Mathematics Solved Model Past Paper 2024 PDF
Part – 1 English
In ‘Seven Ages of Man’, which is the fourth stage of man?
Answer: Soldier
The Synonym of Immaculate is?
Answer: Both B and C (Pure and Natural)
Identify the correct spelling:
Answer: Beginning
Choose the correct spelling:
Answer: Loam
She is not afraid _____ death.
Answer: Of
Change the voice: She has discussed the matter with her teacher.
Answer: The matter has been discussed with her teacher by her.
Complete the analogy: Turtle : Crawl :: Frog : ?
Answer: Hop
The Antonym of Clean is?
Answer: Dirty
Find the correct spelling:
Answer: Existence
The Antonym of Calm is?
Answer: Both A and C (Excited and Agitated)
Change the voice: They have completed the project.
Answer: The project has been completed by them.
We continued working on the assignment last night. Here ‘working’ is?
Answer: Gerund
The Synonym of Prevalent is?
Answer: Both B and C (Widespread and Common)
Where _____ the car?
Answer: Did you park
We are having a social service week _____ school.
Answer: At
Migration out of an area is called _____?
Answer: Emigration
We trust/believe ______ one God.
Answer: In
The Synonym of Tragic is?
Answer: Both B and C (Tragic and Dreadful)
They said, “God helps those who help themselves”.
Answer: They said that God helps those who help themselves.
Everyone must read this book. (Passive Voice)
Answer: This book must be read by everyone.
The poem “About Ben Adhem” is composed by _____?
Answer: Leigh Hunt
Choose the correct spelling:
Answer: Accommodate
His father is anxious ________ his health.
Answer: About
The Antonym of Abandon is?
Answer: Both B and C (Maintain and Keep)
The Antonym of Ability is?
Answer: Both B and C (Incapability and Inability)
The Antonym of “Preliminary” is?
Answer: Final
Part-I: Teaching Methodology
What part of the curriculum includes the actual knowledge and values taught?
Answer: Curriculum Content
Who usually does “Action Research”?
Answer: Teachers
What is the best way to manage a classroom effectively?
Answer: Group learning
What is it called when a child cries just because they see someone else crying?
Answer: Sympathetic imitation
How can we make students more engaged in class?
Answer: Assigning group work
What shows the types of activities needed to learn successfully?
Answer: Learning activities
Why is “Collaborative Research” used?
Answer: Improve practices
Which method uses a digital structure to improve student ability?
Answer: Flipped classroom
Who are Rogers’ counseling methods best suited for?
Answer: Emotionally disturbed individuals
Which education type focuses on social and economic needs to improve lives?
Answer: Non-Formal Education
Which of these is NOT a stage in Piaget’s theory?
Answer: Emotional Intelligence Stage
Why do we use “Aptitude Tests” in guidance?
Answer: It predicts academic performance
What is the main goal of Pakistan’s education policy?
Answer: Literacy and Universal Education
What is the arrangement of curriculum goals, content, and evaluation called?
Answer: Curriculum Design
Which phase is about a new curriculum being accepted and used regularly?
Answer: Implementation phase
What is the best method for the teaching-learning process?
Answer: Project method
At what level does a child recognize a red ball and remember its attributes?
Answer: Concrete Level
Which philosophy focuses on a student’s own personality and individuality?
Answer: Existentialism
What is the very first step in making a curriculum?
Answer: Selection of Goals and objectives
Which principle says subjects like Art and Music are for relaxation?
Answer: Principle of Leisure
What is the lowest level of the affective domain where students just listen?
Answer: Receiving
Who said that personality develops through major conflicts at each life stage?
Answer: Erik Erikson
What is the best way to check a classroom activity?
Answer: Rubrics-based evaluation
In which drama technique does the teacher have total control using scripts?
Answer: Formal drama
Which branch of philosophy studies the meaning of “Value”?
