Edexcel C1 Elmwood Press

Core Mathematics C1 For EdexcelAdvanced Subsidiary

Paper A

Time: 1 hour 30 minutes

Instructions and Information

Candidates may NOT use a calculator in this paper.

Full marks may be obtained for answers to ALL questions.

The booklet ‘Mathematical Formulae and Statistical Tables’, available from Edexcel, may beused.

Advice to Candidates

You must show sufficient working to make your methods clear to an examiner.Answers without working may gain no credit.

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

© Elmwood Press

1. (a) Express30√

5in the form k

√5.

(2)

(b) Express 8− 23 as an exact fraction in its simplest form.

(2)

2. (a) Make x the subject of the equation a(x − b) = x + c.(3)

(b) Solve the equation 2x2 + 7x = 4.(3)

(c) Differentiate 2x5 + x12 with respect to x.

(3)

3. (a) Express x2 − 6x + 10 in the form (x + a)2 + b.(3)

(b) Hence write down the coordinates of the minimum point on the graph ofy = x2 − 6x + 10.

(2)

4. The straight line l has the equation 3x − 2y = 9.

The straight line m is perpendicular to l and passes through the point (3, −2).

Find an equation for m in the form ax + bx + c = 0.(5)

5. The equation x2 + 6x + m = 0 has no real roots for x.

Find the set of values that m can take.(5)

Edexcel C1 paper A page 1

6. (a) An arithmetic series has first term a and common difference d. Prove that the sum of thefirst n terms of the series is

12n[2a + (n − 1)d].

(4)

(b) The tenth term of an arithmetic series is 67 and the sum of the first twenty terms is 1280.

Find the first term a and the common difference d.(6)

7. (a) Solve the simultaneous equations

y = x2 − x + 5, y = 3x + 1.(5)

(b) What can you deduce from the solution to part (a) about the graphs ofy = x2 − x + 5 and y = 3x + 1?

(2)

(c) Hence, or otherwise, find the equation of the normal to the curve y = x2 − x + 5 atthe point (2, 7), giving your answer in the form ax + by + c = 0 where a, b and c

are integers.(4)

8. Given that f(x) = 2x2 − 5x − 3,

(a) find the coordinates of all the points at which the graph of y = f(x) crosses thecoordinate axes.

(3)

(b) Sketch the graph of y = f(x).(2)

(c) The graph of y = f(x) is obtained from the graph of y = 2x2 − 5x by a singletransformation. Describe the transformation fully.

(3)


9. Figure 1

y = x + 1x

y

x

Figure 1 shows a sketch of the curve with equation y = x + 1

x.

Find the coordinates of the two points on the curve where the gradient is zero.(5)

10. PQRS is a rectangle, where P, Q and R are the points (4, 9), (2, k) and (8, 1) respectively.

(a) Find the coordinates of the mid-point of PR.(2)

(b) Find the gradient of the line PQ, giving your answer in terms of k.(2)

(c) Determine the two possible values of k.(4)

(d) Find the area of the rectangle PQRS for the case in which PQRS is a square.(5)

END TOTAL 75 MARKS



Paper B









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1. Write down the exact values of

(a) 3−2,(1)

(b) (5√

3)2,(1)

(c) (13 + 23 + 33)12 .

(2)

2. Giving your answers in the form a + b√

5, where a and b are rational numbers, find

(a) (4 − √5)2,

(3)

(b)1

(4 − √5)

.(3)

3. (a) Calculatedy

dxwhen y = x2 − x + 1.

(2)

(b) Find the coordinates of the point on the curve where the gradient is equal to 9.(3)

(c) Find the equation of the tangent at this point in the form y = mx + c.(3)

4. The rth term of a sequence is defined by

ur = 3 + 4r, r ≥ 1.

(a) Find the first three terms of the sequence.(3)

(b) Calculate200∑

r=1(3 + 4r).

(4)

5.dy

dx= 8 + 1

x2 .

(a) Use integration to find y in terms of x.(3)

(b) Given that y = 12 when x = 1, find the value of y at x = 1

2.

(3)

Edexcel C1 paper B page 1

6. Figure 1

y

x(–1, 0)

(0, –1)

(1, –3)

(3, 0)

Figure 1 shows a sketch of the curve with equation y = f(x).

