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

Explanation

If aₙ means the coefficient of xⁿ in the power series/Taylor series of f(x) = x - x³, then:

f(x) = x - x³ = 0 + 1·x + 0·x² + (-1)·x³ + 0·x⁴ + 0·x⁵ + ...

So:

a₀ = 0

a₁ = 1

a₂ = 0

a₃ = -1

a₄ = 0, a₅ = 0, ... all higher terms = 0

0 is also a finite value.

Last verified on 04-06-2026

Explanation

Formula: Ā × B̄ = |Ā||B̄| sinθ 

For non-zero vectors, |Ā||B̄| is never 0.  

So cross product = 0 only when sinθ = 0 → θ = 0° or 180°

That’s exactly same direction or opposite direction = parallel.  

This works for ALL parallel cases.

So correct answer is = Ā × B̄ = 0̄

Last verified on 04-06-2026

Explanation

x + 2 is a factor = Put x = -2 in polynomial. If answer comes 0, then yes it’s a factor.

x⁴ + 2x³ - x - 2 → 16 - 16 + 2 - 2 = 0 → 0

Last verified on 04-06-2026

Explanation

1. AB is one side → D = A + C - B = (-1+1-6, 2-20-2) = (-6, -20)
2. AC is one side → D = A + B - C = (-1+6-1, 2+2+20) = (4, 24)
3. BC is one side → D = B + C - A = (6+1+1, 2-20-2) = (8, -20)

Explanation

f(x) = e^x - ax makes a U-shape because e^x is always curving up. 

U-shape has only 1 lowest point at x = ln a. 

So the minimum value = f(ln a) = a - a ln a

Last verified on 04-06-2026

Explanation

(1 + i)¹⁰ = (√2)¹⁰ × [cos(45°×10) + i sin(45°×10)]

           = 2⁵ × [cos450° + i sin450°]

           = 32 × [cos90° + i sin90°] 450° = 360° + 90°

           = 32 × [0 + i×1]

           = 32i

Last verified on 04-06-2026

Explanation

System:

x1 + 2x2 + x3 = 9

2x1 + x2 + 3x3 = 7

Matrix form AX = B:

[1 2 1] [x1] [9]

[2 1 3] [x2] = [7]

          [x3]

Augmented matrix [A|B]:

[1 2 1 | 9]

[2 1 3 | 7]

Last verified on 04-06-2026