ANDHERI / BORIVALI / DADAR / THANE / POWAI / CHEMBUR / NERUL / KHARGHAR
IIT – JEE
ASSINGMENT
DATE:
TOPIC: TRIGONOMETRY
1.
f
sin and cos are
(A) p (C) p 2.
If -
p
+
qx + r = 0 , then
q2 - 2pr = 0
(B) (p - r )
=
q2 + r 2
-
q2 + 2pr = 0
2 (D) (p + r )
=
q2 - r 2
£ x<
4
p
and
4
2
1+ tan x 1- tan x
(A) – 1 3.
2
+
2
2
the roots of the equation px
= 1 + sin 2 x ,
(B) 0
If 1< x <
2,
then tan x is equal to (C) 1
(D) 2
the number of solutions of the equation
tan (x ( x - 1) + tan 1 x + tan 1(x ( x + 1) = tan 1 3x is -
1
-
-
(A) 0 4.
(B) 1 2
6.
- 1
(C) 2 5p 8
(D) 3
2
, then x is equal to
(B) – 1 – 1
(C)
1 2
(D)
1
-
2
An Aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°. If after 10seconds the elevation is observed to be 30°, then the uniform speed per hour for the Aeroplane is : km/ sec (A) 2400 3 km/
(B) 240 3 km / hr
(C) 240 3 m/ sec
(D) none of these
Solution of the system of equations x + y = (A) x = (C) x =
7.
2
If (sin x) + (cos x ) = - 1
(A) 1
5.
-
p
2 p
4
p
4
, tan x + tan y = 1 is p
-
np , y = np
(B) x =
-
np , y = 2 np
(D) none of these
4
-
np , y = np
The values of x between 0 & 2 which satisfy the equation common difference of the A.P. is (A)
p
8
(B)
p
4
(D)
3 p 8
2
sin x 8 co cos x
(D)
=
1 are
in A.P. The
5 p 8
4
8.
2 The equation sin x - (K - 2 ) sin x - (K + 3) = 0 possesses a solution if :
9.
(A) K > – 3 (B) K < – 2 (C) - 3 £ K £ If 90, then the maximum value of sin sin is (A) 1
(B)
1 2
(C)
3 2
- 2
(D) K is any positive integer
(D) none of these
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# 1
10.
æp 1 æp 1 - 1 aö - 1 aö ÷ ÷ + cos + tan ç cos ÷ ÷= ç ÷ ç ç è4 2 ø è ø b 4 2 b÷
tan ç ç
(A) 11.
12.
13.
15.
b
(B)
b
(C)
a
(D)
2b a
ö ÷ ÷ ÷ ÷ 2ø
1
The inequality 2sin q + 2cos q ³ 2 holds for (A) 0 £ q < p (B) p £ q < 2p (C) for all real æ ç è
cos ç çq-
If tan( p cos q) = cot(p sin q), then
For
±
1 2 2
0<
f <
(B) p 2
, if x =
xyz
xz
(C)
xyz
x y – z
sin 3A sin A
K- 1
÷ is equal to
÷ 4ø
2
±
(D)
¥
å cos
2n
f
,y=
(D) none of these
q
pö ÷
(C)
±
2 2
¥
å sin
2n
f
, z =
å
n= 0
y
2K =
2
n= 0
(A)
If
1
±
¥
2n
2n
cos f , sin
f , then
n= 0
(B)
xyz
(D)
xy
2
xy y
2
z
x
, then the possible values of ‘K’ are: (B) 1
Find the minimum value of (A) – 4
17.
b
æ ç ç1ç è
(A) 0 16.
a
If cos(a + b)sin(r + d) = cos(a - b)sin(r - d) , then cot a cot bcot r = (A) cot (B) – cot (C) tan (D) – tan
(A) 14.
2a
(C) 2 æ ç è
5 cos q+ 3 cos ç çq+
(B) 3
(D)
3 4
pö ÷ ÷+ 3
ø 3÷
(C) 10
(D) none of these
The value of the determinant
1
a2
a
cos(n - 1)x cosnx cos(n + 1)x (a ¹ 1) is zero if : sin(n - 1)x
sinnx
(A) sin x = 0 18.
(B) cos x = 0
20.
If e- p / 2 < q<
(B)
2
(D) cosx =
(C) a = 0
If sin(x - y),sinx & sin(x + y) are in H.P., then sinxsec (A) 2
19.
sin(n + 1)x
(C) 1
y 2
1+ a
2
2a
=
(D) none of these
p
, then 2 (A) cos(log q) < logcos q (C) cos(log q) £ logcos q
If cos(A - B) =
(B) cos(log q) > logcos q (D) none of these
3 & tan A tanB = 2 , then 5
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# 2
21.