The curve crosses the coordinate axes at the points (−1, 0), (0, −1) and (3, 0).The minimum point on the curve is (1, −3).

On separate diagrams sketch the curve with equation

(a) y = f(x + 1),(3)

(b) y = f(1

2x).

(3)

On each diagram, show clearly the coordinates of the minimum point, and ofeach point at which the curve crosses the coordinate axes.

7. The quadratic equation x2 + kx + 1 = 0 has no real roots for x.

(a) Write down the discriminant of x2 + kx + 1 in terms of k.(2)

(b) Hence find the set of values that k can take.(4)


8. (a) Solve the simultaneous equations

x + y = 8

2x2 + 13 = 12x + y.(6)

(b) Hence, or otherwise, find the set of values of x for which

2x2 − 12x + 13 < 8 − x.(3)

9. (a) f(x) = x2 + 4x + 1

By completing the square, write f(x) in the form (x + a)2 + b where a and b areconstants to be determined.

(3)

(b) Hence solve the equation

x2 + 4x + 1 = 2,(4)

writing your solutions in the form c ± √d.

(c) Write down the coordinates of the minimum point on the graph of y = f(x).(2)

(d) Sketch the graph of y = f(x).(3)

(e) Sketch the graph of y = −f(x).(2)

10. The points A and B have coordinates (−1, 4) and (3, −2) respectively. The straightline l passes through B and is perpendicular to AB.

(a) Find an equation for l, giving your answer in the form ax + by + c = 0, wherea, b and c are integers.

(6)

The line AB crosses the x and y axes at the points C and D respectively.

(b) Calculate the area of triangle OCD, where O is the origin.(6)

END TOTAL 75 MARKS



Paper C









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1. (a) Find, in the form y = mx + c, the equation of the line joining the points (5, 6)and (2, −3).

(3)

(b) Find the coordinates of the points where this line cuts the coordinate axes.(3)

2. (a) Factorise the expression 9x2 + 12x and hence solve the equation 9x2 + 12x = 0.(3)

(b) The function f is defined by f(x) = 9x2 + 12x + c, where c is a constant.

Given that f(x) = 0 has equal roots, find the value of c and hence solve f(x) = 0.(4)

3. f(x) = x3 − x2 − 6x

(a) Fully factorise x3 − x2 − 6x.(3)

(b) Sketch the curve y = f(x), showing the coordinates of any points of intersectionwith the coordinate axes.

(3)

4. The point P with coordinates (3, m) lies on the curve y = x2 + kx. At P the gradientof the curve is 8.

Find the values of the constants k and m.(6)

5. (a) Solve the inequality

2(x − 1) > 5 − x.(2)

(b) Solve the inequality

x2 + 2x − 15 < 0.(3)

Edexcel C1 paper C page 1

6.y

x

A

0 2

2

Figure 1

Figure 1 shows the graph of y = f(x).

(a) Write down the number of solutions that exist for the equation

(i) f(x) = 1.7

(ii) f(x) = 3(2)

(b) Sketch on separate diagrams the graphs below, showing the coordinates of pointA in each case.

(i) y = f(x + 2),

(ii) y = f(2x).(6)

7. The first three terms of an arithmetic series are x, 4x − 9 and 5x respectively.

(a) Show that x = 9.(2)

(b) Find the value of the 51st term of this series.(3)

(c) Show that the sum of the first n terms of the series is 9n2.(4)


8. Figure 2y

x03x + y = 9

y2 = 9(x – 1)

A

P

Q

Figure 2 shows the curve with equation y2 = 9(x − 1) and the line with equation3x + y = 9

The curve crosses the x-axis at the point A, and the line intersects the curve at thepoints P and Q.

(a) Write down the coordinates of A.(1)

(b) Find, using algebra, the coordinates of P and Q.(6)

9. Given that (1 + √5)(3 − √

5) = a + b√

5, where a and b are integers,

(a) find the value of a and the value of b.(2)

Given that5 + √

2

3 + √2

= c + d√

2, where c and d are rational numbers,

(b) find the value of c and the value of d.(3)

(c) Solve the equation

x√

8 − 6 = 2x√2

giving your answer in the form k√

2, where k is an integer.(5)


10. Given that

x2 + 8x + 17 ≡ (x + a)2 + b,

where a and b are constants,

(a) find the value of a and the value of b.(3)

(b) Hence show that the equation x2 + 8x + 17 = 0 has no real roots.(2)

The equation x2 + 8x + k = 0 has equal roots.

