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Test Level 2: Quadratic Equations & Linear Equations - 1 - CAT MCQ


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10 Questions MCQ Test - Test Level 2: Quadratic Equations & Linear Equations - 1

Test Level 2: Quadratic Equations & Linear Equations - 1 for CAT 2024 is part of CAT preparation. The Test Level 2: Quadratic Equations & Linear Equations - 1 questions and answers have been prepared according to the CAT exam syllabus.The Test Level 2: Quadratic Equations & Linear Equations - 1 MCQs are made for CAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test Level 2: Quadratic Equations & Linear Equations - 1 below.
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Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 1

How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 1

ab = a + b + 20
ab - a = b - 1 + 21
a(b - 1) - (b - 1) = 21
(a - 1)(b - 1) = 21
= 1 × 21, 3 × 7, 7 × 3, 21 × 1
∴ a = 2, b = 22
a = 4, b = 8
a = 8, b = 4
a = 22, b = 2
The total number of pairs is 4.

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 2

In a broken calculator, the keys '+' and '×' have their functions switched. For how many ordered pairs (a, b) of integers will it correctly calculate 'a + b' using the labeled '+' key?

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 2

We need a + b = a × b.
⇒ a(1 - b) = -b

Since a is an integer,  is also an integer. Therefore, b must be 0 or 2, and a also must be 0 or 2, i.e. the ordered pairs are (0, 0) and (2, 2).
Hence, there are two such ordered pairs.

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Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 3

Find the value of x + y, if both x and y are real and x2 + y2 + 2x - 10y = -26.  

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 3

x2 + y2 + 2x - 10y + 26 = 0
⇒ (x + 1)2 + (y - 5)2 = 0
Since x and y are both real, we get
(x + 1)2 = 0
⇒ x = -1 and (y - 5)2 = 0
⇒ y = 5
∴ x + y = 4

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 4

a, b and c are three numbers, such that ab = 174375 and ac = 173600. b is greater than c by 1. Find the value of b.

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 4

a, b and c are three numbers.
ab = 174375 ... (1)
ac = 173600 ... (2)
b = c + 1 ... (3)
Dividing (1) ÷ (2), we get:

By soling (3) and (4), we get:
b = 225

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 5

The roots of x3 − ax2 + bx − c = 0 are p, q and r, while the roots of x3 + dx2 + ex − 80 = 0 are p + 4, q + 4 and r + 4. What is the value of 16a + 4b + c?

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 5

p + q + r = a
pq + qr + pr = b
pqr = c
Also, (p + 4) × (q + 4) × (r + 4) = 80 ⇒ pqr + 4(pq + pr + qr) + 16(p + q + r) + 64 = 80
16a + 4b + c = 16

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 6

What is the value of 'ab' in the following polynomial identity?
x6 + 1 = (x2 + 1)(x2 + ax + 1)(x2 + bx + 1)

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 6

If the product (x2 + 1)(x2 + ax + 1)(x2 + bx + 1) is expanded, the coefficient of x2 obtained is ab + 3 (one can see this without expanding the product completely). Since the coefficient of x2 in x6 + 1 is 0, we get ab + 3 = 0. Hence, ab = -3.

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 7

When the sum of two natural numbers is multiplied by each number separately, the products obtained are 2418 and 3666. What is the difference between the two numbers?  

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 7

Suppose the numbers are x and y, then: 
x(x + y) = 3666 … (1)
y(x + y) = 2418 … (2)
Adding 1 and 2, we get
(x + y)2 = 6084
Therefore, x + y = 78 … (3)
Subtracting (2) from (1), we get
x2 - y2 = 1248
Or, (x + y)(x - y) = 1248
Or, 78(x - y) = 1248
Or, x - y = 16
Hence, option 1 is correct.

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 8

The sum of two numbers is 80. If the larger number exceeds four times the smaller number by 5, then find the larger number.  

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 8

Let the two numbers be x and y, such that x > y.
∴ x + y = 80 … (i)
And, x = 4y + 5
Putting x = 4y + 5 in (i), we get
4y + 5 + y = 80
5y = 80 – 5
y = 75/5 = 15
Putting y = 15 in (i), we get
x = 80 – 15
x = 65
∴ The two numbers are 65 and 15.

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 9

Solve for x: 2687x2 + 4248x + 1561 = 0

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 9

The given equation can be written as: 
2687x2 + 2687x + 1561x + 1561 = 0
2687x(x + 1) + 1561(x + 1) = 0
2687x + 1561)(x + 1) = 0

Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 10

One hundred diamonds, each worth Rs. 25,000, are to be divided into two groups, such that one-fourth of the number of diamonds in the first group would be 20 more than one-sixth of the number of diamonds in the second group. What is the ratio of the number of diamonds in the first group to that in the second group?  

Detailed Solution for Test Level 2: Quadratic Equations & Linear Equations - 1 - Question 10

Let the first group have x diamonds, and the second group have 100 - x diamonds.

x = 88
So, the first group has 88 diamonds and the second group has 12 diamonds.
Required ratio = 88/12 =22/3

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