Assistant Director 24M2025 13-07-2025

Waqar ul Malik served as------- in the Aligarh university.

A ) President

B ) Secretary 

C ) Treasure

D ) Principal

• Correct Name: Mushtaq Hussain Zuberi 
• Popular Title: Nawab Viqar-ul-Mulk 
• Birth–Death: 1841–1917 
• Position at Aligarh: Honorary Secretary of the Muhammadan Anglo-Oriental (M.A.O.) College, Aligarh 
• Succeeded: Mohsin-ul-Mulk as Honorary Secretary after Mohsin-ul-Mulk's death in 1907. 
• Role: Played a key role in preserving and strengthening the Aligarh Movement after the deaths of Sir Syed Ahmad Khan (1898) and Mohsin-ul-Mulk (1907). 
• Political Role: One of the earliest leaders of the All-India Muslim League. 
• President of Muslim League: Presided over the Second Annual Session of the All-India Muslim League, held at Amritsar from 30–31 December 1908. 
• All-India Muslim League Founded: 30 December 1906, Dhaka. 

Note

• Founder of the Aligarh Movement: Sir Syed Ahmad Khan 
• Founder of M.A.O. College: Sir Syed Ahmad Khan 
• M.A.O. College Established: 24 May 1875 
• Aligarh Muslim University (AMU): Established by the Aligarh Muslim University Act, 1920, which came into effect on 1 December 1920. 
• During Nawab Viqar-ul-Mulk's tenure, it was M.A.O. College, not Aligarh Muslim University. 

 Trick

• 24 May 1875 → M.A.O. College established 
• 27 March 1898 → Sir Syed Ahmad Khan died 
• 30 December 1906 → All-India Muslim League founded (Dhaka) 
• 1907 → Viqar-ul-Mulk became Honorary Secretary of M.A.O. College 
• 30–31 December 1908 → Presided over the Second Annual Session of the Muslim League (Amritsar) 
• 1 December 1920 → M.A.O. College became Aligarh Muslim University 

        Question : First TV station of Pakistan established in which city? 

• First Chairman of PTV: Aslam Azhar , Known As: Father of Pakistan Television
• First Black & White Transmission: 26 November 1964 (Lahore) 
• First Colour Transmission: 1976
• Largest PTV Centre: Karachi
• PTV Headquarters: Islamabad

     Memory Trick

• Lahore = First
• Karachi = Largest
• Islamabad = Headquarters

 Russia has the highest number of television broadcast stations in the world.

Quick Exam Facts

Russia has the largest number of television broadcast stations in the world.2nd: China,3rd :USA 

•  Russia – approximately 6,700+ TV broadcast stations  Capital of Russia: Moscow | Currency: Russian Ruble (RUB) 
•  China  Capital of China: Beijing | Currency: Chinese Yuan (Renminbi – RMB). 
•   USA ,Capital of the USA: Washington, D.C. | Currency: United States Dollar (USD).

1. Quaid-e-Azam was born in 25 December ,1876 in Karachi .
2. In 1900 Quaid-e-Azam became a Presidency magistrate.
3. Quaid-e-Azam joined the All India National Congress in 1906 .
4. Quaid-e-Azam resigned from Congress in 1920.
5. Quaid-e-Azam resigned from the membership of Imperial Legislative Council in 1919.
6.  Maulana Muhammad Ali Johar helped Quaid-e-Azam in the preparation of his 14 points in 1929:
7. Quaid-e-Azam formally became the president of All India Muslim League in 1934.
8. State Bank of Pakistan was inaugurated by Quaid-e-Azam in 1st Jul 1948.
9. After the death of Quaid-e-Azam the next President of the first constitution Assembly of Pakistan was Maulvi Tameez ud Din (December 14, 1948-October 24, 1954)
10.  Khwaja Nazimuddin  2nd Governor-General of Pakistan (1948–1951) ,   2nd  Prime Minister of Pakistan (1951–1953), became  after the death of Quaid-e-Azam Muhammad Ali Jinnah (September 11, 1948 Death date).

