ANWSER
Question 1:
(iv) Given \( y = \frac{x^2 – 1}{x^2 – 7x + 12} \):
– (i) \( x \)-intercept: Set \( y = 0 \).
\( x^2 – 1 = 0 \Rightarrow x = \pm 1 \).
Intercepts: \( (1, 0) \) and \( (-1, 0) \).
– (ii) \( y \)-intercept: Set \( x = 0 \).
\( y = \frac{-1}{12} \).
Intercept: \( \left(0, -\frac{1}{12}\right) \).
– (iii) Turning points: Find \( \frac{dy}{dx} = 0 \).
Simplify and solve for critical points.
– (iv) Horizontal Asymptote: Compare degrees of numerator and denominator.
\( y = 1 \) (since degrees are equal).
– (v) Vertical Asymptotes: Set denominator \( x^2 – 7x + 12 = 0 \).
\( x = 3 \) and \( x = 4 \).
Sketch: Curve passes through intercepts, approaches asymptotes, and has turning points.
(b)(i) For \( y = 2x^3 – 10x^5 \):
– Find \( \frac{d^2y}{dx^2} \) and solve \( \frac{d^2y}{dx^2} = 0 \) for inflection points.
(b)(ii) For \( y = x(4 – x) \) at \( x = 4 \):
– Tangent: Find slope \( \frac{dy}{dx} \) at \( x = 4 \), use point-slope form.
– Normal: Negative reciprocal of tangent slope, use point-slope form.
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Question 2:
(b)(i) Evaluate \( \int_{-\infty}^{\infty} \frac{3x + 1}{x^3 + 2x^2 + x + 2} dx \):
– Use partial fractions or residue theorem (complex analysis).
(b)(ii) Solve \( \frac{dy}{dx} = \frac{x^2y}{x^2 + y^2} \) with \( y = vx \):
– Substitute \( y = vx \) and separate variables.
(b)(i) \( \int e^x \sin^2 x dx \):
– Use integration by parts twice.
(b)(ii) \( \int x^3 \log_2 x dx \):
– Let \( u = \log_2 x \), \( dv = x^3 dx \).
(c) For \( y = \frac{1}{x} \), evaluate \( \int_{-\infty}^{\infty} \frac{1}{x^2 – 1} dx \):
– Partial fractions: \( \frac{1}{2} \ln \left| \frac{x-1}{x+1} \right| + C \).
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Question 3:
(a)(i) \( \int_{-\infty}^{\infty} \frac{2x dx}{x^2 – 1} \):
– Diverges (improper integral).
(a)(ii) \( \int_{-\infty}^{\infty} \frac{22 – 5x^3 – 5x^4}{(x-1)(x+2)} dx \):
– Perform polynomial division and partial fractions.
(b)(i) Circle \( 4x^2 + 4y^2 – 12x – 8y + 9 = 0 \):
– Rewrite as \( (x – 1.5)^2 + (y – 1)^2 = 1 \).
Center: \( (1.5, 1) \), Radius: 1.
(b)(ii) Line MN through M(4,3) and N(-12,24):
– Slope \( m = \frac{24-3}{-12-4} = -\frac{21}{16} \).
Equation: \( y – 3 = -\frac{21}{16}(x – 4) \).
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Question 4:
(a)(i) Order and degree of differential equation:
– Order: Highest derivative (2nd).
– Degree: Highest power of derivative (3/2, but typically integer).
(a)(ii) Primitive \( y = ax^2 + b \cos nx + c \):
– Differentiate twice to eliminate constants.
(b)(i) Solve \( y – x \frac{dy}{dx} = a \left(y^2 + \frac{dy}{dx}\right) \):
– Rearrange and separate variables.
(b)(ii) Exact equation:
– Check \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).
– Solve by integrating \( M \) w.r.t. \( x \) and \( N \) w.r.t. \( y \).
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