Question

If $a$ and $b$ are two numbers such that $a^2+b^2=7$ and $a^3+b^3=10$, then which is the wrong statement.

• A. greatest value of $\left|a+b\right|$ is $5$
• B. greatest value of $a+b$ is $4$
• C. greatest value of $a+b$ is $1$
• D. least value of $\left|a+b\right|$ is $1$

Answer 1

The correct option is C

greatest value of $a+b$ is $1$

Explanation for the correct option:

Step 1: Framing the equations:

Given, $a^2+b^2=7$ and $a^3+b^3=10$

We know,

$\begin{array}{cccc}& {\left(a+b\right)}^3=a^3+b^3+3ab(a+b)& & \\ \Rightarrow & {\left(a+b\right)}^3=10+3ab(a+b)& \left[\because a^3+b^3=10\right]& \\ \Rightarrow & 10={\left(a+b\right)}^3-3ab(a+b)& & \\ \Rightarrow & 10=\left(a+b\right)\left[{\left(a+b\right)}^2-3ab\right]& & \\ \Rightarrow & 10=\left(a+b\right)\left[a^2+b^2+2ab-3ab\right]& \left[\because {\left(a+b\right)}^2=a^2+b^2+2ab\right]& \\ \Rightarrow & 10=\left(a+b\right)\left[7-ab\right]& \left[\because a^2+b^2=7\right]& ...\left(\mathrm{i}\right)\end{array}$

We also know

$\begin{array}{cc}& {\left(a+b\right)}^2=a^2+b^2+2ab\\ \Rightarrow & 7={\left(a+b\right)}^2-2ab\\ \Rightarrow & 2ab={\left(a+b\right)}^2-7\\ \Rightarrow & ab=\frac{{\left(a+b\right)}^2-7}{2}\end{array}$

Putting $ab$ in $\left(\mathrm{i}\right)$

$\begin{array}{ccc}& \left(a+b\right)\left[7-\frac{{\left(a+b\right)}^2-7}{2}\right]=10& \\ \Rightarrow & \left(a+b\right)\left[21-{\left(a+b\right)}^2\right]=20& \\ \Rightarrow & 21\left(a+b\right)-{\left(a+b\right)}^3=20& \\ \Rightarrow & {\left(a+b\right)}^3-21\left(a+b\right)+20=0& ...\left(\mathrm{ii}\right)\end{array}$

Step 2: Finding the greatest and least value

Let $a+b=x$

Then the above equation becomes

$x^3-21x+20=0$

Substituting $x=1$ in this equation,

$$\begin{gathered} \Rightarrow 1^3-21\left(1\right)+20=0 \\ \Rightarrow 0=0 \\ \Rightarrow \mathrm{LHS}=\mathrm{RHS} \end{gathered}$$

So $x-1$ is a factor of $x^3-21x+20=0$

By long division of polynomials,

$$\begin{gathered} \Rightarrow (x-1)(x^2+x-20)=0 \\ \Rightarrow (x-1)(x+5)(x-4)=0 \end{gathered}$$

Thus,

$$\begin{gathered} x-1=0\Rightarrow x=1 \\ x+5=0\Rightarrow x=-5 \\ x-4=0\Rightarrow x=4 \end{gathered}$$

But, $a+b=x$

So,

Least value of $a+b=-5$

Greatest value of $a+b=4$

Greatest value of $\left|a+b\right|=5$

Least value of $\left|a+b\right|=1$

Hence, options C is the wrong statement.