(c) Find the value of k.(2)

(d) For this value of k, sketch the graph of y = x2 + 8x + k, showing thecoordinates of any points at which the graph meets the coordinate axes.

(4)

END TOTAL 75 MARKS



Paper D









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1. (a) Express

√3 + 3√3 − 1

in the form a√

3 + b, where a and b are integers.(4)

(b) Solve the inequality√

3(x − √3) < x + √

3(3)

2. The line AB has equation 3x − 4y = 8

The point A has coordinates (0, −2) and the point B has coordinates (4, p)

(a) (i) Find the value of p.(1)

(ii) Find the gradient of AB.(2)

(b) The point C has coordinates (3, 1). Show that AC has length k√

2, where k isan integer.

(3)

3. The rth term of an arithmetic series is (3r + 1)

(a) Write down the first three terms of this series.(2)

(b) State the value of the common difference.(1)

(c) Show thatn∑

r=1(3r + 1) = n

2(3n + 5)

(3)

4. Find the set of the values of x for which

(a) 3x + 1 < x − 3(2)

(b) 2x2 + 5x − 3 < 0(4)

(c) both 3x + 1 < x − 3 and 2x2 + 5x − 3 < 0(2)

Edexcel C1 paper D page 1

5. The curve C has the equation y = (x − a)2 where a is a constant.

Given that

dy

dx= 2x − 8

(4)

(a) find the value of a.

(b) Using the same axes, sketch the curve C and the graph of y = x2.(3)

6. f(x) = x2 − kx + 16, where k is a constant.

(a) Find the set of values of k for which the equation f(x) = 0 has no real solutions.(4)

Given that k = 6,

(b) express f(x) in the form (x − a)2 + b, where a and b are constants to be found,(3)

7.dy

dx= 2x + 1

x2

(a) Use integration to find y in terms of x.(3)

(b) Given that y = 5 when x = 1, find the value of y at x = 1

2.

(4)

8. (a) Evaluate (614)− 1

2

(2)

(b) Solve the equation

4x = 2x+3

(3)

(c) Find the value of x such that

2 + x

x= √

2,

giving your answer in the form a + b√

2, where a and b are rational.(4)


9.y

x0

y = f(x)

(2, 3)

(–1, –2)

Figure 1

Figure 1 shows a sketch of the curve with equation y = f(x). The curve has amaximum at (2, 3) and a minimum at (−1, −2).

Showing the coordinates of any turning points, sketch on separate diagrams thecurves with equations

(a) y = f(x + 2) (3)(b) y = f(−x) (3)(c) y = f(2x) (3)

10. The line L has equation

x + y = 4

and the curve C has equation

y = x2 + 2

(a) Sketch on one pair of axes the line L and the curve C. Indicate the coordinatesof their points of intersection with the axes.

(3)

(b) Show that the x-coordinates of the points of intersection of L and C satisfythe equation

x2 + x − 2 = 0(2)

(c) Hence calculate the coordinates of the points of intersection of L and C.(4)

END TOTAL 75 MARKS



Paper E









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1. Solve the equations, for x > 0

(a) x13 = 2

(1)

(b) x−2 = 1

16 (1)

(c) (x8)12 = 81

(1)

2. The number x satisfies the equation

x2 + kx + 25 = 0

where k is a constant.

Find the values of k for which this equation has:

(a) equal roots;(2)

(b) two distinct real roots;(2)

(c) no real roots.(2)

3. (a) Express√

45 in the form a√

5, where a is an integer.(1)

(b) Express (3 − √5)2 in the form b + c

√5, where b and c are integers.

(3)

(c) Given f(x) = (2 + √x)2 + (1 − 2

√x)2,

expand the brackets and write f(x) in its simplest form.(3)

4. An arithmetic series has a common difference of −2.

Given that the sum of the first 10 terms of the series is 910, find

(a) the first term of the series,(3)

(b) the value of n, given that the sum of the first n terms of the series is zero.(4)

Edexcel C1 paper E page 1

5. (a) Solve the equation

5x2 = 3x + 2.(3)

(b) Multiply (2x2 − x − 1) by (3 − 2x), arranging your answer in ascending powers of x.(3)

6. The curve C with equation y = f(x) is such that

dy

dx= 4x + 4√

x, x > 0

(a) Show that, when x = 2, the exact value ofdy

dxis 8 + 2

√2.

