### core maths 3

C3 Algebra 1. Express as a single fraction in its simplest form. (7) 2. (a) Simplify (b) Hence, or otherwise, express as a single fraction in its simplest form.

(3) 3. (a) Show that (b) Show thatx2 + x +1 > O for all values of x. (c) Show that f (x) > O for all values of x, x * -2. 4. (b) Hence, or otherwise, find f'(x) in its simplest form.

5. Given that find the values of the constants a, b, c, d and e. 6.

(a) Express f (x) as a single fraction in its simplest form. (4) (b) Hence show that 7.The function f is defined by The function g is defined by x, x* In 2 (b) Differentiate g(x) to show that c) Find the exact values of x for which g'(x) = 8. Express (a) Simplify fully 9. Given that +1n(x2+2x-15), x * . Edexcel C3 January 2006 Question 2 2. Edexcel C3 June 2006 Question 1 3. Edexcel C3 January 2007 Question 2 4.

Edexcel c3 June 2007 Question 2 5. Edexcel C3 January 2008 Question 1 6. Edexcel C3 June 2008 Question 7.

Edexcel C3 January 2009 Question 2 8. Edexcel CS June 2009 Question 7 9. Edexcel C3 January 2010 Question 1 10.Edexcel C3 June 2010 Question 8 C3 Exponential and Logs – 5, (b) find x in terms of e. 1. A heated metal ball is dropped into a liquid.

As the ball cools, its temperature, T c, t minutes after it enters the liquid, is given by T = 400 e-o. 05t + 25, to O. (a) Find the temperature of the ball as it enters the liquid. (1) (b) Find the value of t for which T = 300, giving your answer to 3 significant figures.

(4) (c) Find the rate at which the temperature of the ball is decreasing at the instant when t = 50. Give your answer in oc per minute to 3 significant fgures. 3) (d) From the equation for temperature T in terms of t, given above, explain why the temperature of the ball can never fall to 20 oc. (1) (Total 9 marks) 2. Find the exact solutions to the equations (a) In x + In 3=1n6, b) ex + 3e-x = 4.

(Total 6 marks) 3. The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula where x is the amount of the drug in the bloodstream in milligrams and D is the dose given in milligrams. A dose of 10 mg of the drug is given. a) Hna tne amount 0T tne arug In tne stream nours arter Give your answer in mg to 3 decimal places.

A second dose of 10 mg is given after 5 hours. tne oose Is glven. (b) Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13. 549 mg to 3 decimal places. (2) No more doses of the drug are given.

At time T hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg. (c) Find the value of T. (Total 7 marks) 4. The point P lies on the curve with equation Y = 4e2x+1. The y -coordinate of P is 8. a) Find, in terms of In 2, the x -coordinate of P. (b) Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are exact constants to be found. (4) 5.

The functions f and gare defined by (a) Write down the range of g. (b) Show that the composite function fg is defined by (c) Write down the range of fg. (d) Solve the equation (Total 10 marks) (x) = 3xex – 1 The curve with equation y = f (x) has a turning point P. (a) Find the exact coordinates of P. The equation f (x) = O has a root between x = 0. 25 and x = 0. (b) Use the iterative formula with xo = 0.

25 to find, to 4 decimal places, the values of xl, x2 and x3. (3) (c) By choosing a suitable interval, show that a root off(x) = O is x = 0. 2576 correct to 4 aeclmal places. ( (Total 11 marks) 7. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation t, t 20 (a) Write down the number of rabbits that were introduced to the island. 1) (b) Find the number of years it would take for the number of rabbits to first exceed 1000.

(2) (c) Find (d) Find P when (Total 8 marks) 8. i) Find the exact solutions to the equations (b) 3xe7x + 2 = 15 (it) The functions f and g are defined by f(x)=e2x+3,x ,x>l (a) Find f-1 and state its domain. (b) Find fg and state its range. (Total 15 marks) 9.

(a) Simplify fully In(2X2 + + 2x C3 Functions ??”15), x Ine Tunctlons T ana gare e n f: x 02x+ln g:x0e2x, xo (a) Prove that the composite function gf is gf:x04e4x, x (b) In the space provided below, sketch the curve with equation y = gf(x), and show he coordinates of the point where the curve cuts the y-axis. (1) (c) Write down the range of gf. d) Find the value of x for which , giving your answer to 3 significant figures. (4) 2. For the constant k, where k > 1, the functions f and gare defined by f: x In (x + k), g: x 12x-kl, x e.

