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### These CAT Quadratic Equations questions/problems with solutions provide you vital practice for the topic. The purpose of these posts is very simple: to help you learn through practice.

Question 1: If the equation x3 – ax2 + bx – a = 0 has three real roots then the following is true
(a) a = 1
(b) a ≠ 1
(c) b = 1
(d) b ≠ l

The given equation x3 – ax2 + bx – a = 0 can be rewritten as:

x (x2 + b) – a(x2 + 1) = 0

In case b = 1, then the equation becomes

(x – a)( x2 + 1) = 0

Now here x2 + 1 = 0 will give imaginary roots.

Hence, if the given equation has three real roots, then b ≠ 1.

Question 2: m is the smallest positive integer such that for any integer n ≥ m, the quantity n3 -7n2 +11n – 5 is positive. What is the value of m ?
(a) 4
(b) 5
(c) 8
(d) None of these

Let y = n3 – 7n2 + 11n – 5

Now n = 1 is a root of the above equation.

Hence, it can be written as

n3 – 7n2 + 11n – 5 = (n – 1)(n2 – 6n + 5) = (n – 1)2(n – 5)

Now, (n – 1)2 is always positive. So the whole expression will be positive if n – 5 > 0 i.e. n > 5.

Since n is an integer so its least value will be n = 6.

Now n ≥ m and m is also an integer.

Hence, the least value of m is 6.

Question 3: All the page numbers from a book are added, beginning at page 1. However, one-page number was mistakenly added twice. The sum obtained was 1000. Which page number was added twice?
(a) 44
(b) 45
(c) 10
(d) 12

Let the total pages be ‘n’. So we have  {n(n+1)}/2 = 100

Since one page was added twice, so 1000 is not the actual sum but it is an increased sum.

We have n(n + 1) = 2000.

Now by hit and trial we can say that the value of n = 44 i.e. initially there were 44 pages and their sum was (44 x 45)/2 = 990

Since the given sum is 1000, so we can say that the page number 10 was added twice.

Question 4: Raman and Manoj attempted to solve a quadratic equation. Raman made a mistake in writing down the constant term. He ended up with the roots (4, 3). Manoj made a mistake in writing down the coefficient of x. He got the roots as (3, 2). What will be the exact roots of the original quadratic equation?
(a) (6, 1)
(b) (–3, –4)
(c) (4, 3)
(d) (–4, –3)

Since Raman made a mistake in constant term, so his product of roots will be wrong but his sum of roots will be correct.

Hence, the sum of roots is 4 + 3 = 7.

Manoj made a mistake in coefficient of ‘x’, so his sum of roots will be wrong but product of roots will be correct.

Hence, the product of roots is 3 × 2 = 6.

Hence, the required equation is x2 – 7x + 6 = 0.

The roots of this equations are 6 and 1.

Question 5: Let p and q be the roots of the quadratic equation x2– (α – 2)x– α1 = 0. What is the minimum possible value of p2 + q2?
(a) 0
(b) 3
(c) 4
(d) 5