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
Which of the following function is surjective but not injective
  • A
    $f : R \to R$ $f (x) = x^4 + 2x^3 - x^2 + 1$
  • B
    $f : R \to R$ $f (x) = x^3 + x + 1$
  • C
    $f : R \to R^+ f (x) =$ $\sqrt {1 + {x^2}} \,$
  • ✓
    $f : R \to R f (x) = x^3 + 2x^2 - x + 1$
✓

Answer

Correct option: D.
$f : R \to R f (x) = x^3 + 2x^2 - x + 1$
d
$f (x) = x^4 + 2x^3 - x^2 + 1 \to A$ polynomial of degree even will always be into say
$f (x) = a_0x^{2n} + a_1 x^{2n-1} + a_2 x^{2n - 2} + .... + a_{2n}$  
$\mathop {Limit}\limits_{x \to \, \pm \,\infty } \,f (x) $

$=\mathop {Limit}\limits_{x \to \, \pm \,\infty } \,$ $[x^{2n} $ $\left( {{a_0}\, + \,\frac{{{a_1}}}{x}\, + \,\frac{{{a_2}}}{{{x^2}}}\, + \,....\, + \frac{{{a_{2n}}}}{{{x^{2n}}}}} \right)$

 $= \left[ \begin{gathered}  \,\infty \,\,\,\,\,if\,{a_0} > 0 \hfill \\  - \,\infty \,\,if\,{a_0}\, < 0 \hfill \\ \end{gathered}  \right.$ 
Hence it will never approach $\infty / - \infty$ 
$f (x) = x^3 + x + 1 \Rightarrow f ‘(x) = 3x^2 + 1$ - injective as well as surjective 
$f (x) =$$\sqrt {1 + {x^2}} \,$ - neither injective nor surjective 
$f (x) = x^3 + 2x^2 - x + 1\Rightarrow f ‘(x) =3x^2 + 4x - 1 \Rightarrow D > 0$ 
Hence $f (x)$ is surjective but not injective.

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

The value of $2 + \frac{1}{{2 + \frac{1}{{2 + ...........\infty }}}}$ is
→
The value of the integral $\int_{0}^{\frac{\pi}{2}} 60 \frac{\sin (6 x)}{\sin x} d x$ is equal to.
→
Consider the system of equations $\mathrm{ax}+\mathrm{by}=0, \mathrm{cx}+\mathrm{dy}=0$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in\{0,1\}$.

$STATEMENT -1$ : The probability that the system of equations has a unique solution is $3 / 8$. and $STATEMENT - 2$: The probability that the system of equations has a solution is $1$ .

→
If the equation ${\sin ^{ - 1}}\sqrt x  + {\cos ^{ - 1}}\sqrt {{x^2} - 1}  + {\tan ^{ - 1}}\left( {\tan \,y} \right) = a$ has at least one solution, then number of integral values of $a$ is 
→
The value of ${\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 + {x^2}}  + \sqrt {1 - {x^2}} }}{{\sqrt {1 + {x^2}}  - \sqrt {1 - {x^2}} }}} \right]\,,\,\left| x \right| < \frac{1}{2},\,x \ne 0\,,$ is equal to
→
A bag contains $3$ white and $7$ red balls. If a ball is drawn at random, then what is the probability that the drawn ball is either white or red
→
Let $P$ be an interior point of a $\triangle A B C$. Let $Q$ and $R$ be the reflections of $P$ in $A B$ and $A C$, respectively. If $Q, A, R$ are collinear, then $\angle A$ equals
→
Let $3, a , b , c$ be in $A.P$. and $3, a -1, b+1, c +9$ be in $G.P$. Then, the arithmetic mean of $a , b$ and $c$ is :
→
Let ${a_1},{a_2},{a_3}, \ldots $ be terms of $A.P.$  If $\frac{{{a_1} + {a_2} + \ldots + {a_p}}}{{{a_1} + {a_2} + \ldots + {a_q}}} = \frac{{{p^2}}}{{{q^2}}},p \ne q$ then $\frac{{{a_6}}}{{{a_{21}}}}$ equals
→
Let $f(x)=2+\cos x$ for all real $x$.

$STATEMENT -1$ : For each real $\mathrm{t}$, there exists a point $\mathrm{c}$ in $[\mathrm{t}, \mathrm{t}+\pi]$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$. because

$STATEMENT -2$: $f(t)=f(t+2 \pi)$ for each real $t$.

→