Questions C1 (1562 questions)

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All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
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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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Edexcel C1 Q2
5 marks Moderate -0.8
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
Edexcel C1 Q3
5 marks Easy -1.3
\(y = 7 + 10x^{\frac{1}{3}}\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find \(\int y \, dx\). [3]
Edexcel C1 Q4
8 marks Moderate -0.8
  1. By completing the square, find in terms of \(k\) the roots of the equation $$x^2 + 2kx - 7 = 0.$$ [4]
  2. Prove that, for all values of \(k\), the roots of \(x^2 + 2kx - 7 = 0\) are real and different. [2]
  3. Given that \(k = \sqrt{2}\), find the exact roots of the equation. [2]
Edexcel C1 Q5
9 marks Moderate -0.8
The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).
  1. On the same axes, sketch the graphs of \(l_1\) and \(l_2\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. [3]
The lines \(l_1\) and \(l_2\) intersect at the point \(A\).
  1. Calculate, as exact fractions, the coordinates of \(A\). [3]
  2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]
Edexcel C1 Q6
8 marks Moderate -0.8
Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays £500. Her payments then increase by £50 each year, so that she pays £550 in the second year, £600 in the third year, and so on.
  1. Find the amount that Anne will pay in the 40th year. [2]
  2. Find the total amount that Anne will pay in over the 40 years. [2]
Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in £890 and his payments then increase by £\(d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
  1. find the value of \(d\). [4]
Edexcel C1 Q7
13 marks Moderate -0.8
\includegraphics{figure_1} The points \(A\) and \(B\) have coordinates \((2, -3)\) and \((8, 5)\) respectively, and \(AB\) is a chord of a circle with centre \(C\), as shown in Fig. 1.
  1. Find the gradient of \(AB\). [2]
The point \(M\) is the mid-point of \(AB\).
  1. Find an equation for the line through \(C\) and \(M\). [5]
Given that the \(x\)-coordinate of \(C\) is 4,
  1. find the \(y\)-coordinate of \(C\), [2]
  2. show that the radius of the circle is \(\frac{5\sqrt{17}}{4}\). [4]
Edexcel C1 Q8
5 marks Standard +0.3
\includegraphics{figure_4} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2\(x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),
  1. express \(h\) in terms of \(x\), [2]
  2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]
Edexcel C1 Q2
5 marks Moderate -0.8
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
Edexcel C1 Q3
5 marks Easy -1.2
  1. Solve the inequality \(3x - 8 > x + 13\). [2]
  2. Solve the inequality \(x^2 - 5x - 14 > 0\). [3]
Edexcel C1 Q4
6 marks Moderate -0.3
  1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{(k^2 - c)}\). [4]
The equation \(x^2 + 2kx + 81 = 0\) has equal roots.
  1. Find the possible values of \(k\). [2]
Edexcel C1 Q5
7 marks Standard +0.3
Solve the simultaneous equations \(x - 3y + 1 = 0\), \(x^2 - 3xy + y^2 = 11\). [7]
Edexcel C1 Q6
7 marks Moderate -0.8
$$\frac{dy}{dx} = 5 + \frac{1}{x^2}.$$
  1. Use integration to find \(y\) in terms of \(x\). [3]
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\). [4]
Edexcel C1 Q8
10 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((4, 6)\) and \((12, 2)\) respectively. The straight line \(l_1\) passes through \(A\) and \(B\).
  1. Find an equation for \(l_1\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(l_2\) passes through the origin and has gradient \(-4\).
  1. Write down an equation for \(l_2\). [1]
The lines \(l_1\) and \(l_2\) intercept at the point \(C\).
  1. Find the exact coordinates of the mid-point of \(AC\). [5]
Edexcel C1 Q9
11 marks Moderate -0.8
A curve \(C\) has equation \(y = x^3 - 5x^2 + 5x + 2\).
  1. Find \(\frac{dy}{dx}\) in terms of \(x\). [2]
The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2. The \(x\)-coordinate of \(P\) is 3.
