M.5 MPT 2025 — Questions and Worked Solutions

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Source: FPSC MPT 2025 — official key plus worked solutions in the paper. Two items (Q1 and Q4) were mis-keyed by FPSC; the correct mathematics is shown. Q7 is flagged because the numbers as extracted do not reproduce the printed answer.

Q1 · Square root of a negative number

Question: Square root of $-25$?

Options: includes (D) None of these

✅ Correct answer: $5i$ (FPSC's key said "(d) None of these")

Solution:

$$\sqrt{-25}=\sqrt{25}\times\sqrt{-1}=5i$$

Concept: the square root of a negative number is imaginary. Same item as 2022 Q3.

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Q2 · Linear equation

Question: $x=-4x+4$. Find $x$.

✅ Answer: (A) 4/5

Solution:

$$x=-4x+4\;\Rightarrow\;5x=4\;\Rightarrow\;x=\frac45$$

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Q3 · Average

Question: The sum of 10 values is 10. Find the average.

✅ Answer: (A) 1

Solution:

$$\bar{x}=\frac{10}{10}=1$$

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Q4 · Exponent law

Question: $x^{\frac12}\times x^{-\frac12}$

✅ Correct answer: (B) 1 (FPSC's key said (A) 0)

Solution:

$$x^{\frac12}\cdot x^{-\frac12}=x^{\frac12-\frac12}=x^{0}=1$$

Concept: exponents add when multiplying; $x^{0}=1$ for $x\neq0$. Same item as 2022 Q6.

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Q5 · Ratio sharing

Question: The ratio of boys to girls is 9 : 5 and the total is 1050. How many boys?

Options: answer is option (C); the other options are not in the extraction.

✅ Answer: (C) 675

Solution:

$$\text{Boys}=\frac{9}{9+5}\times1050=\frac{9}{14}\times1050=675$$

Concept: share = $\frac{a}{a+b}\times\text{Total}$.

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Q6 · Adding a number to a ratio

Question: What number must be added to both terms of 7 : 13 so that the ratio becomes 2 : 3?

✅ Answer: (C) 5

Solution:

$$\begin{aligned} \frac{7+x}{13+x}&=\frac23\\ 21+3x&=26+2x\\ x&=5 \end{aligned}$$

Formula: $x=\dfrac{ad-bc}{c-d}=\dfrac{7\cdot3-13\cdot2}{2-3}=5$.

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Q7 · Mixture of alcohol and water

Question: Alcohol : water = 12 : 5. After adding 14 L of water the ratio becomes 4 : 3. Find the alcohol quantity.

✅ Answer as extracted: (B) 24 litres

Solution with the numbers as extracted:

$$\begin{aligned} \frac{12k}{5k+14}&=\frac43\\ 36k&=20k+56\\ k&=3.5,\qquad \text{Alcohol}=12k=42\ \text{L} \end{aligned}$$

⚠️ These numbers give 42 L, not 24 L. Check the printed values in the booklet; the method above is what is tested.

Concept: keep the alcohol constant; add water to the water part only.

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Q8 · Consecutive integers

Question: The product of two consecutive integers equals 12 plus 12 times their sum. Find the smaller.

✅ Answer: (C) 24

Solution:

$$\begin{aligned} n(n+1)&=12+12(2n+1)\\ n^{2}+n&=24n+24\\ n^{2}-23n-24&=0\\ (n-24)(n+1)&=0 \end{aligned}$$ $$n=24\ \text{(or } n=-1\text{)}$$

Concept: consecutive integers $n,\ n+1$; sum $=2n+1$. Check: $24\times25=600$ and $12+12\times49=600$ ✅.

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Q9 · Ages

Question: Ali was 40 when Asim was born. When will Ali's age be 5 times Asim's age?

✅ Answer: (C) 50

Solution:

$$\begin{aligned} 40+A&=5A\\ 40&=4A\\ A&=10\quad\Rightarrow\quad\text{Ali}=50 \end{aligned}$$

Concept: the age difference (40) never changes.