Tag: decimal representation of rational numbers

Questions Related to decimal representation of rational numbers

If a and b are any two such real numbers that ab $ = 0 $ , then

decimal representation of rational numbers rational and irrational numbers maths

  1. $a = 0, b \leq 0$

  2. $b = 0, a \leq 0$

  3. a = 0 or b = 0 or both

  4. $a = b$ and $b = 0$

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Correct Option: C
Explanation:

if both number are real the either a or b or both should be zero.
then only ab will be 0.
if any real number is multiplied by 0 then result will be zero.
So, answer is
C a = 0 or b = 0 or both

If $f(x)-2f(1-x) = x^2+2$, then what is $f(x)$?

decimal representation of rational numbers rational and irrational numbers maths

  1. $f(x)=-x^2+\dfrac{4}{3}x-\dfrac{3}{8}$

  2. $f(x)=−x^2+\dfrac{4}{3}x−\dfrac{8}{3}$

  3. $f(x)=−x^2+\dfrac{8}{3}x−\dfrac{4}{3}$

  4. $f(x)=−x^2+\dfrac{3}{8}x−\dfrac{3}{4}$

Show answer
Correct Option: B
Explanation:
$f\left(x\right)-2f\left(1-x\right)={x}^{2}+2$      .......$(1)$

Setting $x=1-x$ then we get

$f\left(1-x\right)-2f\left(1-1+x\right)={\left(1-x\right)}^{2}+2$ 

$f\left(1-x\right)-2f\left(x\right)={x}^{2}-2x+3$ 

$2f\left(1-x\right)-4f\left(x\right)=2{x}^{2}-4x+6$    .......$(2)$

Adding $(1)$ and $(2)$ we get

$-3f\left(x\right)=3{x}^{2}-4x+8$ 

$\therefore f\left(x\right)=-{x}^{2}+\dfrac{4}{3}x-\dfrac{8}{3}$