Let f(x)=x^6-2 x^3+x^3+x^2-x-1 and g(x)=x^4-x^3-x^2-1 be two polynomials. Let a, b, c and d be roots

English

Gujarati

Hindi

1.Relation and Function

normal

Let $f(x)=x^6-2 x^3+x^3+x^2-x-1$ and $g(x)=x^4-x^3-x^2-1$ be two polynomials. Let $a, b, c$ and $d$ be the roots of $g(x)=0$. Then, the value of $f(a)+f(b)+f(c)+f(d)$ is

$-5$

$0$

$4$

$5$

(KVPY-2019)

Solution

(b)

Given,

$f(x)=x^6-2 x^5+x^3+x^2-x-1$

$=x^2\left(x^4-x^3-x^2-1\right)$

$-x\left(x^4-x^3-x^2-1\right)+2 x^2-2 x-1$

$\Rightarrow f(x)=\left(x^2-x\right) g(x)+2 x^2-2 x-1$

$\therefore f(a)=2 a^2-2 a-1$

$\therefore \Sigma f(a)=f(a)+f(b)+(c)+f(d)$

$=2 \Sigma a^2-2 \Sigma a-\Sigma 1$

$=2\left((\Sigma a)^2-2 \Sigma a b\right)-2 \Sigma a-4$

$\because a, b, c, d$ are roots of equation

$g(x)=x^4-x^3-x^2-1=0$

then $\Sigma a=1, \Sigma a b=-1$

$\therefore \Sigma f(a)=2\left[(1)^2-2(-1)\right]-2(1)-4$

$=6-2-4=0$

Standard 12

Mathematics

Let $f(\theta)$ is distance of the line $( \sqrt {\sin \theta } )x + ( \sqrt {\cos \theta })y +1 = 0$ from origin. Then the range of $f(\theta)$ is –

normal

Let $f : R \rightarrow R$ be a continuous function such that $f(3 x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to.

hard

(JEE MAIN-2022)

If $f(x)$ satisfies $f(7 -x) = f(7 + x)\ \forall \,x\, \in \,R$ such that $f(x)$ has exactly $5$ real roots which are all distinct such that sum of the real roots is $S$ then $S/7$ is equal to

normal

Define a function $f(x)=\frac{16 x^2-96 x+153}{x-3}$ for all real $x \neq 3$. The least positive value of $f(x)$ is

normal

(KVPY-2017)

Statement $1$ : If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^P$.
Statement $2$ : The total number of selections of $p$ different objects out of $q$ objects is ${}^q{C_p}$.

hard

(AIEEE-2012)

Let $f(x)=x^6-2 x^3+x^3+x^2-x-1$ and $g(x)=x^4-x^3-x^2-1$ be two polynomials. Let $a, b, c$ and $d$ be the roots of $g(x)=0$. Then, the value of $f(a)+f(b)+f(c)+f(d)$ is

Solution

Similar Questions

Let $f(\theta)$ is distance of the line $( \sqrt {\sin \theta } )x + ( \sqrt {\cos \theta })y +1 = 0$ from origin. Then the range of $f(\theta)$ is –

Let $f : R \rightarrow R$ be a continuous function such that $f(3 x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to.

If $f(x)$ satisfies $f(7 -x) = f(7 + x)\ \forall \,x\, \in \,R$ such that $f(x)$ has exactly $5$ real roots which are all distinct such that sum of the real roots is $S$ then $S/7$ is equal to

Define a function $f(x)=\frac{16 x^2-96 x+153}{x-3}$ for all real $x \neq 3$. The least positive value of $f(x)$ is

Statement $1$ : If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^P$. Statement $2$ : The total number of selections of $p$ different objects out of $q$ objects is ${}^q{C_p}$.

Start a Free Trial Now

Statement $1$ : If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^P$.
Statement $2$ : The total number of selections of $p$ different objects out of $q$ objects is ${}^q{C_p}$.