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1

MCQ (Single Correct Answer)

Given $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$ such that $$x=0$$ is the only

real root of $$P'\,\left( x \right) = 0.$$ If $$P\left( { - 1} \right) < P\left( 1 \right),$$ then in the interval $$\left[ { - 1,1} \right]:$$

real root of $$P'\,\left( x \right) = 0.$$ If $$P\left( { - 1} \right) < P\left( 1 \right),$$ then in the interval $$\left[ { - 1,1} \right]:$$

A

$$P(-1)$$ is not minimum but $$P(1)$$ is the maximum of $$P$$

B

$$P(-1)$$ is the minimum but $$P(1)$$ is not the maximum of $$P$$

C

Neither $$P(-1)$$ is the minimum nor $$P(1)$$ is the maximum of $$P$$

D

$$P(-1)$$ is the minimum and $$P(1)$$ is the maximum of $$P$$

We have $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$

$$ \Rightarrow P'\left( x \right) = 4\,{x^3} + 3a{x^2} + 2bx + c$$

But $$P'\left( 0 \right) = 0 \Rightarrow c = 0$$

$$\therefore$$ $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + d$$

As given that $$P\left( { - 1} \right) < P\left( a \right)$$

$$ \Rightarrow 1 - a + b + d\,\, < \,\,1 + a + b + d \Rightarrow a > 0$$

Now $$P'\left( x \right) = 4{x^3} + 3a{x^2} + 2bx = x\left( {4{x^2} + 3ax + 2b} \right)$$

As $$P'\left( x \right) = 0,$$ there is only one solution $$x = 0,$$

therefore $$4{x^2} + 3ax + 2b = 0$$ should not have any real roots i.e. $$D < 0$$

$$ \Rightarrow 9{a^2} - 32b < 0$$

$$ \Rightarrow b > {{9{a^2}} \over {32}} > 0$$

Hence $$a,b > 0 \Rightarrow P'\left( x \right) = 4{x^3} + 3a{x^2} + 2bx > 0$$

$$\forall x > 0$$

$$\therefore$$ $$P(x)$$ is an increasing function on $$\left( {0,1} \right)$$

$$\therefore$$ $$P\left( 0 \right) < P\left( a \right)$$

Similarly we can prove $$P\left( x \right)$$ is decreasing on $$\left( { - 1,0} \right)$$

$$\therefore$$ $$P\left( { - 1} \right) > P\left( 0 \right)$$

So we can conclude that

Max $$P\left( x \right) = P\left( 1 \right)$$ and Min $$P\left( x \right) = P\left( 0 \right)$$

$$ \Rightarrow P\left( { - 1} \right)$$ is not minimum but $$P\left( 1 \right)$$ is the maximum of $$P.$$

$$ \Rightarrow P'\left( x \right) = 4\,{x^3} + 3a{x^2} + 2bx + c$$

But $$P'\left( 0 \right) = 0 \Rightarrow c = 0$$

$$\therefore$$ $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + d$$

As given that $$P\left( { - 1} \right) < P\left( a \right)$$

$$ \Rightarrow 1 - a + b + d\,\, < \,\,1 + a + b + d \Rightarrow a > 0$$

Now $$P'\left( x \right) = 4{x^3} + 3a{x^2} + 2bx = x\left( {4{x^2} + 3ax + 2b} \right)$$

As $$P'\left( x \right) = 0,$$ there is only one solution $$x = 0,$$

therefore $$4{x^2} + 3ax + 2b = 0$$ should not have any real roots i.e. $$D < 0$$

$$ \Rightarrow 9{a^2} - 32b < 0$$

$$ \Rightarrow b > {{9{a^2}} \over {32}} > 0$$

Hence $$a,b > 0 \Rightarrow P'\left( x \right) = 4{x^3} + 3a{x^2} + 2bx > 0$$

$$\forall x > 0$$

$$\therefore$$ $$P(x)$$ is an increasing function on $$\left( {0,1} \right)$$

$$\therefore$$ $$P\left( 0 \right) < P\left( a \right)$$

Similarly we can prove $$P\left( x \right)$$ is decreasing on $$\left( { - 1,0} \right)$$

$$\therefore$$ $$P\left( { - 1} \right) > P\left( 0 \right)$$

So we can conclude that

Max $$P\left( x \right) = P\left( 1 \right)$$ and Min $$P\left( x \right) = P\left( 0 \right)$$

$$ \Rightarrow P\left( { - 1} \right)$$ is not minimum but $$P\left( 1 \right)$$ is the maximum of $$P.$$

