Step-by-Step Guidance for Advanced Level Calculus Exam Questions

Q1. Let $f(x) = x^3 - 3x + 1$ and $g(x) = x^2 + x$. If $f^{-1}(g(x))$ exists, find $g(f(2))$ and $f^{-1}(g(2))$.

Background

Topic: Functions, Inverse Functions, and Composition

This question tests your understanding of function composition, evaluating functions, and finding the inverse of a function.

Key Terms and Formulas

• Function composition: $(f \circ g)(x) = f(g(x))$

• Inverse function: $f^{-1}(y)$ is the value $x$ such that $f(x) = y$

Step-by-Step Guidance

1. First, evaluate $f(2)$ by substituting $x = 2$ into $f(x)$.

2. Next, use the result from $f(2)$ to evaluate $g(f(2))$ by substituting $f(2)$ into $g(x)$.

3. To find $f^{-1}(g(2))$, first evaluate $g(2)$ by substituting $x = 2$ into $g(x)$.

4. Set $f(x) = g(2)$ and solve for $x$ to find $f^{-1}(g(2))$.

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Q2. In triangle OAB, $OA = OB = r$ and $\angle AOB = \theta$. Find the area of the shaded sector and the area of triangle OAB in terms of $r$ and $\theta$.

Background

Topic: Geometry – Sectors and Triangles

This question tests your ability to use formulas for the area of a sector and the area of a triangle given two sides and the included angle.

Key Terms and Formulas

• Area of a sector: $A_{\text{sector}} = \frac{1}{2} r^2 \theta$ (when $\theta$ is in radians)

• Area of a triangle (SAS): $A_{\triangle} = \frac{1}{2} ab \sin C$

Step-by-Step Guidance

1. Write the formula for the area of the sector using $r$ and $\theta$.

2. Write the formula for the area of triangle OAB using $OA$, $OB$, and $\angle AOB$.

3. Substitute the given values ($OA = OB = r$, $\angle AOB = \theta$) into the formulas.

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Q3. The coordinates of points A and B are $A(1, 4)$ and $B(3, 1)$. Find the equation of the perpendicular bisector of AB.

Background

Topic: Coordinate Geometry – Perpendicular Bisector

This question tests your ability to find the midpoint of a segment and the equation of a line perpendicular to a given segment.

Key Terms and Formulas

• Midpoint formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Slope of AB: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Slope of perpendicular bisector: $m' = -\frac{1}{m}$

• Point-slope form: $y - y_1 = m(x - x_1)$

Step-by-Step Guidance

1. Find the midpoint of AB using the midpoint formula.

2. Calculate the slope of AB.

3. Find the negative reciprocal of the slope to get the slope of the perpendicular bisector.

4. Write the equation of the perpendicular bisector using the midpoint and the new slope.

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Q4. Solve the quadratic equation $4x^2 - 3x + 6 = 0$ and the cubic equation $x^3 - 2x^2 + 3x - 5 = 0$.

Background

Topic: Algebra – Solving Quadratic and Cubic Equations

This question tests your ability to solve quadratic equations using the quadratic formula and to find roots of cubic equations.

Key Terms and Formulas

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Cubic equations: May require factoring, synthetic division, or the rational root theorem.

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the quadratic equation and substitute into the quadratic formula.

2. Simplify under the square root and set up the two possible solutions for $x$.

3. For the cubic equation, look for possible rational roots using the rational root theorem.

4. Test possible roots by substitution or synthetic division to factor the cubic equation.

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Q5. A particle moves along a straight line so that its displacement $s$ at time $t$ is given by $s = 2t^3 - 9t^2 + 12t + 5$. Find the velocity and acceleration at any time $t$.

Background

Topic: Calculus – Differentiation and Kinematics

This question tests your ability to find velocity and acceleration by differentiating the displacement function with respect to time.

Key Terms and Formulas

• Velocity: $v(t) = \frac{ds}{dt}$

• Acceleration: $a(t) = \frac{dv}{dt}$

Step-by-Step Guidance

1. Differentiate $s(t)$ with respect to $t$ to find $v(t)$.

2. Differentiate $v(t)$ with respect to $t$ to find $a(t)$.

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Q6. If $y = x^n$, show that $\frac{d^2y}{dx^2} = n(n-1)x^{n-2}$.

Background

Topic: Calculus – Higher Order Derivatives

This question tests your understanding of differentiation rules and how to find the second derivative of a power function.

Key Terms and Formulas

• First derivative: $\frac{dy}{dx} = nx^{n-1}$

• Second derivative: $\frac{d^2y}{dx^2}$ is the derivative of $\frac{dy}{dx}$

Step-by-Step Guidance

1. Differentiate $y = x^n$ with respect to $x$ to find the first derivative.

2. Differentiate the result again with respect to $x$ to find the second derivative.

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