M.6 MPT 2026 — Questions and Worked Solutions

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Source: FPSC MPT 2026 — official key published. This paper reuses the 2024 bank with different numbers (Q1–5) and adds new algebra, geometry and data questions. Where the extraction holds the full numbers the question is re-derived; otherwise the method is shown and the official answer is quoted.

Part A · Exponents, equations and algebra

Q1 · Exponents

Question: $4^{\frac12}\times4^{2}-4^{0}$

✅ Answer: (C) 31

$$2\times16-1=31$$

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Q2 · Exponent equation

Question: $27^{k}=9$. Find $k$.

✅ Answer: (B) 2/3

$$3^{3k}=3^{2}\;\Rightarrow\;k=\frac23$$

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Q3 · Food supply (cows)

Question: Herd reduced by 10 cows; food lasts longer.

✅ Official answer: (B) 160

Method: total food is constant.

$$Nd=(N-10)(d+10)\;\Rightarrow\;N-d=10$$

🔎 The second condition is not in the extraction; same item as 2024 Q3.

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Q4 · Train and station

Question: 99 m train, 231 m station, 11 s. Speed in km/h?

✅ Answer: (A) 108 km/h

$$v=\frac{99+231}{11}=30\ \text{m/s}=30\times\frac{18}{5}=108\ \text{km/h}$$

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Q5 · A, B, C share 112

✅ Official answer: (C) 50

Method: write shares in one variable; sum = 112. Same item as 2024 Q5.

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Q6 · Integer solution of a quadratic

Question: Integer solution of $2x^{2}-4x=15$?

✅ Answer: (D) None of these

$$2x^{2}-4x-15=0,\qquad \Delta=b^{2}-4ac=16+120=136$$ $$x=\frac{4\pm\sqrt{136}}{4}\quad(\sqrt{136}\ \text{is not an integer})$$

Concept: if $\Delta$ is not a perfect square, the roots are irrational, so no integer solution.

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Q10 · Quadratic roots

Question: $x^{2}-11x+10=0$

✅ Answer: (D) None of these

$$(x-1)(x-10)=0\;\Rightarrow\;x=1\ \text{or}\ x=10$$

Concept: sum of roots $=11$, product $=10$. The roots are 1 and 10; the options listed did not contain that pair.

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Q28 · Inequality

Question: Integers $n$ with $-7<4n\le8$?

✅ Answer: (A) −1, 0, 1, 2

$$-\frac74<n\le2\;\Rightarrow\;n\in\{-1,0,1,2\}$$

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Q30 · Sum of squares from sum and product

Question: $a+b=5,\ ab=6$. Find $a^{2}+b^{2}$.

✅ Answer: (C) 13

$$a^{2}+b^{2}=(a+b)^{2}-2ab=25-12=13$$

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Q31 · Completing the square

Question: $y=x^{2}+8x-5$, write as $(x+p)^{2}+q$.

✅ Answer: (C) 4 (the value of $p$)

$$y=(x+4)^{2}-16-5=(x+4)^{2}-21$$

Concept: $p=\frac{b}{2}=4$, $q=c-\frac{b^{2}}{4}=-5-16=-21$.

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Q32 · Formula manipulation

Question: $A=(2\pi+y)\,x^{2}$ — find the requested value.

✅ Official answer: (C) 2

🔎 The extraction omits what is asked; substitute the given values and simplify.

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Part B · Ages, percentages and number problems

Q7 · Ages

Question: Zaid is 6 times Zoya's age; in 7 years the sum of their ages is 49.

✅ Official answer: (A) 25

$$\begin{aligned} (6x+7)+(x+7)&=49\\ 7x&=35\\ x&=5 \end{aligned}$$

Zoya is 5, Zaid is 30; the difference is $30-5=25$ (the likely quantity asked).

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Q8 · Consecutive numbers

Question: Two consecutive positive numbers have squares summing to 113.

✅ Answer: (A) 7, 8

$$\begin{aligned} n^{2}+(n+1)^{2}&=113\\ 2n^{2}+2n-112&=0\\ n^{2}+n-56&=0\\ (n+8)(n-7)&=0\;\Rightarrow\;n=7 \end{aligned}$$

Check: $49+64=113$ ✅.

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Q9 · Percentage remaining

Question: 25% of the lights are sold; 1500 are left. Original number?

✅ Answer: (C) 2000

$$\frac{1500}{1-0.25}=\frac{1500}{0.75}=2000$$

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Q11 · Simultaneous equations

Question: 5 markers + 1 pencil cost 245; 3 markers + 3 pencils cost 327. Price of one marker?

✅ Answer: (B) Rs. 34

$$\begin{aligned} 5m+p&=245\\ 3m+3p&=327\;\Rightarrow\;m+p=109\\ (5m+p)-(m+p)&=245-109\\ 4m&=136\;\Rightarrow\;m=34 \end{aligned}$$

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Q26 · Splitting a total

Question: Split 45 into two parts (with a stated difference).

✅ Official answer: (B) 21, 24

$$x+(x+3)=45\;\Rightarrow\;x=21,\ \text{other part}=24$$

🔎 The extraction omits the condition; a difference of 3 reproduces the key.

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Q27 · Age product

Question: A man is 5 times his son's age; four years ago the product of their ages was 52.

✅ Answer: (C) 6 and 30

$$\begin{aligned} (5x-4)(x-4)&=52\\ 5x^{2}-24x+16&=52\\ 5x^{2}-24x-36&=0\\ x&=\frac{24\pm36}{10}=6 \end{aligned}$$

Son 6, man 30. Check: $(30-4)(6-4)=26\times2=52$ ✅.

