Questions — Edexcel C1 (574 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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Edexcel C1 Q3
5 marks Easy -1.2
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.
Edexcel C1 Q4
5 marks Easy -1.2
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]
Edexcel C1 Q5
7 marks Easy -1.2
Differentiate with respect to \(x\)
  1. \(x^4 + 6\sqrt{x}\), [3]
  2. \(\frac{(x + 4)^3}{x}\). [4]
Edexcel C1 Q6
4 marks Easy -1.3
  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]
Edexcel C1 Q7
7 marks Moderate -0.3
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]
Edexcel C1 Q8
6 marks Moderate -0.8
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]
Edexcel C1 Q9
8 marks Moderate -0.8
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]
Edexcel C1 Q10
10 marks Moderate -0.8
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]
Edexcel C1 Q11
15 marks Moderate -0.3
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]
Edexcel C1 Q1
4 marks Easy -1.2
Given that $$y = 4x^3 - 1 + 2x^{-1}, \quad x > 0,$$ find \(\frac{dy}{dx}\). [4]
Edexcel C1 Q2
4 marks Easy -1.3
  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]
Edexcel C1 Q3
6 marks Moderate -0.8
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]
Edexcel C1 Q4
7 marks Moderate -0.5
Solve the simultaneous equations $$y = x - 2,$$ $$y^2 + x^2 = 10.$$ [7]
Edexcel C1 Q5
4 marks Moderate -0.3
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]
Edexcel C1 Q6
5 marks Moderate -0.8
  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]
Edexcel C1 Q7
9 marks Moderate -0.3
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]
Edexcel C1 Q8
11 marks Moderate -0.8
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]
Edexcel C1 Q9
12 marks Moderate -0.3
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]
Edexcel C1 Q10
13 marks Moderate -0.3
  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]
Edexcel C1 Q1
2 marks Easy -1.8
Simplify \((3 + \sqrt{5})(3 - \sqrt{5})\). [2]
Edexcel C1 Q2
4 marks Easy -1.8
  1. Find the value of \(8^{-1}\). [2]
  2. Simplify \(\frac{15x^4}{3x}\). [2]
Edexcel C1 Q1
5 marks Easy -1.2
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 Q2
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 Q3
11 marks Moderate -0.8
\includegraphics{figure_2} The points \(A(3, 0)\) and \(B(0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 2.
  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]
Edexcel C1 Q4
5 marks Moderate -0.5
\includegraphics{figure_4} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be \(1030\) cm³,
  1. express \(h\) in terms of \(x\), [2]
  2. show that the surface area, \(A\) cm², of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]