(3)

The curve C passes through the point (4, 50).

(b) Using integration, find f(x).(6)

7.y

x0

P(1, 2)

2–2

2

Figure 1

Figure 1 shows the graph of y = f(x) for −2 ≤ x ≤ 2.

Outside this interval f(x) is zero.

(a) Sketch, on separate diagrams, the following graphs. On each graph label the imageof the point P , giving its coordinates

(i) y = 2f(x)(2)

(ii) y = f(x − 1)(2)

(b) The graph of y = f(−x) is obtained from the graph of y = f(x) by a singletransformation. Describe fully the transformation.

(2)


8.y

x0

A(2, 12)

C(p, q)

B(12, 2)

M(4, 6)

Figure 2

The points A(2, 12), B(12, 2) and C(p, q) form the vertices of a triangle ABC, as shownin Figure 2. The point M(4, 6) is the mid-point of AC.

(a) Find the value of p and the value of q.(2)

The line l, which passes through M and is perpendicular to AC, intersects AB at N .

(b) Find an equation for l, in the form ax + by + c = 0, where a, b and c areintegers.

(5)

The line AB has equation x + y = 12.

(c) Find the exact coordinates of N .(2)

9. The curve C has equation y = f(x) and the point P(2, 4) lies on C.

Given that

f ′(x) = 6x2 − 4x − 7

(a) find f(x).(4)

(b) Verify that the point (1, 3) lies on C.(2)

The point Q also lies on C, and the tangent to C at Q is parallel to the tangentto C at P .

(c) Find the x-coordinate of Q.(5)


10. The curve C has equation

y = 2x3 − 7x + 4

x, x �= 0.

The point A with coordinates (1, −1) lies on C.

(a) Show that the gradient of C at A is −5.(2)

(b) Show that an equation for the normal to C at A is

5y = x − 6(4)

The normal to C at A meets the y-axis at the point P .

(c) Find the coordinates of P .(1)

(d) Find the coordinates of another point on C at which the gradient is −5.(4)

END TOTAL 75 MARKS



Paper F









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1. (a) Simplifyn

13 n

13

n.

(1)

(b) Simplify (3√

2 + 1)(√

2 − 1)(1)

(c) Express

√2√

2 + 1in the form a + b

√2, where a and b are integers to be determined.

(3)

2. Solve the simultaneous equations

y = 7 − 3x

xy + 12 = 2x(5)

3. The nth term of a sequence is defined by

un = n2 − 5n + 7, n ≥ 1.

(a) Find the first and second terms of the sequence.(2)

(b) For what value of n is the nth term of the sequence equal to 31?(4)

4. The equation of a curve is y = x2 + 3x − 4.

Find the gradient of the curve at the two points where the curve meets the line y = 6.

y

x0

6

(7)

Edexcel C1 paper F page 1

5. The equation x2 + mx + m = 0 has no real roots for x.

Find the set of values that m can take.(5)

6. Given that

dy

dx= 10x4 + 3

and that y = 2 when x = 1, find the value of y when x = −1.(6)

7. Given that f(x) = 12 + 5x − 2x2

(a) find the coordinates of all points at which the graph of y = f(x) crosses the coordinateaxes.

(3)

(b) Sketch the graph of y = f(x).(2)

(c) The graph of y = f(x) is obtained from the graph of y = 5x − 2x2 by a singletransformation. Describe the transformation fully.

(2)

8. (a) Finddy

dxin each of the following cases:

(i) y = 6x − 5x3

(2)

(ii) y = x2(x − 4)(3)

(iii) y = 2√

x(2)

(b) The equation of a curve is y = 3x + 1

x2 .

Find the coordinates of the point on the curve where the gradient of the curve is equal to 1.

(4)


9. Figure 1

A (1, 2)

C

B (3, 8)

M

The points A and B have coordinates (1, 2) and (3, 8) respectively, and AB is a chordof a circle with center C, as shown in Fig. 1.

(a) Find the gradient of AB.(2)

The point M is the mid-point of AB.

(b) Find an equation for the line through C and M .(5)

Given that the y-coordinate of C is 4,

(c) find the x-coordinate of C,(2)

(d) show that the radius of the circle is 2√

5.(4)

10. A curve has the equation y = x2 − x.

The point P on the curve has x-coordinate 1.

