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10. Algebra

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 10 (Algebra)

Past Paper Weightage

  • High — algebraic equation-solving is one of the most consistently tested quant skills across CSS GSA solved papers.

Key Concepts

  • Linear equations: solve for the unknown by isolating it (do the same operation to both sides) — e.g., 3x + 5 = 20 → x = 5
  • Simultaneous (two-variable) equations: solve via substitution (express one variable in terms of the other) or elimination (add/subtract equations to cancel a variable)
  • Quadratic equations:
ax2+bx+c=0ax^2 + bx + c = 0

solve by factoring, completing the square, or the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • Common algebraic identities:
(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2 a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b) (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
  • Word problems: translate the sentence into an equation first (define variables clearly), then solve

Worked Example

  • Solve: 2x + 7 = 3x − 5 → 12 = x
  • Solve x² − 5x + 6 = 0 → (x−2)(x−3) = 0 → x = 2 or x = 3

Past Paper Style Numericals (Solved)

  1. The sum of two numbers is 25 and their difference is 7. Find the numbers.

    Answer: x+y=25, x−y=7 → 2x=32 → x=16, y=9 (numbers: 16 and 9)

  2. A father's age is 3 times his son's age. After 10 years, he will be twice as old. Find their present ages.

    Answer: F=3S; F+10=2(S+10) → 3S+10=2S+20 → S=10, F=30 (son 10, father 30)

  3. Solve: x² + 2x − 15 = 0

    Answer: (x+5)(x−3)=0 → x = −5 or 3

  4. The length of a rectangle is 4 cm more than its width, and its area is 96 cm². Find its dimensions.

    Answer: w(w+4)=96 → w²+4w−96=0 → (w+12)(w−8)=0 → w=8 → width 8 cm, length 12 cm

Inequalities

  • An inequality compares two expressions using <, >, ≤, or ≥ instead of an equals sign; solved the same way as equations, with one critical exception:
  • Golden rule: when you multiply or divide BOTH sides of an inequality by a negative number, you must flip the inequality sign
    • Example: Solve −2x + 3 > 7
      • Step 1: Subtract 3 from both sides: −2x > 4.
      • Step 2: Divide both sides by −2 — since we're dividing by a negative, FLIP the sign: x < −2.
      • Step 3: Answer: x < −2 (check with a test value: x=−3 gives −2(−3)+3=9>7 ✓, confirming the flipped direction is correct).
  • Compound inequalities (e.g., 3 < 2x+1 ≤ 9): solve by performing the same operation across all three parts simultaneously.
    • Example: Solve 3 < 2x + 1 ≤ 9
      • Step 1: Subtract 1 from all three parts: 2 < 2x ≤ 8.
      • Step 2: Divide all three parts by 2 (positive, no flip needed): 1 < x ≤ 4.
      • Step 3: Answer: 1 < x ≤ 4.

MCQ Traps

  • A quadratic equation generally has two roots — don't stop after finding one
  • The discriminant (b²−4ac): positive → two real roots, zero → one repeated root, negative → no real roots (complex roots) — a common conceptual MCQ

Self-Test Questions

  1. Solve for x: 5(x−2) = 3x + 4
  2. Solve the quadratic: x² − 7x + 12 = 0
  3. Solve simultaneously: 2x + y = 10, x − y = 2
  4. Solve the inequality: −3x + 5 ≤ 11, showing whether the sign flips.
  5. Solve the compound inequality: −4 ≤ 3x − 1 < 8, showing each step.

Past-Paper MCQs (FPSC)

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The FPSC MCQ papers you shared (2024 and 2025) had no direct algebra question.

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me.

