core mathematics c3 - mr. samuel lock -...

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper A

Time: 1 hour 30 minutes Instructions and Information Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.

Full marks may be obtained for answers to ALL questions.

Mathematical formulae and statistical tables are available.

This paper has eight questions. Advice to Candidates You must show sufficient working to make your methods clear to an examiner. Answers without working may gain no credit.

Written by Shaun Armstrong

Solomon Press

These sheets may be copied for use solely by the purchaser’s institute.

Solomon Press C3A page 2

1. Given that

x = sec2 y + tan y,

show that

dd

yx

= 2cos

2 tan 1y

y +. (4)

2. The functions f and g are defined by f : x → 3x − 4, x ∈ ,

g : x → 23x +

, x ∈ , x ≠ −3. (a) Evaluate fg(1). (2) (b) Solve the equation gf(x) = 6. (4) 3. Giving your answers to 2 decimal places, solve the simultaneous equations e2y − x + 2 = 0 ln (x + 3) − 2y − 1 = 0 (8) 4. (a) Use the derivatives of sin x and cos x to prove that

ddx

(tan x) = sec2 x. (4)

The tangent to the curve y = 2x tan x at the point where x = π4 meets the y-axis

at the point P. (b) Find the y-coordinate of P in the form kπ2 where k is a rational constant. (6) 5. (a) Express 3 cos x° + sin x° in the form R cos (x − α)° where R > 0

and 0 < α < 90. (4) (b) Using your answer to part (a), or otherwise, solve the equation 6 cos2 x° + sin 2x° = 0, for x in the interval 0 ≤ x ≤ 360, giving your answers to 1 decimal place where appropriate. (6)


6. y y = f(x)

y = 2 (0, q) (p, 0) O x x = 1 Figure 1 Figure 1 shows the curve with equation y = f(x). The curve crosses the axes at

(p, 0) and (0, q) and the lines x = 1 and y = 2 are asymptotes of the curve. (a) Showing the coordinates of any points of intersection with the axes and the equations of any asymptotes, sketch on separate diagrams the graphs of (i) y = f(x), (ii) y = 2f(x + 1). (6) Given also that

f(x) ≡ 2 11

xx

−−

, x ∈ , x ≠ 1, (b) find the values of p and q, (3) (c) find an expression for f −1(x). (3)

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7. (a) (i) Show that sin (x + 30)° + sin (x − 30)° ≡ a sin x°, where a is a constant to be found. (ii) Hence find the exact value of sin 75° + sin 15°, giving your answer in the form 6b . (6) (b) Solve, for 0 ≤ y ≤ 360, the equation 2 cot2 y° + 5 cosec y° + cosec2 y° = 0. (6)

8. f(x) = 4 3 2

25 9

6x x x

x x+ − −

+ −.

(a) Using algebraic division, show that

f(x) = x2 + A + Bx C+

, where A, B and C are integers to be found. (5) (b) By sketching two suitable graphs on the same set of axes, show that the equation f(x) = 0 has exactly one real root. (3) (c) Use the iterative formula

xn + 1 = 2 + 21

1nx +,

with a suitable starting value to find the root of the equation f(x) = 0 correct to 3 significant figures and justify the accuracy of your answer. (5)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper B




This paper has seven questions. Advice to Candidates You must show sufficient working to make your methods clear to an examiner. Answers without working may gain no credit.


Solomon Press


Solomon Press C3B page 2

1. (a) Simplify

2

27 12

2 9 4x xx x

+ ++ +

. (3)

(b) Solve the equation ln (x2 + 7x + 12) − 1 = ln (2x2 + 9x + 4), giving your answer in terms of e. (4) 2. A curve has the equation y = 3 11x + . The point P on the curve has x-coordinate 3. (a) Show that the tangent to the curve at P has the equation 3x − 4 5 y + 31 = 0. (6) The normal to the curve at P crosses the y-axis at Q. (b) Find the y-coordinate of Q in the form 5k . (3) 3. (a) Use the identities for sin (A + B) and sin (A − B) to prove that

sin P + sin Q ≡ 2 sin 2

P Q+ cos 2

P Q− . (4)

(b) Find, in terms of π, the solutions of the equation sin 5x + sin x = 0, for x in the interval 0 ≤ x < π. (5)