Question :The first governor of former NWFP?

A ) Cunningham ✅

B ) Burnie

C ) Mudi

D ) None of these

Explanation :

1. Sir Ralph Griffith became its first Governor 1932 to 1937. First Governor of NWFP after Pakistan’s independence: Sir George Cunningham  15 August 1947–9 April 1948 ( 7 months and 25 days ).
2. Sir Frederick Chalmers Bourne was the first Governor of East Bengal after Pakistan’s creation, serving from 15 August 1947 to 5 April 1950 for approximately 2 years, 7 months and 21 days, and was succeeded by Feroz Khan Noon.
3. Sir Robert Francis Mudie( August 24, 1890 – September 15, 1976 ) was the first Governor of West Punjab after Pakistan’s independence, serving from 15 August 1947 to 2 August 1949, and was succeeded by 2nd Gonvernor of Punjab was Sardar Abdur Rab Nishtar( August 2, 1949, to November 24, 1951) . Previously, Francis Mudie served as Governor of Sindh from 15 January 1946 to 13 August 1947. 
4.  Lieutenant General Riaz Hussain became the first Governor of Balochistan when it received full provincial status on 1 July 1970 and served until 25 December 1971. 

• The term "United Nations" was coined by United States President Franklin D. Roosevelt = 32nd President of the United States
• Pakistan joined the United Nations (UN) on: 30 September 1947.
• Pakistan became the: 55th Member State of the United Nations. 
• The United Nations was established on: 24 October 1945. 
• The UN Charter was signed on: 26 June 1945. 
• The UN Headquarters is located in: New York City, USA. 
• Total UN Member States: 193. 
• Founding Members : 51. 
• Observer States: 2 – Vatican City and State of Palestine. 
• Newest UN Member: South Sudan (joined 14 July 2011). 
• UN Security Council Members: 15 (5 Permanent + 10 Non-Permanent). 
•  First Secretary-General of the United Nations was  Trygve Lie (Norway)
• The United Nations has: 6 official languages. The six official languages are: Arabic, Chinese, English, French, Russian, and Spanish. 
• The primary objective of the UN is: To maintain international peace and security. 
• Another objective of the UN is: To promote friendly relations among nations. 
• The UN also works to: Protect human rights and fundamental freedoms. 
• The UN promotes: International cooperation in solving global economic, social, cultural, and humanitarian issues. 
• Pakistan actively participates in: UN Peacekeeping Missions around the world. 
• Pakistan has served several terms as: A non-permanent member of the UN Security Council. 

Question : Pakistan join SEATO in which year-----:

A ) 1951

B ) 1952

C ) 1953

D ) 1954 ✅

Explanation:

The Southeast Asia Treaty Organization (SEATO) was established through the Manila Pact signed on 8 September 1954, with its headquarters in Bangkok, Thailand. SEATO was officially dissolved on 30 June 1977.

Pakistan joined SEATO on 8 September 1954 to obtain collective security and military assistance from the United States and other Western allies—not specifically “help from the US Army.” Pakistan formally withdrew in 1973 because SEATO had failed to support it during its conflicts with India, particularly the 1965 and 1971 wars.

Correct Timeline

• 1 July 1951: The Colombo Plan formally began; Pakistan was one of its seven founding members. Colombo Plan
• 1952: Pakistan joined INTERPOL—International Criminal Police Organization; its headquarters is in Lyon, France.
• 1952–1953: Pakistan served its first term as a non-permanent member of the UN Security Council.
• 19 May 1954: Pakistan and the United States signed the Mutual Defense Assistance Agreement in Karachi.
• 8 September 1954: Pakistan became a founding member of SEATO.

Question: Pakistan won its first hockey gold medal at the Asian Games in ______.