  • CSS 2024 Q6(a): The sum of four numbers is 105. When 3 is added to one number, twice another, five times the third and the fourth all become equal. Find them in ascending order. (5)
    1. Let a + 3 = 2b = 5c = d = k
    2. a = k − 3, b = k/2, c = k/5, d = k
    3. Sum: (k − 3) + k/2 + k/5 + k = 105 → 2.7k − 3 = 105 → k = 40
    4. 8, 20, 37, 40
  • CSS 2024 Q6(d): The digits of a three-digit number add to 15; tens + units = 12; and tens − units = 2. Find the number. (5)
    1. Hundreds digit = 15 − 12 = 3
    2. t + u = 12 and t − u = 2 → t = 7, u = 5
    3. The number is 375
  • CSS 2022 Q7(d): Sara's mother is 6 times her age and her brother Ali is twice her age. In three years the sum of their ages is 72. Find their present ages. (5)
    1. Sara = s, mother = 6s, Ali = 2s
    2. (s + 3) + (6s + 3) + (2s + 3) = 72 → 9s + 9 = 72 → s = 7
    3. Sara 7, Ali 14, mother 42
  • CSS 2021 Q6(a): A man is 3 times as old as his son. In 10 years the sum of their ages will be 76. How old was the man when his son was born? (5)
    1. Son = s, man = 3s; (s + 10) + (3s + 10) = 76 → 4s = 56 → s = 14
    2. Man = 42; age when son was born = 42 − 14 = 28
  • CSS 2019 Q7(b): A farmer's hens and rabbits have 70 heads and 196 legs. How many more hens than rabbits? (5)
    1. h + r = 70 and 2h + 4r = 196
    2. 2h + 4r − 2(h + r) = 2r = 196 − 140 = 56 → r = 28, h = 42
    3. Hens − rabbits = 14
  • CSS 2017 Q7(C): Two tickets Rawalpindi–Islamabad and three Rawalpindi–Murree cost Rs. 770; three Islamabad and two Murree cost Rs. 730. Find each fare. (5)
    1. 2x + 3y = 770 and 3x + 2y = 730 (x = Islamabad, y = Murree)
    2. Add: 5x + 5y = 1,500 → x + y = 300; subtract: x − y = −40
    3. Islamabad Rs. 130, Murree Rs. 170 (check: 260 + 510 = 770)
  • CSS 2022 Q7(a): Three consecutive odd numbers add to 273. (5)
    • Answer: 89, 91, 93 (see the Integers chapter)
  • CSS 2025 Q7(c): A rectangular camp site has sides (3x + 6) m and (x + 1) m and diagonal (4x + 1) m. Find its area and perimeter. (5)
    1. Pythagoras: (3x + 6)² + (x + 1)² = (4x + 1)²
    2. 10x² + 38x + 37 = 16x² + 8x + 1 → x² − 5x − 6 = 0 → (x − 6)(x + 1) = 0 → x = 6
    3. Sides 24 m and 7 m, diagonal 25 m
    4. Area = 168 m²; perimeter = 62 m
  • CSS 2016 Q13(b): Y = mX + C is the equation of a straight line. Draw a graph of X and Y and relate the equation to the slope and intercept. (5)
    • m = slope: the change in Y for each 1 unit change in X (rise ÷ run); positive m rises, negative m falls
    • C = Y-intercept: the value of Y where the line crosses the Y-axis (X = 0)
    • Example graph Y = 2X + 1: points (0, 1), (1, 3), (2, 5); the line passes through (0, 1) with slope 2
    • Steps: plot the intercept, then move 1 right and m up; join with a straight line
  • CSS 2023 Q8(c): Alpha-ray counts Nc rise linearly with energy Eα: (0.25, 17,500), (0.45, 23,500), (0.65, 29,500), (1, 40,000), (1.4, 52,000). Graph it and find Eα when Nc = 31,600. (5)
    1. Slope = (23,500 − 17,500) ÷ (0.45 − 0.25) = 30,000 counts per MeV
    2. Intercept: 17,500 − 30,000 × 0.25 = 10,000, so Nc = 30,000 Eα + 10,000
    3. For 31,600: Eα = (31,600 − 10,000) ÷ 30,000 = 0.72 MeV
    • Graph: plot the five points and draw the straight line; read across at 31,600
  • CSS 2021 Q7(c): For y = x³ − 3x − 10, complete the table, graph it, and find y at x = 1.8 and x when y = 10. (5)
    1. Missing values: x = −2 → −12; x = −1 → −8; x = 1 → −12; x = 2 → −8
    2. y at x = 1.8: 5.832 − 5.4 − 10 = −9.57
    3. y = 10: x³ − 3x − 20 = 0 → x ≈ 3.08

CSS Exam Perspective

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  • Most repeated areas: forming equations from word problems (ages, digits, heads and legs, tickets), simultaneous equations, graphs of lines
  • Tip: define the unknown clearly in the first line, and always check the answer against the original statement