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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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]
Edexcel C1 Q5
5 marks Moderate -0.8
  1. Given that \(8 = 2^k\), write down the value of \(k\). [1]
  2. Given that \(4^x = 8^{2-x}\), find the value of \(x\). [4]
Edexcel C1 Q6
7 marks Moderate -0.8
The equation \(x^2 + 5kx + 2k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k(25k - 8) \geq 0\). [2]
  2. Hence find the set of possible values of \(k\). [4]
  3. Write down the values of \(k\) for which the equation \(x^2 + 5kx + 2k = 0\) has equal roots. [1]
Edexcel C1 Q7
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 Q8
12 marks Moderate -0.8
The points \(A(-1, -2)\), \(B(7, 2)\) and \(C(k, 4)\), where \(k\) is a constant, are the vertices of \(\triangle ABC\). Angle \(ABC\) is a right angle.
  1. Find the gradient of \(AB\). [2]
  2. Calculate the value of \(k\). [2]
  3. Show that the length of \(AB\) may be written in the form \(p\sqrt{5}\), where \(p\) is an integer to be found. [3]
  4. Find the exact value of the area of \(\triangle ABC\). [3]
  5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [2]
Edexcel C1 Q9
5 marks Moderate -0.8
Given that \(2^x = \frac{1}{\sqrt{2}}\) and \(2^y = 4\sqrt{2}\),
  1. find the exact value of \(x\) and the exact value of \(y\), [3]
  2. calculate the exact value of \(2^{y-x}\). [2]
Edexcel C1 Q10
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 Q11
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 Q12
8 marks Easy -1.2
Initially the number of fish in a lake is 500 000. The population is then modelled by the recurrence relation $$u_{n+1} = 1.05u_n - d, \quad u_0 = 500000.$$ In this relation \(u_n\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),
  1. calculate \(u_1\), \(u_2\) and \(u_3\) and comment briefly on your results. [3]
Given that \(d = 100000\),
  1. show that the population of fish dies out during the sixth year. [3]
  2. Find the value of \(d\) which would leave the population each year unchanged. [2]
Edexcel C1 Q13
6 marks Moderate -0.3
  1. Find the sum of all the integers between 1 and 1000 which are divisible by 7. [3]
  2. Hence, or otherwise, evaluate \(\sum_{r=1}^{142} (7r + 2)\). [3]
Edexcel C1 Q14
5 marks Moderate -0.8
Given that \(f(x) = 15 - 7x - 2x^2\),
  1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
  2. Sketch the graph of \(y = f(x)\). [2]
Edexcel C1 Q15
8 marks Moderate -0.3
  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 Q16
13 marks Standard +0.3
\includegraphics{figure_3} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 3.
  1. Find the coordinates of the centre of the circle. [2]
  2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
  3. Find an equation of tangent to the circle at \(B\). [3]
The line \(l\) passes through \(A\) and the origin.
  1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]
Edexcel C1 Q17
5 marks Easy -1.3
  1. Solve the inequality $$3x - 8 > x + 13.$$ [2]
  2. Solve the inequality $$x^2 - 5x - 14 > 0.$$ [3]
Edexcel C1 Q18
10 marks Moderate -0.8
  1. An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n-1)d].$$ [4]
A company made a profit of £54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £\(d\). This model predicts total profits of £619 200 for the 9 years 2001 to 2009 inclusive.
  1. Find the value of \(d\). [4]
Using your value of \(d\),
  1. find the predicted profit for the year 2011. [2]