# IPMAT Question Paper 2020 | IPM Indore Quants

###### IPMAT Sample Paper | IPMAT Question Paper | Question 4

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 $$left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b & b^{2} & b^{3}-1 \\ c & c^{2} & c^{3}-1\end{array}\right| \\$ = 0 where a, b and c are distinct real numbers. If a= 3, then the value of a b c is ## 🎉 Fabulous 3,500 off on XAT '22 crash course Valid until 1st Dec #### 2IIM : Best Online CAT Coaching. ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2021Online Batches Available Now! ### Explanatory Answer $\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b & b^{2} & b^{3}-1 \\ c & c^{2} & c^{3}-1\end{array}\right| \\$ = 0 Performing the following row operations R3 = R3 - R2 R2 = R2 - R1 we get, $\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b-a & b^{2}-a^{2} & b^{3}-a^{3} \\ c-a & c^{2}-b^{2} & c^{3}-c^{3}\end{array}\right| \\$ = 0 $\left|\begin{array}{ccc}a & a^{2} & a^{3}-1 \\b-a & [b-a][b+a] & [b-a][b^{2}+ab+a^{2}] \\ c-a & [c+a][c-a] & [c-a][c^{2}+cb+b^{2}]\end{array}\right| \\$ = 0$b-a)(c-b)$$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 R2 = R2 - R3$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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