# Component

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 2

Written Paper Section I Question 2 - 2010 HSC

Differentiate and integrate functions. Solve an inequality. Determine the gradient of a tangent to a curve.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 2

Written Paper Section I Question 2 - 2002 HSC

Find the equation of a tangent to a curve. Differentiate and integrate functions. Use the sine rule to find the exact value of a ratio.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 4

Written Paper Section I Question 4 - 2001 HSC

Find the values of a constant for which a specified quadratic equation has no real roots. Prove expressions for the size of specified angles in a diagram. Sketch a trigonometric curve. Represent on a diagram the region bounded by the curve and a straight line. Find the exact value of the integral representing the region.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 5

Written Paper Section I Question 5 - 2010 HSC

Show that the surface area of a cylinder has a minimum value for a particular value of its radius. Prove trigonometric identities and find the exact value of an integral. Find the ordinates at two x-values for areas under a curve enclosed by the x-axis, a given ordinate and the ordinates to be found, and the curve.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 5

Written Paper Section I Question 5 - 2002 HSC

Find the number of terms and the sum of an arithmetic series. Calculate a sector angle to the nearest degree. Find the vertex and focus of a parabola.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 5

Written Paper Section I Question 5 - 2001 HSC

State the domain and range of a given function. Solve numerical problems involving logarithms. Find the length of the radius of a given sector. Calculate the area of a given cross-section using the trapezoidal rule, and the approximate volume of water that flows past this section.

#### Student Responses (13)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2### Question 6

Written Paper Section I Question 6 - 2010 HSC

Show that a specified graph has no stationary points. Determine values for which the graph is concave down and for which it is concave up. Sketch the graph. Find an angle in radians, the length of an interval, and the area of a shaded region. Prove two specified triangles are congruent.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 7

Written Paper Section I Question 7 - 2010 HSC

Determine the velocity and displacement of a particle and the time that it first comes to rest. Find the equation of a tangent to a parabola. Show a specified line is vertical and another specified line is a tangent to the parabola.

#### Student Responses (11)

Band 5/6 response 1 Band 5/6 response 3 Band 5/6 response 2 Band 4/5 response 1 Band 4/5 response 2 Band 3/4 response 1 Band 3/4 response 2 Band 2/3 response 1 Band 2/3 response 2 Band 1/2 response 1 Band 1/2 response 2### Question 8

Written Paper Section I Question 8 - 2010 HSC

Calculate a population that is growing exponentially. Determine the probability that two coins tossed together will both show tails. Find values that determine the equation of a sine graph. Draw a graph on the same set of axes as the given graph. Find the values of a coefficient in the equation of a function for which the function is an increasing function.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 8

Written Paper Section I Question 8 - 2002 HSC

Find the value of constants in an exponential decay equation. Calculate when one-eighth of a quantity undergoing exponential decay remains. Sketch the graph of a trigonometric function modelling the motion of a particle. Find when and where the particle is at rest and describe the motion.

#### Student Responses (13)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 1/2 response 1 Band 1/2 response 2### Question 9

Written Paper Section I Question 9 - 2001 HSC

Verify the size of a specified angle and hence that two triangles within the given diagram are similar. Deduce an equation. Use the cosine rule to deduce the exact value of the cosine ratio of angles within the given diagram. Given an expression for the time rate of change of a quantity, find the initial rate. Find an expression for a volume. Verify and solve a given equation for a particular value of the volume.

#### Student Responses (12)

Band 5/6 response 1 Band 5/6 response 2 Band 4/5 response 1 Band 4/5 response 2 Band 3/4 response 1 Band 3/4 response 2 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 10

Written Paper Section I Question 10 - 2010 HSC

Show two specified triangles are similar. Show relationships involving the side lengths of the similar triangles. Find an expression for the volume of a solid of revolution. Apply the expression found in solving a problem involving a hemispherical container.

#### Student Responses (13)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1### Question 10

Written Paper Section I Question 10 - 2002 HSC

Verify equations related to sectors of a circle and graph a related piecemeal function. Differentiate a complex expression. Verify changes in a quantity and describe a related situation, giving reasons for the answer.

#### Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3### Question 10

Written Paper Section I Question 10 - 2001 HSC

Apply geometric series concepts to financial situations. Find two expressions to describe travel situations and use the expressions to solve a real-world problem.