(A)
cos A cosB =
(C)
cos A cosB = -
5
- 1 5
2
a
2
1 If sin é ê2cos {cot(2tan ë -
-
a
cot
-
p
a2
-
1
q
cos a + tan
(C) 1
2 + cot 8 + cot
(C)
2
1
-
equals :
b2
a
(D)
18 + cot
-
1
cot
2
32 + .... is
a
equal to
(D) 0
p
=
2
If å cos
1
xi
(C)
2+1
(D) none of these
(C)
n(n + 1) 2
(D) none of these
å x i is
0, then
=
i 1 =
(B) 2n æ1ö
æ1ö
æ1 ö
th - 1 - 1 - 1 ÷ ÷ If tan- 1 çç ÷ + tan ç + tan ç + ....n term = tan q, then q = ÷ ÷ ÷ ç ç ÷ ÷ ÷ ç ç ç è3 ø è7ø è13ø
n n+ 1
n+ 1
(B)
n+ 2
(C)
n n+ 2
(D)
If a1, a2, … an is an A.P. with common difference d, where each é æ d ö æ d ö 1ç 1ç ÷+ ÷ ê ÷ ÷ tan tan ç + tan ç ÷ ÷ ê ç ç ÷ ÷ è1+ a1a2 ø è1+ a2a3 ø ê ë (n - 2)d (n - 1)d (A) (B) 1+ a1a n 1 + a1a n -
-
sin (A)
1
-
(cos(sin x)) -
p
4
1
+
-
(B) 1
If a sin x
-
-
bcos
1
1
-
cos
x
=
p
(C)
2
p
2
(C) -
1
2 p 3
3
1
d an
-
x
0 then
value of
ö ÷ ÷ ÷ is a ÷ 1 nø (D) none of these
1, is equal to:
3 p
(D) 0
4 -
n+ 2
1
p ab + c(b - a)
a+ b p ab + c(a - b) a+ b
,then sin- 1 x + sin- 1 y =
(B) -
æ ç ç ç è1+
n- 1
c, then a sin x + bcos x is equal to:
(D) p
1
ai
(n - 1)d 1- a1a n
-
sin 1 x < cos -
(C)
(B)
If cos- 1 x + cos- 1 y = (A)
... + tan
-
(sin(cos 1 x))where 0
(A) 0
30.
5
2
2n -
(A)
29.
- 1
x)} ù ú 0 , then one of the values of x is:
(B)
(A) N
28.
5
û
=
27.
2pq ab
4
1
i 1
26.
p
(B)
2n
25.
2
The sum of the infinite series p
then
cos
(B)
(A) – 1 24.
a,
2
1
=
sin A sinB =
(D)
-
sin
(A)
23.
(B) z cos A cosB
æ æö pö 1 q÷ ÷ If cos 1 ç + cos ç ÷ ÷ ç ç ÷ ÷= ç çb ø èa ø è (A)
22.
1
p
3
(C)
p
6
(D)
x holds for
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# 3
31.
(A) all values of x
æ 1 ö÷ (B) x Î çç0, ÷ çè 2 ÷ ø
æ1 ö (C) x Î çç ,1 çè 2 ø
(D) x
=
0.75
The length of the shadow of a rod inclined at 10° to the vertical towards the sun is 2.05m when the elevation of the sum is 38°. The length of the rod is: (A) 2.05m (B) 205 tan 42 cm
(C)
32.
205
sin38 °
cm
(D) none of the above
cos48 °
The upper
3 4
th portion
tan
of a vertical pole subtends an angle
-
1
æ3 ö ÷ ç ÷at a point in the horizontal ç ÷ ç è5 ø
plane through its foot and at a distance 40m from the foot. A possible height of the vertical pole is: (A) 20m (B) 40 m (C) 60m (D) 80 cm 33.
The longer side of a parallelogram is 10cm and the shorter side is 6cm. If the longer diagonal makes an angle of 30° with the longer side, the length of the longer diagonal is: (A) (C)
34.
5 5
3 3
13 11
cm
(B)
11
cm
A spherical balloon of radius ‘r’ subtends an angle at the eye of the observer, while the angle of elevation of its centre is . Then the height of the centre of the balloon is æ a a ö (A) r sin sin b (B) r ççcos ec ÷ sin b ÷ ÷ ç è 2 2ø (D) none of these
If upper part of a tree broken over by the wind makes an angle of 30° with the ground, and the distance from the root to the point where the top of the tree touches the ground is 10m, the height of the tree is: (A) 20 3m
36.
3
(D) none of these
cm
(C) r sin b 35.
4
(B) 10 3m
(C) 15 3m
(D) none of these
The least positive non-integral solution of the equation sin p(x 2 + x) = sin p x2 is (A) rational (B) irrational of the form
q
q- 1
(C) irrational of the form (D) Irrational of the form 37.
38.
If e sin x
-
e
-
sin x
-
4
=
4 q+1
4 0 , then x =
, where q is odd integer , where q is an even integer.
(A) 0
(B) sin- 1 {log e (2 -
(C) 1
(D) none of these
The values of x, 0 £ x £ (A)
p
6
(C) both (A) and (B)
p 2
2
5)}
which satisfy the equation 16sin x + 16cos (B)
2
x
=
10 are:
p
3
(D) none of these
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# 4
39.
2
Find the values of x in ( – , ) which satisfy the equation: 271+|cosx|cos (A) 0
40.
±
p
2
(C)
1 2
,- 1
1
(B) é p
In the interval êê ë
,
p
2 2
2
, - 1, 0
44.
