Download X 2014 Note that answer key might contain answers of some

M. Prakash Academy: X 2014
Note that answer key might contain answers of some questions which
are not in the set. Please ignore those answers.
Trigonometric Ratios, Trigonometric Equations
One Correct Option
1 Let α, β be such that π < α − β < 3π.
α−β
21
In sin α + sin β = − 65
, cos α + cos β = − 27
,
then
cos
is
65
2
3
3
6
6
(A) − √130
(B) √130
(C) √65
(D) − √65 .
√
√
2 If u = a2 cos2 θ + b2 sin2 θ + a2 sin2 θ + b2 cos2 θ, then the
difference between maximum and minimum values of u2 is
(A) 2(a2 + b2 )
(B) 2(a2 − b2 )
(C) (a + b)2
(D) (a − b)2 .
3 If cos θ + cos2 θ + cos3 θ = 1 and sin6 θ = a + b sin2 θ + c sin4 θ,
then a + b + c =
(A) 0
(B) 1
(C) −1
(D) 2.
3pi
5 If tan(π cos θ) = cot(π sin θ), 0 < θ < 4 , then sin θ + π4 =
(A) √12
(B) − √12
(C) 2√1 2
(D) − 2√1 2 .
8 The least value of
13
(A) 24
(B) 61
48
n−1
rπ X
cos2
10
=
n
r=1
(A)
n
2
(B)
1
2
n−1
2
11 If 0 < A < π4 , then
n−2
(D) n+1
.
2
2
√
√
√1+sin 2A+√1−sin 2A =
1+sin 2A− 1−sin 2A
(C)
(B) − tan A
(A) tan A
=
cos x
b
(C) cot A
(D) − cot A.
1
ak
= k, then bc + ck
+ 1+bk
=
(A) ka
(B) k12
(C) k a − a1
(D) k1 a + a1 .
1
1
16 sec2 α−cos2 α + csc2 α−sin2 α · cos2 α sin2 α =
12 If
sin x
a
+ 32 csc2 θ + 38 sec2 θ is
(C) 61
(D) 61
.
25
24
=
tan x
c
1+cos2 α sin2 α
2−cos2 α sin2 α
2 α sin2 α
(C) 1−cos
2−cos2 α sin2 α
17 If x = y cos 2π
=
3
4
(A) 3
(B) 1
(A)
1+cos2 α sin2 α
2+cos2 α sin2 α
2 α sin2 α
(D) 1−cos
2+cos2 α sin2 α
4π
, then xy + yz +
3
(B)
z cos
zx =
(C) −1
(D) 0.
√
18 If sin x + sin y = 3(cos y − cos x) and sin 3y + sin 3x = k, then
k will have
(A) a single value
(C) three values
(B) two values
(D) infinity of values.
19 If cos x + cos 2x + cos 3x = 3, then sin x + sin 2x + sin 3x =
(A) 3
(B) 1
(C) 0
(D) none.
23 If A and B are two angles satisfying 0 < A, B < π2 and
A + B = π3 , then the minimum value of sec A + sec B is
(A) √23
(B) √43
(C) √13
(D) N.O.T.
24 If x + y + z = π, tan x tan z = 2 and tan y tan z = 18, then tan2 z
is equal to
(A) 15
(B) 16
(C) 19
(D) 20.
26 If A, B, C are angles of an acute angled triangle, then the
minimum value of tan4 A + tan4 B + tan4 C will be
√
(A) 729
(B) 27
(C) 81 3
(D) N.O.T.
27 If A is an obtuse angle, then
sin3 A−cos3 A
sin A
+ √1+tan
2 A − 2 tan A cot A is always equal to
sin A−cos A
(A) 1
(B) −1
(C) 2
(D) N.O.T.
28 The
number of roots of the equation 3 sin2 x = 8 cos x in
− π2 , π2 is
(A) 1
(B) 2
(C) 3
(D) 4.
√
29 The solution set of |4 sin x − 1| < 5, |x| < π is
π
π
(A) −π, − 4π
∪ − π5 , 10
∪ 9π
,π
(B) − 9π
, − 10
∪ 3π
, 7π
5 10
10
10
10
π 3π
(C) −π, − 9π
∪ − 10
(D) − 7π
, 10 ∪ 7π
,π
, 9π .
10
10
10 10
30 The number of solutions of
2
2
3sin 2x+2 cos x + 31−sin 2x+2 sin x = 28 in [0, 2π] is
(A) 3
(B) 4
(C) 5
(D) 6.
34 If α, β, γ, δ satisfy the equation tan x +
tan α tan β tan γ tan δ =
(A) 12
(B) − 12
(C) 1
(D) − 31 .
π
4
= 3 tan 3x, then
35 The number of roots of the equation sec2 θ + csc2 θ = 8 in
[0, π] is
(A) 1
(B) 2
(C) 0
(D) 4.