4. The curve with equation y = 52x ln 4

x , x > 0 crosses the x-axis at the point P. (a) Write down the coordinates of P. (1) The normal to the curve at P crosses the y-axis at the point Q. (b) Find the area of triangle OPQ where O is the origin. (6) The curve has a stationary point at R. (c) Find the x-coordinate of R in exact form. (3)


5. f(x) ≡ 2x2 + 4x + 2, x ∈ , x ≥ −1. (a) Express f(x) in the form a(x + b)2 + c. (2) (b) Describe fully two transformations that would map the graph of y = x2, x ≥ 0 onto the graph of y = f(x). (3) (c) Find an expression for f −1(x) and state its domain. (4) (d) Sketch the graphs of y = f(x) and y = f −1(x) on the same diagram and state the relationship between them. (4) 6. f(x) = e3x + 1 − 2, x ∈ . (a) State the range of f. (1) The curve y = f(x) meets the y-axis at the point P and the x-axis at the point Q. (b) Find the exact coordinates of P and Q. (4) (c) Show that the tangent to the curve at P has the equation y = 3ex + e − 2. (4) (d) Find to 3 significant figures the x-coordinate of the point where the tangent to the curve at P meets the tangent to the curve at Q. (4)

Turn over


7. (a) Solve the equation π − 3 arccos θ = 0. (2) (b) Sketch on the same diagram the curves y = arccos (x − 1), 0 ≤ x ≤ 2 and y = 2x + , x ≥ −2. (5) Given that α is the root of the equation

arccos (x − 1) = 2x + , (c) show that 0 < α < 1, (3) (d) use the iterative formula

xn + 1 = 1 + cos 2nx + with x0 = 1 to find α correct to 3 decimal places. (4)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper C






Solomon Press


Solomon Press C3C page 2

1. (a) Express

24

2 3 1x

x x+

+ + − 2

2 1x +

as a single fraction in its simplest form. (3) (b) Hence, find the values of x such that

24

2 3 1x

x x+

+ + − 2

2 1x + = 1

2. (3)

2. (a) Prove, by counter-example, that the statement “cosec θ − sin θ > 0 for all values of θ in the interval 0 < θ < π” is false. (2) (b) Find the values of θ in the interval 0 < θ < π such that cosec θ − sin θ = 2, giving your answers to 2 decimal places. (5) 3. Solve each equation, giving your answers in exact form. (a) ln (2x − 3) = 1 (3) (b) 3e y + 5e−y = 16 (5) 4. Differentiate each of the following with respect to x and simplify your answers. (a) ln (3x − 2) (2)

(b) 2 11x

x+

− (3)

(c) 32x e2x (3)


5. y (−3, 2)

O x y = f(x)

(2, −4)

Figure 1 Figure 1 shows the curve y = f(x) which has a maximum point at (−3, 2) and a

minimum point at (2, −4). (a) Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of (i) y = f( | x | ), (ii) y = 3f(2x). (7) (b) Write down the values of the constants a and b such that the curve with equation y = a + f(x + b) has a minimum point at the origin O. (2) 6. The function f is defined by f(x) ≡ 4 − ln 3x, x ∈ , x > 0. (a) Solve the equation f(x) = 0. (2) (b) Sketch the curve y = f(x). (2) (c) Find an expression for the inverse function, f −1(x). (3) The function g is defined by g(x) ≡ e2 − x, x ∈ . (d) Show that fg(x) = x + a − ln b, where a and b are integers to be found. (3)

Turn over


7. (a) Express 4 sin x + 3 cos x in the form R sin (x + α) where R > 0 and 0 < α < π2 . (4) (b) State the minimum value of 4 sin x + 3 cos x and the smallest positive value of x for which this minimum value occurs. (3) (c) Solve the equation 4 sin 2θ + 3 cos 2θ = 2, for θ in the interval 0 ≤ θ ≤ π, giving your answers to 2 decimal places. (6) 8. The curve C has the equation y = x + e1 − 4x, x ≥ 0. (a) Find an equation for the normal to the curve at the point ( 1

4 , 32 ). (4)

The curve C has a stationary point with x-coordinate α where 0.5 < α < 1. (b) Show that α is a solution of the equation

x = 14 [1 + ln (8 x )]. (3)

