Edexcel C1 (Core Mathematics 1)

1. Write down the value of \(16^{-1}\). [1]
2. Find the value of \(16^{-\frac{1}{2}}\). [2]

1. Write down the value of \(8^{-1}\). [1]
2. Find the value of \(8^{-\frac{2}{3}}\). [2]

Factorise completely $$x^3 - 4x^2 + 3x.$$ [3]

Find \(\int (6x^2 + 2x + x^{-2}) \, dx\), giving each term in its simplest form. [4]

Given that $$y = 4x^3 - 1 + 2x^{-1}, \quad x > 0,$$ find \(\frac{dy}{dx}\). [4]

Simplify \((3 + \sqrt{5})(3 - \sqrt{5})\). [2]

1. Given that \(y = 5x^3 + 7x + 3\), find
   1. \(\frac{dy}{dx}\), [3]
   2. \(\frac{d^2y}{dx^2}\). [1]
2. Find \(\int \left(1 + 3\sqrt{x} - \frac{1}{x^2}\right) dx\). [4]

Given that \(y = 6x - \frac{4}{x^2}\), \(x \neq 0\),

1. find \(\frac{dy}{dx}\), [2]
2. find \(\int y \, dx\). [3]

The sequence of positive numbers \(u_1, u_2, u_3, \ldots\) is given by $$u_{n+1} = (u_n - 3)^2, \quad u_1 = 1.$$

1. Find \(u_2\), \(u_3\) and \(u_4\). [3]
2. Write down the value of \(u_{20}\). [1]

Find the set of values of \(x\) for which $$x^2 - 7x - 18 > 0.$$ [4]

1. Express \(\sqrt{108}\) in the form \(a\sqrt{3}\), where \(a\) is an integer. [1]
2. Express \((2 - \sqrt{3})^2\) in the form \(b + c\sqrt{3}\), where \(b\) and \(c\) are integers to be found. [3]

1. Find the value of \(8^{-1}\). [2]
2. Simplify \(\frac{15x^4}{3x}\). [2]

Given that the equation \(kx^2 + 12x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\). [4]

\(x^2 - 8x - 29 = (x + a)^2 + b\), where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\). [3]
2. Hence, or otherwise, show that the roots of $$x^2 - 8x - 29 = 0$$ are \(c \pm d\sqrt{5}\), where \(c\) and \(d\) are integers to be found. [3]

The line \(L\) has equation \(y = 5 - 2x\).

1. Show that the point \(P(3, -1)\) lies on \(L\). [1]
2. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

On separate diagrams, sketch the graphs of

1. \(y = (x + 3)^2\), [3]
2. \(y = (x + 3)^2 + k\), where \(k\) is a positive constant. [2]

Show on each sketch the coordinates of each point at which the graph meets the axes.

Given that \(f(x) = \frac{1}{x}\), \(x \neq 0\),

1. sketch the graph of \(y = f(x) + 3\) and state the equations of the asymptotes. [4]
2. Find the coordinates of the point where \(y = f(x) + 3\) crosses a coordinate axis. [2]

Solve the simultaneous equations $$x + y = 2$$ $$x^2 + 2y = 12.$$ [6]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve passes through the origin \(O\) and through the point \((6, 0)\). The maximum point on the curve is \((3, 5)\). On separate diagrams, sketch the curve with equation

1. \(y = 3f(x)\), [2]
2. \(y = f(x + 2)\). [3]

On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.

Given that \(y = 2x^2 - \frac{6}{x}\), \(x \neq 0\),

1. find \(\frac{dy}{dx}\), [2]
2. find \(\int y \, dx\). [3]

A sequence \(a_1, a_2, a_3, \ldots\) is defined by $$a_1 = 3,$$ $$a_{n+1} = 3a_n - 5, \quad n \geq 1.$$

1. Find the value \(a_2\) and the value of \(a_3\). [2]
2. Calculate the value of \(\sum_{r=1}^5 a_r\). [3]

Solve the simultaneous equations $$y = x - 2,$$ $$y^2 + x^2 = 10.$$ [7]

The \(r\)th term of an arithmetic series is \((2r - 5)\).

1. Write down the first three terms of this series. [2]
2. State the value of the common difference. [1]
3. Show that \(\sum_{r=1}^n (2r - 5) = n(n - 4)\). [3]

Solve the simultaneous equations $$x - 2y = 1,$$ $$x^2 + y^2 = 29.$$ [6]

1. Write \(\sqrt{45}\) in the form \(a\sqrt{5}\), where \(a\) is an integer. [1]
2. Express \(\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}\) in the form \(b + c\sqrt{5}\), where \(b\) and \(c\) are integers. [5]

Differentiate with respect to \(x\)

1. \(x^4 + 6\sqrt{x}\), [3]
2. \(\frac{(x + 4)^3}{x}\). [4]

The equation \(2x^2 - 3x - (k + 1) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\). [4]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve crosses the \(x\)-axis at the points \((2, 0)\) and \((4, 0)\). The minimum point on the curve is \(P(3, -2)\). In separate diagrams sketch the curve with equation

1. \(y = -f(x)\), [3]
2. \(y = f(2x)\). [3]

On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.

