IPMAT 2020 Question Paper IPM Indore Quantitative Ability. Solve questions from IPMAT 2020 Question Paper from IPM Indore and check the solutions to get adequate practice. The best way to ace IPMAT is by solving IPMAT Question Paper. To solve other IPMAT Sample papers, go here: **IPM Sample Paper**

Question 4 : Suppose \\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b & b^{2} & b^{3}-1 \\ c & c^{2} & c^{3}-1\end{array}\right| \\) = 0 where a, b and c are distinct real numbers. If a= 3, then the value of a b c is

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\\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b & b^{2} & b^{3}-1 \\ c & c^{2} & c^{3}-1\end{array}\right| \\) = 0

Performing the following row operations

R_{3} = R_{3} - R_{2}

R_{2} = R_{2} - R_{1}

we get,

\\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b-a & b^{2}-a^{2} & b^{3}-a^{3} \\ c-a & c^{2}-b^{2} & c^{3}-c^{3}\end{array}\right| \\) = 0

\\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b-a & [b-a][b+a] & [b-a][b^{2}+ab+a^{2}] \\ c-a & [c+a][c-a] & [c-a][c^{2}+cb+b^{2}]\end{array}\right| \\) = 0

(b-a)(c-b)\\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\1 & b+a & b^{2}+ab+a^{2} \\ 1 & c+a & c^{2}+cb+b^{2}\end{array}\right| \\) = 0

Perform R_{2} = R_{2} - R_{3}

(b-a)(c-b)\\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\0 & a-c & ab+a^{2}-c^{2}-cb \\ 1 & c+a & c^{2}+cb+b^{2}\end{array}\right| \\) = 0

(b-a)(c-b)\\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\0 & a-c & [a-c][a+b+c] \\ 1 & c+a & c^{2}+cb+b^{2}\end{array}\right| \\) = 0

(b-a)(c-b)(a-c)\\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\0 & 1 & a+b+c \\ 1 & c+a & c^{2}+cb+b^{2}\end{array}\right| \\) = 0

Solving which we get (b-a)(c-b)(a-c)(abc - 1) = 0

Since a, b, c are distinct, only (abc-1) = 0

abc - 1 = 0

abc = 1

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