Explanation

log(x² y³) - 2log(x) 

Use 2 log rules:

1. k·log(a) = log(aᵏ) → 2log(x) = log(x²)

2. log(A) - log(B) = log(A/B)

So: log(x² y³) - log(x²) = log[(x² y³)/x²] = log(y³)

And log(y³) = 3log(y)

Last verified on 09-06-2026

Explanation

For polynomials with 2+ variables, degree = highest sum of exponents in any single term.

Check each term of 2x⁴ y³ + x² y² + 8x:

1. 2x⁴ y³ → 4 + 3 = 7

2. x² y² → 2 + 2 = 4

3. 8x → 1

Highest is 7, so degree = 7

Last verified on 09-06-2026

Explanation

For equal roots, discriminant D = 0

For 3x² + 2x + m = 0:  

a = 3, b = 2, c = m  

D = b² - 4ac = 2² - 4(3)(m) = 4 - 12m

Set D = 0:  

4 - 12m = 0  

12m = 4  

m = 1/3

Last verified on 09-06-2026

Explanation

Given x + 1/x = 3

Square both sides:

(x + 1/x)² = 3²  

x² + 2·x·(1/x) + 1/x² = 9  

x² + 2 + 1/x² = 9

So x² + 1/x² = 9 - 2 = 7

Last verified on 09-06-2026

Explanation

A complex number is z = x + iy, where x = real part, y = imaginary part.

1. Purely imaginary → real part is 0, so x = 0. Then z = 0 + iy = iy. Ex: 3i, -5i

2. Imaginary in general just means y ≠ 0. But if a question says "is imaginary" without "purely", they usually mean "not real", so y ≠ 0.

"x + iy is imaginary if x = 0" → purely imaginary.

If they meant "non-real", then condition is y ≠ 0.

Last verified on 09-06-2026

Explanation

Let roots be r and r+2, since they differ by 2.

For x² + kx + 4 = 0:

1. Sum of roots = -b/a = -k → r + (r+2) = 2r + 2 = -k → r = (-k-2)/2

2. Product of roots = c/a = 4 → r(r+2) = 4

Plug r in: [(-k-2)/2] × [(-k-2)/2 + 2] = 4  

→ [(-k-2)/2] × [(2-k)/2] = 4  

→ (-k-2)(2-k) = 16

Simplify: k² - 4 = 16 → k² = 20 → k = ±2√5

So k can be 2√5 or -2√5.

Last verified 09-06-2026

Explanation

• The length of Sir Creek is about 96 km (60 miles).
• It is a tidal estuary (inlet channel of water).
• It is located on the border of India and Pakistan.
• It is part of the Indus River Delta system.
• It flows into the Arabian Sea.

Last verified on 02-05-2026

Explanation

• The Indus Valley Civilization (IVC) flourished around 2500 BC in South Asia.
• Major cities include Harappa, Mohenjo-Daro, and Dholavira.
• It developed advanced town planning, drainage systems, and trade networks.
• The civilization declined around 1900 BC.
• Early development: 3300 BC (Early Harappan phase).

Last verified on 08-06-2026

Explanation

• First Introduced: 1933 by Chaudhary Rahmat Ali.
• Pamphlet: Titled "Now or Never" meaning Are We to Live or Perish Forever?

Additional information:

• First Use of Pakistan: Word appeared for the first time in this pamphlet.
• Based On: Two-nation theory of Syed Ahmed Khan.

Regions Represented (Pakistan):

1. Punjab
2. Afghanistan
3. Kashmir
4. Sindh
5. Baluchistan

Last verified on 24-03-2026

Explanation

• Hamoon-e-Mashkel (Hamun-e-Mashkel) is a saltwater lake located in Washuk District of Balochistan, Pakistan.
• Type: Saltwater (Saline) Lake
• Region: Balochistan Plateau
• Nearby border: Iran
• Nature: Often seasonal (depends on rainfall and water inflow)

Last verified on 08-06-2026