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

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.

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.

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

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

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

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.

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.

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.

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

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