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QUESTION: 1

A function f(x) is defined as below

x ≠ 0 and f(0) = a, f(x) is continuous at x = 0 if a equals

Solution:

QUESTION: 2

, -1 ≤ x < 0 is continuous in the interval [–1, 1], then `p' is equal to:

Solution:

lim f(x) = -½

Apply L hospital

lim(x → 0) p/2(1+px)^{½} + p/2(1-px)^{½ }= -½

⇒ 2p = -1

p = -1/2

QUESTION: 3

Let f(x) = when – 2 ≤ x ≤ 2. Then (where [ * ] represents greatest integer function)

Solution:

At (x = 0) f(0) = |(0+½) .[0]| = 0

f(1) = |(1+½) .[1]| = 3/2

f(2) = |(2+½) .[2]| = 5

f(-1) = |(-1+½) .[-1]| = ½

LHL at x = 2

lim x-->2 f(x) = |(x+½) .[2]|

=|(5/2) * 1| = 5/2

f(2) = limx-->2 f(x) ( continuous at x=2)

QUESTION: 4

Let f(x) = sgn (x) and g(x) = x (x^{2} – 5x + 6). The function f(g(x)) is discontinuous at

Solution:

QUESTION: 5

If y = where t = , then the number of points of discontinuities of y = f(x), x ∈ R is

Solution:

QUESTION: 6

The equation 2 tan x + 5x – 2 = 0 has

Solution:

Let f(x) = 2tan x + 5x – 2

Now, f(0) = – 2 < 0

and f(π/4 ) = 2 + 5π/4 – 2

= 5π/5 > 0,

the line intersects the x-axis at x = ⅖

since 0 < ⅖ < π/4

Thus, f(x) = 0 has atleast real solution root in ( 0, π/4)

QUESTION: 7

If f(x) = , then indicate the correct alternative(s)

Solution:

QUESTION: 8

The function f (x) = 1 + | sin x l is

Solution:

QUESTION: 9

If f(x) = be a real valued function then

Solution:

QUESTION: 10

The function f(x) = sin^{-1} (cos x) is

Solution:

QUESTION: 11

Let f(x) be defined in [–2, 2] by f(x) = then f(x)

Solution:

QUESTION: 12

If f(x) is differentiable everywhere, then

Solution:

Even if f(x) is differentiable everywhere, |f(x)| need not be differentiable everywhere. An example being where the function changes its sign from negative to positive. f(x)=x at x=0B.

∣f∣= {f if f>0 −f if f<0}

So,∣f∣^{2} = {f^{2} if f>0 −f^{2} if f<0}

Differentiating ∣f∣^{2},

{2ff′ if f>0 −2ff′ if f<0}

At f=0, LHD=RHD=∣f(0)∣^{2}=0, so function is differentiable.

QUESTION: 13

Let f(x + y) = f(x) f(y) all x and y. Suppose that f(3) = 3 and f'(0) = 11 then f'(3) is given by

Solution:

f(x + y)=f(x) f(y) wrt x we get:

f′(x+y) (1 + y′) = f(x) f′(y) (y′) + f′(x) f(y)

Now on substituting x = 0 and y = 3, we get :

f′(3) (1 + 0)=f(0) f′(3) 0+ f(3) f′(0)

f′(3) = 3 × 11 = 33

QUESTION: 14

If f : R → R be a differentiable function, such that f(x + 2y) = f(x) + f(2y) + 4xy x, y ∈ R, then

Solution:

QUESTION: 15

Let f(x) = x – x^{2} and g(x) = . Then in the interval [0, ¥)

Solution:

QUESTION: 16

Let [x] denote the integral part of x ∈ R and g(x) = x – [x]. Let f(x) be any continuous function with f(0) = f(1) then the function h(x) = f(g(x))

Solution:

QUESTION: 17

If f (x) = [x sin p x] { where [x] denotes greatest integer function}, then f (x) is

Solution:

QUESTION: 18

If f(x) = , then f(x) is

Solution:

QUESTION: 19

The functions defined by f(x) = max {x^{2}, (x – 1)^{2}, 2x (1 – x)}, 0 £ x £ 1

Solution:

QUESTION: 20

Let f(x) = x^{3} – x^{2} + x + 1 and g(x) = then

Solution:

QUESTION: 21

Let f²(x) be continuous at x = 0 and f²(0) = 4 then value of is

Solution:

QUESTION: 22

Let f : R → R be a function such that f , f(0) = 0 and f¢(0) = 3, then

Solution:

QUESTION: 23

Suppose that f is a differentiable function with the property that f(x + y) = f(x) + f(y) + xy and f(h) = 3 then

Solution:

QUESTION: 24

If a differentiable function f satisfies f x, y Î R, find f(x)

Solution:

QUESTION: 25

Let f : R → R be a function defined by f(x) = Min {x + 1, |x| + 1}. Then which of the following is true ?

Solution:

f(x) = Min {x + 1,∣x∣ + 1}

x = 0 ⇒ ∣x∣ = x

f(x) = min {x + 1,x + 1}

= x + 1

x < 0,∣x∣ = −x

f(x) = min {x + 1,−x + 1}

−x + 1> x + 1

x < 0

⇒x < 0, f(x) = x + 1

x > 0,f (x) = x + 1

⇒ f(x) = x + 1 for all x

∴f(x) is differential everywhere (continuous and constant slope)

QUESTION: 26

The function f : R /{0} → R given by vf(x) = can be made continuous at x = 0 by defining f(0) as

Solution:

QUESTION: 27

Function f(x) = (|x – 1| + |x – 2| + cos x) where x Î [0, 4] is not continuous at number of points

Solution:

QUESTION: 28

Let f(x + y) = f(x) f(y) for all x, y, where f(0) ≠ 0. If f'(0) = 2, then f(x) is equal to

Solution:

QUESTION: 29

A function f : R → R satisfies the equation f(x + y) = f(x) . f(y) for all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f'(0) = 2 then f'(x) =

Solution:

QUESTION: 30

Let f(x) = [cos x + sin x], 0 < x < 2p where [x] denotes the greatest integer less than or equal to x. the number of points of discontinuity of f(x) is

Solution:

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