Answer: Axiology
Part – II Mathematics
Q: Population growth problem
A: Model:
dP/dt = kP
Solution:
P = 500 eᵏᵗ
Given:
e¹⁰ᵏ = 1.15 → k = ln(1.15)/10
After 30 years:
P = 500(1.15)³ ≈ 760
Growth rate at t=30:
dP/dt = kP
Q: Find principal argument of
−2 / (1 + √3 i)
A: Simplify:
= (-1/2 + √3/2 i)
Argument = 2π/3
Q: Lagrangian formalism in polar coordinates
A:
L = ½ m (ṙ² + r²θ̇²) − V
Equations:
mr̈ − mrθ̇² = Fᵣ
d/dt(mr²θ̇) = 0
In an examination, 80% of candidates passed English and 85% of candidates passed in Mathematics. If 73% candidates passed in both these subjects, then what percent of candidates failed in both the subjects?
Answer: 8%
Solution:
Use the formula for union of two sets:
Percent passed at least one subject = %English + %Math – %both = 80 + 85 – 73 = 92%.
Percent failed in both subjects = 100 – 92 = 8%.
So the correct answer is 8%.
If 8 people work for 60 days, how many days will it take for 6 people to complete the same work?
Answer: 80 days
Solution: Work × Days × People = Constant.
8 × 60 = 6 × x → x = (8×60)/6 = 480/6 = 80 days.
The product of 513 with itself is?
Answer: 263169
Solution: 513 × 513 = 263169.
Half of 1 percent written as a decimal is?
Answer: 0.005
Solution: 1% = 0.01 → half of 0.01 = 0.005.
15 men can do a work in 12 days. How long it will take 9 men to do the same work?
Answer: 20 days
Solution: Work × Men × Days = Constant
15 × 12 = 9 × x → x = 180/9 = 20 days.
Angle of measure of 72° can be drawn by?
Answer: Protector only
Solution: A protractor (protector) is used to measure specific angles directly. Compass alone cannot draw an exact 72° angle, so only a protractor is sufficient.
The smallest 4-digit number using 0, 3, 5, and 8, is?
Answer: 3058
Solution: Smallest 4-digit number cannot start with 0. Arrange digits in ascending order: 3,0,5,8 → 3058.
The bridge is completed in 20 days by 80 men. How long will it take 70 men to complete the work?
Answer: 22.86 days ≈ 23 days
Solution: 80 × 20 = 1600 man-days. 1600 ÷ 70 ≈ 22.86 days.
Which of the following is a factor of 36?
Answer: All of these ( 4,6,12)
Solution: Factors of 36: 1,2,3,4,6,9,12,18,36 → 4,6,12 are all factors.
What is the measure of each interior angle of a regular hexagon?
Answer: 120°
Solution: Interior angle = (n–2)×180/n = (6–2)×180/6 = 720/6 = 120°.
The set of integers is a subset of?
Answer: Rational Numbers
Solution: Every integer can be written as a fraction (like 5 = 5/1), so integers are part of rational numbers.
A rectangular front-wall of a hospital has a perimeter of 294 meters. If its length is 85 meters, find its area.
Answer: 5270 m²
Solution:
Perimeter of rectangle = 2(length + width) → 294 = 2(85 + width)
Divide both sides by 2 → 147 = 85 + width
Subtract 85 → width = 147 – 85 = 62
Area = length × width = 85 × 62 = 5270 m²
The difference of two positive integers is 3 and the sum of their squares is 89. The integers are?
Answer: 5 and 8
Solution: Let the numbers be x and y → y – x = 3 → y = x + 3
x² + y² = 89 → x² + (x + 3)² = 89 → 2x² + 6x + 9 = 89 → 2x² + 6x – 80 = 0 → x² + 3x – 40 = 0
Factor: (x + 8)(x – 5) = 0 → x = 5 → y = 8
60 men can do a work in 40 days. How long will it take 40 men to complete the same work?
Answer: 60 days
Solution: Work = men × days = constant → 60 × 40 = 2400 man-days
For 40 men: 2400 ÷ 40 = 60 days
Digital number 9 can be represented in binary number as?
Answer: 1001
Solution: Convert 9 to binary: 9 ÷ 2 = 4 remainder 1, 4 ÷ 2 = 2 remainder 0, 2 ÷ 2 = 1 remainder 0, 1 ÷ 2 = 0 remainder 1 → 1001
While watching television from 7 PM to 8 PM you count 20 commercials each 30 seconds long. What nearest percent of the hour is taken by the commercials?
