Questions C1 (1562 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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OCR MEI C1 2013 June Q10
12 marks Moderate -0.8
The circle \((x - 3)^2 + (y - 2)^2 = 20\) has centre C.
  1. Write down the radius of the circle and the coordinates of C. [2]
  2. Find the coordinates of the intersections of the circle with the \(x\)- and \(y\)-axes. [5]
  3. Show that the points A\((1, 6)\) and B\((7, 4)\) lie on the circle. Find the coordinates of the midpoint of AB. Find also the distance of the chord AB from the centre of the circle. [5]
OCR MEI C1 2013 June Q11
12 marks Moderate -0.8
You are given that \(\text{f}(x) = (2x - 3)(x + 2)(x + 4)\).
  1. Sketch the graph of \(y = \text{f}(x)\). [3]
  2. State the roots of \(\text{f}(x - 2) = 0\). [2]
  3. You are also given that \(\text{g}(x) = \text{f}(x) + 15\).
    1. Show that \(\text{g}(x) = 2x^3 + 9x^2 - 2x - 9\). [2]
    2. Show that \(\text{g}(1) = 0\) and hence factorise \(\text{g}(x)\) completely. [5]
OCR MEI C1 2013 June Q12
12 marks Standard +0.3
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{1}{x - 2}\).
  1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\). [3]
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
  3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x - 2}\). [4]
Edexcel C1 Q1
4 marks Easy -1.8
  1. Express \(\frac{21}{\sqrt{7}}\) in the form \(k\sqrt{7}\). [2]
  2. Express \(8^{-1}\) as an exact fraction in its simplest form. [2]
Edexcel C1 Q2
4 marks Moderate -0.5
Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]
Edexcel C1 Q3
5 marks Moderate -0.3
Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]
Edexcel C1 Q4
6 marks Moderate -0.8
  1. Solve the inequality $$x^2 + 3x > 10.$$ [3]
  2. Find the set of values of \(x\) which satisfy both of the following inequalities: $$3x - 2 < x + 3$$ $$x^2 + 3x > 10$$ [3]
Edexcel C1 Q5
7 marks Standard +0.3
The sequence \(u_1, u_2, u_3, ...\) is defined by the recurrence relation $$u_{n+1} = (u_n)^2 - 1, \quad n \geq 1.$$ Given that \(u_1 = k\), where \(k\) is a constant,
  1. find expressions for \(u_2\) and \(u_3\) in terms of \(k\). [3]
Given also that \(u_2 + u_3 = 11\),
  1. find the possible values of \(k\). [4]
Edexcel C1 Q6
8 marks Moderate -0.3
  1. By completing the square, find in terms of the constant \(k\) the roots of the equation $$x^2 + 4kx - k = 0.$$ [4]
  2. Hence find the set of values of \(k\) for which the equation has no real roots. [4]
Edexcel C1 Q7
9 marks Standard +0.3
  1. Describe fully a single transformation that maps the graph of \(y = \frac{1}{x}\) onto the graph of \(y = \frac{3}{x}\). [2]
  2. Sketch the graph of \(y = \frac{3}{x}\) and write down the equations of any asymptotes. [3]
  3. Find the values of the constant \(c\) for which the straight line \(y = c - 3x\) is a tangent to the curve \(y = \frac{3}{x}\). [4]
Edexcel C1 Q8
10 marks Moderate -0.3
The points \(P\) and \(Q\) have coordinates \((7, 4)\) and \((9, 7)\) respectively.
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(m\) has gradient \(8\) and passes through the origin, \(O\).
  1. Write down an equation for \(m\). [1]
The lines \(l\) and \(m\) intersect at the point \(R\).
  1. Show that \(OP = OR\). [5]
Edexcel C1 Q9
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$\text{f}'(x) = 6 - 4x - 3x^2,$$
  1. find an expression for \(y\) in terms of \(x\), [5]
  2. show that \(AB = k\sqrt{7}\), where \(k\) is an integer to be found. [6]
Edexcel C1 Q10
11 marks Standard +0.3
A curve has the equation \(y = x + \frac{3}{x}\), \(x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate \(1\).
  1. Show that the gradient of the curve at \(P\) is \(-2\). [3]
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = mx + c\). [4]
  3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]
Edexcel C1 Q1
3 marks Easy -1.2
Express \(\sqrt{50} + 3\sqrt{8}\) in the form \(k\sqrt{2}\). [3]
Edexcel C1 Q2
4 marks Easy -1.8
Differentiate with respect to \(x\) $$3x^2 - \sqrt{x} + \frac{1}{2x}.$$ [4]
Edexcel C1 Q3
4 marks Easy -1.2
A sequence is defined by the recurrence relation $$u_{n+1} = u_n - 2, \quad n > 0, \quad u_1 = 50.$$
  1. Write down the first four terms of the sequence. [1]
  2. Evaluate $$\sum_{r=1}^{20} u_r.$$ [3]
Edexcel C1 Q4
6 marks Moderate -0.8
  1. Find the value of the constant \(k\) such that the equation $$x^2 - 6x + k = 0$$ has equal roots. [2]
  2. Solve the inequality $$2x^2 - 9x + 4 < 0.$$ [4]
Edexcel C1 Q5
7 marks Moderate -0.3
Solve the simultaneous equations \begin{align} x + y &= 2
3x^2 - 2x + y^2 &= 2 \end{align} [7]
Edexcel C1 Q6
7 marks Moderate -0.5
Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{2}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]
Edexcel C1 Q7
10 marks Moderate -0.3
The first three terms of an arithmetic series are \((12 - p)\), \(2p\) and \((4p - 5)\) respectively, where \(p\) is a constant.
  1. Find the value of \(p\). [2]
  2. Show that the sixth term of the series is 50. [3]
  3. Find the sum of the first 15 terms of the series. [2]
  4. Find how many terms of the series have a value of less than 400. [3]
Edexcel C1 Q8
10 marks Moderate -0.8
$$f(x) = 2x^2 + 3x - 2.$$
  1. Solve the equation \(f(x) = 0\). [2]
  2. Sketch the curve with equation \(y = f(x)\), showing the coordinates of any points of intersection with the coordinate axes. [2]
  3. Find the coordinates of the points where the curve with equation \(y = f(\frac{1}{2}x)\) crosses the coordinate axes. [3]
When the graph of \(y = f(x)\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants.
  1. Find the values of \(a\), \(b\) and \(c\). [3]
Edexcel C1 Q9
11 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [3]
The line \(l\) is the tangent to \(C\) at \(O\).
  1. Find an equation for \(l\). [4]
  2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]
Edexcel C1 Q10
13 marks Moderate -0.3
The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). [2]
The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).
  1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]
The line \(l_2\) crosses the \(x\)-axis at the point \(C\).
  1. Find the coordinates of \(C\). [1]
The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).
  1. Show that \(D\) has coordinates \((5, 4)\). [5]
  2. Find the area of triangle \(ACD\). [2]
Edexcel C1 Q1
4 marks Easy -1.3
  1. Express \(\frac{18}{\sqrt{3}}\) in the form \(k\sqrt{3}\). [2]
  2. Express \((1 - \sqrt{3})(4 - 2\sqrt{3})\) in the form \(a + b\sqrt{3}\) where \(a\) and \(b\) are integers. [2]
Edexcel C1 Q2
4 marks Moderate -0.3
Solve the equation $$3x - \frac{5}{x} = 2.$$ [4]