Unit 6

In a science experiment, Marta adds chemicals to a solution following this sequence: she adds 2 ml on the first day, 5 ml on the second day, 8 ml on the third day, and so on, increasing the amount by 3 ml each day.

On which day will she add exactly 44 ml to the solution?

The first term of an arithmetic sequence is 12. The common difference is 7. Calculate the sum of the first 30 terms of this sequence.

Sophie decides to save money for a holiday by following a plan: she saves $15 in the first week and then increases how much she saves by $4 each week. She needs to save at least $350. Explain how she can work out the minimum number of weeks needed to reach her goal. Show your reasoning clearly.

Chords AP and BP intersect inside a circle at point P. If AP = 4 cm, BP = 6 cm, and CP = 8 cm, find DP.

In a circle, points A, B, C, D lie on the circumference. Angle ABC = 80°. Find angle CDA. Give a reason for your answer.

In the circle below, O is the centre. The distance from O to the chord AB is 5 cm. The radius of the circle is 13 cm. Find the length of the chord AB.

Name the theorem that states, “The angle in a semicircle is a right angle.”

Which of the following statements is always true for any cyclic quadrilateral?

In the diagram, O is the centre of the circle. The radius of the circle is 5 cm. AB and AC are chords of the circle and angle BAC = 32°.

Explain, with reasons, why the points A, B, C, and D in a circle are called concyclic points. Give a geometrical property relating their angles.