(a) On separate axes, sketch the graph off and the graph of g. On each sketch state, in terms of k, the coordinates of points where the graph meets the coordinate axes. (5) (b) Write down the range of f. (c) Find in terms of k, giving your answer in its simplest form. (2) The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 is parallel to the line with equation 2x+ 1.

d) Find the value of k. (Total 12 marks) 3. The function f is defined by f: x In(4-2x), x < 2 and x 0. (a) Show that the inverse function off is defined by and write down the domain of f-1 . (b) Write down the range of f-1 . (c) Sketch the graph of y = f-l(x). State the coordinates of the points of intersection with the x and y axes. (4) The graph of y = x + 2 crosses the graph of y = f-l(x) at x = k.

The iterative formula is used to find an approximate value for k. (d) Calculate the values of xl and x2, giving your answers to 4 decimal places. 2) (e) Find the value of k to 3 decimal places. Total 13 marks) 4. The functions f and g are defined by (a) Hna tne exact value 0T Tg(4) (b) Find the inverse function f-l(x), stating its domain.

(c) Sketch the graph of y = Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the y-axis. (3) (d) Find the exact values of x for which (a) Find the inverse function f-1 . (b) Show that the composite function gf is (c) Solve gf(x) = O.

(d) Use calculus to find the coordinates of the stationary point on the graph of y = gf(x). (5) 6. The function f is defined by (b) Find the range off. c) Find f-1 (x). State the domain of this inverse function. (3) (d) Solve . 7. The functions f and gare defined by The functions f and g are defined by 9.

The function f is defined by (a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph cuts or meets the axes. (2) (b) Solve f = (x) = 15 + x. (c) Find fg(2).

(d) Find the range of g. C3 Transformations The figure above shows the graph of y = f(x), -5 Ox 0 5. The point M (2, 4) is the maximum turning point of the graph. Sketch, on separate diagrams, the graphs of a) y = f(x) + 3, (c) y = f(lxl).

now on eacn grapn tne coorolnates 0T any maximum turning points / marks) 2. The functions f and g are defined by g:x0e2x, x 0 The figure above shows part of the curve with equation y = f(x), x 0 , where f is an increasing function of x. The curve passes through the points P(O, -2) and Q(3, O) as shown.

In separate diagrams, sketch the curve with equation f -I(x), (c) y = f(3x). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes. 4. For the constant k, where k > 1, the functions f and gare defined by f: x In (x + k), .I ne Olagram aoove snows a sketcn 0T tne curve wltn equation y = I ne curve passes through the origin O and the points A(5, 4) and 8(-5, -4). In separate diagrams, sketch the graph with equation (c) y=2f(x+ 1). On each sketch, show the coordinates of the points corresponding to A and B. The diagram above shows the graph of .

The graph consists of two line segments that meet at the point P. The graph cuts the y-axis at the point Q and the x-axis at the points (-3, O) and R. (b)y = f (-x). Given that lx+ 1 1, (c) find the coordinates of the points P, Q and R, (d) solve . 7.

The figure above shows the graph of y = f 1 < x < 9. The points T(3, 5) and S(7, 2) are turning points on the graph. (a) y = 2f(x) - 4, Indicate on each diagram the coordinates of any turning points on your sketch. 8. The figure above shows a sketch of part of the curve with equation y = f(x), x The curve meets the coordinate axes at the points A(O, 1 - k) and 1 B, where k is a constant and k as shown in the diagram above.

On separate diagrams, sketch the curve with equation (b)y = f-l(x) Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.Given that f(x) = e2x – k, (c) state the range of f, (d) find f-l(x), (e) write down the domain of f-1 . 9. Sketch the graph of y = In, stating the coordinates of any points of intersection with the axes. (Total 3 marks) 10. The diagram above shows a sketch of the graph of y = f(x).

The graph intersects the y-axis at the point (O, 1) and the point A(2, 3) is the maximum turning point. Sketch, on separate axes, the graphs of y = (x +2)+3, y = 2f(2X). On each sketch, show the coordinates of the point at which your graph intersects the y-axis and the coordinates of the point to which A is transformed.Total 9 marks) 11. The diagram above shows a sketch of the curve with the equation y = f(x), x. The curve has a turning point at A(3, – 4) and also passes through the point (O, 5). (a) Write down the coordinates of the point to which A is transformed on the curve with equation (i) (b) Sketch the curve with equation On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis. The curve with equation y = f(x) is a translation of the curve with equation y = x2.

(c)Find f(x). (d) Explain why the function f does not have an inverse.