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AQA C1 2009 June Q4
17 marks Moderate -0.8
4
  1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
    1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
    2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
    3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
    4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
  2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
    1. State the \(y\)-coordinate of the point \(B\).
    2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
    3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).
AQA C1 2009 June Q5
11 marks Moderate -0.8
5 A circle with centre \(C\) has equation $$( x - 5 ) ^ { 2 } + ( y + 12 ) ^ { 2 } = 169$$
  1. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
    1. Verify that the circle passes through the origin \(O\).
    2. Given that the circle also passes through the points \(( 10,0 )\) and \(( 0 , p )\), sketch the circle and find the value of \(p\).
  3. The point \(A ( - 7 , - 7 )\) lies on the circle.
    1. Find the gradient of \(A C\).
    2. Hence find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
AQA C1 2009 June Q6
10 marks Moderate -0.3
6
    1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
    3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
  2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
    1. Expand \(( x - 5 ) ^ { 2 }\).
    2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
    3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.
AQA C1 2009 June Q7
9 marks Standard +0.3
7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).
  1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
  2. The curve \(C\) and the line \(L\) intersect in two distinct points.
    1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
    2. Hence find the possible values of \(k\).
Edexcel C1 Q1
3 marks Easy -1.8
  1. Write down the value of \(16^{-1}\). [1]
  2. Find the value of \(16^{-\frac{1}{2}}\). [2]
Edexcel C1 Q2
8 marks Easy -1.8
  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]
Edexcel C1 Q3
4 marks Moderate -0.8
Given that the equation \(kx^2 + 12x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\). [4]
Edexcel C1 Q4
6 marks Moderate -0.3
Solve the simultaneous equations $$x + y = 2$$ $$x^2 + 2y = 12.$$ [6]
Edexcel C1 Q5
6 marks Easy -1.2
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]
Edexcel C1 Q6
6 marks Moderate -0.8
\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.
Edexcel C1 Q7
10 marks Moderate -0.8
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]
Edexcel C1 Q8
9 marks Moderate -0.8
\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]
Edexcel C1 Q9
11 marks Moderate -0.3
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]
Edexcel C1 Q10
12 marks Moderate -0.3
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]
Edexcel C1 Q1
3 marks Easy -1.8
  1. Write down the value of \(8^{-1}\). [1]
  2. Find the value of \(8^{-\frac{2}{3}}\). [2]
Edexcel C1 Q2
5 marks Easy -1.2
Given that \(y = 6x - \frac{4}{x^2}\), \(x \neq 0\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find \(\int y \, dx\). [3]
Edexcel C1 Q3
6 marks Moderate -0.8
\(x^2 - 8x - 29 = (x + a)^2 + b\), 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]
Edexcel C1 Q4
5 marks Moderate -0.8
\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.
Edexcel C1 Q5
6 marks Moderate -0.5
Solve the simultaneous equations $$x - 2y = 1,$$ $$x^2 + y^2 = 29.$$ [6]
Edexcel C1 Q6
8 marks Moderate -0.8
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]
Edexcel C1 Q7
8 marks Moderate -0.8
  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]
Edexcel C1 Q8
10 marks Moderate -0.8
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]
Edexcel C1 Q9
13 marks Moderate -0.8
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]
Edexcel C1 Q10
11 marks Moderate -0.8
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]
Edexcel C1 Q1
3 marks Easy -1.2
Factorise completely $$x^3 - 4x^2 + 3x.$$ [3]