Find the set of values of \(x\) for which

1. \(3(2x + 1) > 5 - 2x\), [2]
2. \(2x^2 - 7x + 3 > 0\), [4]
3. both \(3(2x + 1) > 5 - 2x\) and \(2x^2 - 7x + 3 > 0\). [2]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve passes through the points \((0, 3)\) and \((4, 0)\) and touches the \(x\)-axis at the point \((1, 0)\). On separate diagrams, sketch the curve with equation

1. \(y = f(x + 1)\), [3]
2. \(y = 2f(x)\), [3]
3. \(y = f\left(\frac{1}{2}x\right)\). [3]

On each diagram show clearly the coordinates of all the points at which the curve meets the axes.

1. Expand and simplify \((4 + \sqrt{3})(4 - \sqrt{3})\). [2]
2. Express \(\frac{26}{4 + \sqrt{3}}\) in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [2]

1. Show that \((4 + 3\sqrt{x})^3\) can be written as \(16 + k\sqrt{x} + 9x\), where \(k\) is a constant to be found. [2]
2. Find \(\int (4 + 3\sqrt{x})^3 \, dx\). [3]

The curve \(C\) has equation \(y = 4x^2 + \frac{5-x}{x}\), \(x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate \(1\).

1. Show that the value of \(\frac{dy}{dx}\) at \(P\) is \(3\). [5]
2. Find an equation of the tangent to \(C\) at \(P\). [3]

This tangent meets the \(x\)-axis at the point \((k, 0)\).

1. Find the value of \(k\). [2]

1. Show that \(\frac{(3 - \sqrt{x})^2}{\sqrt{x}}\) can be written as \(9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\). [2]

Given that \(\frac{dy}{dx} = \frac{(3 - \sqrt{x})^2}{\sqrt{x}}\), \(x > 0\), and that \(y = \frac{2}{3}\) at \(x = 1\),

1. find \(y\) in terms of \(x\). [6]

On Alice's 11th birthday she started to receive an annual allowance. The first annual allowance was £500 and on each following birthday the allowance was increased by £200.

1. Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was £1200. [1]
2. Find the amount of Alice's annual allowance on her 18th birthday. [2]
3. Find the total of the allowances that Alice had received up to and including her 18th birthday. [3]

When the total of the allowances that Alice had received reached £32 000 the allowance stopped.

1. Find how old Alice was when she received her last allowance. [7]

An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a\) km and common difference \(d\) km. He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period. Find the value of \(a\) and the value of \(d\). [7]

The curve \(C\) has equation \(y = f(x)\), \(x \neq 0\), and the point \(P(2, 1)\) lies on \(C\). Given that $$f'(x) = 3x^2 - 6 - \frac{8}{x^3},$$

1. find \(f(x)\). [5]
2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers. [4]

\includegraphics{figure_2} The points \(A(1, 7)\), \(B(20, 7)\) and \(C(p, q)\) form the vertices of a triangle \(ABC\), as shown in Figure 2. The point \(D(8, 2)\) is the mid-point of \(AC\).

1. Find the value of \(p\) and the value of \(q\). [2]

The line \(l\), which passes through \(D\) and is perpendicular to \(AC\), intersects \(AB\) at \(E\).

1. Find an equation for \(l\), in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
2. Find the exact \(x\)-coordinate of \(E\). [2]

The line \(l_1\) passes through the point \((9, -4)\) and has gradient \(\frac{1}{3}\).

1. Find an equation for \(l_1\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]

The line \(l_2\) passes through the origin \(O\) and has gradient \(-2\). The lines \(l_1\) and \(l_2\) intersect at the point \(P\).

1. Calculate the coordinates of \(P\). [4]

Given that \(l_1\) crosses the \(y\)-axis at the point \(C\),

1. calculate the exact area of \(\triangle OCP\). [3]

The curve with equation \(y = f(x)\) passes through the point \((1, 6)\). Given that $$f'(x) = 3 + \frac{5x^2 + 2}{x^4}, \quad x > 0,$$ find \(f(x)\) and simplify your answer. [7]

The equation \(x^2 + 2px + (3p + 4) = 0\), where \(p\) is a positive constant, has equal roots.

