# Band

### Component: Applications of calculus to the physical world

### 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.

### Question 7

Written Paper Section I Question 7 - 2002 HSC

Explain why a given geometric series has a limiting sum and find the limiting sum. Given a formula for a quantity in terms of time, find the initial amount of the quantity, when a quarter is left, and the rate of change when a quarter is left. Explain a given probability and find other probabilities, including in relation to complementary events.

### Question 7

Written Paper Section I Question 7 - 2001 HSC

Find the volume of a solid of revolution. Find probabilities in relation to multi-stage events. Calculate the initial displacement of a particle moving in a straight line. Show that a given equation represents an equivalent statement of the equation for the displacement of the particle, and hence find expressions for the velocity and acceleration of the particle. Determine the limiting velocity of the particle.

### 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.

### 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.

### Question 8

Written Paper Section I Question 8 - 2001 HSC

Calculate values of constants in equation modelling exponential population growth. Find the probability of a single-stage event and of a multi-stage event. Determine the maximum and minimum values of the rate of change of a quantity. Sketch the graph of the quantity as a function of time and identify any points on the graph where the concavity changes.

### Question 9

Written Paper Section I Question 9 - 2002 HSC

Sketch a logarithmic function and use Simpson's rule to approximate a related integral. Apply geometric series to a financial situation. Find an equation for the speed of a car in terms of time. Determine the distance a jet is behind the car after a given time and the time for the jet to catch up with the car.

### 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.

### 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.

### Component: Basic arithmetic and algebra

### Question 1

Written Paper Section I Question 1 - 2010 HSC

Use basic arithmetic and algebra. Determine the equation of a circle. Find the derivative of a function. Find the limiting sum of a geometric series. Determine the domain of a function.

### Question 1

Written Paper Section I Question 1 - 2002 HSC

Evaluate an arithmetic expression. Solve linear, quadratic, and simultaneous equations. Differentiate a function and integrate a function.

### Question 1

Written Paper Section I Question 1 - 2001 HSC

Solve basic arithmetic and algebra problems. Find a primitive of a function.

### Question 3

Written Paper Section I Question 3 - 2001 HSC

Evaluate a definite integral. Find the value of a constant in a formula and apply the formula. Differentiate functions. Verify a given equation for a triangle by using the cosine rule, and hence an unknown side-length of the triangle.

### Question 4

Written Paper Section I Question 4 - 2002 HSC

Solve and graph an absolute value inequation. Solve a simple trigonometric equation for a specified range. Calculate the length of a side and the area of a given triangle. Show that a given pair of coordinates represents the point of intersection of two curves. Find the size of the shaded area bounded by the two curves.

### Question 7

Written Paper Section I Question 7 - 2002 HSC

Explain why a given geometric series has a limiting sum and find the limiting sum. Given a formula for a quantity in terms of time, find the initial amount of the quantity, when a quarter is left, and the rate of change when a quarter is left. Explain a given probability and find other probabilities, including in relation to complementary events.

### Component: Geometrical applications of differentiation

### 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.

### 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.

### 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.

### Question 6

Written Paper Section I Question 6 - 2002 HSC

Sketch a function representing a semi-circle and state the range of the function. Given the gradient function of a curve, determine the equation of the curve. Sketch the curve, labelling turning points and the y-intercept. Determine for what values of the independent variable x the curve is concave up. Calculate the volume of a container formed by rotating part of a given curve.

### Question 6

Written Paper Section I Question 6 - 2001 HSC

Calculate a term and sum of the given arithmetic series. Find the decimal value of a pronumeral in an alternative expression of an exponential equation. Find the coordinates of two stationary points on a given curve. Determine the values of x for which the curve is concave up and give reasons for the answer. Find the possible values of a constant for which an equation related to the curve has three real solutions.