(a) Find an equation for the normal to the curve at P , giving your answer in the formy = mx + c.

(6)

(b) Find the coordinates of the point where the normal to the curve at P intersects the curveagain.

(4)

END TOTAL 75 MARKS



Paper G









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1. (a) Given that 27 = 3m, write down the value of m.(1)

(b) Given that 9x = 271−x , find the value of x.(3)

2. The sum of an arithmetic series isn∑

r=1(27 + 3r)

(a) Write down the first two terms of the series.(2)

(b) Find the common difference of the series.(1)

Given that n = 20,

(c) find the sum of the series.(3)

3. Find the set of values for x for which

(a) 4x + 8 > 3 − x (1)

(b) 2x2 + 5x − 3 < 0 (4)

(c) both 4x + 8 > 3 − x and 2x2 + 5x − 3 < 0.(2)

4. (a) By completing the square, find the exact roots of the equation

x2 + 4x + 1 = 0.(3)

(b) By completing the square, find in terms of the constant k, the roots of the equation

x2 + 2kx + 5 = 0(4)

5. The gradient of a curve is given by

dy

dx= 6x2 − 3x

The curve passes through the point (−1, 2). Find the equation of the curve.(6)

Edexcel C1 paper G page 1

6. The function f is defined for all real values of x by

f(x) = (1 + x)(1 − 2x)

(a) (i) Find the coordinates of the points where the graph of y = f(x) cuts thecoordinate axes.

(3)

(ii) Sketch the graph of y = f(x).(2)

(b) The graph of y = f(x) is translated by 3 units in the positive y-direction to give thegraph of y = g(x). Find an expression for g(x) in the form ax2 + bx + c, where a, b

and c are integers.(2)

7. The diagram shows a part of the graph of

y

x0

P

y = 3 + 2x – 4x4

(a) (i) Finddy

dx.

(2)

(ii) Show that the x-coordinate of the stationary point P is 12 .

(3)

(iii) Find the y-coordinate of P .(2)


8. For the curve C with equation y = x3 − 3x2 + 2x

(a) finddy

dx,

(2)

The point A, on the curve C, has x-coordinate 2.

(b) Find an equation for the normal to C at A, giving your answer in the formax + by + c = 0, where a, b and c are integers.

(5)

9. The points A, B and C have coordinates (2, 8), (6, 6) and (8, 10) respectively.

(a) Show that AB and BC are perpendicular.(2)

(b) Find an equation of the line BC.(3)

(c) The equation of the line AC is 3y = x + 22 and M is the mid-point of AB.(3)

(i) Find an equation of the line through M parallel to AC.(3)

(ii) This line intersects BC at the point T . Find the coordinates of T .(2)

10. The function f is defined for all real values of x by

f(x) = (x2 + 3)(x − 2).

(a) Find f(−3) and f(3).(2)

(b) Show that the curve with equation y = f(x) crosses the x-axis at only onepoint and state the x-coordinate of this point.

(3)

(c) Write down the y-coordinate of the point where the curve y = f(x) crossesthe y-axis.

(1)

(d) Differentiate f(x) with respect to x to obtain f ′(x).(3)

(e) Show that the equation f ′(x) = 0 has no real roots.(3)

(f ) Sketch the curve y = f(x).(2)

END TOTAL 75 MARKS



Paper H









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1. (a) Write down the value of 2713

(1)(b) Find the value of 16− 1

2

(2)(c) Given that 64 = 4x , write down the value of x.

(1)(d) Given that 64(1−y) = 4y , find the value of y.

(3)

2. (a) Given that y = 3x3 + 8x − 7, find

(i)dy

dx,

(3)

(ii)d2y

dx2 .(1)

(b) Find �(

5 + 6√

x + 2

x2

)

dx(4)

3. Given that the equation kx2 + 12x + 9 = 0, where k is a positive constant, has equal roots,find the value of k.

(4)


x − 2y = 7

x2 + 4y2 = 37(6)

5. The sequence u1, u2, u3, . . . is defined by

un = 2n − n

k

where k is a constant.