  1. Find the \(x\)-coordinate of \(Q\). [2]
  2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [3]
This tangent intersects the coordinate axes at the points \(R\) and \(S\).
  1. Find the length of \(RS\), giving your answer as a surd. [4]
Edexcel C1 Q10
11 marks Moderate -0.5
\includegraphics{figure_1} The points \(A(3, 0)\) and \(B(0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 1.
  1. Write down the gradient of \(AB\) and hence the gradient of \(BC\). [3]
The point \(C\) has coordinates \((8, k)\), where \(k\) is a positive constant.
  1. Find the length of \(BC\) in terms of \(k\). [2]
Given that the length of \(BC\) is 10 and using your answer to part (b),
  1. find the value of \(k\), [4]
  2. find the coordinates of \(D\). [2]
OCR C1 2013 January Q1
5 marks Moderate -0.8
  1. Solve the equation \(x^2 - 6x - 2 = 0\), giving your answers in simplified surd form. [3]
  2. Find the gradient of the curve \(y = x^2 - 6x - 2\) at the point where \(x = -5\). [2]
OCR C1 2013 January Q2
6 marks Easy -1.3
Solve the equations
  1. \(3^n = 1\), [1]
  2. \(t^{-3} = 64\), [2]
  3. \((8p^6)^{\frac{1}{3}} = 8\). [3]
OCR C1 2013 January Q3
5 marks Moderate -0.3
  1. Sketch the curve \(y = (1 + x)(2 - x)(3 + x)\), giving the coordinates of all points of intersection with the axes. [3]
  2. Describe the transformation that transforms the curve \(y = (1 + x)(2 - x)(3 + x)\) to the curve \(y = (1 - x)(2 + x)(3 - x)\). [2]
OCR C1 2013 January Q4
6 marks Moderate -0.3
  1. Solve the simultaneous equations $$y = 2x^2 - 3x - 5, \quad 10x + 2y + 11 = 0.$$ [5]
  2. What can you deduce from the answer to part (i) about the curve \(y = 2x^2 - 3x - 5\) and the line \(10x + 2y + 11 = 0\)? [1]
OCR C1 2013 January Q5
6 marks Easy -1.3
  1. Simplify \((x + 4)(5x - 3) - 3(x - 2)^2\). [3]
  2. The coefficient of \(x^2\) in the expansion of $$(x + 3)(x + k)(2x - 5)$$ is \(-3\). Find the value of the constant \(k\). [3]
OCR C1 2013 January Q6
10 marks Easy -1.3
  1. The line joining the points \((-2, 7)\) and \((-4, p)\) has gradient 4. Find the value of \(p\). [3]
  2. The line segment joining the points \((-2, 7)\) and \((6, q)\) has mid-point \((m, 5)\). Find \(m\) and \(q\). [3]
  3. The line segment joining the points \((-2, 7)\) and \((d, 3)\) has length \(2\sqrt{13}\). Find the two possible values of \(d\). [4]
OCR C1 2013 January Q7
8 marks Easy -1.3
Find \(\frac{dy}{dx}\) in each of the following cases:
  1. \(y = \frac{(3x)^2 \times x^4}{x}\), [3]
  2. \(y = ^3\sqrt{x}\), [3]
  3. \(y = \frac{1}{2x^3}\). [2]
OCR C1 2013 January Q8
7 marks Standard +0.3
The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]
OCR C1 2013 January Q9
9 marks Moderate -0.3
A circle with centre \(C\) has equation \(x^2 + y^2 - 2x + 10y - 19 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle. [3]
  2. Verify that the point \((7, -2)\) lies on the circumference of the circle. [1]
  3. Find the equation of the tangent to the circle at the point \((7, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
OCR C1 2013 January Q10
10 marks Standard +0.3
Find the coordinates of the points on the curve \(y = \frac{1}{3}x^3 + \frac{9}{x}\) at which the tangent is parallel to the line \(y = 8x + 3\). [10]