(c) Use the iteration formula

xn + 1 = 14 [1 + ln (8 nx )],

with x0 = 1 to find x1, x2, x3 and x4, giving the value of x4 to 3 decimal places. (3) (d) Show that your value for x4 is the value of α correct to 3 decimal places. (2) (e) Another attempt to find α is made using the iteration formula

xn + 1 = 8 2164 e nx − ,

with x0 = 1. Describe the outcome of this attempt. (2)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper D






Solomon Press


Solomon Press C3D page 2

1. The function f is defined by f(x) ≡ 2 + ln (3x − 2), x ∈ , x > 2

3 . (a) Find the exact value of ff(1). (2) (b) Find an expression for f −1(x). (3) 2. Find, to 2 decimal places, the solutions of the equation 3 cot2 x − 4 cosec x + cosec2 x = 0 in the interval 0 ≤ x ≤ 2π. (6) 3. (a) Given that y = ln x, find expressions in terms of y for (i) log2 x,

(ii) ln 2

ex . (4)

(b) Hence, or otherwise, solve the equation

log2 x = 4 − ln 2

ex ,

giving your answer to 2 decimal places. (4) 4. (a) Use the identities for (sin A + sin B) and (cos A + cos B) to prove that

sin 2 sin 2cos 2 cos 2

x yx y

++

≡ tan (x + y). (4)

(b) Hence, show that

tan 52.5° = 6 − 3 − 2 + 2. (5)


5. f(x) = 3 − 13

xx

−−

+ 211

2 5 3x

x x+

− −, x ∈ , x < −1.

(a) Show that

f(x) = 4 12 1

xx

−+

. (5)

(b) Find an equation for the tangent to the curve y = f(x) at the point where x = −2, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (5) 6. A curve has the equation y = e3x cos 2x.

(a) Find dd

yx

. (2)

(b) Show that 2

2dd

yx

= e3x (5 cos 2x − 12 sin 2x). (3)

The curve has a stationary point in the interval [0, 1]. (c) Find the x-coordinate of the stationary point to 3 significant figures. (4) (d) Determine whether the stationary point is a maximum or minimum point and justify your answer. (2) 7. (a) Sketch on the same diagram the graphs of y = 4a2 − x2 and y = 2x − a,

where a is a positive constant. Show, in terms of a, the coordinates of any points where each graph meets the coordinate axes. (6)

(b) Find the exact solutions of the equation 4 − x2 = 2x − 1. (6)

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8. y

y = 2x − 3 ln (2x + 5) O x Q

P Figure 1 Figure 1 shows the curve with equation y = 2x − 3 ln (2x + 5) and the normal

to the curve at the point P (−2, −4). (a) Find an equation for the normal to the curve at P. (4) The normal to the curve at P intersects the curve again at the point Q with x-coordinate q. (b) Show that 1 < q < 2. (3) (c) Show that q is a solution of the equation x = 12

7 ln (2x + 5) − 2. (2) (d) Use the iterative formula xn + 1 = 12

7 ln (2xn + 5) − 2, with x0 = 1.5, to find the value of q to 3 significant figures and justify the accuracy of your answer. (5)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper E






Solomon Press


Solomon Press C3E page 2

1. Express

3 2

22

4x xx

+−

× 22

2 5 3x

x x−

− −

as a single fraction in its simplest form. (5) 2. (a) Prove that, for cos x ≠ 0, sin 2x − tan x ≡ tan x cos 2x. (5) (b) Hence, or otherwise, solve the equation sin 2x − tan x = 2 cos 2x, for x in the interval 0 ≤ x ≤ 180°. (5) 3. f(x) = x2 + 5x − 2 sec x, x ∈ , − π

2 < x < π2 . (a) Show that the equation f(x) = 0 has a root in the interval [1, 1.5]. (2) A more accurate estimate of this root is to be found using iterations of the form xn + 1 = arccos g(xn). (b) Find a suitable form for g(x) and use this formula with x0 = 1.25 to find x1, x2, x3 and x4. Give the value of x4 to 3 decimal places. (6) The curve y = f(x) has a stationary point at P. (c) Show that the x-coordinate of P is 1.0535 correct to 5 significant figures. (3) 4. (a) Differentiate each of the following with respect to x and simplify your answers.