Given that u1 = u2

(a) find the value of k,(3)

(b) find the value of u4(2)

Edexcel C1 paper H page 1

6. The curve with equation y = f(x) passes through the point (4, −1).

Given that

f ′(x) = 3x12 + 5

find f(x).(6)

7. Figure 1

y

x0 3 12

P(6, 3)

Figure 1 shows a sketch of the curve with equation y = f(x). The curvecrosses the x-axis at the points (3, 0) and (12, 0). The maximum point onthe curve is P(6, 3).

In separate diagrams sketch the curve with equation

(a) y = f(x + 2)(3)

(b) y = f(3x)(3)

On each diagram, give the coordinates of the points at which the curvecrosses the x-axis, and the coordinates of the image of P under the giventransformation.


8. (a) Given that y = x3 − 4x2 + 5x − 2, finddy

dx.

(2)

P is the point on the curve where x = 3.

(b) Calculate the y-coordinate of P .(1)

(c) Calculate the gradient at P .(2)

(d) Find the equation of the tangent at P .(2)

(e) Find the equation of the normal at P .(2)

(f ) Find the values of x for which the curve has a gradient of 5.(3)

9. The curve C has equation y = x2 − 3 and the straight line l has equationy = 3 − x.

(a) Sketch C and l on the same axes.(3)

(b) Write down the coordinates of the points at which C meets the coordinate axes.(2)

(c) Using algebra, find the coordinates of the points at which l intersects C.(4)

10. The points A(1, 2), B(3, −2) and C(k, 0), where k is a constant, are the verticesof �ABC. Angle ABC is a right angle.

(a) Find the gradient of AB.(2)

(b) Calculate the value of k.(3)

(c) Show that the length of AB may be written in the form p√

5, where p is aninteger to be found.

(3)

(d) Find the exact value of the area of �ABC.(4)

END TOTAL 75 MARKS



Paper I









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1. (a) Simplifya

14 a

18

a12 (1)

(b) Express

(2

3

)−2

as an exact fraction in its simplest form(2)

(c) Simplify√

20 + 3√

45, giving your answer in simplified surd form.(2)

(d) Given that4n × 25n

16n= 2kn, find the value of k.

(3)

2. (a) Differentiate with respect to x

5x2 − 1

3x (3)

(b) Find �(

1

x2 − x

)

dx.(3)

(c) Find �(x3 + √x) dx.

(3)

3. Find the gradient of the straight line l with equation 3y − 2x + 5 = 0(1)

Find an equation of the straight line which passes through the origin and which isperpendicular to l.

(3)

4. Find the coordinates of the points of intersection of the line y = 2x + 3 and the curvey = x2 − 2x + 5, giving your answers as surds.

(5)

5. The terms of a sequence are given by

un = (2n + k)2, n ≥ 1,

where k is a positive constant.

(a) Write down the values of u1 and u2, in terms of k.(2)

Given that u2 = 2u1,

(b) find the value of k,(2)

(c) show that u3 = 4(11 + 6√

2).(2)

Edexcel C1 paper I page 1

6. (a) Express x2 − 4x + 7 in the form (x + a)2 + b where a and b are constants to bedetermined. Hence show that the value of x2 − 4x + 7 is positive for all valuesof x.

(4)

(b) Sketch the graph of y = x2 − 4x + 7.

Mark the axis of symmetry and give its equation.

State the coordinates of the lowest point of the curve.(3)

(c) Solve the inequality x2 − 4x + 7 < 12(2)

7. The first term of an arithmetic series is −7 and the eighth term of the series is 14.

(a) Find the common difference and the sum of the first thirty terms of the series.(4)

(b) Find the value of n for which the nth term of the series is 212.(2)

(c) Find the value of n for which the sum of the first n terms is 114.(4)

8. (a) Sketch the graph of y = 1

x, where x �= 0, showing the parts of the graph

corresponding to both positive and negative values of x.(2)

(b) Describe fully the geometrical transformation that transforms the curve

y = 1

xto the curve y = 1

x + 2.

(2)

(c) Describe fully the geometrical transformation that transforms the curve

y = 1

xto the curve y = 1

x+ 4.

(2)

(d) Hence sketch the curve y = 1

x + 2and curve y = 1

x+ 4.

(3)

(e) Differentiate1

xwith respect to x and hence find the gradient of the curve

y = 1

x+ 4 at the point (2, 41

2).(3)


9. The curve C has equation y = f(x). Given that

dy

dx= 6x2 − 4x + 1

and that C passes through the point P(1, 0)

(a) find y in terms of x.(4)

(b) Find an equation of the tangent to C at P .(3)

The tangent to C at the point Q is parallel to the tangent at P .