2

MCQ (Single Correct Answer)

Let $$f\left( x \right) = x\left| x \right|$$ and $$g\left( x \right) = \sin x.$$

**Statement-1:** gof is differentiable at $$x=0$$ and its derivative is continuous at that point.

**Statement-2:** gof is twice differentiable at $$x=0$$.

A

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

B

Statement-1 is true, Statement-2 is false

C

Statement-1 is false, Statement-2 is true

D

Statement-1 is true, Statement-2 is true Statement-2 is a correct explanation for Statement-1

Given that $$f\left( x \right) = x\left| x \right|\,\,$$ and $$\,\,g\left( x \right) = \sin x$$

So that go

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

$$ = g\left( {x\left| x \right|} \right) = \sin x\left| x \right|$$

$$ = \left\{ {\matrix{ {\sin \left( { - {x^2}} \right),} & {if\,\,\,x < 0} \cr {\sin \left( {{x^2}} \right),} & {if\,\,\,x \ge 0} \cr } } \right.$$

$$ = \left\{ {\matrix{ { - \sin \,{x^2},} & {if\,\,\,x < 0} \cr {\sin \,\,{x^2},} & {if\,\,\,x \ge 0} \cr } } \right.$$

$$\therefore$$ $$\left( {go\,f} \right)'\,\,\left( x \right) = \left\{ {\matrix{ { - 2x\,\,\cos \,{x^2},\,\,\,\,if\,\,\,\,x < 0} \cr {2x\,\cos \,{x^2},\,\,\,if\,\,\,\,x \ge 0} \cr } } \right.$$

Here we observe

$$L\left( {gof} \right)'\left( 0 \right) = 0 = R\left( {gof} \right)'\left( 0 \right)$$

$$ \Rightarrow $$ go $$f$$ is differentiable at $$x=0$$

and $$\left( {go\,f} \right)'$$ is continuous at $$x=0$$

Now $$\left( {go\,f} \right)''\left( x \right) = \left\{ {\matrix{ { - 2\cos {x^2} + 4{x^2}\sin {x^2},x < 0} \cr {2\cos {x^2} - 4{x^2}\sin {x^2},x \ge 0} \cr } } \right.$$

Here $$L\left( {gof} \right)''\left( 0 \right) = - 2$$ and $$R\left( {go\,f} \right)''\left( 0 \right) = 2$$

As $$L{\left( {go\,f} \right)^{''}}\left( 0 \right) \ne R\left( {go\,f} \right)''\,\,\left( 0 \right)$$

$$ \Rightarrow go\,f\left( x \right)$$ is not twice differentiable at $$x=0.$$

$$\therefore$$ Statement - $$1$$ is true but statement $$-2$$ is false.

So that go

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

$$ = g\left( {x\left| x \right|} \right) = \sin x\left| x \right|$$

$$ = \left\{ {\matrix{ {\sin \left( { - {x^2}} \right),} & {if\,\,\,x < 0} \cr {\sin \left( {{x^2}} \right),} & {if\,\,\,x \ge 0} \cr } } \right.$$

$$ = \left\{ {\matrix{ { - \sin \,{x^2},} & {if\,\,\,x < 0} \cr {\sin \,\,{x^2},} & {if\,\,\,x \ge 0} \cr } } \right.$$

$$\therefore$$ $$\left( {go\,f} \right)'\,\,\left( x \right) = \left\{ {\matrix{ { - 2x\,\,\cos \,{x^2},\,\,\,\,if\,\,\,\,x < 0} \cr {2x\,\cos \,{x^2},\,\,\,if\,\,\,\,x \ge 0} \cr } } \right.$$

Here we observe

$$L\left( {gof} \right)'\left( 0 \right) = 0 = R\left( {gof} \right)'\left( 0 \right)$$

$$ \Rightarrow $$ go $$f$$ is differentiable at $$x=0$$

and $$\left( {go\,f} \right)'$$ is continuous at $$x=0$$

Now $$\left( {go\,f} \right)''\left( x \right) = \left\{ {\matrix{ { - 2\cos {x^2} + 4{x^2}\sin {x^2},x < 0} \cr {2\cos {x^2} - 4{x^2}\sin {x^2},x \ge 0} \cr } } \right.$$

Here $$L\left( {gof} \right)''\left( 0 \right) = - 2$$ and $$R\left( {go\,f} \right)''\left( 0 \right) = 2$$

As $$L{\left( {go\,f} \right)^{''}}\left( 0 \right) \ne R\left( {go\,f} \right)''\,\,\left( 0 \right)$$

$$ \Rightarrow go\,f\left( x \right)$$ is not twice differentiable at $$x=0.$$

$$\therefore$$ Statement - $$1$$ is true but statement $$-2$$ is false.