36
of values of x in [0, 2π] such that
√ The number
√
3 sin x + 2 · cos x = 0 is
(A) 0
(B) 1
(C) 2
(D) 4.
37 The number of solutions of | cot x| = cot x + csc x, 0 ≤ x ≤ 2π
is
(A) 0
(B) 1
(C) 2
(D) 3.
38 The number of values of x in [0, 3π] such that
2 sin2 x + 5 sin x − 3 = 0 is
(A) 1
(B) 2
(C) 4
(D) 6.
40 The number of solutions of the equation
sin 2x + cos 2x + sin x + cos x + 1 = 0 in [0, 2π] is
(A) 2
(B) 3
(C) 4
(D) 6.
41 If tan 5θ = cot 2θ, then θ =
π
π
π
(A) nπ + 14
(B) 2nπ ± 14
(C) (2n + 1) 14
π
(D) (2n − 1) 14
.
51
number of solutions of sin 5θ cos 3θ = sin 9θ · cos 7θ in
πThe
0, 2 is
(A) 5
(B) 10
(C) 4
(D) 9.
√n
π
π
52 The positive integer n such that sin 2n
+ cos 2n
= 2 is
(A) 4
(B) 5
(C) 6
(D) 8.
59 If cos4 x + 4 sin4 x = 1; then general value of x equals to
q
−1
1
(A) nπ
(B)
nπ
±
sin
, nπ
(C) 2nπ
(D) 2nπ ± π4 .
2
5
3
Asnswer Keys
One Option Correct
1A
2D
3A
9C
10 C 11 C
17 D 18 A 19 C
25 A 26 B 27 B
33 A 34 D 35 D
41 C 42 A 43 A
45 C 46 C 47 B
53 A 54 C 55 C
4B
12 D
20 D
28 B
36 B
44 C
48 D
56 A
5C
13 B
21 B
29 C
37 B
6B
7A
14 D 15 C
22 A 23 B
30 B 31 A
38 C 39 B
8D
16 D
24 B
32 C
40 C
49 C
57 D
50 C
58 A
52 C
60 C
51 D
59 B
Integer Answer Type
1 If A + B = 225◦ , then value of (1 + tan A)(1 + tan B) is
2 (sin θ + csc θ)2 + (cos θ + sec θ)2 is greater than or equal to
3 The greatest value fo ‘a0 such that the equation
cos 2x + a sin x = 2a − 7 possesses a solution is
4 The angle θ whose cosine equal to its tangent is given by
sin θ = n sin 18◦ . Then n is equal to
5 The maximum value of 4 sin2 x + 3 cos2 x + sin x2 + cos x2 is
√
a + 2, then a is
Asnswer Keys
Integer Answer Type
12 29 36 42 54
Properties of Triangles and Radii of Circles
One Correct Option
1 In a ∆ABC if s−a
= s−b
= s−c
then
11
12
13
(A) a, b, c are in A.P.
(B) a, b, c are in G.P.
1
(C) Ar(∆ABC) = 216
(D) tan2 A2 = 20
.
33
4 If three sides a, b, c of triangle ABC are in A.P., then the
value of cot A2 cot C2 is
(A) 1
(B) 2
(C) 3
(D) 4.
√
9 If the sides of a triangle are in α, cos α, 1 + sin α cos α,
0 < α < π2 , the largest angle is
(A) 60◦
(B) 90◦
(C) 120◦
(D) 150◦ .
12 The internal bisectors of the angles of triangle ABC meet
=
the circumcricle at D, E, F respectively. Then BC
EF
(A) sin A2
(B) cos A2
(C) 2 sin A2
(D) 2 cos A2 .
14 If r is inradius and R is the circumradius of a right angled
triangle, its area is
(A) r(r + 2R)
(B) 2(2r + R)
(C) rR
(D) R(2r + R).
15 In a triangle ABC if r = 2, R = 5, r3 = 12, then C =
(A) 30◦
(B) 45◦
(C) 60◦
(D) 90◦ .
16 In a triangle ABC, D and E are the midpoints of BC, CA
respectively. If AD = 5, BC = BE = 4, then CA =
√
√
√
(C) 2 7
(D) 5 5.
(A) 5
(B) 7
19 In a triangle ABC if a = 2b, A − B = 90◦ , then tan C =
(A) − 43
(B) 34
(C) 23
(D) 23 .
21 In a triangle ABC if r = 3, R = 6, r1 = 9, then A =
(A) 30◦
(B) 45◦
(C) 60◦
(D) 90◦ .
22 In a triangle ABC, if a, b, c are in A.P. Then a possible
value of B is
(A) 45◦
(B) 75◦
(C) 90◦
(D) 120◦ .
23 In a triangle ABC, D is the midpoint of BC and AD is
perpendicular to AC. Then
(A) a2 = 3b2 + c2
(B) b2 = 3(a2 + c2 )
(C) a2 = b2 + c2
(D) b2 = 5c2 − a2 .
