Question

In the expanded form of ${{\left( -2x+5y-3z \right)}^{2}}$, coefficient of $xy$ is given by
A. -10
B. 10
C. -20
D. 20

Answer 1

Hint:Use the formula ${{\left( a+b+c \right)}^{2}}=\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac \right)$ to expand ${{\left( -2x+5y-3z \right)}^{2}}$. After expanding ${{\left( -2x+5y-3z \right)}^{2}}$, we will get the coefficient of $xy$, the coefficient is defined as the numbers that are multiplied with variables.

Complete step by step answer:
Before proceeding with the question, we must know what is the coefficient of a variable. The coefficient of a variable is defined as the number to which the variable has been multiplied. We must know the formula ${{\left( a+b+c \right)}^{2}}=\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac \right)$ for expanding ${{\left( -2x+5y-3z \right)}^{2}}$.
In this question, we have to find the coefficient of $xy$ after expansion of ${{\left( -2x+5y-3z \right)}^{2}}$.
We will first expand ${{\left( -2x+5y-3z \right)}^{2}}$by using the formula ${{\left( a+b+c \right)}^{2}}=\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac \right)$ where $a=-2x,b=5y,c=-3z$.
After expanding ${{\left( -2x+5y-3z \right)}^{2}}$we get,
$\Rightarrow {{\left( -2x+5y-3z \right)}^{2}}=\left( {{\left( -2x \right)}^{2}}+{{\left( 5y \right)}^{2}}+{{\left( -3z \right)}^{2}}+2\times \left( -2x \right)\times \left( 5y \right)+2\left( 5y \right)\times \left( -3z \right)+2\times \left( -2x \right)\times \left( -3z \right) \right)$
Multiplying the terms and simplifying, we will get,
$\Rightarrow {{\left( -2x+5y-3z \right)}^{2}}=\left( 4{{x}^{2}}+25{{y}^{2}}+9{{z}^{2}}-20xy-30yz+12xz \right)$
$\therefore {{\left( -2x+5y-3z \right)}^{2}}=\left( 4{{x}^{2}}+25{{y}^{2}}+9{{z}^{2}}-20xy-30yz+12xz \right)$
After expanding we can see that the number $-20$ has been multiplied with $xy$.
Therefore, we get the coefficient of $xy$ as $-20$ after the expansion of ${{\left( -2x+5y-3z \right)}^{2}}$.
Hence, the correct answer is option C.

Note: We must be very careful with the signs while expanding ${{\left( -2x+5y-3z \right)}^{2}}$. We must know the formula ${{\left( a+b+c \right)}^{2}}=\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac \right)$ for the expansion of ${{\left( -2x+5y-3z \right)}^{2}}$. The mistake that can be committed is while applying the formula and adding the terms. Also, if we don’t remember the formula, we can write it as $\left( a+b+c \right)\left( a+b+c \right)$and then perform the multiplication. There is one more method to expand the given expression. We can consider $\left( a+b \right)$ as one term and $c$ as second term and then apply the formula for${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ to it. Even though we have many options to expand the given expression, the best way to solve the question fast is by using the ${{\left( a+b \right)}^{2}}$ formula.