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M.8 Analysis — Concepts Repeated Across MPTs (2022–2026)

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What this page tells you: which concepts repeat in the FPSC MPT quantitative section (2022, 2023 Special, 2024, 2025, 2026), which exact questions were reused, and where to spend your time for 2027.

1️⃣ Concepts that appear in (almost) every paper

⭐⭐⭐⭐⭐ Exponents and surds — 2022, 2024, 2025, 2026

aman=am+n,a0=1,amn=(an)ma^{m}a^{n}=a^{m+n},\quad a^{0}=1,\quad a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^{m}

⭐⭐⭐⭐⭐ Ratio, proportion, sharing and mixtures — 2023, 2024, 2025, 2026

Share=aa+b×T,a+xb+x=cd\text{Share}=\frac{a}{a+b}\times T,\qquad \frac{a+x}{b+x}=\frac{c}{d}

⭐⭐⭐⭐⭐ Averages — 2022, 2023, 2025, 2026

xˉ=∑xn\bar{x}=\frac{\sum x}{n}

⭐⭐⭐⭐ Ages — 2023, 2025, 2026 (several items in 2026)

difference of ages is constant;sum after t years=S+2t\text{difference of ages is constant};\quad \text{sum after }t\text{ years}=S+2t

⭐⭐⭐⭐ Percentages and "remaining" problems — 2022, 2023, 2026

Original=R1−a100\text{Original}=\frac{R}{1-\frac{a}{100}}

⭐⭐⭐⭐ Linear equations and algebra — 2022, 2023, 2025, 2026

ax+b=cx+d  ⇒  x=d−ba−cax+b=cx+d\;\Rightarrow\;x=\frac{d-b}{a-c}

⭐⭐⭐⭐ Quadratics and identities — 2024, 2025, 2026

a2+b2=(a+b)2−2ab,x1+x2=−ba, x1x2=caa^{2}+b^{2}=(a+b)^{2}-2ab,\qquad x_{1}+x_{2}=-\frac{b}{a},\ x_{1}x_{2}=\frac{c}{a}

⭐⭐⭐ Geometry and mensuration — 2022 (vocabulary), 2023, 2026 (five items)

Psemicircle=πr+2r,TSAcylinder=2πr(r+h),n=360∘exterior angleP_{\text{semicircle}}=\pi r+2r,\quad \text{TSA}_{\text{cylinder}}=2\pi r(r+h),\quad n=\frac{360^{\circ}}{\text{exterior angle}}

⭐⭐⭐ Speed, distance and time (trains) — 2024, 2026

v=Ltrain+Lplatformt,m/s×185=km/hv=\frac{L_{\text{train}}+L_{\text{platform}}}{t},\qquad \text{m/s}\times\frac{18}{5}=\text{km/h}

⭐⭐⭐ Work and time — 2024, 2026

M1D1=M2D2,1T=1a+1bM_{1}D_{1}=M_{2}D_{2},\qquad \frac1T=\frac1a+\frac1b

⭐⭐ Series and patterns — 2023, 2026

test differences, then ratios, then squares / alternating terms\text{test differences, then ratios, then squares / alternating terms}

⭐⭐ Sets and Venn diagrams — 2026 (two items)

∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A\cup B|=|A|+|B|-|A\cap B|

⭐⭐ Probability — 2024 block, 2026

P(neither)=1−P(A∪B)P(\text{neither})=1-P(A\cup B)

⭐ Standard form and simple interest — 2023 only (but classic repeaters)

