Questions C2 (1550 questions)

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Edexcel C2 Q10
11 marks Standard +0.3
  1. Given that $$(2 + x)^5 + (2 - x)^5 = A + Bx^2 + Cx^4,$$ find the values of the constants \(A\), \(B\) and \(C\). [6]
  2. Using the substitution \(y = x^2\) and your answers to part (a), solve, $$(2 + x)^5 + (2 - x)^5 = 349.$$ [5]
Edexcel C2 Q11
4 marks Easy -1.2
A circle \(C\) has equation $$x^2 + y^2 - 10x + 6y - 15 = 0.$$
  1. Find the coordinates of the centre of \(C\). [2]
  2. Find the radius of \(C\). [2]
Edexcel C2 Q12
6 marks Moderate -0.3
$$f(x) = ax^3 + bx^2 - 7x + 14, \text{ where } a \text{ and } b \text{ are constants.}$$ Given that when \(f(x)\) is divided by \((x - 1)\) the remainder is 9,
  1. write down an equation connecting \(a\) and \(b\). [2]
Given also that \((x + 2)\) is a factor of \(f(x)\),
  1. Find the values of \(a\) and \(b\). [4]
Edexcel C2 Q13
7 marks Moderate -0.8
$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),
  1. Find the value of \(c\), [2]
  2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
  3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]
Edexcel C2 Q14
8 marks Standard +0.3
Find the values of \(\theta\), to 1 decimal place, in the interval \(-180 \leq \theta < 180\) for which $$2 \sin^2 \theta ° - 2 \sin \theta ° = \cos^2 \theta °.$$ [8]
Edexcel C2 Q15
10 marks Standard +0.3
\includegraphics{figure_6} Figure 1 shows a gardener's design for the shape of a flower bed with perimeter \(ABCD\). \(AD\) is an arc of a circle with centre \(O\) and radius 5 m. \(BC\) is an arc of a circle with centre \(O\) and radius 7 m. \(OAB\) and \(ODC\) are straight lines and the size of \(\angle AOD\) is \(\theta\) radians.
  1. Find, in terms of \(\theta\), an expression for the area of the flower bed. [3]
Given that the area of the flower bed is 15 m\(^2\),
  1. show that \(\theta = 1.25\), [2]
  2. calculate, in m, the perimeter of the flower bed. [3]
The gardener now decides to replace arc \(AD\) with the straight line \(AD\).
  1. Find, to the nearest cm, the reduction in the perimeter of the flower bed. [2]
Edexcel C2 Q16
12 marks Moderate -0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1 - r^n)}{1 - r}.$$ [4]
The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find
  1. the value of \(r\) and the value of \(a\), [5]
  2. the sum to infinity of the series. [3]
Edexcel C2 Q17
12 marks Moderate -0.3
\includegraphics{figure_7} Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6x - x^2 - 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(R\) is bounded by the line and the curve.
  1. Calculate the area of \(R\). [7]
Edexcel C2 Q18
6 marks Moderate -0.8
Given that \(p = \log_4 16\), express in terms of \(p\),
  1. \(\log_4 2\). [2]
  2. \(\log_4 (8q)\). [4]
Edexcel C2 Q19
11 marks Standard +0.8
$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of \(f(x)\),
  1. prove that \(n = 6k + 2\). [3]
Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero,
  1. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). [4]
Using these values of \(k\) and \(n\),
  1. expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]
Edexcel C2 Q20
3 marks Moderate -0.8
$$f(x) = 4x^3 + 3x^2 - 2x - 6.$$ Find the remainder when \(f(x)\) is divided by \((2x + 1)\). [3]
Edexcel C2 Q21
9 marks Moderate -0.3
A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).
  1. Write down an equation of the circle \(C\). [2]
  2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
  3. Find the distance between these two points. [2]
Edexcel C2 Q22
7 marks Standard +0.3
\includegraphics{figure_8} The shape of a badge is a sector \(ABC\) of a circle with centre \(A\) and radius \(AB\), as shown in Fig 1. The triangle \(ABC\) is equilateral and has a perpendicular height 3 cm.
