Mathematics_Today_-_October_2016

1. The value of tan22 2 2tan tan tan
   16

2

16

3

16

5

16

π πππ+++

++tan^226 tan

16

7

16

ππ is equal to

(a) 24 (b) 34
(c) 44 (d) none of these

2. If fx e()= sin()xx−[ ] cosπx, then f(x) is ([x] denotes
   the greatest integer function)
   (a) non-periodic
   (b) periodic with no fundamental period
   (c) periodic with period 2
   (d) periodic with period π


3. Which of the following homogeneous functions
   are of degree zero?

(a) x
y

y

x

y

x

x

y

ln + ln (b) xx yyx y()()−+^

(c) xy
xy^22 +

(d) all of these

4. Match the following.
   Column-I Column-II
   A. limln(cos )
   x

x

→ 0 x

P. – 1

B. xlim lnx ln

x

→ x

⎛⎝⎜ − ⎞⎠⎟

1

1

Q. −

1

2

C. lim

sin

x tan

xx

→ xx

−
0 − R.^0
(a) A → P, B → R, C → Q
(b) A → Q, B → P, C → R

(c) A → R, B → P, C → Q

(d) none of these

5. If θ is small and positive number, then which of the
   following is/are correct?
   (a) sinθ
   θ

= 1 (b) θ < sinθ < tanθ^

(c) tanθ sin

θ

θ

θ

> (d) none of these

6. If y

x

x

=cos− cos

cos

(^133) , then prove that
dy
dx x x
=^3
cos cos 3
.

7. Let a ∈ R, then prove that a function f : R → R
   is differentiable at a if a function φ : R → R satisfies
   f (x) – f (a) = φ(x)(x – a) ∀ x ∈ R and φ is continuous at ‘a’.


8. If β, γ ∈ (0, π) such that cosα + cos(α + β) +
   cos(α + β + γ) = 0 and sinα + sin(α + β) + sin(α +
   β + γ) = 0. Then evaluate f ′(β) and lim ( ),
   x

gx

→γ

where

f (x) = sin2x(1 + cos2x)–1 and gx

xx

xx

() sin cos

sin cos

= + −.

++

1

1

9. Find the area of the triangle formed with vertices

( , ), lim

cos

00 2 , 00 , lim tan

2

x x^0

x

x

x

→ →

⎡ −

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

π ⎟

π

and

xx

⎛ x

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

1

,

where [·] denotes the greatest integer function.

10. Prove that the straight lines whose direction
    cosines are given by the relations al + bm + cn = 0 and
    fmn + gnl + hlm = 0 are perpendicular if f
    a

g

b

h

c

++=0.^

Math Archives, as the title itself suggests, is a collection of various challenging problems related to the topics of IIT-JEE Syllabus. This section

is basically aimed at providing an extra insight and knowledge to the candidates preparing for IIT-JEE. In every issue of MT, challenging

problems are offered with detailed solution. The readers’ comments and suggestions regarding the problems and solutions offered are

always welcome.



 

By : Prof. Shyam Bhushan, Director, Narayana IIT Academy, Jamshedpur. Mob. : 09334870021