Mathematics_Today_-_October_2016

8. Five speakers A, B, C, D and E have been asked to
   deliver a lecture in a meeting. In how many ways
   can their lectures be arranged so that C delivers
   lecture just after A?
   (a) 48 (b) 24
   (c) 60 (d) none of these


9. The equation 34 sinxx+=cos has
   (a) infinitely many solutions
   (b) no solution (c) two solutions
   (d) only one solution


10. If 522
    2

cos θ++= <<cos^2 θ 10 , when 0 θπ, then

the values of θ are

(a) π π

3

± (b)

π

3

3

5

,cos−^1 ⎛

⎝⎜

⎞

⎠⎟

(c) cos−⎛

⎝⎜

⎞

⎠⎟±

1 3

5

π (d) π π

3

3

5

,cos−^1 ⎛

⎝⎜

⎞

⎠⎟

−

11. The most general solution of the equation
    logcosθθtanθθ+=logsin cot 0 , is

(a) nnZπ+π ∈

4

, (b) nnZπ−∈π

4

,

(c) 2

4

nnZπ−∈π, (d)^2 nnZπ 4

+π, ∈

12. The general solution of 22
    2

−cosx= tanx is

(a) (), 21

2

nnZ+ π ∈ (b) (), 41

2

nnZ+ π ∈

(c) 2nπ, n∈Z (d) (4n + 1)π, n∈Z

13. The general solution of
    sinxxcos { ,aa}
    aR

+= − +

∈

min 1^246

is

(a) nππn nZ

2

1

4

+() ,−∈ (b)^21 nnZπ n 4

+() ,−∈π

(c) nnZπ+()1−∈n+ π

4

(^1) ,
(d) nnZπ+()1−−∈nππ 44 ,

14. The equation cos^4 θ + sin^4 θ + λ = 0 has real solution
    for θ, if
    (a)^3
    4

≤≤λ 1 (b) −≤ ≤− 1 1

2

λ

(c) 0 ≤ λ ≤ 1 (d) λ < –1

15. In a triangle ABC, if ∠BC= ∠ =

ππ

34

, and

D divides BC internally in the ratio 1 : 3, then

sin

sin

∠

∠

BAD

CAD

equals

(a)^1

3

(b)^1

3

(c)^1

6

(d)^2

3

16. The ratio of the sides of a triangle is 4 : 5 : 7, then the
    triangle must be
    (a) right-angled
    (b) acute-angled
    (c) obtuse-angled
    (d) right-angled and isosceles


17. In a triangle ABC, (a + b + c)(b + c – a) = λbc, if
    (a) λ < 0 (b) 0 ≤ λ ≤ 4
    (c) 0 ≤ λ < 4 (d) 0 < λ ≤ 4


18. If two sides of a triangle are 23 2− and 2 3 2+
    and their included angle is 60°, then the other
    angles are
    (a) 75°, 45° (b) 105°, 15°
    (c) 60°, 60° (d) 90°, 30°


19. In a triangle ABC, if and cm,BC==ππa=+
    43

,() 31

then the area of the triangle is

(a)^31

2

• cm (^2) (b)^33
  2


• cm 2
  (c)^3
  2
  cm^2 (d)^31
  2
  − cm 2

20. Two adjacent sides of a cyclic quadrilateral are 3
    and 5 and the angle between them is 60°. If the third
    side is 2, then the remaining fourth side is
    (a) 2 (b) 3 (c) 5 (d) 4


21. The circles x^2 + y^2 + 6x + 6y = 0 and
    x^2 + y^2 – 12x – 12y = 0
    (a) cut orthogonally
    (b) touch each other internally
    (c) intersect at two points
    (d) touch each other externally


22. If the ends of the diameter of a circle are the points
    (0, 0) and a
    a

3

3

⎛ ,^1

⎝⎜

⎞

⎠⎟, then through which of the

following points the circle passes?

(a)

a

a

⎛⎝⎜ ,^1 ⎞⎠⎟

(b)

a

a

2

2

⎛ , 1

⎝⎜

⎞

⎠⎟

(c)

1

2

2

a

⎛⎝⎜ ,a ⎞⎠⎟

(d)

1

a

⎛⎝⎜ ,a⎞⎠⎟