Algebra for Business: Linear & Quadratic Equations, Inequalities and Applications Questions

Question 1

Define a linear equation.

Accepted Answer

A linear equation is an equation of first degree (highest power of variable is 1).

Question 2

What are simultaneous equations?

Accepted Answer

Simultaneous equations are two or more equations involving the same variables solved together.

Question 3

Write the discriminant of a quadratic equation.

Accepted Answer

Discriminant D = b^2 − 4ac for ax^2 + bx + c = 0.

Question 4

What is an inequality?

Accepted Answer

An inequality shows a range of values using signs like <, >, ≤, ≥.

Question 5

State the break-even condition.

Accepted Answer

Break-even occurs when total revenue equals total cost (profit = 0).

Question 6

Give the quadratic formula.

Accepted Answer

x = (−b ± √(b^2 − 4ac)) / (2a).

Question 7

Solve: 5x − 3 = 22.

Accepted Answer

5x − 3 = 22 ⇒ 5x = 25 ⇒ x = 5.
So, x = 5.

Question 8

Solve the system: 2x + y = 11 and x + y = 7.

Accepted Answer

To solve simultaneous linear equations, we can use elimination.

Example:
2x + y = 11
x + y = 7

Subtract the second equation from the first:
(2x + y) − (x + y) = 11 − 7
x = 4

Substitute x = 4 into x + y = 7:
4 + y = 7 ⇒ y = 3

Therefore, the solution is x = 4 and y = 3.

Question 9

Find the solution set of the inequality 3x + 2 ≤ 14.

Accepted Answer

3x + 2 ≤ 14 ⇒ 3x ≤ 12 ⇒ x ≤ 4.
So the solution set is all real x such that x ≤ 4.

Question 10

A firm has fixed cost ₹10,000 and variable cost ₹200 per unit. If total cost is ₹50,000, find output Q.

Accepted Answer

Total cost C = F + vQ.
50,000 = 10,000 + 200Q ⇒ 200Q = 40,000 ⇒ Q = 200 units.

Question 11

Explain the meaning of discriminant and what D>0, D=0, D<0 indicate.

Accepted Answer

Discriminant D = b^2 − 4ac decides nature of roots.
If D>0: two distinct real roots.
If D=0: two equal real roots (repeated).
If D<0: no real roots (complex roots).

Question 12

A product sells at ₹500 per unit, variable cost ₹300 per unit, fixed cost ₹40,000. Find break-even units.

Accepted Answer

Contribution per unit = 500 − 300 = ₹200.
BEP units = Fixed cost / contribution = 40,000/200 = 200 units.

Question 13

Derive the break-even quantity formula using revenue and cost equations.

Accepted Answer

Break-even point is the level of sales/output at which profit is zero.

Profit Π = Revenue − Cost.
At break-even, Π = 0, so Revenue = Cost.

If revenue R = pQ and cost C = F + vQ (fixed + variable), then:
pQ = F + vQ
(p − v)Q = F
Q = F/(p − v)

This shows break-even units depend on fixed cost and contribution per unit (p−v).

Question 14

Explain how simultaneous equations are used in business pricing problems with an example.

Accepted Answer

In business, we often have two unknowns like price of two products. If we know two different total bills, we can form two linear equations and solve them.

Example: 2 notebooks + 1 pen cost ₹110 and 1 notebook + 1 pen cost ₹70.
Let notebook price = N and pen price = P.
2N + P = 110
N + P = 70
Subtract: N = 40. Substitute: 40 + P = 70 ⇒ P = 30.

Thus, simultaneous equations help determine unknown prices accurately from observed totals.

Question 15

Explain quadratic equations and solve one quadratic using formula. Mention how to interpret roots in business context.

Accepted Answer

A quadratic equation is of the form ax^2 + bx + c = 0 (a ≠ 0).

Quadratic formula:
x = [−b ± √(b^2 − 4ac)] / (2a)

Example: Solve x^2 − 5x + 6 = 0.
Here a=1, b=−5, c=6.
Discriminant D = b^2 − 4ac = 25 − 24 = 1.
Roots:
x = [5 ± √1]/2 = (5 ± 1)/2
So x = 3 or x = 2.

Interpretation: If x represents quantity, we choose the root that satisfies the business condition (e.g., x>0).

Algebra for Business: Linear & Quadratic Equations, Inequalities and Applications

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