Questions — Edexcel C1 (574 questions)

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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]
Edexcel C1 Q19
14 marks Easy -1.2
\(f(x) = 9 - (x - 2)^2\)
  1. Write down the maximum value of \(f(x)\). [1]
  2. Sketch the graph of \(y = f(x)\), showing the coordinates of the points at which the graph meets the coordinate axes. [5]
The points \(A\) and \(B\) on the graph of \(y = f(x)\) have coordinates \((-2, -7)\) and \((3, 8)\) respectively.
  1. Find, in the form \(y = mx + c\), an equation of the straight line through \(A\) and \(B\). [4]
  2. Find the coordinates of the point at which the line \(AB\) crosses the \(x\)-axis. [2]
The mid-point of \(AB\) lies on the line with equation \(y = kx\), where \(k\) is a constant.
  1. Find the value of \(k\). [2]
Edexcel C1 Q20
14 marks Moderate -0.3
The curve \(C\) has equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 3x^2 - 20x + 29$$ and that \(C\) passes through the point \(P(2, 6)\),
  1. find \(y\) in terms of \(x\). [4]
  2. Verify that \(C\) passes through the point \((4, 0)\). [2]
  3. Find an equation of the tangent to \(C\) at \(P\). [3]
The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
  1. Calculate the exact \(x\)-coordinate of \(Q\). [5]
Edexcel C1 Q21
5 marks Easy -1.3
\(y = 7 + 10x^{\frac{3}{2}}\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find \(\int y \, dx\). [3]
Edexcel C1 Q22
8 marks Moderate -0.8
  1. Given that \(3^x = 9^{y-1}\), show that \(x = 2y - 2\). [2]
  2. Solve the simultaneous equations \begin{align} x &= 2y - 2,
    x^2 &= y^2 + 7. \end{align} [6]
Edexcel C1 Q23
11 marks Moderate -0.8
The straight line \(l_1\) with equation \(y = \frac{3}{2}x - 2\) crosses the \(y\)-axis at the point \(P\). The point \(Q\) has coordinates \((5, -3)\). The straight line \(l_2\) is perpendicular to \(l_1\) and passes through \(Q\).
  1. Calculate the coordinates of the mid-point of \(PQ\). [3]
  2. Find an equation for \(l_2\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integer constants. [4]
The lines \(l_1\) and \(l_2\) intersect at the point \(R\).
  1. Calculate the exact coordinates of \(R\). [4]
Edexcel C1 Q24
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 Q25
7 marks Moderate -0.8
Find the set of values for \(x\) for which
  1. \(6x - 7 < 2x + 3\), [2]
  2. \(2x^2 - 11x + 5 < 0\), [4]
  3. both \(6x - 7 < 2x + 3\) and \(2x^2 - 11x + 5 < 0\). [1]
Edexcel C1 Q26
8 marks Moderate -0.8
In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \((280 + x)\) phones will be sold in the second month, \((280 + 2x)\) in the third month, and so on. Using this model with \(x = 5\), calculate
    1. the number of phones sold in the 36th month, [2]
    2. the total number of phones sold over the 36 months. [2]
The shop sets a sales target of 17 000 phones to be sold over the 36 months. Using the same model,
  1. find the least value of \(x\) required to achieve this target. [4]
Edexcel C1 Q27
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 Q28
7 marks Moderate -0.8
For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),
  1. find \(\frac{dy}{dx}\). [2]
The point \(A\), on the curve \(C\), has \(x\)-coordinate 1.
  1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
Edexcel C1 Q29
6 marks Easy -1.2
The sum of an arithmetic series is $$\sum_{r=1}^{n} (80 - 3r).$$
  1. Write down the first two terms of the series. [2]
  2. Find the common difference of the series. [1]
Given that \(n = 50\),
  1. find the sum of the series. [3]