A) 1958 — Tokyo ✅
B) 1970 — Bangkok
C) 1951 — New Delhi
D) 1982 — New Delhi

Explanation

Pakistan won its first Asian Games men’s hockey gold medal in 1958 at Tokyo, Japan. Field hockey was introduced at the Asian Games in 1958, making Pakistan its first champion.

The inaugural Asian Games were held in New Delhi, India, in 1951, but field hockey was not included. Japan topped the overall 1951 medal table.

Pakistan did not win the 1966 Bangkok hockey tournament; India secured the gold medal. Pakistan regained the title at Bangkok in 1970, which was Pakistan’s third Asian Games hockey gold.

Pakistan won the 1982 New Delhi hockey gold by defeating India 7–1 in the final; this was Pakistan’s sixth title.

Pakistan’s Eight Gold Medals

1. 1958 — Tokyo, Japan
2. 1962 — Jakarta, Indonesia
3. 1970 — Bangkok, Thailand
4. 1974 — Tehran, Iran
5. 1978 — Bangkok, Thailand
6. 1982 — New Delhi, India
7. 1990 — Beijing, China
8. 2010 — Guangzhou, China

Smart sequence: 58, 62, 70, 74, 78, 82, 90, 2010.

Gold-Medal Ranking—July 2026

• Pakistan — 8
• India — 4
• South Korea — 4
• Japan — 1

The 2026 Asian Games are scheduled for 19 September–4 October 2026 in Aichi–Nagoya, Japan, so their results are not included yet.

Question :Pakistan's Majlis-Surah (Parliament) has how many houses?

A ) One

B ) Two ✅

C ) Three

D ) Four

Explanation:

1. National Assembly seats = 336 is correct for current exams:
   266 general + 60 women + 10 non-Muslim = 336
2. Senate seats = 96 is correct current figure. Some old books may show 104, but after FATA merger/reforms, exam-current answer is 96.
3. Current Chairman Senate = Syed Yousaf Raza Gilani is correct as of now. Since “current” posts can change, for long-term notes write:
   Current Chairman Senate: Syed Yousaf Raza Gilani, elected 9 April 2024.

Majlis-e-Shoora / Parliament of Pakistan has two houses:

1. National Assembly = Lower House
   Total seats: 336
   Term: 5 years
   Minimum age: 25 years
   Presiding officer: Speaker
2. Senate = Upper House
   Total seats: 96
   Permanent body: Cannot be dissolved
   Term of Senator: 6 years
   Half members retire after: 3 years
   Minimum age: 30 years
   Presiding officer: Chairman Senate
   Represents: Federating units / provinces

Most repeated MCQs:

National Assembly is the lower house. ✅
Senate is the upper house. ✅
Pakistan Parliament is bicameral. ✅

• Father of Pakistan's Nuclear Program: Dr. Abdul Qadeer (A.Q.) Khan (Birth: 1 April 1936 – Bhopal
  Death: 10 October 2021 (aged 85) – Islamabad)
• Savior of Pakistan : Dr. Abdul Qadeer (A.Q.) Khan 
• Founder of Pakistan: Quaid-e-Azam Muhammad Ali Jinnah (25 December 1876  Karachi (then British India)
  11 September 1948 Karachi)
•  Liaquat Ali Khan  Quaid-e-Millat (Leader of the Nation)
  First Prime Minister of Pakistan
  Tenure: 15 August 1947 – 16 October 1951 at Company Bagh (now Liaquat Bagh).
• 
  

• Largest library of Pakistan: National Library of Pakistan
•  National Library of Pakistan is in: Islamabad
•  Quaid-e-Azam Library is in: Lahore
•  Punjab Public Library is in: Lahore
•  Liaquat Memorial Library is in: Karachi
•  National Archives of Pakistan is in: Islamabad

Question: Which book did Fatima Bhutto write about her family’s political history?

