ALLEN JEE-Mathematics 1

NURTURE COURSE
ADDITIONAL EXERCISE
ON
COMPOUND ANGLE
[STRAIGHT OBJECTIVE TYPE]
1. In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch A
each other and also the sides of the triangle. Area of the triangle is -

(A) 4 + 2 3 (B) 6 + 4 3 [JEE 2005 Screening]

B C
7 3 7 3
(C) 12 + (D) 3 +
4 4
4
2. If tan q = - , then sin q is- [AIEEE-2002]
3
4 4 4 4 4 4
(A) - but not (B) – or (C) but not – (D) None of these
5 5 5 5 5 5
4xy
3. sec2 q = is true if and only if - [AIEEE-2003]
(x + y)2
(A) x + y ¹ 0 (B) x = y,x ¹ 0 (C) x = y (D) x ¹ 0, y ¹ 0
1 - tan 2 15°
4. The value of = [AIEEE-2002]
1 + tan 2 15°
3
(A) 1 (B) 3 (C) (D) 2
2
p
5. If a is a root of 25 cos2 q + 5 cos q – 12 = 0, < a < p, then sin 2a = [AIEEE-2002]
2
24 24 13 13
(A) (B) - (C) (D) -
25 25 18 18
6. Which of the following number ( s ) is / are rational ? [JEE 1998]
(A) sin15° (B) cos15° (C) sin15°cos15° (D) sin15°cos75°
7. Let a,b be such that p < a-b < 3p.
node06\B0B0-BA\Kota\JEE(Advanced)\Enthuse\Maths\Additional Exercise_(Star)\Eng.p65

21 27 a-b
If sin a + sin b = – and cos a + cos b = - , then the value of cos is- [AIEEE-2004]
65 65 2
3 3 6 -6
(A) - (B) (C) (D)
130 130 65 65
8. Let A 0 A1 A2 A 3 A 4 A 5 be a regular hexagon inscribed in a circle of unit radius Then the product of the
lengths of the line segments A 0 A1 , A 0 A 2 & A 0 A 4 is - [ JE E 19 98 ]
3 3 3
(A) (B) 3 3 (C) 3 (D)
4 2

E
2 Additional Exercise on Compound Angle ALLEN
1 1
9. If q and f are acute angles satisfying sin q = , cos f = , then q + f Î [JEE 2004 Screening]
2 3
æ p pù æ p 2p ö æ 2 p 5p ö æ 5p ö
(A) ç , ú (B) ç , ÷ (C) ç , ÷ (D) ç , p ÷
è 3 2û è2 3 ø è 3 6 ø è 6 ø
10. Let a and b be the roots of the quadratic equation x2 sin q – x (sin q cos q + 1) + cos q = 0
æ n ( -1)n ö
¥

(0 < q < 45º), and a < b. Then å çè a + b n ÷ø is equal to :- [JEE(Main)-2019]

n= 0

1 1 1 1 1 1 1 1
(1) + (2) + (3) - (4) -
1 - cos q 1 + sin q 1 + cos q 1 - sin q 1 - cos q 1 + sin q 1 + cos q 1 - sin q
11. The roots of 5x2 – 7x + k = 0 are sinA and cosA the value of k is -

12 49
(1) (2) (3) 7 (4) 1
5 10

( )
12. The roots of quadratic equation x 2 + 2 2 sin12° x + 1 - sin 24° = 0 are-

(1) both real and distinct (2) real and equal

(3) imaginary (4) one real and other imaginary
p p
13. Let - < q < - . Suppose a1 and b1 are the roots of the equation x2 – 2xsecq + 1 = 0 and a2
6 12
and b2 are the roots of the equation x2 + 2xtanq – 1 = 0. If a1 > b1 and a2 > b2, then a1 + b2
equals
[JEE(Advanced)-2016, 3(–1)]
(A) 2(secq – tanq) (B) 2secq (C) –2tanq (D) 0
n
14. Let n be an odd integer. If sin nq = å br sin r q , for every value of q , then - [JEE 1998]
r =0

(A) b0 = 1, b1 = 3 (B) b0 = 0, b1 = n
(C) b0 = -1, b1 = n (D) b0 = 0, b1 = n2 - 3n + 3
15. Let f(q) = sin q(sin q + sin3q). Then f(q) - [JEE 2000 Screening, 1M out of 35]
(A) ³ 0 only when q ³ 0 (B) £ 0 for all real q
(C) ³ 0 for all real q (D) £ 0 only when q £ 0
node06\B0B0-BA\Kota\JEE(Advanced)\Enthuse\Maths\Additional Exercise_(Star)\Eng.p65

p
16. If a + b = and b + g = a then tan a equals - [JEE 2001 Screening, 1M out of 35M]
2
(A) 2(tan b + tan g) (B) tan b + tan g (C) tan b + 2tan g (D) 2 tan b + tan g

