Download App

STD 12 - 1. relation and function — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 1. relation and function4 Marks
MCQ
Let $P(x)$ be a polynomial with real coefficients such that $P\left(\sin ^2 x\right)=P\left(\cos ^2 x\right)$ for all $x \in[0, \pi / 2)$. Consider the following statements:

$I.$ $P(x)$ is an even function.

$II.$ $P(x)$ can be expressed as a polynomial in $(2 x-1)^2$

$III.$ $P(x)$ is a polynomial of even degree.

Then,

  • A
    all are false
  • B
    only $I$ and $II$ are true
  • ✓
    only $II$ and $III$ are true
  • D
    all are true
✓

Answer

Correct option: C.
only $II$ and $III$ are true
c
(c)

We have,

$P\left(\sin ^2 x\right) =P\left(\cos ^2 x\right), x \in\left[0, \frac{\pi}{2}\right)$

$P\left(\sin ^2 x\right) =P\left(1-\sin ^2 x\right)$

$P(x) =P(1-x), x \in[0,1]$

$P^{\prime}(x) =-P^{\prime}(1-x)$

So, $P^{\prime}(x)$ is symmetric about line $x=\frac{1}{2}$

So, $P^{\prime}(x)$ has highest degree is odd. $\Rightarrow P(x)$ has highest degree is even.

Hence, option $(c)$ is correct.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Consider three vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$. Let $|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=3$ and $\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}$. If $\alpha \in\left[0, \frac{\pi}{3}\right]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of $27|\overrightarrow{c}-\overrightarrow{a}|^2$ is equal to :
→
If $f(x)=\left\{\begin{array}{ccc}\frac{1}{|x|} & ; & |x| \geq 1 \\ a x^{2}+b & ; & |x|<1\end{array}\right.$ is differentiable at every point of the domain, then the values of $a$ and $b$ are respectively
→
An online exam is attempted by $50$ candidates out of which $20$ are boys. The average marks obtained by boys is $12$ with a variance $2 .$ The variance of marks obtained by $30$ girls is also $2 .$ The average marks of all $50$ candidates is $15 .$ If $\mu$ is the average marks of girls and $\sigma^{2}$ is the variance of marks of $50$ candidates, then $\mu+\sigma^{2}$ is equal to ...... .
→
The solution of $3\tan (A - {15^o}) = \tan (A + {15^o})$ is
→
The function $f(x) = {{\log x} \over x}$ is increasing in the interval
→
If $1 + \cot \theta = {\rm{cosec}}\theta $, then the general value of $\theta $ is
→
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
→
Let the function $f, g$ and $h$ be defined as follows :

$f(x)\, = \left\{ {\begin{array}{*{20}{c}}{x\,\sin \,\left( {\frac{1}{x}}\right)\,\,\,\,\,\,\,for\,\, - 1 \le x \le 1\,\,and\,\,x \ne \,0}\\
{0\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,0}
\end{array}} \right.$

$g(x)\, = \left\{ {\begin{array}{*{20}{c}}{{x^2}\,\sin \,\left( {\frac{1}{x}} \right)\,\,\,\,\,\,\,for\,\, - 1 \le x \le 1\,\,and\,\,x \ne \,0}\\{0\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,0}\end{array}} \right.$ $h (x) = | x |^3$ for $- 1 \le x \le 1$ Which of these functions are differentiable at $x = 0$ ?

→
If the line $3x - 4y = \lambda $ touches the circle ${x^2} + {y^2} - 4x - 8y - 5 = 0$, then $\lambda $ is equal to
→
Two numbers $x$ $\&$ $y$ are chosen at random (without replacement) from the set $\{1, 2, 3, ......, 1000\}$. Then the probability that $|x^4 - y^4|$ is divisible by $5$, is -
→