  1. Find, in surd form, the length \(AB\). [2]
  2. Find, in terms of \(\pi\), the area of the badge. [2]
  3. Prove that the perimeter of the badge is \(\frac{2\sqrt{3}}{3}(\pi + 6)\) cm. [3]
Edexcel C2 Q23
8 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]
  3. Calculate the coordinates of the stationary point of \(f(x)\). [3]
Edexcel C2 Q24
10 marks Moderate -0.8
$$f(x) = 5 \sin 3x°, \quad 0 \leq x \leq 180.$$
  1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis. [3]
  2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
  3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]
Edexcel C2 Q25
11 marks Standard +0.3
Given that \(f'(x) = (2x^3 - 3x^{-2})^2 + 5\), \(x > 0\),
  1. Find, to 3 significant figures, the value of \(x\) for which \(f'(x) = 5\). [3]
  2. Show that \(f'(x)\) may be written in the form \(Ax^6 + \frac{B}{x^4} + C\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
  3. Hence evaluate \(\int_1^2 f'(x) \, dx\). [5]
Edexcel C2 Q26
12 marks Moderate -0.3
\includegraphics{figure_9} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.
  1. Show that the volume, \(V\) cm\(^3\), of the tray is given by $$V = 4x(x^2 - 45x + 500).$$ [3]
  2. State the range of possible values of \(x\). [1]
  3. Find the value of \(x\) for which \(V\) is a maximum. [4]
  4. Hence find the maximum value of \(V\). [2]
  5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]
Edexcel C2 Q27
6 marks Moderate -0.3
  1. Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \((1 + 3x)^n\), where \(n > 2\). [2]
Given that the coefficient of \(x^3\) in this expansion is ten times the coefficient of \(x^2\),
  1. find the value of \(n\), [2]
  2. find the coefficient of \(x^4\) in the expansion. [2]
Edexcel C2 Q28
10 marks Moderate -0.3
  1. Given that \(3 + 2 \log_2 x = \log_2 y\), show that \(y = 8x^2\). [3]
  2. Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log_2 x = \log_2 (14x - 3).$$ [3]
  3. Show that \(\log_2 \alpha = -2\). [1]
  4. Calculate \(\log_2 \beta\), giving your answer to 3 significant figures. [3]
Edexcel C2 Q29
7 marks Moderate -0.3
$$f(x) = x^3 + ax^2 + bx - 10, \text{ where } a \text{ and } b \text{ are constants.}$$ When \(f(x)\) is divided by \((x - 3)\), the remainder is 14. When \(f(x)\) is divided by \((x + 1)\), the remainder is \(-18\).
  1. Find the value of \(a\) and the value of \(b\). [5]
  2. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
Edexcel C2 Q30
6 marks Moderate -0.8
  1. Using the factor theorem, show that \((x + 3)\) is a factor of $$x^3 - 3x^2 - 10x + 24.$$ [2]
  2. Factorise \(x^3 - 3x^2 - 10x + 24\) completely. [4]
Edexcel C2 Q31
10 marks Moderate -0.8
  1. Differentiate with respect to \(x\) $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5]
Edexcel C2 Q32
13 marks Moderate -0.3
  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]
An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06. Using this alternative model and again taking the profit in 2001 to be £54 000,
  1. find the predicted profit for the year 2011. [3]
Edexcel C2 Q33
13 marks Moderate -0.3
  1. Solve, for \(0° < x < 180°\), the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [7]
  2. In the triangle \(ABC\), \(AC = 18\) cm, \(\angle ABC = 60°\) and \(\sin A = \frac{1}{3}\).
    1. Use the sine rule to show that \(BC = 4\sqrt{3}\). [4]
    2. Find the exact value of \(\cos A\). [2]
Edexcel C2 Q34
6 marks Moderate -0.8
  1. Using the substitution \(u = 2^x\), show that the equation \(4^x - 2^{(x + 1)} - 15 = 0\) can be written in the form \(u^2 - 2u - 15 = 0\). [2]
  2. Hence solve the equation \(4^x - 2^{(x + 1)} - 15 = 0\), giving your answers to 2 decimals places. [4]