Component

Component: Integration

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 1

Question 1

Written Paper Section I Question 1 - 2001 HSC

Solve basic arithmetic and algebra problems. Find a primitive of a 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 2

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

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 3

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.

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 3

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.

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

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.

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

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.

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

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

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 (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 6

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.

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

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.

Student Responses (14)

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 2 Band 3/4 response 1 Band 3/4 response 2 Band 2/3 response 2 Band 2/3 response 1 Band 2/3 response 2 Band 1/2 response 1 Band 1/2 response 2

Question 9

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.

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 9

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.

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

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.

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