### 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.

### Question 8

Written Paper Section I Question 8 - 2001 HSC

Calculate values of constants in equation modelling exponential population growth. Find the probability of a single-stage event and of a multi-stage event. Determine the maximum and minimum values of the rate of change of a quantity. Sketch the graph of the quantity as a function of time and identify any points on the graph where the concavity changes.

### Question 9

Written Paper Section I Question 9 - 2010 HSC

Apply geometric series to financial situations. Determine the values for which a given function is increasing and at its maximum. Find a further value of the function. Draw a graph of the function.

### 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.

### Component: Integration

### Question 1

Written Paper Section I Question 1 - 2001 HSC

Solve basic arithmetic and algebra problems. Find a primitive of a function.

### 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.

### 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.

### Question 3

Written Paper Section I Question 3 - 2010 HSC

Find the midpoint, gradient and length of an interval and the equation of a line. Prove two given triangles are similar. Find the perpendicular distance of a point from a line. Sketch a curve. Obtain an approximation to a definite integral using the trapezoidal rule and compare the approximation to the exact value of the definite integral.

### Question 3

Written Paper Section I Question 3 - 2001 HSC

Evaluate a definite integral. Find the value of a constant in a formula and apply the formula. Differentiate functions. Verify a given equation for a triangle by using the cosine rule, and hence an unknown side-length of the triangle.

### Question 4

Written Paper Section I Question 4 - 2010 HSC

Find a term and a sum of an arithmetic series. Find the area between two curves. Calculate probabilities. Prove a relationship expressed in function notation.

### Question 4

Written Paper Section I Question 4 - 2002 HSC

Solve and graph an absolute value inequation. Solve a simple trigonometric equation for a specified range. Calculate the length of a side and the area of a given triangle. Show that a given pair of coordinates represents the point of intersection of two curves. Find the size of the shaded area bounded by the two curves.

### 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.

### 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.

### Question 6

Written Paper Section I Question 6 - 2002 HSC

Sketch a function representing a semi-circle and state the range of the function. Given the gradient function of a curve, determine the equation of the curve. Sketch the curve, labelling turning points and the y-intercept. Determine for what values of the independent variable x the curve is concave up. Calculate the volume of a container formed by rotating part of a given curve.

### Question 7

Written Paper Section I Question 7 - 2001 HSC

Find the volume of a solid of revolution. Find probabilities in relation to multi-stage events. Calculate the initial displacement of a particle moving in a straight line. Show that a given equation represents an equivalent statement of the equation for the displacement of the particle, and hence find expressions for the velocity and acceleration of the particle. Determine the limiting velocity of the particle.

### Question 9

Written Paper Section I Question 9 - 2010 HSC

Apply geometric series to financial situations. Determine the values for which a given function is increasing and at its maximum. Find a further value of the function. Draw a graph of the function.

### Question 9

Written Paper Section I Question 9 - 2002 HSC

Sketch a logarithmic function and use Simpson's rule to approximate a related integral. Apply geometric series to a financial situation. Find an equation for the speed of a car in terms of time. Determine the distance a jet is behind the car after a given time and the time for the jet to catch up with the car.

### 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.

### Component: Linear functions

### Question 2

Written Paper Section I Question 2 - 2001 HSC

Find the equation of a tangent to a curve. Show the equation and length of a line are as given. Calculate the perpendicular distance of the line from the origin and the area of a parallelogram. Find the perpendicular distance of another line from the origin.

### Question 3

Written Paper Section I Question 3 - 2010 HSC

Find the midpoint, gradient and length of an interval and the equation of a line. Prove two given triangles are similar. Find the perpendicular distance of a point from a line. Sketch a curve. Obtain an approximation to a definite integral using the trapezoidal rule and compare the approximation to the exact value of the definite integral.

### Question 3

Written Paper Section I Question 3 - 2002 HSC

Calculate the value of an investment earning compound interest. Determine the value of a pronumeral using geometric reasoning. Find the midpoint of an interval, the coordinates of a particular point, the point of intersection of two lines, and the area of a specified triangle. Show a given equation is the equation of the perpendicular bisector of an interval.

