Explanation

Given: x² - 7x + 10 = 0

Step 1: Find original roots

x² - 7x + 10 = (x - 2)(x - 5) = 0

So roots are 2 and 5

Step 2: Multiply roots by 3

New roots = 2 × 3 = 6, 5 × 3 = 15

Step 3: Form new equation

If roots are α and β, equation is x² - (α+β)x + αβ = 0

Here α+β = 6 + 15 = 21

     αβ = 6 × 15 = 90

Therefore new equation = x² - 21x + 90 = 0

Last verified on 15-05-2026

Explanation

Let the two factors be (x - a) and (x - b)

Then:

1. (x - a)(x - b) = x^2 - (a + b)x + ab

2. Compare with x^2 - px + 48:

   ab = 48

   a + b = p

3. Given: a - b = 2 => a = b + 2

Substitute:

(b + 2)b = 48

b^2 + 2b - 48 = 0

(b + 8)(b - 6) = 0

So b = 6 or b = -8

If b = 6, then a = 8, p = a + b = 14

If b = -8, then a = -6, p = a + b = -14

Last verified on 15-05-2026

Explanation

(a + bi)(a − bi) = a² + b²

Given:

(a + bi)(a − bi) = 41

Therefore:

a² + b² = 41

Last verified on 15-05-2026

Explanation

• Karezes are traditional underground water channels used in arid areas.
• In Balochistan, some karezes are still functioning today.
• According to the chapter, they are found in Zhob, Loralai, Pishin, Quetta, and Mastung districts.
• These systems are used for irrigation and drinking water supply in dry regions.
• They are an ancient and sustainable water management method.

Last verified on 14-05-2026

Explanation

• Afghanistan was the only country that voted against Pakistan admission to the United Nations in 1947.
• Pakistan officially joined the UN on 30 September 1947, shortly after independence on 14 August 1947.
• Afghanistan opposed Pakistan mainly because of the Durand Line dispute and the issue of Pashtunistan.

Additional information:

• UN Headquarters: New York City
• Founded: 24 October 1945
• Founder Nations: 51 original member states
• Official Languages: Arabic, Chinese, English, French, Russian, Spanish

Last verified on 14-05-2026

Explanation

• The Hindu Kush range starts near the Pamir Mountains where the borders of Afghanistan, Pakistan, and China meet.
• Starting Point: Pamir Mountains / Pamir Plateau
• Famous Name: “Roof of the World”
• Length: About 800 km (500 miles)
• Region: Pakistan and Afghanistan
• Highest Peak in Hindu Kush: Tirich Mir
• Height of Tirich Mir: 7,708 meters (25,289 ft)

Last verified on 14-05-2026

Explanation

• Passed: July 1947
• Royal Assent: 18 July 1947
• Independence Date: 14–15 August 1947
• Last Viceroy: Lord Mountbatten
• Pakistan’s Governor-General: Muhammad Ali Jinnah
• Pakistan’s First Prime Minister: Liaquat Ali Khan
• India’s First Prime Minister: Jawaharlal Nehru

Last verified on 14-05-2026

Explanation

• Date: 24 June 1945
• Presided by: Lord Wavell
• Place: Viceregal Lodge, Simla
• Main parties: Congress & All India Muslim League
• Purpose: Formation of Executive Council and constitutional settlement

Last verified on 14-05-2026

Explanation

• In 1971, Pakistan faced a major political crisis.
• Tensions between East and West Pakistan had already increased.
• Political unrest turned into armed conflict.
• Military action was taken to control the situation.
• This ultimately led to the separation of East Pakistan.

Last verified on 14-05-2026

Explanation

• Pakistan was originally made of two wings: East and West Pakistan.
• Both regions had cultural, political, and geographical differences.
• Governance style played an important role in unity or division.
• Excess control from the center reduced trust between regions.
• This increased dissatisfaction in East Pakistan.

Last verified on 14-05-2026