PAF GD Pilot Mathematics Syllabus

Give the formula for percentage increase and percentage decrease.

% increase = (increase ÷ original) × 100. % decrease = (decrease ÷ original) × 100, where increase/decrease is the change from the original value.

What is 20% of 150, and how is it calculated?

30. Calculation: (20/100) × 150 = 0.20 × 150 = 30.

Define a ratio and how it is written.

A ratio compares two quantities of the same kind by division, written a : b or a/b. Example: 4 boys to 6 girls = 4 : 6 = 2 : 3 in simplest form.

What is a proportion?

A statement that two ratios are equal, written a : b :: c : d or a/b = c/d. The terms a and d are extremes; b and c are means.

State the cross-multiplication property of proportion.

If a/b = c/d, then a × d = b × c (product of extremes = product of means).

What is the difference between direct and inverse proportion?

In direct proportion, as one quantity increases the other increases (x/y constant). In inverse proportion, as one increases the other decreases (x × y constant).

How do you divide an amount in a given ratio? Example: Rs 600 in 2:3.

Add the ratio parts (2+3=5), divide the amount by the total (600/5=120), then multiply by each part: 2×120 = Rs 240 and 3×120 = Rs 360.

Define cost price (CP) and selling price (SP).

Cost price is the price at which an item is bought. Selling price is the price at which it is sold. Profit occurs when SP > CP; loss when SP < CP.

State the profit and loss formulas in terms of CP and SP.

Profit = SP − CP (when SP > CP). Loss = CP − SP (when CP > SP).

Give the formula for profit percent and loss percent.

Profit% = (Profit ÷ CP) × 100. Loss% = (Loss ÷ CP) × 100. Both percentages are always calculated on the cost price.

How do you find SP when CP and profit% are known?

SP = CP × (100 + Profit%)/100. For a loss, SP = CP × (100 − Loss%)/100.

How do you find CP when SP and profit% are known?

CP = SP × 100/(100 + Profit%). For a loss, CP = SP × 100/(100 − Loss%).

State the formula for Simple Interest.

SI = (P × R × T)/100, where P = principal, R = rate per annum (%), T = time in years. Amount A = P + SI.

State the formula for Compound Interest (amount and CI).

Amount A = P(1 + R/100)^T. Compound Interest CI = A − P = P[(1 + R/100)^T − 1].

What is the key difference between simple and compound interest?

Simple interest is calculated only on the original principal each period. Compound interest is calculated on the principal plus accumulated interest, so it grows faster.

How is compound interest calculated when compounded half-yearly?

Halve the rate and double the time: A = P(1 + (R/2)/100)^(2T), since interest is added twice per year.

Define marked price (list price) and discount.

Marked price (MP) is the price printed/listed on an item before reduction. Discount is the reduction given on the marked price: Discount = MP − SP.

State the formula for discount percent and selling price after discount.

Discount% = (Discount ÷ MP) × 100. SP = MP × (100 − Discount%)/100 = MP − Discount.

What is markup, and how does it relate to cost price?

Markup is the amount added to the cost price to set the marked/selling price. Markup% = (Markup ÷ CP) × 100, where Markup = MP − CP.

Define a variable, a constant, and a coefficient in algebra.

A variable is a symbol for an unknown (e.g. x). A constant is a fixed value (e.g. 5). A coefficient is the number multiplying a variable (in 7x, 7 is the coefficient).

What is the difference between a monomial, binomial, and trinomial?

A monomial has one term (3x), a binomial has two terms (x + 2), and a trinomial has three terms (x² + 3x + 2). They are classified by the number of terms.

What are like terms? Give an example.

Terms having the same variables raised to the same powers; only their coefficients may differ. Example: 3x² and 5x² are like terms and can be added to 8x².

How do you factorize by taking out the highest common factor? Example: 6x² + 9x.

Find the HCF of the terms and write it outside a bracket. 6x² + 9x → HCF is 3x → 3x(2x + 3).

Factorize a difference of two squares: a² − b².

a² − b² = (a + b)(a − b). Example: x² − 16 = (x + 4)(x − 4).

How do you factorize a trinomial of the form x² + bx + c?

Find two numbers whose product is c and whose sum is b, then split the middle term. Example: x² + 5x + 6 = (x + 2)(x + 3), since 2×3=6 and 2+3=5.

What is a linear equation in one variable, and how many solutions does it have?

An equation of the form ax + b = 0 (a ≠ 0) where the highest power of the variable is 1. It has exactly one solution: x = −b/a.

Solve the linear equation 2x + 5 = 17 and state the method.

x = 6. Method: subtract 5 from both sides (2x = 12), then divide both sides by 2 (x = 6) — keep the equation balanced by doing the same to both sides.

What is the general form of a quadratic equation, and what makes it quadratic?

ax² + bx + c = 0, where a ≠ 0. It is quadratic because the highest power of the variable is 2, giving up to two solutions (roots).

State the quadratic formula.

For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / (2a). The expression b² − 4ac is the discriminant.

What does the discriminant (b² − 4ac) tell you about the roots?

If b² − 4ac > 0: two distinct real roots. If = 0: one real (repeated) root. If < 0: no real roots (roots are imaginary/complex).

State the algebraic identity for (a + b)² and (a − b)².

(a + b)² = a² + 2ab + b². (a − b)² = a² − 2ab + b².

State the identity for (a + b)(a − b) and for (a + b)³.

(a + b)(a − b) = a² − b². (a + b)³ = a³ + 3a²b + 3ab² + b³.

What is the difference between complementary and supplementary angles?

Complementary angles add up to 90°. Supplementary angles add up to 180°.

Define acute, right, obtuse, straight, and reflex angles by their measures.

Acute: less than 90°. Right: exactly 90°. Obtuse: between 90° and 180°. Straight: exactly 180°. Reflex: between 180° and 360°.

What are vertically opposite angles, and what is their key property?

The pair of opposite angles formed when two lines intersect. Vertically opposite angles are always equal.

When a transversal cuts two parallel lines, what is true of alternate angles and co-interior angles?

Alternate angles (interior or exterior) are equal. Co-interior (allied) angles are supplementary, adding to 180°.