1
17. If sin (a + b) = 1, sin (a-b) = , then tan (a+ 2b)tan (2a + b) = [AIEEE-2002]
2
(A) 1 (B) –1 (C) zero (D) None of these
18. If y = sec2 q + cos2 q, q ¹ 0, then- [AIEEE-2002]
(A) y = 0 (B) y < 2 (C) y > –2 (D) y > 2

E
 ALLEN JEE-Mathematics 3

19. If u = a 2 cos2 q + b 2 sin 2 q + a 2 sin 2 q + b 2 cos 2 q where a, b Î ¡+ then the difference between the
maximum and minimum values of u2 is given by- [AIEEE-2004]

(A) 2(a2 + b2) (B) 2 a 2 + b 2 (C) (a + b)2 (D) (a – b)2

[MORE THAN ONE CORRECT T YPE]
20. Which of the following option(s) is/are correct
(A) 16(cos266º – sin26º) (cos248º – sin212º) = 1
(B) tan9º + tan36º + tan36º tan9º = 1
(C) tan8º tan35º + tan8º tan47º + tan35ºtan47º = 1
cos 25° cos 70° cos85°
(D) + + =1
sin 70°.sin 85° sin 25°.sin 85° sin 25°.sin 70°
21. If the perpendicular sides of a right angled triangle are {cos2a + cos2b + 2cos(a + b)} and
{sin2a + sin2b + 2sin(a + b)}, then the length of the hypotenuse is :
a -b a+b
(A) 2[1 + cos(a – b)] (B) 2[1 – cos(a + b)] (C) 4 cos2 (D) 4 sin 2
2 2

22. If A,B,C are the angles of a triangle such that cos2 A + cos2 B + sin 2 C - 2 cos A - 2 sin C + 1 = 0 ,
then
(A) sin2A + sin2B + cos2C = 1 (B) sin2A+ sin2B + cos2C = 2
(C) DABC is a right angled D (D) D ABC is an issosceles triangle.
23. Two parallel chords are drawn on the same side of the centre of a circle of radius R . It is found that
they subtend an angle of q and 2 q at the centre of the circle . The perpendicular distance between the
chords is
3q q æ qö æ qö
(A) 2 R sin sin (B) ç1 - cos ÷ ç1 + 2 cos ÷ R
2 2 è 2ø è 2ø

æ qö æ qö 3q q
(C) ç1 + cos ÷ ç1 - 2 cos ÷ R (D) 2 R sin sin
è 2ø è 2ø 4 4

24. For a positive integer n, let fn (q) = çæ tan q ÷ö (1 + sec q)(1 + sec 2q)(1 + sec 4q)....(1 + sec2n q) . Then [JEE 99, 3M]
è 2ø

p æ ö p æ ö p æ p ö
(A) f2 æç ö÷ = 1 (B) f3 ç ÷ = 1 (C) f4 ç ÷ = 1 (D) f5 çè =1
è 16 ø è 32 ø è 64 ø 128 ÷ø
node06\B0B0-BA\Kota\JEE(Advanced)\Enthuse\Maths\Additional Exercise_(Star)\Eng.p65

[SUBJECTIVE TYPE]

25. In any traingle ABC , prove that : cot A + cot B + cot C = cot A cot B cot C .
2 2 2 2 2 2
[JEE 2000 Mains, 3M out of 100]
26. Let a, b, g be distinct real numbers such that
aa2 + ba + c = (sinq)a2 + (cosq)a
ab2 + bb + c = (sinq)b2 + (cosq)b
ag2 + bg + c = (sinq)g2 + (cosq)g
(where a, b, c Î R)

E
4 Additional Exercise on Compound Angle ALLEN
27. Given that 3 sin x + 4 cos x = 5 where x Î (0,p/2). Find the value of 2 sinx + cosx + 4 tanx.
2p p 2p
28. Prove that 4cos .cos - 1 = 2cos .
7 7 7