(i) 1 cos x− (ii) x3 ln x (6) (b) Given that

x = 13 2y

y+

−,

find and simplify an expression for dd

yx

in terms of y. (5)


5. (a) Express 3 sin θ + cos θ in the form R sin (θ + α) where R > 0 and 0 < α < π2 . (4)

(b) State the maximum value of 3 sin θ + cos θ and the smallest positive value of θ for which this maximum value occurs. (3) (c) Solve the equation

3 sin θ + cos θ + 3 = 0, for θ in the interval −π ≤ θ ≤ π, giving your answers in terms of π. (5) 6. The function f is defined by f(x) ≡ 3 − x2, x ∈ , x ≥ 0. (a) State the range of f. (1) (b) Sketch the graphs of y = f(x) and y = f −1(x) on the same diagram. (3) (c) Find an expression for f −1(x) and state its domain. (4) The function g is defined by

g(x) ≡ 83 x−

, x ∈ , x ≠ 3.

(d) Evaluate fg(−3). (2) (e) Solve the equation f −1(x) = g(x). (3)

Turn over


7. T

18

12

O 10 60 70 120 t Figure 1 Figure 1 shows a graph of the temperature of a room, T °C, at time t minutes. The temperature is controlled by a thermostat such that when the temperature falls

to 12°C, a heater is turned on until the temperature reaches 18°C. The room then cools until the temperature again falls to 12°C.

For t in the interval 10 ≤ t ≤ 60, T is given by T = 5 + Ae−kt, where A and k are constants. Given that T = 18 when t = 10 and that T = 12 when t = 60, (a) show that k = 0.0124 to 3 significant figures and find the value of A, (6) (b) find the rate at which the temperature of the room is decreasing when t = 20. (4) The temperature again reaches 18°C when t = 70 and the graph for 70 ≤ t ≤ 120 is a

translation of the graph for 10 ≤ t ≤ 60. (c) Find the value of the constant B such that for 70 ≤ t ≤ 120 T = 5 + Be−kt. (3)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper F






Solomon Press


Solomon Press C3F page 2

1. Solve the equation 3 cosec θ ° + 8 cos θ ° = 0 for θ in the interval 0 ≤ θ ≤ 180, giving your answers to 1 decimal place. (6) 2. The functions f and g are defined by f : x → 1 − ax, x ∈ , g : x → x2 + 2ax + 2, x ∈ , where a is a constant. (a) Find the range of g in terms of a. (3) Given that gf(3) = 7, (b) find the two possible values of a. (4) 3. (a) Solve the equation ln (3x + 1) = 2 giving your answer in terms of e. (3) (b) Prove, by counter-example, that the statement “ln (3x2 + 5x + 3) ≥ 0 for all real values of x” is false. (5) 4. A curve has the equation x = 1 2y y− . (a) Show that

dd

yx

= 1 2

1 3y

y−

−. (5)

The point A on the curve has y-coordinate −1. (b) Show that the equation of tangent to the curve at A can be written in the form

3 x + py + q = 0 where p and q are integers to be found. (3)


5. (a) Sketch the graph of y = 2 + sec (x − π6 ) for x in the interval 0 ≤ x ≤ 2π. Show on your sketch the coordinates of any turning points and the equations of any asymptotes. (5) (b) Find, in terms of π, the x-coordinates of the points where the graph crosses the x-axis. (5) 6. y (3, 6)

(0, 4) y = f(x)

( 32− , 0) O x

Figure 1 Figure 1 shows the curve y = f(x) which has a minimum point at ( 3

2− , 0), a maximum point at (3, 6) and crosses the y-axis at (0, 4).

Sketch each of the following graphs on separate diagrams. In each case, show the coordinates of any turning points and of any points where the graph meets the coordinate axes. (a) y = f( | x | ) (3) (b) y = 2 + f(x + 3) (4) (c) y = 1

2 f(−x) (4)

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7. f(x) = 1 + 42 5

xx −

− 215

2 7 5x x− +, x ∈ , x < 1.