3

MCQ (Single Correct Answer)

How many real solutions does the equation

$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?

$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?

A

$$7$$

B

$$1$$

C

$$3$$

D

$$5$$

Let $$f\left( x \right) = {x^7} + 14{x^5} + 16{x^3} + 30x - 560$$

$$ \Rightarrow f'\left( x \right) = 7{x^6} + 70{x^4} + 48{x^2} + 30 > 0,\,\forall x \in R$$

$$ \Rightarrow f$$ is an increasing function on $$R$$

Also $$\mathop {\lim }\limits_{x \to \infty } \,\,f\left( x \right) = \infty $$ and $$\mathop {\lim }\limits_{x \to - \infty } \,\,f\left( x \right) = - \infty $$

$$ \Rightarrow $$ The curve $$y = f\left( x \right)$$ crosses $$x$$-axis only once.

$$\therefore$$ $$f\left( x \right) = 0$$ has exactly one real root.

$$ \Rightarrow f'\left( x \right) = 7{x^6} + 70{x^4} + 48{x^2} + 30 > 0,\,\forall x \in R$$

$$ \Rightarrow f$$ is an increasing function on $$R$$

Also $$\mathop {\lim }\limits_{x \to \infty } \,\,f\left( x \right) = \infty $$ and $$\mathop {\lim }\limits_{x \to - \infty } \,\,f\left( x \right) = - \infty $$

$$ \Rightarrow $$ The curve $$y = f\left( x \right)$$ crosses $$x$$-axis only once.

$$\therefore$$ $$f\left( x \right) = 0$$ has exactly one real root.

4

MCQ (Single Correct Answer)

Suppose the cubic $${x^3} - px + q$$ has three distinct real roots

where $$p>0$$ and $$q>0$$. Then which one of the following holds?

where $$p>0$$ and $$q>0$$. Then which one of the following holds?

A

The cubic has minima at $$\sqrt {{p \over 3}} $$ and maxima at $$-\sqrt {{p \over 3}} $$

B

The cubic has minima at $$-\sqrt {{p \over 3}} $$ and maxima at $$\sqrt {{p \over 3}} $$

C

The cubic has minima at both $$\sqrt {{p \over 3}} $$ and $$-\sqrt {{p \over 3}} $$

D

The cubic has maxima at both $$\sqrt {{p \over 3}} $$ and $$-\sqrt {{p \over 3}} $$

Let $$y = {x^3} - px + q$$

$$ \Rightarrow {{dy} \over {dx}} = 3{x^2} - p$$

For $${{dy} \over {dx}} = 0 \Rightarrow 3{x^2} - p = 0$$

$$ \Rightarrow x = \pm \sqrt {{p \over 3}} $$

$${{{d^2}y} \over {d{x^2}}} = 6x$$

$${\left. {{{{d^2}y} \over {d{x^2}}}} \right|_{x = \sqrt {{p \over 3}} }} = + ve\,\,\,\,$$ and

$$\,\,\,\,\,\,\,\,\,\,$$ $${\left. {\,\,\,{{{d^2}y} \over {d{x^2}}}} \right|_{x = - \sqrt {{p \over 3}} }} = - ve$$

$$\therefore$$ $$y$$ has ninima at $$x = \sqrt {{p \over 3}} $$

and maxima at $$x = - \sqrt {{p \over 3}} $$

$$ \Rightarrow {{dy} \over {dx}} = 3{x^2} - p$$

For $${{dy} \over {dx}} = 0 \Rightarrow 3{x^2} - p = 0$$

$$ \Rightarrow x = \pm \sqrt {{p \over 3}} $$

$${{{d^2}y} \over {d{x^2}}} = 6x$$

$${\left. {{{{d^2}y} \over {d{x^2}}}} \right|_{x = \sqrt {{p \over 3}} }} = + ve\,\,\,\,$$ and

$$\,\,\,\,\,\,\,\,\,\,$$ $${\left. {\,\,\,{{{d^2}y} \over {d{x^2}}}} \right|_{x = - \sqrt {{p \over 3}} }} = - ve$$

$$\therefore$$ $$y$$ has ninima at $$x = \sqrt {{p \over 3}} $$

and maxima at $$x = - \sqrt {{p \over 3}} $$

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