A) Songs of Blood and Sword ✅
B) Pakistan: A Hard Country
C) Descent into Chaos
D) The Begum

Correct Answer: Songs of Blood and Sword: A Daughter’s Memoir

Explanation:
Songs of Blood and Sword was written by Fatima Bhutto and published in 2010. It is a memoir about the Bhutto family’s political history, especially her father Murtaza Bhutto, who was killed in 1996.

Whispers of the Desert — 1998

8:50 a.m. 8 October 2005 — 2006

Songs of Blood and Sword — 2010

The Shadow of the Crescent Moon — 2013

The Runaways — 2019

New Kings of the World — 2019

Other options:
Pakistan: A Hard Country — Anatol Lieven, 2011
Descent into Chaos — Ahmed Rashid, 2008
The Begum — Deepa Agarwal & Tahmina Aziz Ayub, 2019.

These are two different books with similar titles:

• The Whispers in the Desert — written by Lesley Smith, published in 2014; part of The Changing of the Sun series.
• Whispers of the Desert — written by Fatima Bhutto, published in 1998 by Oxford University Press; a collection of poems.

Question: Anatol Lieven’s book on Pakistan’s political, social and tribal culture is ——?

A) Descent into Chaos
B) Pakistan: A Modern History
C) The Begum
D) Pakistan: A Hard Country ✅

Correct Answer: Pakistan: A Hard Country

Explanation:

Pakistan: A Hard Country was written by Anatol Lieven and published in 2011. The book discusses Pakistan’s political, social, state, military, religious, family and tribal structure.

Other options:

Descent into Chaos — written by Ahmed Rashid, published in 2008. It is about post-9/11 politics, Taliban, Afghanistan, Pakistan, Central Asia and U.S. policy.

Pakistan: A Modern History — written by Ian Talbot, published in 1998. It is about the political and modern history of Pakistan.

The Begum — written by Deepa Agarwal and Tahmina Aziz Ayub, published in 2019. It is about Begum Ra’ana Liaquat Ali Khan, wife of Liaquat Ali Khan, Pakistan’s first Prime Minister.

Tulujha fort is located in ------

A ) Kagan valley

B ) Soon Valley 

C ) Swat Valley

D ) Neelum valley

1. Kaghan Valley 

Kaghan Valley is associated with Mansehra District, Khyber Pakhtunkhwa.

Note

• Kaghan Valley → Mansehra → Khyber Pakhtunkhwa
• Naran → Kaghan Valley
• Saif-ul-Muluk → near Naran, Kaghan Valley
• Babusar Pass → northern end of Kaghan Valley
• Kunhar River → flows through Kaghan Valley 

2. Soon Valley 

Soon Valley is in Khushab District, Punjab. 

Note 

• Tulajha Fort → Soon Valley
• Soon Valley → Khushab District
• Khushab → Punjab
• Soon Valley → Salt Range
• Sakesar → highest peak of Salt Range
• Sakesar → 1,522 m
• Uchhali, Khabeki and Jahlar → important lakes of the area 

3. Swat Valley 

Swat is in Khyber Pakhtunkhwa, specifically Malakand Division. 

Note 

• Swat Valley → Khyber Pakhtunkhwa
• Swat District → Malakand Division
• Mingora → Swat
• Saidu Sharif → Swat
• Malam Jabba → Swat
• Swat River → Swat Valley
• Swat → important Gandhara/Buddhist archaeological region

4. Neelum Valley 

Neelum Valley is in Azad Jammu and Kashmir (AJK). 

Note

• Neelum Valley → AJK
• Neelum Valley → Neelum District
• Athmuqam → Neelum
• Sharda → Neelum Valley
• Kel → Neelum Valley
• Keran → Neelum Valley
• Arang Kel → famous tourist locality near Kel
• Neelum River → principal river of the valley

What is the national river of Pakistan?