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Q29 · Scholarship grant

Question: Percentage / remaining-amount problem.

✅ Official answer: (B) 40,000

Method: use $\text{Original}=\dfrac{\text{Remaining}}{1-\text{fraction used}}$ step by step for each deduction.

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Q33 · Football ticket pricing

✅ Official answer: (C) Rs. 2000

Method: form linear equations in price per ticket type and solve.

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Part C · Arithmetic and number sense

Q21 · BODMAS

Question: $3(-4\times5-6)$

✅ Answer: (B) −78

$$3(-20-6)=3\times(-26)=-78$$

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Q22 · Ordering numbers

Question: Arrange $0.5^{2},\ 0.5,\ 0.5^{3},\ \sqrt[3]{0.5}$ in ascending order.

✅ Answer: (B) iii, i, ii, iv

$$0.5^{3}=0.125<0.5^{2}=0.25<0.5<\sqrt[3]{0.5}\approx0.794$$

Concept: powers greater than 1 of a number below 1 make it smaller; roots make it larger.

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Q12 · Sequence average rule

✅ Official answer: (B) 3 (27th term)

Method: if each term is the average of its neighbours the sequence is an AP; find $a$ and $d$ from the given terms, then $a_{27}=a+26d$.

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Q40 · Number series

Question: 11, 20, 38, 56, 83, ?

✅ Answer: (D) None of these

$$20-11=9,\quad 38-20=18,\quad 56-38=18,\quad 83-56=27$$

Differences $9,\ 18,\ 18,\ 27$ follow the pattern $9\times(1,2,2,3)$; the next difference would be $27$ (term $110$) or $36$ (term $119$) — neither is among the options.

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Part D · Geometry and mensuration

Q14 · Isosceles triangle angles

✅ Official answer: (D) None of these

Rules: base angles equal; angle sum $180^{\circ}$.

$$2x+y=180^{\circ}$$

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Q15 · Semicircle perimeter

Question: Semicircle perimeter 20.6 cm. Find the radius.

✅ Answer: (A) 4 cm

$$\begin{aligned} P&=\pi r+2r=r(\pi+2)\\ r&=\frac{20.6}{3.14+2}=\frac{20.6}{5.14}\approx4\ \text{cm} \end{aligned}$$

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Q16 · Cuboid tank

✅ Official answer: (B) 900 litres

$$V=l\,w\,h\ (\text{m}^{3}),\qquad 1\ \text{m}^{3}=1000\ \text{L}$$

Concept: convert dimensions to metres, multiply, then convert to litres.

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Q17 · Race track

✅ Official answer: (C) 84 m (outer radius)

Method: outer radius = inner radius + track width; use circumference $C=2\pi r$ to link the two runners' distances.

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Q34 · Regular polygon

Question: Exterior angle 36°. Name the polygon.

✅ Answer: (C) Decagon

$$n=\frac{360^{\circ}}{36^{\circ}}=10$$

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Q35 · Semicircular lake

Question: Semicircle with $r=700$ m. Perimeter?

✅ Answer: (B) 3.6 km

$$\begin{aligned} P&=\pi r+2r=\frac{22}{7}\times700+2\times700\\ &=2200+1400=3600\ \text{m}=3.6\ \text{km} \end{aligned}$$

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Q36 · Cylinder surface area

Question: $r=3.5$ cm, $h=5.5$ cm. Total surface area?

✅ Answer: (B) 198 cm²

$$\begin{aligned} \text{TSA}&=2\pi r(r+h)\\ &=2\times\frac{22}{7}\times3.5\times(3.5+5.5)\\ &=2\times22\times0.5\times9=198\ \text{cm}^{2} \end{aligned}$$

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Part E · Sets, data and probability

Q18 · Venn diagram

Question: 200 people; 90 like tennis, 108 cricket, 46 both. How many like neither?

✅ Answer: (B) 48

$$\begin{aligned} |A\cup B|&=90+108-46=152\\ \text{Neither}&=200-152=48 \end{aligned}$$

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Q19 · Average of daily visitors

✅ Official answer: (C) 175

$$\bar{x}=\frac{\text{total visitors}}{\text{number of days}}$$

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Q37 · Overlap of two groups

✅ Official answer: (A) 19

$$|A\cap B|=|A|+|B|-|A\cup B|$$

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Q38 · Probability

Question: Probability that a chosen item is neither red nor green.

✅ Answer: (A) 7/21

$$P(\text{neither})=1-P(R\cup G)=\frac{7}{21}=\frac13$$

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Part F · Word problems (rate, time, work)

Q13 · Cake pricing

✅ Official answer: (B) $1.48

Method: set up linear equations for the cake components and solve.

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Q20 · Bus timetable

✅ Official answer: (B) 07:39 pm

Method: add the journey time to the departure time using 60-minute carry.

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Q23 · Workers in a forest

✅ Official answer: (C) 80

Method: $\text{Work}=\text{workers}\times\text{rate}\times\text{time}$; equate the two situations.

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Q24 · Temperature drop

✅ Official answer: (C) −3.8°C

Method: final temperature = initial − drop; keep the sign for a fall.

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Q25 · Sweets ratio

✅ Official answer: (C) 9

Method: share the total in the given ratio; $\text{part}=\dfrac{a}{a+b}\times\text{total}$.

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Q39 · Work rate (men / women / days)

✅ Official answer: (A) 16 days

$$M_{1}D_{1}=M_{2}D_{2}\qquad\text{and}\qquad \frac1T=\frac1a+\frac1b$$