Mathematics_Today_-_October_2016

SECTION-I

Single Correct Answer Type

1. If AxBx

r

rr

r r

= ⎛−

⎝⎜

⎞

⎠⎟

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

=

=

∞

=

∞

∑ ∑

1

2

22

0 0

sin , sin , then

number of solution in [–2π, 2π] of

A:B = 4sin^2 x : (1 + cos2x) is

(a) 2 (b) 4 (c) 8 (d) none of these

2. lim

()

x

r

rx

→∞ x

=

+

+

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

=

∑^2013

1

2013

2013 20132013

(a) 2014 (b) 2013 (c)^1

2013

(d) none of these

3. If

dx

xx

xxce

(^323)
1
−
∫ =+α⎡⎣⎢γγlog | − |⎥⎤⎦+ , then
(a) α, β, γ are in A.P.
(b) α, β, γ are in G.P.
(c) α = β = γ (d) all of these

4. If f(x) be an identity function, then equation

fx r

r

{}() (− + )− =

=

∑^20120

1

1

3

has

(a) no real roots (b) real and equal roots

(c) real and different roots

(d) none of these

5. If sin x : sin y : sin z = cos A : cos B : cos C then

sin cos

sin cos

(^22) Ax
xA
−
−
⎛
⎝
⎜⎜
⎞
⎠
∑ ⎟⎟=
(a)
sin cos
sin cos
(^22) xA
xA
−
()( )
∑ ∑
∑∑
(b)
sin cos
sin cos
Ax
xA
∑
∑∑−
(c)
(sin cos )
sin cos
Ax
Ax
∑ −
∑
2
(d)
sin cos
(sin cos )
xA
xA
∑∑
∑
()−()
−
22

6. If xx xrr r pxx
   r

r r

r

()++++=<()

=

∞

=

∞

∑∑^12

00

|| , (^1) then
pr
r=
∑ =
0
671
(a) 0 (b) 2012 (c) 2015 (d) 2013

7. If ∫(tan)^1 +=xxdx−^2 xfx+^1 ()+c, then f(x) =
   (a) x tan x (b) cot x
   (c) tan x (d) none of these


8. If xsin^2 α + y + z = 0, x + ysin^2 β + z = 0 &
   x + y + zsin^2 γ = 0 (α ≠ β ≠ γ ≠ (2n + 1)π/2, where
   n ∈ I) have a non-trivial solution, then ∑tan^2 α =
   (a) 0 (b) 1 (c) 2 (d) none of these


9. If ai > 0 (i = 1, 2, 3, 4) so that 502a 1 + 503a 2 + 504a 3

+ 505a 4 = 2014 and 256 a 1 a 2 a 3 a 4 ≥ ar

r=

∑

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

1

4 4

,then

arr

r=

∑ =

1

4

(a) 2014 (b) 1 (c) 4 (d) none of these

10. If x and y be two real variables satisfying

xyt

t

(^22) +=^2 −^1 and xyt
t
444
2
+=+^1 , then which
of the following is(are) true?
(a) y^2 + x–2 = 0 (b)
xydx
dy
(^3) =
(c) xdy + ydx = 0 (d) none of these
Duration : 30 minutes