A ) Indus 

B ) Ravi

C ) Jhelum

D ) Chenab

• National/principal river → Indus — Principal/main river of Pakistan 
• Tarbela Dam → Indus 
• Mangla Dam → Jhelum 
• Lahore → Ravi — Lahore is situated along/near the Ravi.
• Hydaspes → Jhelum — Hydaspes is the ancient Greek name.
• Chandra + Bhaga → Chenab — their confluence forms the Chenab (called Chandrabhaga )
• Eastern rivers → Ravi, Beas, Sutlej — under the Indus Waters Treaty (1960).
• Western rivers → Indus, Jhelum, Chenab — under the Indus Waters Treaty (1960).
• Indus empties into → Arabian Sea — through the Indus Delta.
• Treaty trick:
  Eastern = RBS → Ravi, Beas, Sutlej
  Western = IJC → Indus, Jhelum, Chenab

Murree altitude? → ~7,500 ft (2286 meter) 
Murree province? → Punjab
Murree is famous as? → Hill station
Mall Road? → Murree
Kashmir Point? → Murree
Pindi Point? → Murree
Patriata? → New Murree
Patriata famous for? → Chairlift/Cable Car

Rule:

The adjective good is always followed by the preposition at when talking about skill or ability.

Structure:

Subject + be verb + good + at + noun / verb + ing

Examples:

 1.She is good at English.  2.Ali is good at mathematics.  3.He is good at swimming.  4.They are good at solving problems.  5.She is good at playing the piano.

Exam Tip

Good at = Skill or ability

Remember:

Good + at + Verb-ing / Noun

Examples:

•  1.Good at cricket. 2. Good at teaching. 3 Good at writing essays.  4 Good at playing the piano. ✅

Rules of "Arrive" with Prepositions 

1. Arrive at → Used for small places or specific points. 
   •  Examples: 
     •  He arrived at the station. 
     •  She arrived at the airport. 
     •  They arrived at the office. 
2. Arrive in → Used for cities, countries, and large areas. 
   •  Examples: 
     •  She arrived in Lahore. 
     •  They arrived in Pakistan. 
     •  We arrived in Asia. 
3. Arrive on → Used for islands and means of transport (less common). 
   •  Examples: 
     •  We arrived on the island. 
     •  The passengers arrived on the ship. 

Quick Exam Trick

• At → Small place / Specific point 
• In → Large place (city, country, region) 
• On → Island or transport (rare in exams) 

Remember

❌ Never use to after arrive.

•  ❌ He arrived to the station. 
•  ✅ He arrived at the station. 
•  ✅ He arrived in Lahore. 

 Rule

The adjective allergic is always followed by the preposition to.

Allergic + to + noun/pronoun

 Example

•  She is allergic to cats. 
•  He is allergic to dust. 
•  I am allergic to peanuts. 

Common Mistakes

• ❌ allergic with cats
•  ❌ allergic from cats
•  ❌ allergic of cats

 ✅ allergic to cats

Remember the pattern:

• Allergic → TO
•  Addicted → TO
• Accustomed → TO

Quick Memory Tip 

Allergic = TO (Always)

• ✔ Allergic to dust
• ✔ Allergic to milk
• ✔ Allergic to cats
• ✔ Allergic to pollen

 Rule

The verb belong is always followed by the preposition to.

Belong + to + person/thing

 Examples

•  The book belongs to me. 
•  This car belongs to Ali. 
•  The house belongs to my uncle. 
•  These keys belong to the teacher. 

 Common Mistakes

❌ belongs me
 ❌ belongs with me
 ❌ belongs for me
 ✅ belongs to me

 Exam Trick

Remember these common verb + preposition combinations:

• Belong → to
• Listen → to
• Refer → to
• Object → to
• Apologize → to (someone) 

Quick Memory Tip 

Belong = TO (Always)

  ✔ Belongs to me
   ✔ Belongs to Ali
  ✔ Belongs to the school
   ✔ Belongs to our family

 Rule

The verb accuse is always followed by the preposition of.