(a) Show that

f(x) = 3 21

xx

+−

. (5)

(b) Find an expression for the inverse function f −1(x) and state its domain. (5) (c) Solve the equation f(x) = 2. (2) 8. A curve has the equation y = x2 − 4 ln x+ . (a) Show that the tangent to the curve at the point where x = 1 has the equation 7x − 4y = 11. (5) The curve has a stationary point with x-coordinate α. (b) Show that 0.3 < α < 0.4 (3) (c) Show that α is a solution of the equation

x = 141

2 (4 ln )x −+ . (2) (d) Use the iteration formula

xn + 1 = 141

2 (4 ln )nx −+ , with x0 = 0.35, to find x1, x2, x3 and x4, giving your answers to 5 decimal places. (3)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper G






Solomon Press


Solomon Press C3G page 2

1. A curve has the equation y = (3x − 5)3. (a) Find an equation for the tangent to the curve at the point P (2, 1). (4) The tangent to the curve at the point Q is parallel to the tangent at P. (b) Find the coordinates of Q. (3) 2. (a) Use the identities for cos (A + B) and cos (A − B) to prove that 2 cos A cos B ≡ cos (A + B) + cos (A − B). (2) (b) Hence, or otherwise, find in terms of π the solutions of the equation

2 cos (x + π2 ) = sec (x + π6 ), for x in the interval 0 ≤ x ≤ π. (7) 3. Differentiate each of the following with respect to x and simplify your answers. (a) ln (cos x) (3) (b) x2 sin 3x (3)

(c) 62 7x −

(4)

4. (a) Express 2 sin x° − 3 cos x° in the form R sin (x − α)° where R > 0 and 0 < α < 90. (4) (b) Show that the equation cosec x° + 3 cot x° = 2 can be written in the form 2 sin x° − 3 cos x° = 1. (1) (c) Solve the equation cosec x° + 3 cot x° = 2, for x in the interval 0 ≤ x ≤ 360, giving your answers to 1 decimal place. (5)


5. (a) Show that (2x + 3) is a factor of (2x3 − x2 + 4x + 15). (2) (b) Hence, simplify

2

3 22 3

2 4 15x x

x x x+ −

− + +. (4)

(c) Find the coordinates of the stationary points of the curve with equation

y = 2

3 22 3

2 4 15x x

x x x+ −

− + +. (6)

6. The population in thousands, P, of a town at time t years after 1st January 1980 is

modelled by the formula P = 30 + 50e0.002t. Use this model to estimate (a) the population of the town on 1st January 2010, (2) (b) the year in which the population first exceeds 84 000. (4) The population in thousands, Q, of another town is modelled by the formula Q = 26 + 50e0.003t. (c) Show that the value of t when P = Q is a solution of the equation t = 1000 ln (1 + 0.08e−0.002t). (3) (d) Use the iteration formula tn + 1 = 1000 ln (1 + 0.0020.08e nt− ) with t0 = 50 to find t1, t2 and t3 and hence, the year in which the populations of these two towns will be equal according to these models. (4)

Turn over


7. y y = f(x) (a, 0) O x

(0, b) Figure 1 Figure 1 shows the graph of y = f(x) which meets the coordinate axes at the points

(a, 0) and (0, b), where a and b are constants. (a) Showing, in terms of a and b, the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of (i) y = f −1(x), (ii) y = 2f(3x). (6) Given that

f(x) = 2 − 9x + , x ∈ , x ≥ −9, (b) find the values of a and b, (3) (c) find an expression for f −1(x) and state its domain. (5)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper H






Solomon Press


Solomon Press C3H page 2

1. The functions f and g are defined by f : x → 2 − x2, x ∈ ,

g : x → 32 1

xx −

, x ∈ , x ≠ 12 .

(a) Evaluate fg(2). (2) (b) Solve the equation gf(x) = 1

2 . (4) 2. Giving your answers to 1 decimal place, solve the equation 5 tan2 2θ − 13 sec 2θ = 1, for θ in the interval 0 ≤ θ ≤ 360°. (7) 3. (a) Simplify

2

22 3 92 7 6

x xx x

+ −− +

. (3)

(b) Solve the equation ln (2x2 + 3x − 9) = 2 + ln (2x2 − 7x + 6), giving your answer in terms of e. (4)


( π2 , −1)

4. y

x = π O x ( 3π

2 , −5) y = f(x) Figure 1 Figure 1 shows the graph of y = f(x). The graph has a minimum at ( π2 , −1),

a maximum at ( 3π2 , −5) and an asymptote with equation x = π.

