Algebra: Expression Positive, Negative And Zero

1.(a)Express f (x) = x2 – 6x + 14 in the form f (x) = (x – h)2 + k, where h and k are to be determined. (b)Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation y – x2 – 6x + 14. (Total 4 marks)

2.The diagram shows the graph of the function y = ax2 + bx + c.

Complete the table below to show whether each expression is positive, negative or zero. Expression|positive|negative|zero|
a||||
c||||
b2 – 4ac||||
b||||

(Total 4 marks)
3.The diagram shows the parabola y = (7 – x)(l + x). Points A and C are the x-intercepts and point B is the maximum point.

Find the coordinates of A, B, and C.
(Total 4 marks)

4.Find the sum of the arithmetic series
17 + 27 + 37 +…+ 417.
(Total 4 marks)
5.Consider the arithmetic sequence 2, 5, 8, 11, …..
(a)Find u101.
(3)
(b)Find the value of n so that un = 152.
(3)
(Total 6 marks)

6.Gwendolyn added the multiples of 3, from 3 to 3750 and found that 3 + 6 + 9 + … + 3750 = s.
Calculate s.
(Total 6 marks)
7.Find the coefficient of a5b7 in the expansion of (a + b)12. (Total 4 marks)

8.Find the term containing x10 in the expansion of (5 + 2×2)7. (Total 6 marks)
9.The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12. Find the first term, a, and the common difference, d, of the sequence. (Total 4 marks)

10.Consider the arithmetic series 2 + 5 + 8 +….
(a)Find an expression for Sn, the sum of the first n terms. (b)Find the value of n for which Sn = 1365.
(Total 6 marks)
11.Find the sum to infinity of the geometric series
(Total 3 marks)

12.The first and fourth terms of a geometric series are 18 and respectively. Find
(a)the sum of the first n terms of the series;
(4)
(b)the sum to infinity of the series.
(2)
(Total 6 marks)

13.Find the coefficient of x7 in the expansion of (2 + 3x)10, giving your answer as a whole number. (Total 3 marks)

14.Find the coefficient of the x3 term in the expansion of (Total 6 marks)