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Edexcel C1 Q8
10 marks Moderate -0.3
  1. Describe fully the single transformation that maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = \text{f}(x - 1)\). [2]
  2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
  3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]
Edexcel C1 Q9
10 marks Moderate -0.8
A store begins to stock a new range of DVD players and achieves sales of £1500 of these products during the first month. In a model it is assumed that sales will decrease by £\(x\) in each subsequent month, so that sales of £\((1500 - x)\) and £\((1500 - 2x)\) will be achieved in the second and third months respectively. Given that sales total £8100 during the first six months, use the model to
  1. find the value of \(x\), [4]
  2. find the expected value of sales in the eighth month, [2]
  3. show that the expected total of sales in pounds during the first \(n\) months is given by \(kn(51 - n)\), where \(k\) is an integer to be found. [3]
  4. Explain why this model cannot be valid over a long period of time. [1]
Edexcel C1 Q10
12 marks Moderate -0.3
The curve \(C\) with equation \(y = \text{f}(x)\) is such that $$\frac{\text{d}y}{\text{d}x} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that \(C\) passes through the points \((0, -2)\) and \((2, 18)\),
  1. show that \(k = 2\) and find an equation for \(C\), [7]
  2. show that the line with equation \(y = x - 2\) is a tangent to \(C\) and find the coordinates of the point of contact. [5]
Edexcel C1 Q1
3 marks Standard +0.3
Find in exact form the real solutions of the equation $$x^4 = 5x^2 + 14.$$ [3]
Edexcel C1 Q2
3 marks Moderate -0.8
Express $$\frac{2}{3\sqrt{5} + 7}$$ in the form \(a + b\sqrt{5}\) where \(a\) and \(b\) are rational. [3]
Edexcel C1 Q3
4 marks Easy -1.3
  1. Solve the equation $$x^{\frac{3}{2}} = 27.$$ [2]
  2. Express \((2\frac{1}{4})^{-\frac{3}{2}}\) as an exact fraction in its simplest form. [2]
Edexcel C1 Q4
5 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]
Edexcel C1 Q5
6 marks Moderate -0.3
Given that $$y = \frac{x^4 - 3}{2x^2},$$
  1. find \(\frac{dy}{dx}\), [4]
  2. show that \(\frac{d^2y}{dx^2} = \frac{x^4 - 9}{x^4}\). [2]
Edexcel C1 Q6
8 marks Moderate -0.8
  1. Sketch on the same diagram the curve with equation \(y = (x - 2)^2\) and the straight line with equation \(y = 2x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Find the set of values of \(x\) for which $$(x - 2)^2 > 2x - 1.$$ [3]
Edexcel C1 Q7
10 marks Moderate -0.8
A curve has the equation \(y = \frac{x}{2} + 3 - \frac{1}{x}\), \(x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2.
  1. Find the gradient of the curve at \(A\). [4]
  2. Show that the tangent to the curve at \(A\) has equation $$3x - 4y + 8 = 0.$$ [3]
The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
  1. Find the coordinates of \(B\). [3]
Edexcel C1 Q8
11 marks Moderate -0.8
The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A(5, 3)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).
  1. Find the coordinates of \(B\). [3]
  2. Find the coordinates of the mid-point of \(AB\). [2]
  3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]
Edexcel C1 Q9
12 marks Moderate -0.3
The third term of an arithmetic series is \(5\frac{1}{2}\). The sum of the first four terms of the series is \(22\frac{3}{4}\).
  1. Show that the first term of the series is \(6\frac{1}{4}\) and find the common difference. [7]
  2. Find the number of positive terms in the series. [3]
  3. Hence, find the greatest value of the sum of the first \(n\) terms of the series. [2]
Edexcel C1 Q10
13 marks Moderate -0.3
The curve \(C\) has the equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 8x - \frac{2}{x^3}, \quad x \neq 0,$$ and that the point \(P(1, 1)\) lies on \(C\),
  1. find an equation for the tangent to \(C\) at \(P\) in the form \(y = mx + c\), [3]
  2. find an equation for \(C\), [5]
  3. find the \(x\)-coordinates of the points where \(C\) meets the \(x\)-axis, giving your answers in the form \(k\sqrt{2}\). [5]
Edexcel C1 Q1
3 marks Easy -1.2
Evaluate $$\sum_{r=1}^{20} (3r + 4).$$ [3]
Edexcel C1 Q2
4 marks Easy -1.2
  1. Express \(x^2 + 6x + 7\) in the form \((x + a)^2 + b\). [3]
  2. State the coordinates of the minimum point of the curve \(y = x^2 + 6x + 7\). [1]
Edexcel C1 Q3
6 marks Moderate -0.8
The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\).
  1. Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [3]
  2. Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
Edexcel C1 Q4
7 marks Moderate -0.3
Find the pairs of values \((x, y)\) which satisfy the simultaneous equations $$3x^2 + y^2 = 21$$ $$5x + y = 7$$ [7]
Edexcel C1 Q5
7 marks Moderate -0.8
  1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
  3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]
Edexcel C1 Q6
8 marks Moderate -0.3
The curve with equation \(y = x^2 + 2x\) passes through the origin, \(O\).
  1. Find an equation for the normal to the curve at \(O\). [5]
  2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again. [3]
Edexcel C1 Q7
8 marks Moderate -0.8
Given that $$y = \sqrt{x} - \frac{4}{\sqrt{x}},$$
  1. find \(\frac{dy}{dx}\). [3]
  2. find \(\frac{d^2y}{dx^2}\). [2]
  3. show that $$4x\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} - y = 0.$$ [3]
Edexcel C1 Q8
9 marks Moderate -0.3
  1. Prove that the sum of the first \(n\) positive integers is given by $$\frac{1}{2}n(n + 1).$$ [4]
  2. Hence, find the sum of
    1. the integers from 100 to 200 inclusive,
    2. the integers between 300 to 600 inclusive which are divisible by 3.
    [5]
Edexcel C1 Q9
10 marks Moderate -0.3
  1. Express each of the following in the form \(p + q\sqrt{2}\) where \(p\) and \(q\) are rational.
    1. \((4 - 3\sqrt{2})^2\)
    2. \(\frac{1}{2 + \sqrt{2}}\) [5]
    1. Solve the equation $$y^2 + 8 = 9y.$$
    2. Hence solve the equation $$x^3 + 8 = 9x^{\frac{1}{2}}.$$ [5]
Edexcel C1 Q10
13 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\). The curve meets the \(x\)-axis at the origin and at the point \(A\). Given that $$\text{f}'(x) = 3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}},$$
  1. find f\((x)\). [5]
  2. Find the coordinates of \(A\). [2]
The point \(B\) on the curve has \(x\)-coordinate 2.
  1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]
Edexcel C1 Q1
3 marks Moderate -0.8
Find the value of \(y\) such that $$4^{y + 3} = 8.$$ [3]
Edexcel C1 Q2
4 marks Easy -1.2
Find $$\int \left( 3x^2 + \frac{1}{2x^2} \right) dx.$$ [4]