(c) Calculate the x-coordinate of Q.(5)

END TOTAL 75 MARKS



Paper J









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1. (a) Express (√

3 + 2)2 in the form a + b√

3, where a and b are integers.(2)

(b) Hence express(√

3 + 2)2

(√

3 + 1)in the form p + q

√3, where p and q are rational

numbers.(4)

2. The point A has coordinates (3, 10) and the point B has coordinates (7, −2).

The mid-point of AB is P .

Find the equation of the straight line which passes through P and which isperpendicular to the line 4y + 2x = 11.

Give your answer in the form y = mx + c.(5)


x + y − 3 = 0

x2 + 3xy + y2 = 11.(7)

4. f(x) = (3x2 − 1)2

x3 , x �= 0

(a) Show that f(x) = 9x − 6x−1 + x−3.(2)

(b) Hence, or otherwise, differentiate f(x) with respect to x.(3)

5. Given that

f(x) = x2 + x + 2

(a) express f(x) in the form (x + a)2 + b, where a and b are rational numbers.(3)

The curve C with equation y = f(x) meets the y-axis at P and has a minimum point at Q.

(b) Sketch the graph of C, showing the coordinates of P and Q.(4)

Edexcel C1 paper J page 1

6. Figure 1

y

xA B

y = f(x)

Figure 1 shows the curve with equation y = f(x) which crosses the x-axis at theorigin and at the points A and B.

Given that

f ′(x) = 3x2 − 2x − 2

(a) find an expression for y in terms of x,(5)

(b) find the coordinates of the points A and B.(5)

7. The width of a rectangular field is x metres, x > 0. The length of the field is 40 mmore than its width. Given that the perimeter of the pitch must be less than 200 m,

(a) form a linear inequality in x.(2)

Given that the area of the field must be greater than 500 m2,

(b) form a quadratic inequality in x.(2)

(c) by solving your inequalities, find the set of possible values of x.(4)


8. f(x) = (x − 2)(x + 4).

(a) Solve the equation f(x) = 0.(1)

(b) Sketch the curve with equation y = f(x), showing the coordinates of any points ofintersection with the coordinate axes.

(3)

(c) Sketch the curve with equation y = f(2x), showing the coordinates of any points ofintersection with the coordinate axes.

(3)

When the graph of y = f(x) is stretched by a scale factor of 3 parallel to the y-axis itmaps onto the graph with equation y = ax2 + bx + c, where a, b, and c are constants.

(d) Find the values of a, b and c.(3)

9. (a) By completing the square, find in terms of m, the roots of the equation

x2 + 2mx − 1 = 0(4)

(b) Prove that, for all values of m, the roots of x2 + 2mx − 1 = 0 are real and different.(2)

(c) Given that m = √3, find the exact roots of the equation.

(2)

10. The curve C has equation y = x3 + 2 + 4

x, x �= 0. The point P on C has x-coordinate 2.

(a) Show that the value ofdy

dxat P is 11.

(5)

(b) Find an equation of the tangent to C at P .(3)

This tangent meets the y-axis at the point (0, k)


END TOTAL 75 MARKS



Paper K









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© Elmwood Press

1. Find the integer n such that

2√

20 − √45 = √

n.(4)

2. The points A and B have coordinates (3, −1) and (5, 3) respectively.

The straight line which passes through A and B meets the x-axis at P and the y-axis at Q.Find the area of the triangle OPQ, where O is the origin.

(6)

3. Given that 2x = 4√2

and 2y = 8√

2

(a) find the exact value of x and the exact value of y,(3)

(b) calculate the exact value of 2y−x .(2)

4. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 9.(3)

(b) Hence, or otherwise, evaluate111∑

r=1(9r + 1).

(3)

5. y = 2x + √x − 5

(a) Finddy

dx.

(2)

(b) Find∫

y dx.(3)

6. Given that

f ′(x) = 2 − 8

x2 , x �= 0,

(a) find an expression for f(x).(3)

Given also that

f(4) = 3f(1),

(b) find f(8).(5)

Edexcel C1 paper K page 1

7. (a) Express 2x2 + 4x − 3 in the form

a[(x + p)2 + q],stating the values of the constants a, p and q.

