Explanation

Equation 3x² + 7xy + 2y² = 0 represents a pair of lines through origin.

Factor it:

3x² + 6xy + xy + 2y² = 0  

3x(x + 2y) + y(x + 2y) = 0  

(3x + y)(x + 2y) = 0

So lines are: y = -3x and y = -x/2  

Slopes: m₁ = -3, m₂ = -1/2

Angle between lines:

tan θ = |(m₁ - m₂)/(1 + m₁m₂)|  

= |(-3 + 1/2) / (1 + 3/2)|  

= |(-5/2) / (5/2)| = | -1 | = 1

tan θ = 1 → θ = 45°

So the acute angle between the two lines is 45°.

Last verified on 05-06-2026

Explanation

∂²u/∂x² + ∂²u/∂y² = f(x,y) is the 2D Poisson equation.  

If f(x,y) = 0, then it becomes Laplace's equation: ∇²u = 0

Last verified on 05-06-2026

Explanation

Jacobian J = ∂(x,y)/∂(r,θ) = |∂x/∂r ∂x/∂θ|

                               |∂y/∂r ∂y/∂θ|

∂x/∂r = cosθ, ∂x/∂θ = -r sinθ  

∂y/∂r = sinθ, ∂y/∂θ = r cosθ

J = cosθ × r cosθ - (-r sinθ) × sinθ  

  = r cos²θ + r sin²θ  

  = r(cos²θ + sin²θ) = r

Last verified on 05-06-2026

Explanation

Sₙ = n/(n³+0) + n/(n³+1) + ... + n/(n³+n-1) = Σₖ₌₀ⁿ⁻¹ n/(n³ + k)

Each term ≤ n/n³ = 1/n²  

There are n terms → Sₙ ≤ n × 1/n² = 1/n → 0 as n→∞

Lower bound: n/(n³+n-1) × n = n²/(n³+n-1) → 0

By squeeze theorem, limit = 0

Last verified on 05-06-2026

Explanation

Orthogonal matrix definition: Aᵀ = A⁻¹  

Multiply both sides by A → AᵀA = I

Last verified on 05-06-2026

Explanation

• “Of Muslim States” requires the superlative degree of the adjective.
• Greatest is the superlative form of great (great → greater → greatest).

Last verified on 05-06-2026

Explanation

• The sentence is in Simple Present Tense: Lion does not eat grass.
• Passive voice structure: Object + is/am/are + not + past participle + by + subject.
• Therefore, “Grass is not eaten by a lion, however hungry he may be.” is correct.

Last verified on 05-06-2026

Explanation

Let width = W, length = L. Fence for 3 sides: L + 2W = x → L = x - 2W  

Area A = L×W = (x - 2W)W = xW - 2W²  

For max, dA/dW = x - 4W = 0 → W = x/4  

Then L = x - 2(x/4) = x/2  

Max area = L×W = (x/2)(x/4) = x²/8

Last verified on 05-06-2026

Explanation

Sum of coefficients = put a = 1, b = 1 in expression  

(a - b)¹⁰ → (1 - 1)¹⁰ = 0¹⁰ = 0

Last verified on 05-06-2026

Explanation

Inflection point: y'' = 0  

y' = 3x² - 6x + 4  

y'' = 6x - 6 = 0 → x = 1  

At x = 1: y = 1 - 3 + 4 = 2  

Slope m = y' at x=1 = 3 - 6 + 4 = 1  

Tangent: y - 2 = 1(x - 1) → y = x + 1

Last verified on 05-06-2026