$ $A_{1}$ $A_{2}$ $A_{3}$ $A_{4}$ $A_{5}$ be a regular hexagon inscribed in a circle of unit redius. Then the product of the lengths of the line segments $A_{0}$ $A_{1}$ , $A_{0}$ $A_{2}$ , and $A_{0}$ $A_{4}$ is

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

36. In a triangle ABC, the altitudes from A, B, C are in H.P., then sinA, sin B, sinC are in

a)		A.P
b)		G.P
c)		H.P
d)		A.G.P

37. If the sides of a triangle are sin $α$ ,cos $α$ , $\sqrt{1 + \sin α \cos α}$ , 0 $< α$ $< \frac{π}{2}$ , the largestangle is

a)		$60^{\circ}$
b)		$90^{\circ}$
c)		$120^{\circ}$
d)		$150^{\circ}$

38. In a triangle ABC medians AD, BE are drawn If AD = 4, $∠ DAB$ = $\frac{π}{6}$ . $∠ ABE$ = $\frac{π}{3}$ , then the area of the triangle is

a)		$\frac{8}{3}$
b)		$\frac{16}{3}$
c)		$\frac{32}{3}$
d)		$\frac{32}{3 \sqrt{3}}$

39. In $Δ ABC$ , if $a \cos^{2} \frac{C}{2} + c \cos^{2} \frac{A}{2}$ = $\frac{3 b}{2}$ ,then a, b, c

a)		are in A.P
b)		are in G.P
c)		are in H.P
d)		satisfy a + b = c

40. The sum of the radii of inscribed and circumsribed circles of an n- side regular polygon of side a is

a)		a cot $(\frac{π}{n})$
b)		$\frac{a}{2}$ cot $(\frac{π}{2 n})$
c)		a cot $(\frac{π}{2 n})$
d)		$\frac{a}{4}$ cot $(\frac{π}{2 n})$

41. In $Δ ABC$ , if $r_{1}$ $>$ $r_{2}$ $>$ $r_{3}$ , then

a)		a $>$ b $>$ c
b)		a $<$ b $<$ c
c)		a $>$ b $<$ c
d)		a $<$ b $>$ c

42. There exists a triangle ABC satisfying the conditions

a)		$b \sin A$ = a, A $< \frac{π}{2}$
b)		$b \sin A >$ a, A $> \frac{π}{2}$
c)		$b \sin A >$ a, A $< \frac{π}{2}$
d)		$b \sin A \pm$ a, A $< \frac{π}{2}$ , b $>$ a

43. In a triangle ABC, A $>$ B. If A and B satisfy the equation $3 \sin x - 4 \sin^{3} x - k$ = 0, 0 $<$ k $<$ 1 , then C =

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

44. If the vertices of a triangle are rational points, which of the points are always irrational?

a)		centroid
b)		incentre
c)		orthocentre
d)		circumcentre

45. In a triangle PQR, if sin P, sin Q, sin R are in A.P, then

a)		altitudes are in A.P
b)		altitudes are in H.P
c)		medians are in G.P
d)		medians are in A.P

46. In a triangle PQR, R = $\frac{π}{2}$ . If tan $\frac{P}{2}$ and tan $\frac{Q}{2}$ are the roots of the equation ${ax}^{2} + bx + c = 0 (a \neq 0)$ , then

a)		a + b = 0
b)		b + c = a
c)		a + c = b
d)		b = c

47. If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the longest side to the perimeter is

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

48. Which of the folowing data do not determine an acute angled triangle?

a)		a, sin A, sin B
b)		a, b, c
c)		a, sin B, R
d)		a, sin A, R

49. In a triangle ABC , if C = $\frac{π}{2}$ , then 2(r+R) =

a)		a + b
b)		b + c
c)		c + a
d)		a + b + c

50. In a triangle ABC, 2ac sin $\frac{1}{2} (A - B + C)$ =

a)		$a^{2} + b^{2} - c^{2}$
b)		$c^{2} + a^{2} - b^{2}$
c)		$b^{2} - c^{2} - a^{2}$
d)		$c^{2} - a^{2} - b^{2}$