BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 1 Lynn is selling cushions as part of an enterprise project. On her first attempt, she sold 80 cushions at the cost of 15 each. She hopes to sell more cushions next time. Her adviser suggests that she can expect to sell 10 more cushions for every 1 that she lowers the price. (a) the number of cushions sold c can be modelled by the equation , where p is the price of each cushion and H is a constant. Determine the value of H.

To model her total revenue, r, Lynn multiplies the number of cushions sold by the price of each cushion. She writes this as . (b) Rearrange r into the form , where A, B and C are constants to be found. (c) Using your answer to part b or otherwise, show that Lynn can increase her revenue by 122.50 through lowering her prices, and state the optimum selling price of a cushion. BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A

2 The graph of is shown where are real constants (a) Find the coordinates of the y intercept (b) Find the values of BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 3 (a) Find the equation of the line l, which goes through the point P (5, 9) and has

gradient 2. (b) The circle C has equation . Show that l is a tangent to C. A line is a tangent to a circle if it touches it once only (rather than intersecting it twice or not touching it at all). (c) Find, as a surd, the length from P to the point where l touches the circle. BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 4

Evaluate the following definite integrals: (a) (b) (c) (d) (e) (f) (g) (h)

BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 5 Solve the following equations on the interval (a) (b) (c) BHASVIC MTHS

A1 DOUBLES ASSIGNMENT 10A 6 A cubic graph is defined as (a) By considering the factors of 8, or otherwise, express as the product of three linear factors (b) Sketch the graph of BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A

7 Solve these equations for 0 < < 2 (a) (b) (c) cosec BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A

8 Given that , show that . BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 9 The curve C has the equation . (a) Find the coordinates of the points where C crosses the x-axis. (b) Find the exact coordinates of the stationary point of C.

(c) Determine the nature of the stationary point. (d) Sketch the curve C. BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 10 Given that y = 10 at x = 4, find y in terms of x, giving each term in its simplest form.

BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 11 The region is bounded by the curve , the and axis and the normal to the curve at the point . (a) Sketch the curve (b) Find the equation of the normal (c) Find the area of R.

BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 12 A rectangular box, with no top, is made from thin card. The volume of the box is 500 cm3. The base of the box is a square with sides of length x cm. (a) Show that the area, A cm2, of card used to make such an open box is given by .

(b) find the minimum amount of card needed to make this box BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 13 A quadratic function is defined by where k is a constant. It is given that the equation has two distinct real roots. (a) Find the set of values k can take. For the case where

(b) Express in the form stating the values of a and b, and hence write down the least value taken by (c) solve the equation expressing your answer in terms of surds simplified as far as possible. BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 14 For each of the following circles, find the lengths along the tangents from the

given point to the circle: (a) from the point (0, 0) (b) from the point (8, 2) (c) from the point (-2, 3) BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 1 - Answers (a)

(b) (c) Old revenue is ; new revenue is 1322.50; different is 122.50. The best selling price of a cushion is 11.50. BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 2 - Answers (a) (0,12) (b) b= 1, c= -13, d= -1, e= 12

BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 3 - Answers (a) (c) BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A

4 Answers (a) (b) 10 (c) (d) (e) (f) (g) (h)

BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 5- Answers (a) 161.5, 318.5 (b) 71.6, 108.4, 251.6, 288.4 (c) 326.6 BHASVIC MTHS

A1 DOUBLES ASSIGNMENT 10A 6 - Answers (a) BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 7 - Answers (a)

(b) (c) BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 8 - Answers , so BHASVIC MTHS

A1 DOUBLES ASSIGNMENT 10A 9 - Answers (a) (1, 0) and (4,0) (b) (c) maximum (need to give a reason) (d) y O

x BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 10 - Answers BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A

11 - Answers (b) (c) BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 12 - Answers (b) 300cm2

BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A 13 - Answers or (you must include a sketch) hence least value is BHASVIC MTHS A1 DOUBLES ASSIGNMENT 10A

14 - Answers (a) (b) (c)