Questions C2 (1550 questions)

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OCR C2 Q7
9 marks Moderate -0.3
7. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for
  1. \(3 ^ { x + 1 }\),
  2. \(3 ^ { 2 x - 1 }\).
    (b) Hence, or otherwise, solve the equation $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$
OCR C2 Q8
11 marks Standard +0.3
  1. Given that $$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
  2. Evaluate $$\int _ { 2 } ^ { \infty } \frac { 6 } { x ^ { \frac { 5 } { 2 } } } \mathrm {~d} x$$ giving your answer in its simplest form.
OCR C2 Q9
11 marks Standard +0.3
9. The second and fifth terms of a geometric series are - 48 and 6 respectively.
  1. Find the first term and the common ratio of the series.
  2. Find the sum to infinity of the series.
  3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).
Edexcel C2 Q1
4 marks Easy -1.2
Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((3 + 2x)^5\), giving each term in its simplest form. [4]
Edexcel C2 Q2
6 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((5, -1)\) and \((13, 11)\) respectively.
  1. Find the coordinates of the mid-point of \(AB\). [2]
Given that \(AB\) is a diameter of the circle \(C\),
  1. find an equation for \(C\). [4]
Edexcel C2 Q3
7 marks Moderate -0.3
Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(3^x = 5\), [3]
  2. \(\log_2(2x + 1) - \log_2 x = 2\). [4]
Edexcel C2 Q4
7 marks Moderate -0.3
  1. Show that the equation $$5 \cos^2 x = 3(1 + \sin x)$$ can be written as $$5 \sin^2 x + 3 \sin x - 2 = 0.$$ [2]
  2. Hence solve, for \(0 \leq x < 360°\), the equation $$5 \cos^2 x = 3(1 + \sin x),$$ giving your answers to 1 decimal place where appropriate. [5]
Edexcel C2 Q5
8 marks Moderate -0.8
\(f(x) = x^3 - 2x^2 + ax + b\), where \(a\) and \(b\) are constants. When \(f(x)\) is divided by \((x - 2)\), the remainder is 1. When \(f(x)\) is divided by \((x + 1)\), the remainder is 28.
  1. Find the value of \(a\) and the value of \(b\). [6]
  2. Show that \((x - 3)\) is a factor of \(f(x)\). [2]
Edexcel C2 Q6
8 marks Moderate -0.3
The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive. For this series, find
  1. the common ratio, [2]
  2. the first term, [2]
  3. the sum of the first 50 terms, giving your answer to 3 decimal places, [2]
  4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places. [2]
Edexcel C2 Q7
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 8\) cm, \(AC = 11\) cm and \(\angle BAC = 0.7\) radians. The arc \(BD\), where \(D\) lies on \(AC\), is an arc of a circle with centre \(A\) and radius 8 cm. The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(BC\) and \(CD\) and the arc \(BD\). Find
  1. the length of the arc \(BD\), [2]
  2. the perimeter of \(R\), giving your answer to 3 significant figures, [4]
  3. the area of \(R\), giving your answer to 3 significant figures. [5]
Edexcel C2 Q8
12 marks Moderate -0.3
\includegraphics{figure_2} The line with equation \(y = 3x + 20\) cuts the curve with equation \(y = x^2 + 6x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
  1. Use calculus to find the exact area of \(S\). [7]
Edexcel C2 Q9
12 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the stage is \(2x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2x\) metres. The perimeter of the stage is 80 m.
  1. Show that the area, \(A\) m², of the stage is given by $$A = 80x - \left(2 + \frac{\pi}{2}\right)x^2.$$ [4]
  2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value. [4]
  3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\). [2]
  4. Calculate, to the nearest m², the maximum area of the stage. [2]
Edexcel C2 Q10
12 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows part of a curve \(C\) with equation \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).