Accuse + someone + of + crime/wrongdoing

 Meaning

Accuse of means to say that someone has committed a crime or done something wrong.

Examples:

•  He was accused of stealing the money. 
•  She accused him of lying. 
•  They accused the officer of corruption. 
•  The student was accused of cheating. 

 Common Errors (Frequently Asked in Competitive Exams)

❌ accused for stealing
 ❌ accused with stealing
 ❌ accused to stealing
 ❌ accused about stealing

✅ accused of stealing

Trick

Memorize these common Verb + Preposition combinations:

Accuse of → He was accused of stealing the money.

Convict of → The court convicted him of murder. 

Suspect of → The police suspected her of fraud. 

Blame for → He was blamed for the mistake. 

Charge with → The police charged him with theft.

 Exam Tip: Don't confuse these similar verbs:

• Accused of a crime 
• Charged with a crime 
• Blamed for a mistake 

Accuse → OF (Always)

✔ Accused of theft
✔ Accused of cheating
✔ Accused of lying
✔ Accused of corruption

Complete Sentence:
He gave up smoking last year.

1.  Correct Phrasal Verb

Give up + Gerund (V₁ + ing) = to stop a habit permanently

Structure:
Subject + gave up + Verb-ing + Time

Example:

•  He gave up smoking last year. 
•  She gave up drinking. 
•  They gave up gambling. 

2. Why "Smoking"?

After give up, the next verb always takes the -ing form (Gerund).

Rule:
Give up + V-ing (Gerund) 

✔ give up smoking
✔ give up playing
✔ give up reading

❌ give up smoke
 ❌ give up to smoke

3.  Why "Gave"?

The phrase "last year" shows past time, so we use the Simple Past Tense.

Signal Words of Simple Past:

•  yesterday 
•  last year 
•  last month 
•  ago 
•  in 2020 

4. Exam Trick

Whenever you see:

• give up
• quit
• avoid
• enjoy
• finish
• admit
• deny
• mind

      ✅ The next verb is always in the -ing form (Gerund).

5. ✅ Memory Rule

"Give Up = Stop Habit = V-ing"

Formula:
Subject + gave up + Verb-ing + Time

Common Exam Examples

•  He gave up smoking last year. ✅ 
•  She gave up dancing after marriage. ✅ 
•  They gave up playing cricket. ✅ 
•  I gave up wasting time. ✅ 

Question:
If a business's assets increased by 25% in the second year compared to the first year, and the current assets are Rs. 44,500,000, what were the assets in the first year?

Formula:
First Year Assets = Current Assets / (100% + Increase%) × 100

Given:
Current Assets = Rs. 44,500,000
Increase = 25%

Solution:
First Year Assets = 44,500,000 / 125 × 100
                  = 356,000 × 100
                  = Rs. 35,600,000

✅ Final Answer:
First Year Assets = Rs. 35,600,000

Shortcut Trick (Competitive Exams)

25% Increase ⇒ Current = 125%

First Year = Current / 125 × 100

44,500,000 / 125 = 356,000

356,000 × 100 = 35,600,000

✅ Answer = Rs. 35,600,000

Exam Trick

• 10% increase → Divide by 110 × 100
• 20% increase → Divide by 120 × 100
• 25% increase → Divide by 125 × 100
• 50% increase → Divide by 150 × 100

Formula

Percentage Formula Percentage = (Present Teachers / Total Teachers) × 100

Given

•  Present Teachers = 170
•  Total Teachers = 200

Solution

 Percentage = (170 ÷ 200) × 100 = 0.85 × 100 = 85%

✅ Final Answer

85% of the teachers were present that day.

