1 Mark Question

Question 11 Mark

Write the value of $25^3-75^3+50^3$

Answer

The given expression is
$25^3-75^3+50^3$
Let $a =25, b=-75$ and $c =50$. Then the given expression becomes $25^3-75^3+50^3= a ^3+ b ^3+ c ^3$
Note that
$a+b+c=25+(-75)+50$
$=25-75+50$
$=0$
Recall the formula
$a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
when $a+b+c=0$, this becomes
$a^3+b^3+c^3-3 a b c=0 \cdot\left(a^2+b^2+c^2-a b-b c-c a\right)$
$=0$
$a^3+b^3+c^3=3 a b c$
So, we have the new formula
$a^3+b^3+c^3=3 a b c \text {, when } a+b+c=0$
Using the above formula, the value of the given expression is
$a^3+b^3+c^3=3 a b c$
$25^3-75^3+50^3=3 \cdot(25) \cdot(-75) \cdot(50)$
$25^3-75^3+50^3=-281250$

View full question & answer→

Question 21 Mark

Write the value of $\Big(\frac{1}{2}\Big)^3+\Big(\frac{1}{3}\Big)^3-\Big(\frac{5}{6}\Big)^3.$

Answer

The given expression is $\left(\frac{1}{2}\right)^3+\left(\frac{1}{3}\right)^3-\left(\frac{5}{6}\right)^3$
Let $a =\frac{1}{2}, b=\frac{1}{3}$ and $c =-\frac{5}{6}$.
Then the given expression becomes $\left(\frac{1}{2}\right)^3+\left(\frac{1}{3}\right)^3-\left(\frac{5}{6}\right)^3= a ^3+ b ^3+ c ^3$
Note that $a+b+c=\frac{1}{2}+\frac{1}{3}+\left(\frac{5}{6}\right)=\frac{1}{2}+\frac{1}{3}-\frac{5}{6}=0$
Recall the formula $a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2\right.$ $\left.+b^2+c^2-a b-b c-c a\right)$
when $a+b+c=0$,
this becomes $a^3+b^3+c^3-3 a b c=0 .\left(a^2+b^2+c^2-a b-b c-c a\right)=0 a^3$
$+b^3+c^3=3 a b c$
So, we have the new formula $a^3+b^3+c^3=3 a b c$,
when $a+b+c=0$ Using the above formula,
the value of the given expression is $a^3+b^3+c^3=3 a b c\left(\frac{1}{2}\right)^3+\left(\frac{1}{3}\right)^3-\left(\frac{5}{6}\right)^3=3 \cdot\left(\frac{1}{2}\right) \cdot\left(\frac{1}{3}\right) \cdot\left(-\frac{5}{6}\right)$
$\left(\frac{1}{2}\right)^3+\left(\frac{1}{3}\right)^3-\left(\frac{5}{6}\right)^3=\left(-\frac{5}{12}\right)$

View full question & answer→

Question 31 Mark

Write the value of $30^3+20^3-50^3$.

Answer

The given expression is
$30^3+20^3-50^3$
Let $a=30, b=20$ and $c=-50$. Then the given expression becomes $30^3+20^3-50^3=a^3+b^3+c^3$
Note that
$a+b+c=30+20+(-50)$
$=30+20-50$
$=0$
Recall the formula
$a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
when $a+b+c=0$, this becomes
$a^3+b^3+c^3-3 a b c=0 \cdot\left(a^2+b^2+c^2-a b-b c-c a\right)$
$=0$
$a^3+b^3+c^3=3 a b c$
So, we have the new formula
$a^3+b^3+c^3=3 a b c \text {, when } a+b+c=0$
Using the above formula, the value of the given expression is
$a^3+b^3+c^3=3 a b c$
$30^3+20^3-50^3=3 \cdot(30) \cdot(20) \cdot(-50)$
$30^3+20^3-50^3=-90000$

View full question & answer→

Question 41 Mark

Write the value of $48^3-30^3-18^3$.

Answer

The given expression is
$48^3-30^3-18^3$
Let $a =48, b=-30$ and $c =-18$. Then the given expression becomes $48^3-30^3-18^3=a^3+b^3+c^3$
Note that
$a+b+c=48+(-30)+(-18)$
$=48-30-18$
$=0$
Recall the formula
$a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
when $a+b+c=0$, this becomes
$a^3+b^3+c^3-3 a b c=0 \cdot\left(a^2+b^2+c^2-a b-b c-c a\right)$
$=0$
$a^3+b^3+c^3=3 a b c$
So, we have the new formula
$a^3+b^3+c^3=3 a b c \text {, when } a+b+c=0$
Using the above formula, the value of the given expression is
$a^3+b^3+c^3=3 a b c$
$48^3-30^3-18^3=3 \cdot(48) \cdot(-30) \cdot(-18)$
$48^3-30^3-18^3=77760$

View full question & answer→

Maths 1 Mark Question

Take a timed test