
If the values a + b + c = 8, ab + bc + ca = 17 and abc = 10 are given, then find the value of $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$?
(a) 94
(b) 84
(c) 68
(d) 88
Answer
483.6k+ views
Hint: We start solving the problem by assuming the value of $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$ as ‘d’. We now multiply each term of $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$ present in brackets. After multiplication we substitute values given in the problem to get the required result.
Complete step by step answer:
We have given values of a + b + c, ab + bc + ca and abc are 8, 17 and 10. We need to find the value of $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$.
Let us first multiply all the given terms in $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$. Let us assume this value is ‘d’.
So, $d=\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$.
$\Rightarrow d=\left( 2.2+2.a+2.b+a.b \right).\left( 2+c \right)$.
$\Rightarrow d=\left( 4+2a+2b+ab \right).\left( 2+c \right)$.
$\Rightarrow d=\left( 4.2+4.c+2a.2+2a.c+2b.2+2b.c+ab.2+ab.c \right)$.
$\Rightarrow d=8+4c+4a+2ac+4b+2bc+2ab+abc$.
$\Rightarrow d=8+4a+4b+4c+2ab+2bc+2ca+abc$.
$\Rightarrow d=8+4.\left( a+b+c \right)+2.\left( ab+bc+ca \right)+abc$ ---(1).
We substitute the values a + b + c = 8, ab + bc + ca = 17 and abc = 10 in the equation (1).
d = 8 + 4(8) + 2(17) + 10.
d = 8 + 32 + 34 + 10.
d = 84.
∴ The value of $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)=84$.
So, the correct answer is “Option B”.
Note: Alternatively, we can calculate the values of ‘a’, ‘b’ and ‘c’ to substitute in the given equation $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$. We should not make any calculation mistakes while solving this problem to get the accurate result. Similarly, we can expect to find the polynomial having the roots ‘a’, ‘b’ and ‘c’.
Complete step by step answer:
We have given values of a + b + c, ab + bc + ca and abc are 8, 17 and 10. We need to find the value of $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$.
Let us first multiply all the given terms in $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$. Let us assume this value is ‘d’.
So, $d=\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$.
$\Rightarrow d=\left( 2.2+2.a+2.b+a.b \right).\left( 2+c \right)$.
$\Rightarrow d=\left( 4+2a+2b+ab \right).\left( 2+c \right)$.
$\Rightarrow d=\left( 4.2+4.c+2a.2+2a.c+2b.2+2b.c+ab.2+ab.c \right)$.
$\Rightarrow d=8+4c+4a+2ac+4b+2bc+2ab+abc$.
$\Rightarrow d=8+4a+4b+4c+2ab+2bc+2ca+abc$.
$\Rightarrow d=8+4.\left( a+b+c \right)+2.\left( ab+bc+ca \right)+abc$ ---(1).
We substitute the values a + b + c = 8, ab + bc + ca = 17 and abc = 10 in the equation (1).
d = 8 + 4(8) + 2(17) + 10.
d = 8 + 32 + 34 + 10.
d = 84.
∴ The value of $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)=84$.
So, the correct answer is “Option B”.
Note: Alternatively, we can calculate the values of ‘a’, ‘b’ and ‘c’ to substitute in the given equation $\left( 2+a \right).\left( 2+b \right).\left( 2+c \right)$. We should not make any calculation mistakes while solving this problem to get the accurate result. Similarly, we can expect to find the polynomial having the roots ‘a’, ‘b’ and ‘c’.
Recently Updated Pages
Master Class 9 General Knowledge: Engaging Questions & Answers for Success

Master Class 9 English: Engaging Questions & Answers for Success

Master Class 9 Science: Engaging Questions & Answers for Success

Master Class 9 Social Science: Engaging Questions & Answers for Success

Master Class 9 Maths: Engaging Questions & Answers for Success

Class 9 Question and Answer - Your Ultimate Solutions Guide

Trending doubts
Name the states which share their boundary with Indias class 9 social science CBSE

Which of the following is the most important sentence class 9 english CBSE

On an outline map of India mark the Karakoram range class 9 social science CBSE

Why did India adopt the multiparty system class 9 social science CBSE

What occurs in the minerals of the apatite family APhosphorus class 9 chemistry CBSE

Who is eligible for RTE class 9 social science CBSE
