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

Question

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

Vedantu Content Team · Accepted Answer

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’.

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

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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

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NCERT Book Solutions

Reference Book Solutions

Question Papers

Exams

Quick Links

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