3
(D)
±
p
6
(C)
1
-
2
,1,0
(D) -
1 2
, - 1, 0
(B) a unique solution (D) infinity many solutions
The general solution for the equation cos7 x + sin 4 x = 1 is (A)
43.
, the equation logsinq (cos 2q) = 2 has:
(A) no solution (C) two solutions 42.
p
±
93
=
sin 2 q cos 2 q x æ p ö÷ If f(x) = cos2 q x sin 2 q , where qÎ çç0, ÷ ÷ then roots of f(x) = 0 are: ç è ø 2 x sin 2 q cos 2 q (A)
41.
(B)
x|cos3 x|+ .....¥
If
x
(B)
3 cos x + 4 sin x = 5 ,
x
then
2n
(C)
x=
p
(D) none of these
2
æx ö ÷ tan ç ÷ ç ÷= ç è2 ø 1
(A) 0
(B)
(C) Both (A) & (B)
(D) none of these
2
The inequality sin 1(sin 5) > x 2 - 4x holds if -
(A) x = 2 -
9 - 2p
(B)
x= 2+
9 - 2p
(C) x Î (2 -
45.
(D) x > 2 + 9 - 2p 9 - 2p , 2 + 9 - 2p ) 3p 1 1 1 - 1 - 1 - 1 If sin a + sin b+ sin g = , then 1 + 2000 + 2001 + 2002 is equal to: 2 a b g (A) 0
46.
(B) 4 (C) – 1 æ pö ÷ The values of x ç 0< x£ ÷, which satisfy the equation ç ç è ø 2÷ (A)
p
3
(B)
(C) no real value of x 47.
2 cos
2
p 2
2
2
(A)
2
x
- 2
. sin x = x + x
are:
p
12
(D) none of these
The smallest positive number ‘p’ for which the equation cos p sin x solution for
48.
(D) none of these
sin p cos x has a
[0, 2 ] is;
(B)
2
(C)
p
2 2
(D)
2
If x & y are the solutions of the equation 12 sin x + 5 cos x = 2y 2 - 8y + 21 , then the value of xy is (A) 0
(B) 1
(C) – 1
(D) p -
2 tan
- 1
æ5 ç ç ç è
ö ÷ ÷ ÷ ø 12
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# 5
49.
The arithmetic mean of the roots of the equation 4 cos 3 x - 4 cos 2 x - cos(p + x) - 1 = 0 in the interval (0, 315) is equal to (A) 49 (B) 50
50.
The most general values of for which (A) n p + (- 1)n (C)
51.
2n p +
p
4
p
-
4
sin f
56.
(B) n p + (- 1)n
(B)
sin q sin f
(C)
If
(B) G.P.
A B C
If
cot A cot B
180, then
sin a
+
- 6a + 11} are given by
p
4
sin q 1 - cos q
(D)
sin q 1 - cos f
2
g=
(C) 2
0 and cos a
(B)
–1
°
(D) none of these
cos g = 0 ,
(C) -
The value of cot 1 cot 2 ..........cot 89 is (A) 1 (B) – 1 °
(D) none of these
+ cos b+ cos(a - b) + cos(b- g) + cos( g- a ) is : 3
sin b+ sin
then the value of
1
(D) 0
2
°
(C) both (A) & (B)
(D) none of these
sec 1 (sin x) is real if -
p
2
(B) x Î [- 1, 1] (D) x = nq, n Î Z
,n Î I
æ cos If the numerical value of tan ç ç ç è
-
(A) a + b = 20 sin
- 1
6x + sin
- 1
If 0 < A <
æ4 ö ÷ ç ÷+ ç ç è5 ÷ ø
(B) a – b = 11 6 3x = -
(A) x = 0
59.
4
+
(C) H.P.
(B) – 1
(C) x = (2n + 1)
58.
p
2
If sin(y + z - x), sin(z + x - y) and sin(x + y - z) are in A.P., then tan x, tan y & tan z are in
(A) x Î (- ¥ , ¥ )
57.
{1, a
100
(D) none of these
sin q
(A) 55.
aÎ R
4
(A) 1 54.
(D)
sin q- cos q = min
p
(A) A.P. 53.
51
ysin q x sin f x and tan f = , then = 1 - x cos f 1 - y cos q y
If tan q = (A)
52.
(C)
(B)
p 2 x
tan
-
1
ö a æ2 ö÷ ÷ ç is , then ÷ ÷ ç ÷ø ç ÷ b è3 ø
(D) a + b= 23
(D) 2a = 3b
if: 1
=
12
(C) x =
-
1 12
(D) none of these
ææ1 ö ö 3 ÷tan 2A÷ ç , then tan- 1 ç + tan 1 (cot A) + tan 1 (cot A) is equal to: ÷ ÷ ç ç ÷ ÷ ç çè 2 ø è ø 4
p
(A) 0 (C) Both (A) & (B)
(B) (D) none of these
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# 6
60.
The greatest value of (sin (A)
5 p 4
2
(B)
-
1
x 2 ) + (cos 1 x) 2 is {1, a 2 - 6a + 11} -
p
2
4
(C) 0
(D) none of these
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# 7