1. Find the value of \(p\). [4]
2. For this value of \(p\), solve the equation \(x^2 + 2px + (3p + 4) = 0\). [2]

The curve \(C\) has equation \(y = 4x + 3x^{-1} - 2x^2\), \(x > 0\).

1. Find an expression for \(\frac{dy}{dx}\). [3]
2. Show that the point \(P(4, 8)\) lies on \(C\). [1]
3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3y - x + 20.$$ [4]

The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).

1. Find the length \(PQ\), giving your answer in a simplified surd form. [3]

The gradient of the curve \(C\) is given by $$\frac{dy}{dx} = (3x - 1)^2.$$ The point \(P(1, 4)\) lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\). [4]
2. Find an equation for the curve \(C\) in the form \(y = f(x)\). [5]
3. Using \(\frac{dy}{dx} = (3x - 1)^2\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2x\). [2]

An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]

Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).

1. Find the amount Sean repays in the 21st month. [2]

Over the \(n\) months, he repays a total of £5000.

1. Form an equation in \(n\), and show that your equation may be written as $$n^2 - 150n + 5000 = 0.$$ [3]
2. Solve the equation in part (c). [3]
3. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem. [1]

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = (x - 1)(x^2 - 4).$$ The curve cuts the \(x\)-axis at the points \(P\), \((1, 0)\) and \(Q\), as shown in Figure 2.

1. Write down the \(x\)-coordinate of \(P\) and the \(x\)-coordinate of \(Q\). [2]
2. Show that \(\frac{dy}{dx} = 3x^2 - 2x - 4\). [3]
3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point \((-1, 6)\). [2]

The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point \((-1, 6)\).

1. Find the exact coordinates of \(R\). [5]

Given that \(f(x) = (x^2 - 6x)(x - 2) + 3x\),

1. express \(f(x)\) in the form \(a(x^2 + bx + c)\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Hence factorise \(f(x)\) completely. [2]
3. Sketch the graph of \(y = f(x)\), showing the coordinates of each point at which the graph meets the axes. [3]

Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 \(\square\) Row 2 \(\square\square\) Row 3 \(\square\square\square\) She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. [3]

Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.

1. Find the total number of sticks Ann uses in making these 10 rows. [3]

Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the \((k + 1)\)th row,

1. show that \(k\) satisfies \((3k - 100)(k + 35) < 0\). [4]
2. Find the value of \(k\). [2]

Given that $$f(x) = x^2 - 6x + 18, \quad x \geq 0,$$

1. express \(f(x)\) in the form \((x - a)^2 + b\), where \(a\) and \(b\) are integers. [3]

The curve \(C\) with equation \(y = f(x)\), \(x \geq 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).

1. Sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). [4]

The line \(y = 41\) meets \(C\) at the point \(R\).

1. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q\sqrt{2}\), where \(p\) and \(q\) are integers. [5]

The curve \(C\) has equation \(y = \frac{1}{3}x^3 - 4x^2 + 8x + 3\). The point \(P\) has coordinates \((3, 0)\).

1. Show that \(P\) lies on \(C\). [1]
2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [5]

Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).

1. Find the coordinates of \(Q\). [5]

\(x^2 + 2x + 3 \equiv (x + a)^2 + b\).

1. Find the values of the constants \(a\) and \(b\). [2]
2. Sketch the graph of \(y = x^2 + 2x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes. [3]
3. Find the value of the discriminant of \(x^2 + 2x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). [2]

The equation \(x^2 + kx + 3 = 0\), where \(k\) is a constant, has no real roots.

1. Find the set of possible values of \(k\), giving your answer in surd form. [4]

The curve \(C\) with equation \(y = f(x)\), \(x \neq 0\), passes through the point \((3, 7\frac{1}{2})\). Given that \(f'(x) = 2x + \frac{3}{x^2}\),

1. find \(f(x)\). [5]
2. Verify that \(f(-2) = 5\). [1]
3. Find an equation for the tangent to \(C\) at the point \((-2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

1. On the same axes sketch the graphs of the curves with equations
   1. \(y = x^2(x - 2)\), [3]
   2. \(y = x(6 - x)\), [3]
   and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
2. Use algebra to find the coordinates of the points where the graphs intersect. [7]

The line \(l_1\) passes through the points \(P(-1, 2)\) and \(Q(11, 8)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [4]

The line \(l_2\) passes through the point \(R(10, 0)\) and is perpendicular to \(l_1\). The lines \(l_1\) and \(l_2\) intersect at the point \(S\).

1. Calculate the coordinates of \(S\). [5]
2. Show that the length of \(RS\) is \(3\sqrt{5}\). [2]
3. Hence, or otherwise, find the exact area of triangle \(PQR\). [4]