Question

y is the sum of three numbers, one of which is a constant, the 2nd varies as x and the 3rd varies inversely as x. The values of y at x = 1, -1 and 3 are respectively 6, -4 and 8. Then, y is equal to

• A. \(1 + x - \frac{1}{x}\)
• B. \(1 + 2x + \frac{3}{x}\)
• C. \(2 + x + \frac{1}{x}\)
• D. \(2 - x + \frac{1}{x}\)

Hint:

Form equations carefully and eliminate variables systematically.

Answer 1

The Correct Option is B

Concept: \[ y = a + bx + \frac{c}{x} \]

Step 1: Substitute given values.
At \(x=1\): \[ a + b + c = 6 \quad (1) \] At \(x=-1\): \[ a - b - c = -4 \quad (2) \] At \(x=3\): \[ a + 3b + \frac{c}{3} = 8 \quad (3) \]

Step 2: Solve equations.
Add (1) and (2): \[ 2a = 2 \Rightarrow a = 1 \] Substitute into (1): \[ 1 + b + c = 6 \Rightarrow b + c = 5 \quad (4) \] Substitute into (3): \[ 1 + 3b + \frac{c}{3} = 8 \Rightarrow 3b + \frac{c}{3} = 7 \] Multiply by 3: \[ 9b + c = 21 \quad (5) \]

Step 3: Find b and c.
Subtract (4) from (5): \[ (9b + c) - (b + c) = 21 - 5 \Rightarrow 8b = 16 \Rightarrow b = 2 \] \[ c = 5 - b = 3 \]

Step 4: Final expression.
\[ y = 1 + 2x + \frac{3}{x} \]