25 In a triangle ABC, if 4 = a2 − (b − c)2 , then tan A =
(A)
15
16
(B)
1
2
(C)
8
17
2
(D)
8
.
15
28 In a triangle ABC if b + c2 = 3a2 , then cot B + cot C =
(A) 0
(B) cot A
(C) 2 cot A
(D) 1.
30 In a triangle ABC, if 3s2 = r12 , then A =
(A) 30◦
(B) 60◦
(C) 120◦
(D) 150◦ .
41 In a ∆ABC, if a = 3, b = 4, c = 5, then r1 + r2 + r3 =
(A) 9
(B) 10
(C) 11
(D) 12.
50 In a ∆ABC, a = 90◦ , B = 30◦ and D is the foot of perpendicular from A on BC. If AC = 2, then BD =
(A) 1
(B) 2
(C) 3
(D) 4.
51 In a ∆ABC, the length of the altitude from A to BC is
2rr1
2 r3
1
2 r3
(B) r2r2 +r
(C) r2rr
(D) r2r2 −r
.
(A) r+r
1
3
1 −r
3
52 In a ∆ABC, if the median and altitude from A trisect
angle A, then |B − C| =
◦
(C) 30◦
(D) 45◦ .
(A) 15◦
(B) 22 21
3
2
55 In ∆ABC,
P cos Aif the sides a, b, c are roots of x −11x +38x−40 =
=
0, then
a
3
9
(A) 4
(B) 43
(C) 16
(D) 16
.
9
Asnswer Keys
One Option Correct
1A
2A
3A
9C
10 A 11 B
17 C 18 D 19 B
25 D 26 C 27 B
33 C 34 D 35 A
41 C 42 D
43 D 44 D 45 C
51 C 52 C 53 C
4C
5D
12 C 13 C
20 C 21 C
28 B 29 B
36 D 37 C
6A
7B
8A
14 A 15 D 16 C
22 A 23 A 24 C
30 C 31 A 32 B
38 C 39 C 40 D
46 B
54 C
48 C
56 B
47 A
55 D
49 D
57 C
50 C
58 C
Integer Answer Type
1 The sides of a triangle are 3 consecutive numbers and the
largest angle is double the least, then the greatest side is
2 The number of triangles can be constructed with the data
a = 5, b = 7, sin A = 43 , is
5 In a right angled ∆ABC, sin2 A + sin2 B + sin2 C is equal to
6 Points D, E are taken on the side BC of a triangle ABC,
such that BD = DE = EC. If ∠BAD = x, ∠DAE = y, ∠EAC =
sin(y+z)
z, then the value of sin(x+y)
is equal to
sin x sin z
7 If in a triangle ABC, right angled at B, s − a = 3, s − c = 2,
then the value of a is
8 If length of the sides AB and AC of a ∆ABC are 3 cm and 6
cm respectively and if cosine of angle BAC is 81 , then length
of the angle bisector of angle BAC is
9 If I is the incentre of ∆ABC and AD is the angle bisector
of angle BAC so that AI : ID = 2 : 1, then value of tan B2 tan C2
is equal to n1 , where n is
10 If the sum of the squares of the sides of a triangle is
equal to twice the square of its circumdiameter, then sin2 A+
sin2 B + sin2 C is equal to
11 In a triangle ABC, the median to the side BC is of length
√ 1 √ and it divides the angle A into angles of 30◦ and 45◦ .
11−6 3
Find the length of the side BC.
12 In a ∆ABC, find side c, when a = 5, b = 4 and
cos(A − B) = 31/32.
13 In a ∆ABC, the tangent of half the difference of two angles is one-third the tangent of half the sum of the angles.
The ratio of the sides opposite to the angles is k : 2, then
find k.
14 Two sides of√a triangle are a, b given by the roots of the
equation x2 − 2 3x + 2 = 0. The angle between the sides is
π/3. Find the value of c2 .
15 Find the value of
of ∆ABC.
b−c
r1
+ c−a
+ a−b
, if a, b, c are length of sides
r2
r3
16 If A0 , A1 , A2 , A3 , A4 and A5 be the consecutive vertices of
a regular hexagon inscribed in a unit circle. Then find the
product of length of A0 A1 , A0 A2 and A0 A4 .
17 If the area of cyclic quadrilateral ABCD is
√ 3 3
.
4
The ra√
dius of the circle circumscribing it is 1. If AB = 1, BD = 3
then evaluate BC · CD.
18 The two adjacent sides of a cyclic quadrilateral are 2
◦
and 5 and the angle
√ between them is 60 . If the area of the
quadrilateral is 4 3, Find the sum of remaining two sides.
20 If x, y, z are perpendicular from the circumcentre of the
abc
. Find
sides of the ∆ABC respectively, then xa + yb + zc = kxyz
the value of k
Asnswer Keys
Integer Answer Type
16
20
33
41
11 2 12 6 13 1 14 6
52
64
15 0 16 3
74
83
17 2 18 5
93
10 2
19 3 20 4