Answer: 17%
Solution: Total commercial time = 20 × 30 sec = 600 sec = 10 min
Percent of hour = 10 ÷ 60 × 100 = 16.67% ≈ 17%
What number is 25% of 150?
Answer: 37.5
Solution: 25% of 150 = 150 × 0.25 = 37.5
If 15 men can manufacture 27 baskets in an hour, how many baskets would 45 men manufacture in 40 minutes?
Answer: 54
Solution: Baskets per man per hour = 27 ÷ 15 = 1.8
For 45 men in 1 hour: 45 × 1.8 = 81 baskets
In 40 minutes (2/3 hour): 81 × 2/3 = 54 baskets
A number whose 7 percent is 42 is?
Answer: 600
Solution: 7% of number = 42 → number × 0.07 = 42 → number = 42 ÷ 0.07 = 600
a³ – b³ , a² – b² , a² + ab + b² and x are in a proportion. The value of x is?
Answer: a + b
Solution: Proportion: (a³ – b³)/(a² – b²) = (a² + ab + b²)/x → simplify → x = a + b
Distance between points (2,3) and (7,3) is?
Answer: 5
Solution: Distance = √((x₂ – x₁)² + (y₂ – y₁)²) = √((7–2)² + (3–3)²) = √25 = 5
Find x, if log₁₂(12³) = x?
Answer: 3
Solution: logₐ(aᵏ) = k → log₁₂(12³) = 3
7/4 ÷ [77/8 ÷ 30/5] = ?
Answer: 12/11
Solution: First, 77/8 ÷ 30/5 = 77/8 × 5/30 = 385/240 = 77/48
Then 7/4 ÷ 77/48 = 7/4 × 48/77 = 336/308 = 12/11
A bag contains 6 black and 8 white balls. 1 ball is drawn at random. Probability it is white?
Answer: 4/7
Solution: Total balls = 6 + 8 = 14 → Probability white = 8 ÷ 14 = 4/7
If 70 men can work in 35 days, 40 men complete this work in how many days?
Answer: 61.25 days
Solution: Total work = 70 × 35 = 2450 man-days → 40 men: 2450 ÷ 40 = 61.25 days
If A’s salary is 25% higher than B’s, B’s salary is what percent lower than A’s?
Answer: 20%
Solution: Let B = 100 → A = 125 → Difference = 25 → 25 ÷ 125 × 100 = 20%
40 cows eat 40 bags of husk in 40 days. In how many days will 1 cow eat 1 bag?
Answer: 40 days
Solution: Husk per cow per day = 40 ÷ (40 × 40) = 1/40 → 1 bag → 40 days
Perimeter of a rhombus with side 6 cm?
Answer: 24 cm
Solution: Perimeter = 4 × side = 4 × 6 = 24 cm
In a quadratic equation, the highest power of variable?
Answer: 2
Solution: Standard quadratic: ax² + bx + c → highest power = 2
Product of two numbers is 2500. LCM is 125. HCF is?
Answer: 20
Solution: Product = LCM × HCF → HCF = 2500 ÷ 125 = 20
80 men dug 4 holes in 12 months. How many men needed for 6 holes?
Answer: 360
Solution:
Step 1: Total man-months for 80 men digging 4 holes = 80 × 12 = 960
Step 2: Man-months needed per house = 960 ÷ 4 = 240
Step 3: Total man-months for 6 houses = 240 × 6 = 1440
Step 4: Number of men needed in 4 months = 1440 ÷ 4 = 360
Answer: 360 men.
How many two-digit numbers are divisible by 6?
Answer: 15
Solution: Two-digit numbers divisible by 6: 12, 18, …, 96 → (96–12)/6 + 1 = 15 numbers
A fort had food for 150 men for 45 days. After 10 days, 25 men left. Remaining food will last?
Answer: 42 days
Solution: Initial food = 150 × 45 = 6750 man-days
After 10 days: 150 × 10 = 1500 man-days used → remaining = 5250
Remaining men = 150 – 25 = 125 → 5250 ÷ 125 = 42 days
A fraction where numerator degree > denominator degree is called?