(4)

(b) Sketch the graph of y = 2x2 + 4x − 3, stating the coordinates of the vertex.(3)

(c) Solve the equation 2x2 + 4x − 3 = 3.(3)

8. The points A, B and C have coordinates (5, 2), (1, 10) and (9, 4) respectively.

y

x0

A

C

B

(a) Find an equation for the straight line BC in the form ax + by = c, where a, b and c areintegers.

(3)

(b) Prove that the triangle ABC is right-angled and find its area.(3)

(c) Determine an equation for the straight line which passes through A and which isperpendicular to BC.

(3)

9. The gradient of the curve C is given by

dy

dx= 3(x + 2)2

The point P(0, 4) lies on C.

(a) Find an equation of the normal to C at P.(4)

(b) Find an equation for the curve C in the form y = f(x).(5)


10. f(x) = 4 − (x − 1)2

(a) Write down the maximum value of f(x).(1)

(b) Sketch the graph of y = f(x), showing the coordinates of the points at which thegraph meets the coordinate axes.

(5)

The points A and B on the graph of y = f(x) have coordinates (−2, −5) and (2, k)respectively.


(d) Find, in the form y = mx + c, an equation of the straight line through A and B.(4)

(e) Find the coordinates of the point at which the line AB crosses the x-axis.(2)

END TOTAL 75 MARKS



Paper L









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© Elmwood Press

1. The points P, Q and R have coordinates (2, 1), (6, 2) and (−1, 5) respectively.

Find an equation for the straight line which passes through R and is parallel to PQ.Give your answer in the form ax + by = c, where a, b and c are integers.

(4)

2. (a) Solve the inequality

3x − 8 > x + 13.(2)

(b) Solve the inequality

(x − 6)(x + 1) < 8(4)

3. (a) Express 2x2 + 12x + 13 in the form a(x + b)2 + c.(4)

(b) Find the equation of the line of symmetry of the curve

y = 2x2 + 12x + 13.(3)

4. The equation x2 + 3kx + k = 0, where k is a constant, has real roots.

(a) Prove that k(9k − 4) ≥ 0.(2)

(b) Hence find the set of possible values of k.(4)

(c) Write down the values of k for which the equation x2 + 3kx + k = 0 has equal roots.(1)

5. (a) Given that 16 = 2m, write down the value of m.(1)

(b) Given that 4n = 83−n, find the value of n.(4)

(c) (i) Given that u14 = y, show that the equation

u14 = 2 + 3u− 1

4

may be written as

y2 − 2y − 3 = 0.(3)

(ii) Hence solve the equation u14 = 2 + 3u− 1

4 .(2)

Edexcel C1 paper L page 1

6. (a) Evaluate

20∑

r=1(5t + 1)

(3)

(b) A firm sold 20000 phones in the year 2005. A model for the future assumes that saleswill increase in an arithmetic sequence with common difference x. This model predictsthat total sales for the 10 years from 2005 to 2014 inclusive will be 312 500 phones.

(i) Find the value of x.(4)

Using your value of x,

(ii) find the predicted sales for the year 2012.(2)

7. Given that

dy

dx= 8x3 + 1

x2 , x �= 0,

(a) find the value of x for whichdy

dx= 0,

(2)

(b) findd2y

dx2 .(3)

Given also that y = 2 when x = 1

2,

(c) find the value of y when x = 1.(6)

8. f(x) = x3 − 2x2 − 13x − 10.

(a) Show that

(x + 2)(x + 1)(x − 5) ≡ x3 − 2x2 − 13x − 10.(3)

(b) Sketch the curve y = f(x), showing the coordinates of any points of intersectionwith the coordinate axes.

(3)

(c) Sketch on separate diagrams the curves

(i) y = f(x + 1),

(ii) y = f(−x).(4)


9. A curve has the equation y = 4

x2 + x2.

The point P on the curve has coordinates (−2, 5)

(a) Show that the gradient of the curve at P is −3.(3)

(b) Find an equation for the tangent to the curve at P , giving your answer in the formax + by + c = 0.

(4)

This tangent intersects the coordinate axes at the points A and B.

(c) Show that the length of AB is1

3

√10.

(4)

END TOTAL 75 MARKS


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Edexcel C1 Elmwood Press