C3 papers June 2007 to 2008

Transcription

1 physicsandmathstutor.com June 007 C3 papers June 007 to Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04*

2 physicsandmathstutor.com June 007 x x 1. f ( x) =, x >. x + x + 3x 4x 6 (a) Show that f( x) =. x 1 (7) (b) Hence, or otherwise, find f (x) in its simplest form. 4 *N6109A044*

3 physicsandmathstutor.com June A curve C has equation y = x e x. (a) Find d y dx, using the product rule for differentiation. (b) Hence find the coordinates of the turning points of C. (c) Find d y dx. (d) Determine the nature of each turning point of the curve C. 6 *N6109A064*

4 physicsandmathstutor.com June f(x) = x 3 + 3x 1. (a) Show that the equation f(x) = 0 can be rewritten as x = 1. 3 x (b) Starting with x 1 = 0.6, use the iteration x n+ 1 = 1 3 x to calculate the values of x, x 3 and x 4, giving all your answers to 4 decimal places. n (c) Show that x = is a root of f(x) = 0 correct to 3 decimal places. 8 *N6109A084*

5 physicsandmathstutor.com June The functions f and g are defined by 1 f: x ln( x 1), x, x>, g: x, x 3 x, x 3. (a) Find the exact value of fg. (b) Find the inverse function f 1 (x), stating its domain. (c) Sketch the graph of y = g( x). Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the y-axis. (d) Find the exact values of x for which x 3 =3. 10 *N6109A0104*

6 physicsandmathstutor.com June (a) Express 3 sin x + cos x in the form R sin(x + α) where R > 0 and 0 < α < π. (b) Hence find the greatest value of (3 sin x + cos x) 4. (c) Solve, for 0 < x < π, the equation 3 sin x + cos x = 1, giving your answers to 3 decimal places. (5) 14 *N6109A0144*

7 physicsandmathstutor.com June (a) Prove that sin θ cosθ + = cosec θ, θ 90 n. cosθ sin θ (b) On the axes on page 0, sketch the graph of y = cosec θ for 0 < θ < 360. (c) Solve, for 0 < θ < 360, the equation sinθ cosθ + =3, cosθ sinθ giving your answers to 1 decimal place. (6) 18 *N6109A0184*

8 physicsandmathstutor.com June 007 Question 7 continued y O θ 0 *N6109A004*

9 physicsandmathstutor.com June The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula 1 8, x= De t 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) Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 10 mg is given after 5 hours. (b) Show that the amount of the drug in the bloodstream 1 hour after the second dose is mg to 3 decimal places. 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. *N6109A04*

10 physicsandmathstutor.com January 008 C3 January Given that 4 x 3x + x+ 1 dx + e ( ax + bx + c) + ( x 1) ( x 1), find the values of the constants a, b, c, d and e. *H6315RB04*

11 physicsandmathstutor.com January 008. A curve C has equation x y = e tan x, x ( n+ 1) π. (a) Show that the turning points on C occur where tan x = 1. (b) Find an equation of the tangent to C at the point where x = 0. (6) 4 *H6315RB044*

12 physicsandmathstutor.com January f( x) = ln( x+ ) x+ 1, x>, x. (a) Show that there is a root of f( x ) = 0 in the interval < x < 3. (b) Use the iterative formula to calculate the values of x1, x and x 3 giving your answers to 5 decimal places. (c) Show that x =.505 is a root of f( x ) = 0 correct to 3 decimal places. 6 *H6315RB064*

13 4. physicsandmathstutor.com January 008 y A (5, 4) O x B ( 5, 4) Figure 1 Figure 1 shows a sketch of the curve with equation y = f( x). The curve passes through the origin O and the points A(5, 4) and B( 5, 4). In separate diagrams, sketch the graph with equation (a) y = f( x), (b) y = f( x), (c) y = f ( x+ 1). On each sketch, show the coordinates of the points corresponding to A and B. 8 *H6315RB084*

