Explanation

If roots of x² - 9x + 20 = 0 are increased by 1, the new equation is x² - 11x + 30 = 0

Solution:

Step 1: Find the roots of x² - 9x + 20 = 0

Factorize:

x² - 9x + 20 = 0

x² - 5x - 4x + 20 = 0

x(x - 5) - 4(x - 5) = 0

(x - 4)(x - 5) = 0

So the roots are x = 4 and x = 5

Step 2: Increase each root by 1

New roots = 4 + 1 = 5 and 5 + 1 = 6

Step 3: Form the new equation

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

Here α = 5, β = 6

Sum of roots = 5 + 6 = 11

Product of roots = 5 * 6 = 30

Therefore, new equation is:

x² - 11x + 30 = 0

Last verified on 24-05-2026

Explanation

If x² - px + 72 has factors differing by 1, then p = ±17

Solution in older version:

Let the two factors be m and m+1. 

Then,

x² - px + 72 = (x - m)(x - (m+1))

             = (x - m)(x - m - 1)

             = x² - (2m + 1)x + m(m+1)

Compare with x² - px + 72:

1. m(m+1) = 72

2. p = 2m + 1

Solve m(m+1) = 72:

m² + m - 72 = 0

(m + 9)(m - 8) = 0

So m = 8 or m = -9

Case 1: m = 8

Factors are 8 and 9

p = 2_8 + 1 = 17

Case 2: m = -9  

Factors are -9 and -8

p = 2_(-9) + 1 = -17

Last verified on 24-05-2026

Explanation

(a + bi)(a - bi) is the product of conjugates.  

(a + bi)(a - bi) = a² - (bi)² = a² - b²i²  

Since i² = -1:  

= a² - b²(-1) = a² + b²  

Given (a + bi)(a - bi) = 61,  

so a² + b² = 61.

Last verified on 24-05-2026

Explanation

• Balochistan Plateau is mostly arid and mountainous.
• It has very low rainfall and scarce water resources.
• Rough terrain limits large-scale intensive agriculture.
• Farming is mostly limited to small irrigated areas.

Last verified on 24-05-2026

Explanation

• Rainfall varies mainly due to monsoon winds and topography.
• Monsoon brings seasonal moist winds causing rainfall.
• Relief (mountains) forces air to rise, increasing rainfall.
• Together they create uneven rainfall patterns.

Last verified on 24-05-2026

Explanation

• A shared national narrative helps unify diverse ethnic and regional groups.
• It promotes a common identity as Pakistanis.
• It strengthens national cohesion and solidarity.
• It reduces divisions based on language, region, or ethnicity.

Last verified on 24-05-2026

Explanation

• The ideology of Pakistan is based on the Two-Nation Theory.
• It states Muslims and Hindus are separate civilizations.
• Religion, culture, and values shaped this difference.
• This idea justified the demand for a separate homeland.

Last verified on 24-05-2026

Explanation

• Muslim self-determination in South Asia was mainly driven by fear of majority rule.
• Muslims believed they could lose political rights under a Hindu-majority system.
• This concern strengthened the demand for a separate homeland.
• It became a key argument in the Pakistan Movement.

Last verified on 24-05-2026

Explanation

• Quantum theory was first introduced by Max Planck in 1900.
• He proposed that energy is emitted in small packets called “quanta.”
• This idea started modern quantum physics.
• Later scientists like Einstein and Bohr developed it further.

Last verified on 24-05-2026

Explanation

• Photoelectric effect shows light behaves like particles (photons).
• It cannot be explained by wave theory alone.
• Einstein explained it using quantum theory of light.
• Light transfers energy in small packets called photons.

Last verified on 24-05-2026