Mettl Equation Quiz 2

First equation gives: (1 / y) = 2 - x or y = 1 / (2 - x)
Second equation is: y(2x - 3) = –2 or (2x - 3) / (2 - x) = -2
∴ 2x – 3 = – 2(2 - x). This gives 1 = 0.
This is impossible. So there is no solution at all.

(1 / 2x) – (1 / y) = –1 .....(i)
(1 / x) + (1 / 2y) = 8 .....(ii)
Dividing equation (i) by 2
(1 / 4x) – (1 / 2y) = (–1 / 2)
1 / x + 1 / 2y = 8
1 / x {(1 / 4) + 1} = 8 – (1 / 2)
(1 / x) (5 / 4) = 15 / 2 ⇒ x = 1 / 6
Putting x = 1 / 6 in equation (ii)
6 + (1 /2y) = 8 ⇒ 1 /2y = 2 ⇒ y = 1 / 4.

\frac{40}{(u + v)} + \frac{2}{(u - v)} = 5 .....(i)
\frac{25}{(u + v)} - \frac{3}{(u - v)} = 1 .....(ii)
Multiplying equation (i) by 5 and equ. (ii) by 8, we get,
\frac{200}{(u + v)} + \frac{10}{(u - v)} = 25 .....(a)
\frac{200}{(u + v)} - \frac{24}{(u - v)} = 8 ......(b)
Subtracting eq. (b) from (a)
\frac{34}{(u - v)} = 25 - 8
u - v = (34 / 17) = 2 → [A]
Putting the value of u-v in eq. (i) we get
u + v = 10 → [B]
Solving eq. [A] and [B]
u - v = 2; u + v = 10
2u = 12 ⇒ u = 6. So, v = 4.

Let the number be xy = 10x + y
x + y = 8 …….(i)
(10x + y) - (10y + x) = 18
9x – 9y = 18
x – y = 2 …..(ii); Solving equation (i) & (ii)
x = 5, y = 3.
So, the number is 53.

Let the parts be A, B and C.
Acc. To question-
(A / 2) = (B / 3) = (C / 4); C = 2A, B = (3 / 2)A.
A + B + C = 243
A + (3 / 2) A + 2A = 243
2A + 3A + 4A = 486.
9A = 486; A = 54
B = (3 / 2) × 54 = 81
C = 2A = 54 × 2 = 108.

x = - p% of (x - 20).
x = - q% of (x + 10).
p% of (x - 20) = q% of (x + 10)
px - 20p = qx + 10q
x (p - q) = 10q + 20 p = 10 (q + 2p)
(p - q)x = 10q + 20p = 10(q + 2p)
x = \frac{10(q + 2p)}{p - q}

Let a two-digit number be (10x + y) and reversing number be (10y + x)
∴Required sum = 10x + y + 10y + x = 11x + 11y = 11(x + y)
Hence, it's divisible by 11

x + y + z = 13
x² + y² + z² = 69
(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)
⇒(13)² = 69 + 2[xy + z(x + y)]
⇒ 2[xy + z(x + y)] = 169 – 69 = 100
⇒ [xy + z(x + y)] = 100 / 2 = 50.

Given,
5x + 9y = 5 .....(i)
125x³ + 729y³ = 120 ....(ii)
Make whole cube of 1st equation, we get
(5x + 9y)³ = (5)³
⇒125x³ + 729x³ + 3 × 5x × 9y (5x + 9y) = 125
Put the value of given terms in equation
⇒125x³ + 729x³ + 3 × 5x × 9y (5x + 9y) = 125
⇒120 + 135xy × 5 = 125
⇒135xy × 5 = 5
⇒xy = 1/135.

x-axis ⇒ y = 0, putting in equation 3x + 4y = 12
3x = 12 ⇒ x = 4
⇒ Co-ordinates of point of intersection on x-axis = (4, 0)
Putting on y-axis = (0, 3)
∴ (0, 3)
OA = 4
OB = 3
= (1 / 2) × OA × OB = (1 / 2) × 4 × 3 = 6 sq. units