(a) Showing the coordinates of any stationary points, sketch the graph of y = f(x). (3) Given that

f : x → a + b cosec x, x ∈ , 0 < x < 2π, x ≠ π, (b) find the values of the constants a and b, (3) (c) find, to 2 decimal places, the x-coordinates of the points where the graph of y = f(x) crosses the x-axis. (3) 5. The number of bacteria present in a culture at time t hours is modelled by the

continuous variable N and the relationship N = 2000ekt, where k is a constant. Given that when t = 3, N = 18 000, find

(a) the value of k to 3 significant figures, (3) (b) how long it takes for the number of bacteria present to double, giving your answer to the nearest minute, (4) (c) the rate at which the number of bacteria is increasing when t = 3. (3)

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6. (a) Use the derivative of cos x to prove that

ddx

(sec x) = sec x tan x. (4)

The curve C has the equation y = e2x sec x, − π

2 < x < π2 . (b) Find an equation for the tangent to C at the point where it crosses the y-axis. (4) (c) Find, to 2 decimal places, the x-coordinate of the stationary point of C. (3) 7. f(x) = x2 − 2x + 5, x ∈ , x ≥ 1. (a) Express f(x) in the form (x + a)2 + b, where a and b are constants. (2) (b) State the range of f. (1) (c) Find an expression for f −1(x). (3) (d) Describe fully two transformations that would map the graph of y = f −1(x) onto the graph of y = x , x ≥ 0. (2) (e) Find an equation for the normal to the curve y = f −1(x) at the point where x = 8. (4)

8. A curve has the equation y = 2ex + ex, x ≠ 0.

(a) Find dd

yx

. (2) (b) Show that the curve has a stationary point in the interval [1.3, 1.4]. (3) The point A on the curve has x-coordinate 2. (c) Show that the tangent to the curve at A passes through the origin. (4) The tangent to the curve at A intersects the curve again at the point B. The x-coordinate of B is to be estimated using the iterative formula

xn + 1 = 223 3 3 e nx

nx −− + ,

with x0 = −1. (d) Find x1, x2 and x3 to 7 significant figures and hence state the x-coordinate of B to 5 significant figures. (4)

END

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GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper I






Solomon Press


Solomon Press C3I page 2

A

1. Express

22

2 3 5x

x x+ − ÷

3

2x

x x−

as a single fraction in its simplest form. (5) 2. y y = ex + 2

y = 3 + 2ex

B C O x Figure 1 Figure 1 shows the curves y = 3 + 2ex and y = ex + 2 which cross the y-axis at the

points A and B respectively. (a) Find the exact length AB. (3) The two curves intersect at the point C. (b) Find an expression for the x-coordinate of C and show that the y-coordinate

of C is 2

23e

e 2−. (5)

3. f(x) = 2 3

4 1x

x++

, x ∈ , x ≠ 14− .

(a) Find and simplify an expression for f ′(x). (3) (b) Find the set of values of x for which f(x) is increasing. (5)


4. The curve C has the equation y = x2 − 5x + 2 ln 3

x , x > 0. (a) Show that the normal to C at the point where x = 3 has the equation 3x + 5y + 21 = 0. (5) (b) Find the x-coordinates of the stationary points of C. (3) 5. The functions f and g are defined by f(x) ≡ 6x − 1, x ∈ , g(x) ≡ log2 (3x + 1), x ∈ , x > 1

3− . (a) Evaluate gf(1). (2) (b) Find an expression for g−1(x). (3) (c) Find, in terms of natural logarithms, the solution of the equation fg−1(x) = 2. (4) 6. (a) Use the identities for cos (A + B) and cos (A − B) to prove that

cos P − cos Q ≡ −2 sin 2P Q+

sin 2P Q− . (4)

(b) Hence find all solutions in the interval 0 ≤ x < 180 to the equation cos 5x° + sin 3x° − cos x° = 0. (7)