  1. Find the exact area of \(R\). [8]
  2. Use calculus to show that \(y\) is increasing for \(x > 2\). [4]
Edexcel C2 Q1
4 marks Easy -1.2
Find the coordinates of the stationary point on the curve with equation \(y = 2x^2 - 12x\). [4]
Edexcel C2 Q2
6 marks Moderate -0.8
Solve
  1. \(5^x = 8\), giving your answer to 3 significant figures, [3]
  2. \(\log_2(x + 1) - \log_2 x = \log_2 7\). [3]
Edexcel C2 Q3
6 marks Moderate -0.8
  1. Use the factor theorem to show that \((x + 4)\) is a factor of \(2x^3 + x^2 - 25x + 12\). [2]
  2. Factorise \(2x^3 + x^2 - 25x + 12\) completely. [4]
Edexcel C2 Q4
6 marks Moderate -0.3
  1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^{12}\), where \(p\) is a non-zero constant. [2]
Given that, in the expansion of \((1 + px)^{12}\), the coefficient of \(x\) is \((-q)\) and the coefficient of \(x^2\) is \(11q\),
  1. find the value of \(p\) and the value of \(q\). [4]
Edexcel C2 Q5
8 marks Moderate -0.8
Solve, for \(0 \leq x \leq 180°\), the equation $$\sin(x + 10°) = \frac{\sqrt{3}}{2}.$$ [4]
  1. \(\cos 2x = -0.9\), giving your answers to 1 decimal place. [4]
Edexcel C2 Q6
8 marks Moderate -0.8
A river, running between parallel banks, is 20 m wide. The depth, \(y\) metres, of the river measured at a point \(x\) metres from one bank is given by the formula $$y = \frac{1}{10}x(20 - x), \quad 0 \leq x \leq 20.$$
  1. Complete the table below, giving values of \(y\) to 3 decimal places.
    \(x\)048121620
    \(y\)02.7710
    [2]
  2. Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. [4]
Given that the cross-sectional area is constant and that the river is flowing uniformly at 2 m s⁻¹,
  1. estimate, in m³, the volume of water flowing per minute, giving your answer to 3 significant figures. [2]
Edexcel C2 Q7
6 marks Moderate -0.3
In the triangle \(ABC\), \(AB = 8\) cm, \(AC = 7\) cm, \(\angle ABC = 0.5\) radians and \(\angle ACB = x\) radians.
  1. Use the sine rule to find the value of \(\sin x\), giving your answer to 3 decimal places. [3]
Given that there are two possible values of \(x\),
  1. find these values of \(x\), giving your answers to 2 decimal places. [3]
Edexcel C2 Q8
9 marks Moderate -0.3
The circle \(C\), with centre at the point \(A\), has equation \(x^2 + y^2 - 10x + 9 = 0\). Find
  1. the coordinates of \(A\), [2]
  2. the radius of \(C\), [2]
  3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. [2]
Given that the line \(l\) with gradient \(\frac{7}{T}\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),
  1. find an equation of the line which passes through \(A\) and \(T\). [3]
Edexcel C2 Q9
10 marks Moderate -0.3
  1. A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac{a(1-r^n)}{1-r}.$$ [4]
Mr King will be paid a salary of £35 000 in the year 2005. Mr King's contract promises a 4% increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
  1. Find, to the nearest £100, Mr King's salary in the year 2008. [2]
Mr King will receive a salary each year from 2005 until he retires at the end of 2024.
  1. Find, to the nearest £1000, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024. [4]
Edexcel C2 Q1
8 marks Moderate -0.8
\(f(x) = 2x^3 + x^2 - 5x + c\), where \(c\) is a constant. Given that \(f(1) = 0\),
  1. find the value of \(c\), [2]
  2. factorise \(f(x)\) completely, [4]
  3. find the remainder when \(f(x)\) is divided by \((2x - 3)\). [2]
Edexcel C2 Q2
6 marks Moderate -0.8
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^9\), where \(p\) is a constant. [2]
The first 3 terms are 1, 36x and \(qx^2\), where \(q\) is a constant.
  1. Find the value of \(p\) and the value of \(q\). [4]
Edexcel C2 Q3
7 marks Moderate -0.8
\includegraphics{figure_1} In Figure 1, \(A(4, 0)\) and \(B(3, 5)\) are the end points of a diameter of the circle \(C\). Find
  1. the exact length of \(AB\), [2]
  2. the coordinates of the midpoint \(P\) of \(AB\), [2]
  3. an equation for the circle \(C\). [3]