### Component: Logarithmic and exponential functions

### Question 1

Written Paper Section I Question 1 - 2002 HSC

Evaluate an arithmetic expression. Solve linear, quadratic, and simultaneous equations. Differentiate a function and integrate a function.

### 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.

### 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.

### Question 3

Written Paper Section I Question 3 - 2010 HSC

Find the midpoint, gradient and length of an interval and the equation of a line. Prove two given triangles are similar. Find the perpendicular distance of a point from a line. Sketch a curve. Obtain an approximation to a definite integral using the trapezoidal rule and compare the approximation to the exact value of the definite integral.

### Question 4

Written Paper Section I Question 4 - 2010 HSC

Find a term and a sum of an arithmetic series. Find the area between two curves. Calculate probabilities. Prove a relationship expressed in function notation.

### 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.

### 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.

### Question 6

Written Paper Section I Question 6 - 2001 HSC

Calculate a term and sum of the given arithmetic series. Find the decimal value of a pronumeral in an alternative expression of an exponential equation. Find the coordinates of two stationary points on a given curve. Determine the values of x for which the curve is concave up and give reasons for the answer. Find the possible values of a constant for which an equation related to the curve has three real solutions.

### Question 9

Written Paper Section I Question 9 - 2002 HSC

Sketch a logarithmic function and use Simpson's rule to approximate a related integral. Apply geometric series to a financial situation. Find an equation for the speed of a car in terms of time. Determine the distance a jet is behind the car after a given time and the time for the jet to catch up with the car.

### 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.

### Component: Plane geometry

### Question 3

Written Paper Section I Question 3 - 2010 HSC

### Question 3

Written Paper Section I Question 3 - 2002 HSC

Calculate the value of an investment earning compound interest. Determine the value of a pronumeral using geometric reasoning. Find the midpoint of an interval, the coordinates of a particular point, the point of intersection of two lines, and the area of a specified triangle. Show a given equation is the equation of the perpendicular bisector of an interval.

### 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.

### 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.

### 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.

### 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.

### Component: Probability

### Question 4

Written Paper Section I Question 4 - 2010 HSC

Find a term and a sum of an arithmetic series. Find the area between two curves. Calculate probabilities. Prove a relationship expressed in function notation.

### Question 7

Written Paper Section I Question 7 - 2002 HSC

Explain why a given geometric series has a limiting sum and find the limiting sum. Given a formula for a quantity in terms of time, find the initial amount of the quantity, when a quarter is left, and the rate of change when a quarter is left. Explain a given probability and find other probabilities, including in relation to complementary events.

### Question 7

Written Paper Section I Question 7 - 2001 HSC

Find the volume of a solid of revolution. Find probabilities in relation to multi-stage events. Calculate the initial displacement of a particle moving in a straight line. Show that a given equation represents an equivalent statement of the equation for the displacement of the particle, and hence find expressions for the velocity and acceleration of the particle. Determine the limiting velocity of the particle.

### 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.

### Question 8

Written Paper Section I Question 8 - 2001 HSC

Calculate values of constants in equation modelling exponential population growth. Find the probability of a single-stage event and of a multi-stage event. Determine the maximum and minimum values of the rate of change of a quantity. Sketch the graph of the quantity as a function of time and identify any points on the graph where the concavity changes.

### Component: Real functions

### Question 1

Written Paper Section I Question 1 - 2010 HSC

Use basic arithmetic and algebra. Determine the equation of a circle. Find the derivative of a function. Find the limiting sum of a geometric series. Determine the domain of a function.

### Question 4

Written Paper Section I Question 4 - 2010 HSC

### Question 4

Written Paper Section I Question 4 - 2002 HSC

Solve and graph an absolute value inequation. Solve a simple trigonometric equation for a specified range. Calculate the length of a side and the area of a given triangle. Show that a given pair of coordinates represents the point of intersection of two curves. Find the size of the shaded area bounded by the two curves.