x 2 + y2
29. Given x, y Î R, x2 + y2 > 0. If the maximum and minimum value of the expression E =
x 2 + xy + 4 y 2
are M and m, and A denotes the average value of M and m, compute (2016)A.
5
rp 5
rp 1 æ p ö
30. Let x1 = Õ cos and 2 å cos , then show that x1 .x 2 = ç cosec - 1 ÷ , where P denotes
x =
r =1 11 r =1 11 64 è 22 ø
the continued product.
31. If 4 sin x. cos y + 2 sinx + 2 cos y + 1 = 0 where x,y Î [0,2p] find the largest possible value
of the sum (x + y).
32. If tan a = p/q where a = 6b, a being an acute angle, prove that :
1
(pcosec2b - q sec 2b) = p 2 + q 2 .
2
33. Let x > 1, y > 1 and (lnx)2 + (lny)2 = lnx2 + lny2, then find the maximum value of xln y.
34. Let A1,A2,.......,An be the vertices of an n-sided regular polygon such that ;
1 1 1
= + . Find the value of n.
A1 A 2 A1 A 3 A1 A 4
35. In a kite ABCD, AB = AD and CB = CD. If ÐA = 108° and ÐC = 36° then the ratio of the area of
a - b tan 2 36°
DABD to the area of DCBD can be written in the form where a,b and c are relatively
c
prime positive integers. Determine the ordered triplet (a,b,c). Also find the numerical value of this
ratio.
5
36. If (1 + sin t) (1 + cos t) = . Find the value (1– sin t) (1 – cost).
4
¥ ¥
37. Let qÎR and å sin ( 2 q) = a . Find the value å éëcot ( 2 q) - cot ( 2 q) ùû sin ( 2 q) in terms of 'a'.
k =2
k

k =0
3 k k 4 k

38. If A1,A2,A3......An are the vertices of a regular n sided polygon inscribed in a circle of radius R.
If (A1A2)2 + (A1A3)2+ .......+(A1An)2 = 14R2, find the number of sides in the polygon.
node06\B0B0-BA\Kota\JEE(Advanced)\Enthuse\Maths\Additional Exercise_(Star)\Eng.p65

( ) - (3 - 5 )
1/2 1/ 2
39. Prove that : 4sin 27° = 5 + 5 .

40. Determine the smallest positive value of x (in degrees) for which
tan(x + 100°) = tan(x + 50°) tanx tan(x – 50°).
1
41. If the product (sin 1°) (sin 3°) (sin 5°) (sin 7°)..........(sin 89°) = , then find the value of n.
2n
sin 4 a cos4 a 1 sin 8 a cos8 a 1
42. Prove that from the equality + = follows the relation ; + = .
a b a+b a 3
b 3
(a + b)3

E
 ALLEN JEE-Mathematics 5
43. Show that elliminating x & y from the equations, sin x + sin y = a ;
8ab
cos x + cos y = b & tanx + tany = c gives =c.
( )
2
a + b2
2
- 4a 2

44. Prove that a triangle ABC is equilateral if & only if tan A + tan B + tan C = 3 3 [JEE 98, 8M]
45. Prove that the triangle ABC is equilateral iff, cot A + cotB + cot C = 3 .
46. " x Î R, find the range of the function, f(x) = cos x (sin x + sin 2 x + sin 2 a ) ; a Î [0,p]
47. Given that for a,b,c,d Î R, if a sec (200°) – c tan(200°) = d and b sec(200°) + d tan(200°) = c, then find
æ a 2 + b2 + c2 + d 2 ö
the value of ç ÷ sin 20° .
è bd - ac ø
3 + cos x
48. Show that " x Î R can not have any value between -2 2 and 2 2 . What inference can
sin x
sin x
you draw about the values of ?
3 + cos x
1 + sin A cosB 2sin A - 2sin B
49. Show that + =
cosA 1 - sin B sin(A - B) + cosA - cosB
x y x y cos a cos b sin a sin b
50. If cos a + sin a = 1 , cos b + sin b = 1 and + = 0 ; a – b ¹ 2np, n Î ¢ then
a b a b a 2
b2
prove that -
b2 (x2 - a2 )
tan a tan b = and x2 + y2 = a2 + b2
a2 (y2 - b2 )

Additional Questions Answer key

1. B 2. B 3. B 4. C 5. B
6. C 7. A 8. C 9. B 10. 1
11. 1 12. 3 13. C 14. B 15. C
16. C 17. A 18. C 19. D 20. A.B,C

21. A,C 22. B,C,D 23. B,D 24. A,B,C,D 26. 2

23p
node06\B0B0-BA\Kota\JEE(Advanced)\Enthuse\Maths\Additional Exercise_(Star)\Eng.p65

27. 5 29. 1344 31. 33. e4

6

13
34. n=7 35. (1,1,2) ; 5 - 2 36. - 10
4
a 89
37. 38. n=7 40. x = 30° 41.
4 2

é 1 1 ù
46. - 1 + sin 2 a £ y £ 1 + sin 2 a 47. 2 48. ê- , ú
ë 2 2 2 2û