Untitled Document

Question Bank No: 4

1. If sin A = sin B, cos A = cos B, then the value of A in terms of B is

a)		n $π$ + B
b)		n $π$ + ${(- 1)}^{n}$ B
c)		2n $π$ + B
d)		2n $π$ - B

2. If (1 + tan $θ$ ) (1 + tan $Φ$ ) = 2, then $θ$ + $Φ$ =

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

3. The equation 2 $\cos^{2}$ (x/2) $\sin^{2}$ x = $x^{2}$ + 1/ $x^{2}$ 0 $\leq$ x $\leq π$ /2 has

a)		No solution
b)		One real solution
c)		more than one real solution
d)		None of these

4. The number of solutions of $\sin^{2}$ $θ$ + 3 cos $θ$ = 3 in [- $π, π$ ) is

a)		4
b)		2
c)		0
d)		None of these

5. If sin ( $π$ /4 cot $θ$ ) = cos ( $π$ /4 tan $θ$ ), then $θ$ =

a)		n $π$  + $π$ /4
b)		2n $π$  $\pm$ $π$ /4
c)		n $π$  - $π$ /4
d)		2n $π$  $\pm π$ /6

6. $\cos^{- 1}$ (1/2 ) + 2 $\sin^{- 1}$ (1/2) is equal to

a)		$π$ /4
b)		$π$ /6
c)		$π$ /3
d)		2 $π$ /3

7. If x = sin (2 $\tan^{- 1}$ 2) and y = sin (1/2 $\tan^{- 1}$ 4/3), then

a)		x > y and $y^{2}$ = 1 – x
b)		x > y
c)		x > y and $y^{2}$ = x
d)		$y^{2}$ = 1 + x

8. sin ( $\frac{1}{2}$ $\cot^{- 1}$ (- $\frac{3}{4}$ ) is equal to

a)		1/ $\sqrt{5}$
b)		– 2/ $\sqrt{5}$
c)		2/ $\sqrt{5}$
d)		– 1/ $\sqrt{5}$

9. $\tan^{- 1}$ (tan 33 $π$ /4) is equal to

a)		$π$ /4
b)		- $π$ /4
c)		3 $π$ /4
d)		none of these

10. if $\tan^{- 1}$ 2, $\tan^{- 1}$ 3 are two angles of a triangle, then the third angle is

a)		$π$ /4
b)		3 $π$ /4
c)		$π$ /2
d)		none of these

11. The expression $\frac{(a + b + c) (b + c - a) (c + a - b) (a + b - c)}{4 b^{2} c^{2}}$ is equal to

a)		cos 2 A
b)		1 – cos A
c)		$\sin^{2}$ A
d)		1 + cos A

12. If any $Δ$ ABC if 2 cos B = a/c, then the triangle is

a)		right angled
b)		equilateral
c)		isosceles
d)		none of these

13. If in a $Δ$ ABC, (sin A + sin B + sin C) (sin A + sin B – sin C) = 3 sin A sin B then

a)		A = $60^{o}$
b)		B = $60^{o}$
c)		C = $60^{o}$
d)		none of these

14. If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the triangle is

a)		Isosceles
b)		right angled
c)		isosceles right-angled
d)		equilateral

15. If in a $Δ$ ABC, a tan A + b tan B = (a + b) tan A + B, then

a)		A = B
b)		A = -B
c)		A = 2B
d)		B = 2A

16. In a $Δ$ ABC, if $\frac{tanA - tanB}{tanA + tanB}$ = $\frac{c - b}{c}$ then A is equal to

a)		$30^{o}$
b)		$45^{o}$
c)		$60^{o}$
d)		$90^{o}$

17. If one side of a triangle is double the other, and the angles on opposite sides differ by $60^{o}$ , then the triangle is

a)		Right angled
b)		isosceles
c)		equilateral
d)		none of these

18. If the angles A, B. C of triangle ABC are in AP and the sides a, b, c are in GP, then the triangle is

a)		right angles but not isosceles
b)		isosceles but not equilateral
c)		equilateral
d)		scalene

