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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.
Published by Elmwood Press80 Attimore RoadWelwyn Garden CityHerts. AL8 6LP
Tel. 01707 333232
These sheets may be copied for use solely by the purchaser’s institute.
(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)
Edexcel C1 paper A page 2
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
Edexcel C1 paper A page 3
Core Mathematics C1 For EdexcelAdvanced Subsidiary
Paper B
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.
Published by Elmwood Press80 Attimore RoadWelwyn Garden CityHerts. AL8 6LP
Tel. 01707 333232
These sheets may be copied for use solely by the purchaser’s institute.
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)
Edexcel C1 paper G page 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
Edexcel C1 paper G page 3
Core Mathematics C1 For EdexcelAdvanced Subsidiary
Paper H
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.
Published by Elmwood Press80 Attimore RoadWelwyn Garden CityHerts. AL8 6LP
Tel. 01707 333232
These sheets may be copied for use solely by the purchaser’s institute.
(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)
4. Solve the simultaneous equations
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.
Edexcel C1 paper H page 2
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
Edexcel C1 paper H page 3
Core Mathematics C1 For EdexcelAdvanced Subsidiary
Paper I
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.
Published by Elmwood Press80 Attimore RoadWelwyn Garden CityHerts. AL8 6LP
Tel. 01707 333232
These sheets may be copied for use solely by the purchaser’s institute.
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)
Edexcel C1 paper I page 2
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
Edexcel C1 paper I page 3
Core Mathematics C1 For EdexcelAdvanced Subsidiary
Paper J
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.
Published by Elmwood Press80 Attimore RoadWelwyn Garden CityHerts. AL8 6LP
Tel. 01707 333232
These sheets may be copied for use solely by the purchaser’s institute.
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)
3. Solve the simultaneous equations
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)
Edexcel C1 paper J page 2
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)
(c) Find the value of k.(2)
END TOTAL 75 MARKS
Edexcel C1 paper J page 3
Core Mathematics C1 For EdexcelAdvanced Subsidiary
Paper K
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.
Published by Elmwood Press80 Attimore RoadWelwyn Garden CityHerts. AL8 6LP
Tel. 01707 333232
These sheets may be copied for use solely by the purchaser’s institute.
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)
Edexcel C1 paper L page 2
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.