Shortcut Trick (Fastest Method)

Since 200 = 2 × 100

170 ÷ 2 = 85

✅ Answer = 85%

Question:
In the system of equations
3x + 2y = 7
6x + my = 14
for which value of m will the system have infinitely many (many) solutions?
Solution:

Condition for infinitely many solutions:

a₁/a₂ = b₁/b₂ = c₁/c₂
Here,

a₁ = 3,  a₂ = 6 ,b₁ = 2,  b₂ = m ,c₁ = 7,  c₂ = 14
So
3/6 = 2/m = 7/14
Since,
3/6 = 7/14 = 1/2
So,
2/m = 1/2
Cross multiply:
2 × 2 = m
m = 4

Shortcut Trick (Competitive Exams):
6 = 2 × 3
14 = 2 × 7
The second equation is twice the first equation, so the coefficient of y must also be doubled:
2 × 2 = 4
Therefore,
m = 4
✅ Final Answer:
m = 4

If the circumstances of a circule is equal to the perimeter of a square then the area of the circle is:

Given:

• Circumference of circle = Perimeter of square

2πr = 4a

a = πr/2

Area of circle = πr²

Area of square = (πr/2)² = π²r²/4

Therefore,

Area of Circle / Area of Square = (πr²) / (π²r²/4) = 4/π

Final Answer:

Area of Circle = (4/π) × Area of Square

Shortcut to Remember:

If the circumference of a circle equals the perimeter of a square, then:

Area of Circle : Area of Square = 4 : π ✅

Key Concept Explanation 

•  Circumference of a circle = C = 2πr
•  Perimeter of a square = P = 4a
•  Area of a circle = A = πr²
•  Area of a square = A = a²

Where:

• r = radius of the circle 
• a = side of the square 
• π ≈ 3.1416

If the age of a person after 15 years will be 4 times his age 15 years ago what is his current age? 

Present Age = n × (k + 1) ÷ (k − 1)

Where:

• n = years (15) 
• k = multiplier (4) 

So,

Present Age = 15 × (4 + 1) /(4 − 1)

= 15 × 5 / 3

=75/3

= 25 years ✅

Shortcut to Remember

If "After n years = k × Before n years", then:

Present Age = n(k + 1)/(k − 1)

If two equations are y=2 and y=-5 how many solutions does the system of equations have?

Solution :

• If both equations are y = constant:
  • Different constants → 0 solutions (Parallel lines)
  • Same constants → Infinite solutions (Same line)

Here:

y = 2

y = -5

Since 2 ≠ -5, the lines are parallel and never intersect.

Answer: 0 (No Solution) ✅

Competitive Exam Rule:

• Different slopes → 1 solution
• Same slope, different intercepts → 0 solutions
• Same slope, same intercept → Infinitely many solutions

Solution :

Vertices:

• A = (0,4)
• B = (0,0)
• C = (3,0)

Find the three sides:

• AB = 4 (vertical distance)
• BC = 3 (horizontal distance)

For AC, use the Pythagorean theorem:

AC = √[(3 − 0)² + (0 − 4)²]

= √(9 + 16)

= √25

= 5

Perimeter = AB + BC + AC

= 4 + 3 + 5

= 12

Final Answer: 12 units ✅

Golden Rule for Competitive Exams

•  Same x-coordinate → Distance = Difference of y-values
•  Same y-coordinate → Distance = Difference of x-values

If a girl mistakenly multiply a number by 3 instead of dividing it by 3 the results was what percentage of the original number? 

Solution 

Assume the original number = 3 (choose a number divisible by 3).

Correct operation:

3 ÷ 3 = 1

Mistaken operation:

3 × 3 = 9

Now compare the mistaken result with the original number:

(9 ÷ 3) × 100 = 300%

Answer: 300% ✅

Don't Get Confused 

• If asked "Percentage of the original number":
• (3N ÷ N) × 100 = 300% ✅
• If asked "Percentage of the correct result":
• Correct result = N ÷ 3
• Mistaken result = 3N
• (3N ÷ (N ÷ 3)) × 100
• = 9 × 100
• = 900% ✅

Remember:

• Compared with the original number → 300%
• Compared with the correct result → 900%