Joint Entrance Examination

Graduate Aptitude Test in Engineering

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Engineering Mathematics

General Aptitude

1

Let f : ($$-$$1, 1) $$ \to $$ R be a function defined by f(x) = max $$\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}.$$ If K be the set of all points at which f is not differentiable, then K has exactly -

A

one element

B

three elements

C

five elements

D

two elements

f : ($$-$$ 1, 1) $$ \to $$ R

f(x) = max {$$-$$ $$\left| x \right|, - \sqrt {1 - {x^2}} $$}

Non-derivable at 3 points in ($$-$$1, 1)

f(x) = max {$$-$$ $$\left| x \right|, - \sqrt {1 - {x^2}} $$}

Non-derivable at 3 points in ($$-$$1, 1)

2

Let $$f\left( x \right) = \left\{ {\matrix{
{ - 1} & { - 2 \le x < 0} \cr
{{x^2} - 1,} & {0 \le x \le 2} \cr
} } \right.$$ and

$$g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).$$

Then, in the interval (–2, 2), g is :

$$g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).$$

Then, in the interval (–2, 2), g is :

A

non continuous

B

differentiable at all points

C

not differentiable at two points

D

not differentiable at one point

$$\left| {f\left( x \right)} \right| = \left\{ {\matrix{
1 & , & { - 2 \le x < 0} \cr
{1 - {x^2}} & , & {0 \le x < 1} \cr
{{x^2} - 1} & , & {1 \le x \le 2} \cr
} } \right.$$

and $$f\left( {\left| x \right|} \right) = {x^2} - 1,x \in \left[ { - 2,2} \right]$$

Hence $$g(x) = \left\{ {\matrix{ {{x^2}} & , & {x \in \left[ { - 2,0} \right]} \cr 0 & , & {x \in \left[ {0,1} \right)} \cr {2\left( {{x^2} - 1} \right)} & , & {x \in \left[ {1,2} \right]} \cr } } \right.$$

It is not differentiable at x = 1

and $$f\left( {\left| x \right|} \right) = {x^2} - 1,x \in \left[ { - 2,2} \right]$$

Hence $$g(x) = \left\{ {\matrix{ {{x^2}} & , & {x \in \left[ { - 2,0} \right]} \cr 0 & , & {x \in \left[ {0,1} \right)} \cr {2\left( {{x^2} - 1} \right)} & , & {x \in \left[ {1,2} \right]} \cr } } \right.$$

It is not differentiable at x = 1

3

Let [x] denote the greatest integer less than or equal to x. Then $$\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( {\left| x \right| - \sin \left( {x\left[ x \right]} \right)} \right)}^2}} \over {{x^2}}}$$

A

equals $$\pi $$ + 1

B

equals 0

C

does not exist

D

equals $$\pi $$

R.H.L. $$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( {\left| x \right| - \sin \left( {x\left[ x \right]} \right)} \right)}^2}} \over {{x^2}}}$$

(as x $$ \to $$ 0^{+} $$ \Rightarrow $$ [x] $$=$$ 0)

$$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {x^2}} \over {{x^2}}}$$

$$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right)} \over {\left( {\pi {{\sin }^2}x} \right)}} + 1 = \pi + 1$$

L.H.L. $$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( { - x + \sin x} \right)}^2}} \over {{x^2}}}$$

(as x $$ \to $$ 0^{$$-$$} $$ \Rightarrow $$ [x] $$=$$ $$-$$1)

$$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right)} \over {\pi {{\sin }^2}x}}\,.\,{{\pi {{\sin }^2}x} \over {{x^2}}} + {\left( { - 1 + {{\sin x} \over x}} \right)^2} \Rightarrow \pi $$

R.H.L. $$ \ne $$ L.H.L.

(as x $$ \to $$ 0

$$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {x^2}} \over {{x^2}}}$$

$$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right)} \over {\left( {\pi {{\sin }^2}x} \right)}} + 1 = \pi + 1$$

L.H.L. $$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( { - x + \sin x} \right)}^2}} \over {{x^2}}}$$

(as x $$ \to $$ 0

$$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right)} \over {\pi {{\sin }^2}x}}\,.\,{{\pi {{\sin }^2}x} \over {{x^2}}} + {\left( { - 1 + {{\sin x} \over x}} \right)^2} \Rightarrow \pi $$

R.H.L. $$ \ne $$ L.H.L.

4

Let K be the set of all real values of x where the function f(x) = sin |x| – |x| + 2(x – $$\pi $$) cos |x| is not differentiable. Then the set K is equal to :

A

{0, $$\pi $$}

B

$$\phi $$ (an empty set)

C

{ r }

D

{0}

f(x) = sin$$\left| x \right| - \left| x \right|$$ + 2(x $$-$$ $$\pi $$) cosx

$$ \because $$ sin$$\left| x \right|$$ $$-$$ $$\left| x \right|$$ is differentiable function at c = 0

$$ \therefore $$ k = $$\phi $$

$$ \because $$ sin$$\left| x \right|$$ $$-$$ $$\left| x \right|$$ is differentiable function at c = 0

$$ \therefore $$ k = $$\phi $$

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