N=a×10n,SI=PRT100N=a\times10^{n},\qquad SI=\frac{PRT}{100}

2️⃣ Exact questions that were repeated

  • −25\sqrt{-25} → 2022 Q3 and 2025 Q1 — answer 5i5i (mis-keyed in 2025).
  • x=−4x+4x=-4x+4 → 2022 Q4 and 2025 Q2 — answer 45\frac45.
  • Sum of 10 values is 10, find average → 2022 Q5 and 2025 Q3 — answer 11.
  • x12×x−12x^{\frac12}\times x^{-\frac12} → 2022 Q6 and 2025 Q4 — answer 11 (mis-keyed in 2025).
  • 412×42−404^{\frac12}\times4^{2}-4^{0} → 2024 Q1 and 2026 Q1 — answer 3131.
  • 27k=927^{k}=9 → 2024 Q2 and 2026 Q2 — answer 23\frac23.
  • Cows and food stock → 2024 Q3 and 2026 Q3.
  • 99 m train and 231 m station in 11 s → 2024 Q4 and 2026 Q4 — answer 108 km/h.
  • A, B, C share 112 → 2024 Q5 and 2026 Q5 — answer 50.
  • "Sold X%, Y left" → 2023 (40%, 420 left) and 2026 (25%, 1500 left) — same method, different numbers.
  • Consecutive integers → 2025 Q8 and 2026 Q8.

3️⃣ The reuse pattern

  • 2022 ↔ 2025: four consecutive items (square root of a negative number, x=−4x+4x=-4x+4, sum of 10 values, x12×x−12x^{\frac12}\times x^{-\frac12}) appear in both papers.
  • 2024 ↔ 2026: the first five items of the 2026 paper repeat the 2024 opening block (identical numbers in several cases).
  • Implication for 2027: expect the opening block to be drawn from an earlier paper (2023 Special, 2025 or 2026 are the natural pool), and expect the later items to be new numbers on the same rule types.

4️⃣ Traps and key errors seen

  • Mis-keyed by FPSC (2025): −25\sqrt{-25} and x12×x−12x^{\frac12}\times x^{-\frac12} — trust the mathematics, not the key.
  • Unit trap: a train answer left in m/s instead of km/h (option 30 in 2024 Q4).
  • "None of these" is often correct: 2022 Q3, 2025 Q1, 2026 Q6, Q10, Q14, Q40.
  • Wording traps in word problems: "60% of" versus "60% smaller"; age "difference" versus "sum"; time ratio versus speed ratio.

5️⃣ Likelihood for 2027 (based on frequency)

  • Very high: exponents / surds; ratio, sharing and mixtures; ages; percentages with remaining amount; averages.
  • High: linear and quadratic algebra; train speed; work rate; geometry (semicircle, cylinder, polygon angle).
  • Medium: Venn diagrams; series; probability; simultaneous equations.
  • Lower but classic: standard form; simple interest; inequalities; completing the square.

6️⃣ Data-quality note (extraction vs recomputation)

Several items in the earlier extraction do not reproduce the printed answer. Treat these as "check the booklet" items:

  • 2022 Q1: 0.001=0.1%0.001=0.1\%, not 0.01.
  • 2022 Q4: x=45x=\frac45, not 54\frac54.
  • 2022 Q8: x=8x=8, not 4.
  • 2022 Q9: x=6432=512x=64^{\frac32}=512, not 8.
  • 2023 Q1: 400×0.05=20400\times0.05=20 mm (key 200 mm suggests 4000 sheets).
  • 2023 Q8: 3 and 4 years give 698; key 737 fits 2 and 3 years.
  • 2023 Q10 / Q11: 54 is the smaller integer, not the sum; key 12 fits a 3-year gap.
  • 2025 Q7: numbers as extracted give 42 L, not 24 L.

7️⃣ Study plan (7 days)

  • Day 1: exponents, surds, standard form — redo M.1 rules 5 and 6 and the 2022 / 2024 items.
  • Day 2: percentages, ratios, mixtures — M.1 rules 1, 2, 4 and practice sets A and B.
  • Day 3: ages — M.1 rule 3 and practice set C; write every equation in full.
  • Day 4: work, speed, simple interest — M.1 rules 9, 10, 11 and set D.
  • Day 5: algebra and quadratics — M.1 rule 12 and set E.
  • Day 6: geometry, sets, series, probability — M.1 rules 7, 8, 14, 15 and set F.
  • Day 7: timed mock — do M.6 (2026) and M.5 (2025) in 40 minutes and review the mistakes.