IPMAT Question Paper 2020 | IPM Indore Quants

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 23 : A 2 × 2 matrix is filled with four distinct integers randomly chosen from the set {1,2,3,4,5,6} Then the probability that the matrix generated in such a way is singular is

1. \\frac{2}{45})
2. \\frac{1}{45})
3. \\frac{4}{15})
4. \\frac{1}{15})

Explanatory Answer

A matrix is singular if and only if the determinant of the matrix is 0.
In a 2 × 2 matrix,
\\left|\begin{array}{cc}a & c \\d & b\end{array}\right| \\) = ab - cd = 0
Or ab = cd
If one of a,b,c or d is 5, then ab ≠ cd, because in the set {1,2,3,4,5,6} there is only one multiple of 5.
So, we just have the set {1,2,3,4,6} to deal with and out of these we'll have select 4 integers.
We can't pick either 3 or 6. We'll have to pick them both.
That is 3 and 6 have to stay on either side of the equation ab = cd

So, the equation now becomes, 3b = 6d
Clearly, b > d
{b, d} ⊂ {1, 2, 4}
If b = 4, d = 2
If b = 2, d = 1

So, only two such combinations exist.
{a, b, c, d} = {1, 2, 3, 6} or {a, b, c, d} = {4, 2, 3, 6}

We can select four numbers from the six in ⁶C₄ = 15 ways
After selecting the 4 integers, we can arrange them in order in 4! ways.
So, the total number of matrices that can be constructed = 15 × 4!

Once we select {a, b, c, d} = {1, 2, 3, 6} or {a, b, c, d} = {4, 2, 3, 6},
To construct a singular matrix, for example, 2 & 3 should be on the same diagonal. Similarly, 4 & 3 too in case of {a, b, c, d} = {4, 2, 3, 6}.
Only one-third of the arrangments yeild a singular matrix and the others dont.
But the probabilty of getting a singular matrix = \\frac{2 × 4! ÷ 3}{15 × 4!}) = \\frac{2}{45})

The question is " A 2 × 2 matrix is filled with four distinct integers randomly chosen from the set {1,2,3,4,5,6} Then the probability that the matrix generated in such a way is singular is "

Hence, the answer is \\frac{2}{45})

Choicve A is the correct answer.

Verbal Ability for CAT | CAT Reading Comprehension

Verbal Ability for CAT| CAT Para Jumble

Verbal Ability for CAT | CAT Sentence Elimination

Verbal Ability for CAT | CAT Paragraph Completion

Verbal Ability for CAT | CAT Critical Reasoning

Verbal Ability for CAT | CAT Word Usage

Verbal Ability for CAT| CAT Para Summary

Verbal Ability for CAT | CAT Text Completion

Verbal Ability for CAT | CAT Sentence Correction