This EduRev document offers 10 Multiple Choice Questions (MCQs) from the topic Quadratic Equations & Linear Equations (Level - 2). These questions are of Level - 2 difficulty and will assist you in the preparation of CAT & other MBA exams. You can practice/attempt these CAT Multiple Choice Questions (MCQs) and check the explanations for a better understanding of the topic.
Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself:How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?
Explanation
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.
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself: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?
Explanation
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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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself:Find the value of x + y, if both x and y are real and x2 + y2 + 2x - 10y = -26.
Explanation
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
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself: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.
Explanation
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
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself: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?
Explanation
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
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself:What is the value of 'ab' in the following polynomial identity?
x6 + 1 = (x2 + 1)(x2 + ax + 1)(x2 + bx + 1)
Explanation
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.
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself: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?
Explanation
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.
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself:The sum of two numbers is 80. If the larger number exceeds four times the smaller number by 5, then find the larger number.
Explanation
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.
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself:Solve for x: 2687x2 + 4248x + 1561 = 0
Explanation
The given equation can be written as:
2687x2 + 2687x + 1561x + 1561 = 0
2687x(x + 1) + 1561(x + 1) = 0
2687x + 1561)(x + 1) = 0
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Question for Practice Questions Level 2: Quadratic Equations & Linear Equations - 1
Try yourself: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?
Explanation
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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