### 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.

### Question 6

Written Paper Section I Question 6 - 2002 HSC

Sketch a function representing a semi-circle and state the range of the function. Given the gradient function of a curve, determine the equation of the curve. Sketch the curve, labelling turning points and the y-intercept. Determine for what values of the independent variable x the curve is concave up. Calculate the volume of a container formed by rotating part of a given curve.

### Component: Series and applications

### Question 1

Written Paper Section I Question 1 - 2010 HSC

Use basic arithmetic and algebra. Determine the equation of a circle. Find the derivative of a function. Find the limiting sum of a geometric series. Determine the domain of a function.

### Question 3

Written Paper Section I Question 3 - 2002 HSC

Calculate the value of an investment earning compound interest. Determine the value of a pronumeral using geometric reasoning. Find the midpoint of an interval, the coordinates of a particular point, the point of intersection of two lines, and the area of a specified triangle. Show a given equation is the equation of the perpendicular bisector of an interval.

### Question 4

Written Paper Section I Question 4 - 2010 HSC

### 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.

### Question 6

Written Paper Section I Question 6 - 2001 HSC

Calculate a term and sum of the given arithmetic series. Find the decimal value of a pronumeral in an alternative expression of an exponential equation. Find the coordinates of two stationary points on a given curve. Determine the values of x for which the curve is concave up and give reasons for the answer. Find the possible values of a constant for which an equation related to the curve has three real solutions.

### Question 7

Written Paper Section I Question 7 - 2002 HSC

### Question 9

Written Paper Section I Question 9 - 2010 HSC

Apply geometric series to financial situations. Determine the values for which a given function is increasing and at its maximum. Find a further value of the function. Draw a graph of the function.

### Question 9

Written Paper Section I Question 9 - 2002 HSC

### 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.

### Component: Tangent to a curve and derivative of a function

### Question 2

Written Paper Section I Question 2 - 2001 HSC

Find the equation of a tangent to a curve. Show the equation and length of a line are as given. Calculate the perpendicular distance of the line from the origin and the area of a parallelogram. Find the perpendicular distance of another line from the origin.

### Question 3

Written Paper Section I Question 3 - 2001 HSC

Evaluate a definite integral. Find the value of a constant in a formula and apply the formula. Differentiate functions. Verify a given equation for a triangle by using the cosine rule, and hence an unknown side-length of the triangle.

### Component: The quadratic polynomial and the parabola

### Question 1

Written Paper Section I Question 1 - 2001 HSC

Solve basic arithmetic and algebra problems. Find a primitive of a function.

### 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.

### 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.

### 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.

### Question 8

Written Paper Section I Question 8 - 2010 HSC

### Question 9

Written Paper Section I Question 9 - 2001 HSC

### Component: Trigonometric functions

### Question 2

Written Paper Section I Question 2 - 2010 HSC

### Question 2

Written Paper Section I Question 2 - 2002 HSC

### 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.

### Question 5

Written Paper Section I Question 5 - 2010 HSC

### 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.

### Question 5

Written Paper Section I Question 5 - 2001 HSC

### 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.

### 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.

### Question 8

Written Paper Section I Question 8 - 2010 HSC

### 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.

### Question 9

Written Paper Section I Question 9 - 2001 HSC

### 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.

### 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.

### 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.

### Component: Trigonometric ratios

### Question 2

Written Paper Section I Question 2 - 2002 HSC

### Question 3

Written Paper Section I Question 3 - 2001 HSC

### Question 4

Written Paper Section I Question 4 - 2002 HSC

### Question 5

Written Paper Section I Question 5 - 2010 HSC

### Question 9

Written Paper Section I Question 9 - 2001 HSC

### Question 10

Written Paper Section I Question 10 - 2010 HSC

### Question 10

Written Paper Section I Question 10 - 2001 HSC