Explanation

u = x² + y²  

First take ∂u/∂y = 0 + 2y = 2y  

Now take ∂/∂x of 2y = 0, because 2y has no x

So ∂²u/∂x∂y = 0

Last verified on 05-06-2026

Explanation

We know |A × B| = |A||B| sinθ and |A . B| = |A||B| cosθ  

Put them equal: |A||B| sinθ = |A||B| cosθ  

Cancel |A||B|: sinθ = cosθ  

So tanθ = 1 → θ = 45°

Last verified on 05-06-2026

Explanation

RHS all 0 → homogeneous system  

Find det of 3 2 1; 2 1 4; 1 4 1 = -34  

det ≠ 0 → only x1=x2=x3=0 works

det = 0 would mean infinite solutions. Here it’s not 0.

Last verified on 05-06-2026

Explanation

• Vicarious means بدحواس
• Delegated means تفویض کردہ

Last verified on 05-06-2026

Explanation

• Admonish means نصیحت
• Warn means انتباہ
• Counsel means صلاح
• Advise means مشورہ دینا
  Threaten means کسی کو مضبوطی سے متنبہ کرنا یا سرزنش کرنا

Last verified on 05-06-2026

Explanation

Parabola: y = x² - 4  

Circle: x² + y² = 9  

Put y: x² + (x² - 4)² = 9  

x⁴ - 7x² + 7 = 0  

Let t = x²: t² - 7t + 7 = 0  

D = 49 - 28 = 21 > 0  

Both t roots positive => 2 x values each  

Total intersections = 4

Last verified on 05-06-2026

Explanation

For real coefficients, complex roots come in conjugate pairs.

Given roots: 2 + i and 1 - i  

So must also have: 2 - i and 1 + i

Build polynomial:

1) From 2±i: (x - (2+i))(x - (2-i)) = (x-2)² + 1 = x² - 4x + 5  

2) From 1±i: (x - (1+i))(x - (1-i)) = (x-1)² + 1 = x² - 2x + 2  

Multiply: (x² - 4x + 5)(x² - 2x + 2)  

= x⁴ - 2x³ + 2x² - 4x³ + 8x² - 8x + 5x² - 10x + 10  

= x⁴ - 6x³ + 15x² - 18x + 10

Last verified on 05-06-2026  

Last verified on 05-06-2026

Explanation

Given hyperbola: y² = (x/a)² + 1 → y² - x²/a² = 1

This is of form y²/a₀² - x²/b² = 1 with transverse axis along y-axis.

Here a₀² = 1, b² = a²

For hyperbola: c² = a₀² + b², foci at (0, ±c)

Given one focus = (0, √2) → c = √2 → c² = 2

So 2 = 1 + a²  

→ a² = 1  

→ a = ±1

Usually we take a > 0, so a = 1

Last verified on 05-06-2026

Explanation

• L'Hospital's Rule is only used when you get 0/0 or ∞/∞ form after direct substitution.

Conditions for L'Hospital's Rule:

1. Indeterminate form → Must be 0/0 or ∞/∞ at the limit point. If you get 5/2 or 7, don’t use L'Hospital.
2. Both numerator + denominator must be differentiable near the point
3. Denominator’s derivative ≠ 0 near the point

So "indeterminate" is the key condition.

Last verified on 05-06-2026

Explanation

Last verified on 04-06-2026