Answer: Improper fraction
Solution: Numerator degree > denominator degree → Improper fraction
6 men can work in 10 hrs, 4 men can do this work in?
Answer: 15 hrs
Solution: Work = men × hours → 6 × 10 = 60 man-hours → 4 men: 60 ÷ 4 = 15 hrs
How many acres are there in 1 hectare?
Answer: 2.47
Solution: 1 hectare = 2.471 acres ≈ 2.47 acres
What percent of 720 is 36?
Answer: 5%
Solution: (36 ÷ 720) × 100 = 5%
A flagstaff 17.5 m high casts a shadow of 40.25 m. A building casts shadow of 28.75 m. Height of building?
Answer: 12.5 m
Solution: Height/shadow ratio constant → 17.5/40.25 = h/28.75 → h = 17.5 × 28.75 ÷ 40.25 ≈ 12.5 m
A triangle with no equal sides and no equal angles is?
Answer: Scalene triangle
Solution: Scalene → all sides and angles are unequal
20 men can do a work in 8 days. How many men in 10 days?
Answer: 16
Solution: Work = men × days = constant → 20 × 8 = 160 → 160 ÷ 10 = 16 men
250 men can finish work in 20 days working 5 hours/day. To finish in 10 days working 8 hours/day, men required?
Answer: 313 men
Solution: Total man-hours = 250 × 20 × 5 = 25,000
New work hours/day = 10 × 8 = 80 → men = 25,000 ÷ 80 = 312.5 → 313
7 spiders make 7 webs in 7 days. 1 spider makes 1 web in how many days?
Answer: 7 days
Solution: Each spider makes 1 web in 7 days
3/8 + 4/2 = ?
Answer: 19/8
Solution: 4/2 = 16/8 → 3/8 + 16/8 = 19/8
In a cage, there are pigeons and rabbits. 20 heads, 48 feet. Rabbits?
Answer: 4
We have pigeons and rabbits.
Let pigeons = P and rabbits = R.
Total heads = 20
So, P + R = 20
Total feet = 48
Pigeons have 2 feet and rabbits have 4 feet, so:
2P + 4R = 48
From the first equation:
P = 20 − R
Put this value in the second equation:
2(20 − R) + 4R = 48
40 − 2R + 4R = 48
40 + 2R = 48
2R = 8
R = 4
Answer: There are 4 rabbits in the cage.
What number is 20% of 80?
Answer: 16
Solution: 20% of 80 = 0.2 × 80 =16
If 40 men build 4 buildings in 12 months, how many men needed to build 6 houses in 4 months?
Answer: 180 men
Solution: Work = men × months × buildings → 40 × 12 × 4 = 1920 man-months per building
6 buildings in 4 months → men = 1920 ÷ 4 ÷ 6 = 180
A man climbs a hill: ascends 20 yards, slips 5 yards each minute. Time to climb 80 yards?
Answer: 5 minutes
Solution: Net climb per minute = 20 – 5 = 15
Before last step, climb 80 – 20 = 60 → 60 ÷ 15 = 4 minutes
Last 20 yards = 1 minute → total = 5 minutes
Each minute: climbs 20, slips 5 → net 15
Total height 80 → last step doesn’t slip → 80 – 20 = 60 → 60 ÷ 15 = 4 → total minutes = 4 +1 = 5 minutes
Greatest number which exactly divides 1050 and 750?
Answer: 150
Solution: HCF of 1050 and 750 → 1050=2×3×5²×7, 750=2×3×5³ → HCF=2×3×5²=150
Rectangle 82 cm × 75 cm, enclosed by 4 cm wide strip outside. Area of strip?
Answer:1320 cm²
Solution: Outer rectangle: (82+8) × (75+8)=90×83=7470
Original area = 82×75=6150
Area of strip = 7470 – 6150 = 1320 Answer
check 82+8=90, 75+8=83 → 90×83=7470 → 7470–6150=1320
1320 cm²
How many years in a decade?
Answer: 10 years
Solution: By definition, 1 decade = 10 years
