Simultaneous equations

Scientists are comparing h, the average height of a child in cm, with t, the age of the child in years. They suggest the model h = at + b for t ⩾ 2, where a and b are constants.

They find that the average height of a 2 year old child is 87 cm and the average height of a 5 year old child is 108 cm.

Find the values of a and b that are consistent with the scientists’ findings.

Sam is 4 years old.

Use the model to predict Sam’s height.

Comment on the accuracy of this prediction.

Suggest one possible limitation of this model when predicting the average height of a 12 year

old child.

In this question you must show detailed reasoning.

A circle has diameter PQ where P is (- 5, 1) and Q is (5, 1). The line x + 2 y = 12 meets the circle at A and B.

Find the exact length AB.

The diagram shows the curve 5x - 2xy + 2y² - k = 0, where k is a positive integer.

At the points P and Q on the curve, the tangents to the curve are parallel to the y-axis.

Given that the difference in the y-coordinates of P and Q is 3, determine the x-coordinates of

P and Q.

The diagram shows part of the graph of y = x². The normal to the curve at the point A(1, 1) meets the curve again at B. Angle AOB is denoted by α.

Determine the coordinates of B.

Hence determine the exact value of tan α.

A circle has centre C which lies on the x-axis, as shown in the diagram. The line y = x meets the circle at A and B. The midpoint of AB is M.

The equation of the circle is x² - 6x + y² + a = 0, where a is a constant.

In this question you must show detailed reasoning.

Show that the area of triangle ABC is

Find the value of a when the area of triangle ABC is zero.

Give a geometrical interpretation of the case in part (b)(i).

Give a geometrical interpretation of the case where a = 5.

The random variable Y has the distribution N(μ, σ²).

Find P(Y > μ - σ).

Given that P(Y > 45) = 0.2 and P(Y < 2 5) = 0. 3, determine the values of μ and σ.

The random variables U and V have the distributions N(10, 4) and N(12, 9) respectively.

It is given that P(U < b) = P(V > c), where b > 10 and c < 12.

Determine b in terms of c.

The diagram shows the curves y = x² - 2x + 1 and y =-x² + 6x - 5. This diagram is repeated in the Printed Answer Booklet.

On the diagram in the Printed Answer Booklet, draw the line y = 2 x - 2.

The 15th term of an arithmetic sequence is 88. The sum of the first 10 terms is 310.

Determine the first term and the common difference.

In this question you must show detailed reasoning.

The diagram shows the curve with equation 4xy = 2(x² + 4y²) - 9x.

At the point P on the curve the tangent to the curve is parallel to the y-axis and at the point Q on the curve the tangent to the curve is parallel to the x-axis.

A car is travelling on a straight horizontal road. The velocity of the car, vms^-1, at time t seconds as it travels past three points, P, Q and R, is modelled by the equation

v = at² + bt + c,

where a, b and c are constants.

The car passes P at time t = 0 with velocity 8ms^-1.

State the value of c.

The car passes Q at time t = 5 and at that instant its deceleration is 0.12 ms^-2. The car passes R at time t = 18 with velocity 2.96 ms^-1.

Determine the values of a and b.

Find, to the nearest metre, the distance between points P and R.