14 physicsandmathstutor.com January The radioactive decay of a substance is given by ct R = 1000e, t 0. where R is the number of atoms at time t years and c is a positive constant. (a) Find the number of atoms when the substance started to decay. (1) It takes 5730 years for half of the substance to decay. (b) Find the value of c to 3 significant figures. (c) Calculate the number of atoms that will be left when t = 90. (d) In the space provided on page 13, sketch the graph of R against t. 1 *H6315RB014*

15 physicsandmathstutor.com January (a) Use the double angle formulae and the identity cos( A+ B) cos Acos B sin Asin B to obtain an expression for cos 3x in terms of powers of cos x only. (b) (i) Prove that (ii) Hence find, for 0 cos x 1 sin x + + π sec x, x ( n+ 1). 1 + sin x cos x < x <π, all the solutions of cos x 1 sin x + + = sin x cos x 14 *H6315RB0144*

16 physicsandmathstutor.com January A curve C has equation y = 3sin x+ 4cos x, -π x π. The point A(0, 4) lies on C. (a) Find an equation of the normal to the curve C at A. (b) Express y in the form Rsin( x+α ), where R > 0 and 0 < α < π. Give the value of α to 3 significant figures. (c) Find the coordinates of the points of intersection of the curve C with the x-axis. Give your answers to decimal places. (5) 18 *H6315RB0184*

17 physicsandmathstutor.com January The functions f and g are defined by 3 f: x 1 x, x (a) Find the inverse function g: x -1 f. 3 4, x> 0, x x (b) Show that the composite function gf is x gf : x x (c) Solve gf ( x ) = 0. (d) Use calculus to find the coordinates of the stationary point on the graph of y = gf(x). (5) *H6315RB04*

18 C3 June The point P lies on the curve with equation y e x + = 4 1. The y-coordinate of P is 8. physicsandmathstutor.com June 008 (a) Find, in terms of ln, 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. *N30745A04*

19 . physicsandmathstutor.com June 008 f ( x) = 5cos x+ 1sin x Given that f ( x) = Rcos( x α), where R > 0 and 0 < α < π, (a) find the value of R and the value of α to 3 decimal places. (b) Hence solve the equation for 0 x < π. 5cos x+ 1sin x= 6 (5) (c) (i) Write down the maximum value of 5cos x+ 1sin x. (1) (ii) Find the smallest positive value of x for which this maximum value occurs. 4 *N30745A044*

20 3. physicsandmathstutor.com June 008 y P Q 3 R x Figure 1 Figure 1 shows the graph of y = f( x), x. 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, 0) and R. Sketch, on separate diagrams, the graphs of (a) y = f( x), (b) y = f ( x). Given that f( x) = x+ 1, (c) find the coordinates of the points P, Q and R, (d) solve f( x) = 1 x. (5) 8 *N30745A084*

21 physicsandmathstutor.com June The function f is defined by ( x ) f: x 1 1, x 3. x x 3 > x 3 (a) Show that 1 f( x) =, x > 3. x + 1 (b) Find the range of f. (c) Find f 1 (x). State the domain of this inverse function. The function g is defined by g: x x 3, x. (d) Solve fg( x ) = *N30745A014*

22 physicsandmathstutor.com June (a) Given that sin θ + cos θ 1, show that 1 + cot θ cosec θ. (b) Solve, for 0 θ < 180, the equation cot θ 9 cosec θ = 3, giving your answers to 1 decimal place. (6) 16 *N30745A0164*

23 physicsandmathstutor.com June (a) Differentiate with respect to x, (i) e 3 x (sin x+ cos x), (ii) Given that 3x + 6x 7 y =, x 1, ( x + 1) (b) show that (c) Hence find dy 0 = dx ( x+ 1). 3 d d y x and the real values of x for which d y 15 =. dx 4 (5) 18 *N30745A0184*

24 7. physicsandmathstutor.com June f( x) = 3x x 6 (a) Show that f (x) = 0 has a root, α, between x = 1.4 and x = 1.45 (b) Show that the equation f (x) = 0 can be written as x = ( + x 0 x 3),. (c) Starting with x 0 =1.43, use the iteration x n+1 ( xn 3) = + to calculate the values of x 1, x and x 3, giving your answers to 4 decimal places. (d) By choosing a suitable interval, show that α = is correct to 3 decimal places. *N30745A04*