Turn over


7. The function f is defined by f(x) ≡ x2 − 2ax, x ∈ , where a is a positive constant. (a) Showing the coordinates of any points where each graph meets the axes, sketch on separate diagrams the graphs of (i) y = f(x), (ii) y = f( | x | ). (6) The function g is defined by g(x) ≡ 3ax, x ∈ . (b) Find fg(a) in terms of a. (2) (c) Solve the equation gf(x) = 9a3. (4) 8. f(x) = 2x + sin x − 3 cos x. (a) Show that the equation f(x) = 0 has a root in the interval [0.7, 0.8]. (2) (b) Find an equation for the tangent to the curve y = f(x) at the point where it crosses the y-axis. (4)

(c) Find the values of the constants a, b and c, where b > 0 and 0 < c < π2 ,

such that f ′(x) = a + b cos (x − c). (4) (d) Hence find the x-coordinates of the stationary points of the curve y = f(x) in the interval 0 ≤ x ≤ 2π, giving your answers to 2 decimal places. (4)

END

FOR EDEXCEL

GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper J






Solomon Press


Solomon Press C3J page 2

1. (a) Given that cos x = 3 − 1, find the value of cos 2x in the form a + 3b , where a and b are integers. (3) (b) Given that

2 cos (y + 30)° = 3 sin (y − 30)°, find the value of tan y in the form 3k where k is a rational constant. (5) 2. The functions f and g are defined by f(x) ≡ x2 − 3x + 7, x ∈ , g(x) ≡ 2x − 1, x ∈ . (a) Find the range of f. (3) (b) Evaluate gf(−1). (2) (c) Solve the equation fg(x) = 17. (4)

3. f(x) = 4 3 2

213 26 17

3 3x x x x

x x+ − + −

− +, x ∈ .

(a) Find the values of the constants A, B, C and D such that

f(x) = x2 + Ax + B + 2 3 3Cx D

x x+

− +. (4)

The point P on the curve y = f(x) has x-coordinate 1. (b) Show that the normal to the curve y = f(x) at P has the equation x + 5y + 9 = 0. (6)


4. (a) Given that

x = sec 2y , 0 ≤ y < π,

show that

dd

yx

= 2

2

1x x −. (5)

(b) Find an equation for the tangent to the curve y = 3 2cos x+ at the point where x = π3 . (6) 5. f(x) = 5 + e2x − 3, x ∈ . (a) State the range of f. (1) (b) Find an expression for f −1(x) and state its domain. (4) (c) Solve the equation f(x) = 7. (2) (d) Find an equation for the tangent to the curve y = f(x) at the point where y = 7. (4) 6. (a) Prove the identity

2 cot 2x + tan x ≡ cot x, x ≠ 2n π, n ∈ . (5)

(b) Solve, for 0 ≤ x < π, the equation 2 cot 2x + tan x = cosec2 x − 7, giving your answers to 2 decimal places. (6)

Turn over


7. The functions f and g are defined by f : x → 2x − 5, x ∈ , g : x → ln (x + 3), x ∈ , x > −3. (a) State the range of f. (1) (b) Evaluate fg(−2). (2) (c) Solve the equation fg(x) = 3, giving your answers in exact form. (5) (d) Show that the equation f(x) = g(x) has a root, α, in the interval [3, 4]. (2) (e) Use the iteration formula xn + 1 = 1

2 [5 + ln (xn + 3)], with x0 = 3, to find x1, x2, x3 and x4, giving your answers to 4 significant figures. (3) (f) Show that your answer for x4 is the value of α correct to 4 significant figures. (2)

END

FOR EDEXCEL

GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper K






Solomon Press


Solomon Press C3K page 2

1. (a) Find the exact value of x such that 3 arctan (x − 2) + π = 0. (3) (b) Solve, for −π < θ < π, the equation cos 2θ − sin θ − 1 = 0, giving your answers in terms of π. (5) 2. (a) Express

24

9x

x − − 2

3x +

as a single fraction in its simplest form. (4) (b) Simplify

3

28

3 8 4x

x x−

− +. (5)

3. Differentiate each of the following with respect to x and simplify your answers. (a) cot x2 (2) (b) x2 e−x (3)

(c) sin3 2cos

xx+

(4)

4. (a) Find, as natural logarithms, the solutions of the equation e2x − 8ex + 15 = 0. (4) (b) Use proof by contradiction to prove that log2 3 is irrational. (6)