19. In a triangle ABC, the tangent of half the difference of two angles is one third of the tangent of half the sum of the two angles. The ratio of the sides opposite the angle is

a)		2 : 3
b)		1 : 3
c)		2 : 1
d)		3 : 4

20. In a triangle ABC, a = 5, b = 4 and cos (A – B) = 31/32, then the side c is

a)		6
b)		7
c)		9
d)		none of these

21. If A = $60^{o}$ , a = 5, b = 4 $\sqrt{3}$ in $Δ$ ABC, the B =

a)		$30^{o}$
b)		$60^{o}$
c)		$90^{o}$
d)		none of these

22. If A = $30^{o}$ , a = 7, b = 8 in $Δ$ ABC, then B has

a)		One solution
b)		two solutions
c)		no solution
d)		none of these

23. If one angle of a triangle is $30^{o}$ and lengths of the sides adjacent to it are 40 and 40 $\sqrt{3}$ , the triangle is

a)		Right angled
b)		isosceles
c)		(a) and (b)
d)		none of these

24. Two sides of a triangle are $\sqrt{3}$ + 1 and $\sqrt{3}$ – 1 and the included angle is $60^{o}$ . Then the other angles are

a)		$90^{o}$ , $30^{o}$
b)		$105^{o}$ , $15^{o}$
c)		$75^{o}$ , $45^{o}$
d)		none of these

25. The sides of a triangle are 3x + 4y, 4x + 3y and 5x + 5y units where x > 0, y > 0. Then the triangle is

a)		Right angled
b)		equilateral
c)		obtuse angle
d)		none of these

26. If D is the midpoint of the side BC of a triangle ABC and AD is perpendicular to AC. Then

a)		3 $b^{2}$ = $a^{2}$ - $c^{2}$
b)		3 $a^{2}$ = $b^{2}$ - 3 $c^{2}$
c)		$b^{2}$ = $a^{2}$ - $c^{2}$
d)		$a^{2}$ + $b^{2}$ = 5 $c^{2}$

27. If in a triangle ABC, tanA+tanB +tanC=6, then the triangle is

a)		Right angled
b)		obtuse angled
c)		equilateral
d)		acute angled but not equilateral

28. If a = 2. b = 3, c = 5 in $Δ$ ABC, then C =

a)		$π$ /6
b)		$π$ /3
c)		$π$ /2
d)		none of these

29. When the length of the shadow of a pole is equal to a height of the pole, then the elevation of source of light is

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

30. On the level ground, the angle of elevation of the top of the tower is $30^{o}$ , on moving 20 metres nearer; the angle of elevation is $60^{o}$ . Then the height of the tower is

a)		20 $\sqrt{3}$ metres
b)		10 $\sqrt{3}$ metres
c)		10 $\sqrt{3}$ – 1
d)		none of these

31. The angle of elevation of the top of a tower from a point 20 metres away from its base is $45^{o}$ . The height of the tower is

a)		10 m
b)		20 m
c)		40m
d)		20√3

32. A tree is broken by wind and its upper part touches the ground at a point 10 metre from the foot of the tree and makes an angle of $45^{o}$ with the ground. The entire length of the tree is

a)		15m
b)		20m
c)		10 (1 + $\sqrt{2}$ )m
d)		10 [1 + $\sqrt{\frac{3}{2}}$ ] m

33. The angle of elevation of the top of a hill from each of the vertices A, B, C of a horizontal triangle is a. The height of the hill is

a)		b tan a cosec B
b)		$\frac{1}{2}$ a tan a cosec A
c)		$\frac{1}{2}$ c tan a cosec C
d)		none

34. Three vertical poles of heighs h1, h2, h3 at the vertices A, B and C of a $Δ$ ABC subtend angles a, b and y respectively at the circumference of the triangle. If cot a, cot b, cot y are in A.P., the h1, h2, h3 are in

a)		A.P.
b)		G.P.
c)		H.P
d)		none of these

35. The top of a hill observed from the top and bottom of a building of height h is at angles of elevation a and b respectively. The height of the hill is

a)		$\frac{h \cot b}{\cot b - \cot a}$
b)		$\frac{h \cot a}{\cot a - \cot b}$
c)		$\frac{h \tan a}{\tan a - \tan b}$
d)		none of these

36. The tower subtends an angle of $30^{o}$ at a point on the same level as the foot of the tower. At a second point h metres above the first, the depression of the foot of the tower is $60^{o}$ The horizontal distance of the tower from the point is

a)		h cot $60^{o}$
b)		h cot $30^{o}$
c)		h/3 cot $60^{o}$
d)		h/3 cot $30^{o}$

37. The angle of elevation of the top of a T.V. tower from three points A, B , C in a straight line through the foot of the tower are a, 2a, 3a respectively. If AB = a, the height of the tower is

a)		a tan a
b)		a sin a
c)		a sin 2a
d)		a sin 3a