Explanation

Rationalize: 1/(2+i) × (2-i)/(2-i) = (2 - i)/(4 + 1) = (2 - i)/5 = 2/5 - (1/5)i

So imaginary part = -1/5

Last verified on 09-06-2026

Explanation

For real roots, discriminant D ≥ 0  

Given: x² – 2kx + k² – 1 = 0  

Here a=1, b=-2k, c=k² - 1

D = b² − 4ac = (-2k)² − 4(1)(k² − 1)  

D = 4k² − 4k² + 4 = 4

D = 4 > 0 for every k. 

Since D is always positive, roots are real and unequal for all real k.

Last verified on 09-06-2026

Explanation

Factor the first expression:  

a³ + b³ = (a + b)(a² − ab + b²) → sum of cubes formula

Second expression: a² − ab + b²

So LCM = product of common + non-common factors  

= (a + b)(a² − ab + b²)  

= a³ + b³

Last verified on 09-06-2026

Explanation

log₃x = 4 means 3⁴ = x  

3⁴ = 3 × 3 × 3 × 3 = 81

Last verified on 09-06-2026

Explanation

• Balochistan covers about 43.6% (44%) of Pakistan’s total area.
• Area: 347,190 sq km
• Population share: 5–6%
• Capital: Quetta
• Largest province by area in Pakistan: Balochistan

Last verified on 09-06-2026

Explanation

• The Durand Line was drawn in 1893.
• It was established as the official border between British India and Afghanistan.
• British representative: Sir Mortimer Durand
• Afghan ruler: Amir Abdur Rahman Khan
• It later became the international border between Pakistan and Afghanistan.
• Length: Around 2,640 km

Last verified on 09-06-2026

Explanation

• The “Quit India” Movement was launched by Mahatma Gandhi against British rule.
• It was announced on 8 August 1942 at the All India Congress Committee session in Bombay.
• The slogan of the movement was “Do or Die”.
• Key leaders arrested: Gandhi, Nehru, Congress leaders
• Mass arrests: 9 August 1942
• Movement type: Mass civil disobedience
• Goal: End British colonial rule

Last verified on 09-06-2026

Explanation

• The second highest plateau in the world located in Pakistan is the Deosai Plains.
• Location: Gilgit-Baltistan, Pakistan
• Height: ~4,114 meters above sea level
• Area: Around 3,000+ square kilometers
• Famous river/lake: Sheosar Lake
• Status: Deosai National Park
• It is known as the “Land of Giants” due to its high altitude and unique landscape.

Additional information:

• The highest plateau in the world is the Tibetan Plateau.

Last verified on 09-06-2026

Explanation

• The 1956 Constitution of Pakistan abolished the Monarchy system.
• It officially ended the British monarch’s role as head of state.
• Pakistan adopted a republic system with a President instead of a King or Queen.
• The Governor-General post was also replaced by the President.

Additional information:

• Constitution enforced: 23 March 1956
• First Islamic Republic declared: Pakistan
• First President: Iskander Mirza
• System introduced: Parliamentary republic framework

Last verified on 09-06-2026

Explanation

Objectives Resolution (قراردادِ مقاصد)

• Presented By: Prime Minister Liaquat Ali Khan

  Adopted: 12 March 1949

• Adopting Body: Constituent Assembly of Pakistan

• Date of Presentation: 7 March 1949

Additional information:

• Total Members of Assembly: 75

• Members Supporting the Resolution: 21

• Minority Members: 10

• All amendments proposed by minority members were rejected.

• As a result, all 10 minority members voted against the resolution.

Last Updated on 12-01-2026