5. The function f is defined by f : x → 3ex − 1, x ∈ . (a) State the range of f. (1) (b) Find an expression for f −1(x) and state its domain. (4) The function g is defined by g : x → 5x − 2, x ∈ . Find, in terms of e, (c) the value of gf(ln 2), (3) (d) the solution of the equation f −1g(x) = 4. (4) 6. f(x) = 2x2 + 3 ln (2 − x), x ∈ , x < 2. (a) Show that the equation f(x) = 0 can be written in the form

x = 2 − 2

ekx , where k is a constant to be found. (3) The root, α, of the equation f(x) = 0 is 1.9 correct to 1 decimal place. (b) Use the iteration formula

xn + 1 = 2 − 2

e nkx , with x0 = 1.9 and your value of k, to find α to 3 decimal places and justify the accuracy of your answer. (5) (c) Solve the equation f ′(x) = 0. (5)

Turn over


7. y (−45, 7)

y = f(x) O x (135, −1) Figure 1 Figure 1 shows the curve y = f(x) which has a maximum point at (−45, 7) and a

minimum point at (135, −1). (a) Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of (i) y = f( | x | ), (ii) y = 1 + 2f(x). (6) Given that

f(x) = A + 2 2 cos x° − 2 2 sin x°, x ∈ , −180 ≤ x ≤ 180, where A is a constant, (b) show that f(x) can be expressed in the form f(x) = A + R cos (x + α)°, where R > 0 and 0 < α < 90, (3) (c) state the value of A, (1) (d) find, to 1 decimal place, the x-coordinates of the points where the curve y = f(x)

crosses the x-axis. (4)

END

FOR EDEXCEL

GCE Examinations

Advanced Subsidiary Core Mathematics C3 Paper L






Solomon Press


Solomon Press C3L page 2

1. f(x) ≡ 2 32

xx

−−

, x ∈ , x > 2.

(a) Find the range of f. (2) (b) Show that ff(x) = x for all x > 2. (3) (c) Hence, write down an expression for f −1(x). (1) 2. Solve each equation, giving your answers in exact form. (a) e4x − 3 = 2 (3) (b) ln (2y − 1) = 1 + ln (3 − y) (4) 3. The curve C has the equation y = 2ex − 6 ln x and passes through the point P

with x-coordinate 1. (a) Find an equation for the tangent to C at P. (4) The tangent to C at P meets the coordinate axes at the points Q and R.

(b) Show that the area of triangle OQR, where O is the origin, is 93 e−

. (4)

4. (a) Express

10( 3)( 4)

xx x

−− +

− 8( 3)(2 1)

xx x

−− −

as a single fraction in its simplest form. (5) (b) Hence, show that the equation

10( 3)( 4)

xx x

−− +

− 8( 3)(2 1)

xx x

−− −

= 1

has no real roots. (4)


5. Find the values of x in the interval −180 < x < 180 for which tan (x + 45)° − tan x° = 4, giving your answers to 1 decimal place. (9) 6. (a) Sketch on the same diagram the graphs of y = x − a and y = 3x + 5a, where a is a positive constant. Show on your diagram the coordinates of any points where each graph meets

the coordinate axes. (6) (b) Solve the equation x − a = 3x + 5a. (4) 7. (a) Use the identity cos (A + B) ≡ cos A cos B − sin A sin B to prove that

cos x ≡ 1 − 2 sin2 2x . (3)

(b) Prove that, for sin x ≠ 0,

1 cossin

xx

− ≡ tan 2x . (3)

(c) Find the values of x in the interval 0 ≤ x ≤ 360° for which

1 cossin

xx

− = 2 sec2 2x − 5,

giving your answers to 1 decimal place where appropriate. (6)

Turn over


8. A curve has the equation y = (2x + 3) e−x. (a) Find the exact coordinates of the stationary point of the curve. (4) The curve crosses the y-axis at the point P. (b) Find an equation for the normal to the curve at P. (2) The normal to the curve at P meets the curve again at Q. (c) Show that the x-coordinate of Q lies in the interval [−2, −1]. (3) (d) Use the iterative formula

xn + 1 = 3 3ee 2

n

n

x

x−

−,

with x0 = −1, to find x1, x2, x3 and x4. Give the value of x4 to 2 decimal places. (3) (e) Show that your value for x4 is the x-coordinate of Q correct to 2 decimal places. (2)

END

core mathematics c3 - mr. samuel lock -...

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