# Quant Boosters - Rajesh Balasubramanian - Set 5

• Q26) If a, b, c are integers such that – 50 < a, b, c < 50 and a + b + c = 30, what is the maximum possible value of abc?

• abc will be maximum when it is positive. So, a, b, c can all be positive or two of the three can be negative and one positive.

When all are positive, max product is when the numbers are 10, 10 and 10.

When two are negative and one positive, the best–case scenario would be when two negative numbers are as low as possible (magnitudes as high as possible) so that the product can be high. Now, in order for the product to be maximum, the positive number should be as high as possible. So, let the positive number be 49. Then the sum of the two negative numbers should be –19. The best– case scenario would be when numbers are 49, –9, –10.

Product would be 4410. Answer choice (c)

• Q27) a, b, c are distinct natural numbers less than 25. What is the maximum possible value of |a – b| + |b – c| – |c – a|?

• For any two points M, N on the number line representing numbers m, n the distance MN = | m - n|.
So, for three points, P, Q and R on the number line |p – q|, |q – r|, |r – p| are distances between three pairs of points on the number line.

In this case, we are trying to find maximum value of |a – b| + |b – c| – |c – a|. If b lies between a and c, the above value would be zero. So, b should not be between a and c.
The best case scenario would be if a, c were very close to each other and far from b. Let us try b = 24, a = 1, c = 2.
In this case |a – b| + |b – c| – |c – a| = 23 + 22 – 1 = 44. This is the maximum possible value.

We could also have b = 1, a = 24, c = 23,
|a – b| + |b – c| – |c – a| = 23 + 22 – 1 = 44.

• Q28) Consider integers p, q such that – 3 < p < 4, – 8 < q < 7, what is the maximum possible value of p^2 + pq + q^2?

• Trial and error is the best approach for this question. We just need to be scientific about this.
p^2 and q^2 are both positive and depend on |p| and |q|. If p, q are large negative or large positive numbers, p^2 and q^2 will be high.
pq will be positive if p, q have the same sign, and negative if they have opposite signs.

So, for p^2 + pq + q^2 to be maximum, best scenarios would be if both p & q are positive or both are negative.

Let us try two possibilities.
p = – 2, q = – 7: p^2 + pq + q^2 = 4 + 14 + 49 = 67
p = 3, q = 6: p^2 + pq + q^2 = 9 + 15 + 36 = 60
Whenever we have an expression with multiple terms, there are two key points to note.

The equation will be most sensitive to the highest power.
The equation will be more sensitive to the term with the greater value.

In the case, q.

In this question, we have a trade–off between higher value for p^2 and q^2. For q^2, the choice is between 6^2 and (–7)^2. This impact will overshadow the choice for p (where we are choosing between –2 and 3).

So, the maximum value for the expression would be 67.

• Q29) If a, b, c are distinct positive integers, what is the highest value a * b * c can take if a + b + c = 31?

• The sum of three numbers is given; the product will be maximum if the numbers are equal.

(a + b + c)/3 > (abc)^(1/3)

So, if a + b + c is defined, abc will be maximum when all three terms are equal. In this instance, however, with a, b, c being distinct integers, they cannot all be equal.

So, we need to look at a, b, c to be as close to each other as possible.
a = 10, b =10, c = 11 is one possibility, but a, b, c have to be distinct. So, this can be ruled out.
The close options are,
a, b, c : 9, 10, 12; product = 1080
a, b, c : 8, 11, 12: product = 1056
Maximum product = 1080

• Q30) a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

• The roots are a and b:
a + b = p and ab = 12

(a + b)^2 = p^2
(a - b)^2 = (a + b)^2 - 4ab
=> (a - b)^2 = p^2 - 12 * 4 = p^2 - 48

If |a - b| ≥ 12 { Difference between the roots is at least 12}
then, (a - b)^2 ≥ 144

p^2 - 48 ≥ 144
p^2 ≥ 192
P ≥ 8√3 or P ≤ -8√3

102

44

59

45

1

34

71

47