Questions C4 (1219 questions)

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Edexcel C4 Q5
11 marks Standard +0.3
\includegraphics{figure_1} The curve \(C\) has equation \(y = f(x)\), \(x \in \mathbb{R}\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\). Given that $$\frac{dy}{dx} = e^x - 2x^2,$$ and that \(C\) has a single maximum, at \(x = k\),
  1. show that \(1.48 < k < 1.49\). [3]
Given also that the point \((0, 5)\) lies on \(C\),
  1. find \(f(x)\). [4]
The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).
  1. Use integration to find the exact area of \(R\). [4]
Edexcel C4 Q6
8 marks Standard +0.3
When \((1 + ax)^n\) is expanded as a series in ascending powers of \(x\), the coefficients of \(x\) and \(x^2\) are \(-6\) and \(27\) respectively.
  1. Find the value of \(a\) and the value of \(n\). [5]
  2. Find the coefficient of \(x^3\). [2]
  3. State the set of values of \(x\) for which the expansion is valid. [1]
Edexcel C4 Q7
12 marks Standard +0.3
Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l_1\) and \(l_2\), along which they travel are \begin{align} \mathbf{r} &= 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})
\text{and} \quad \mathbf{r} &= 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k}), \end{align} where \(\lambda\) and \(\mu\) are scalars.
  1. Show that the submarines are moving in perpendicular directions. [2]
  2. Given that \(l_1\) and \(l_2\) intersect at the point \(A\), find the position vector of \(A\). [5]
The point \(B\) has position vector \(10\mathbf{j} - 11\mathbf{k}\).
  1. Show that only one of the submarines passes through the point \(B\). [3]
  2. Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance \(AB\). [2]
Edexcel C4 Q8
13 marks Standard +0.8
In a chemical reaction two substances combine to form a third substance. At time \(t\), \(t \geq 0\), the concentration of this third substance is \(x\) and the reaction is modelled by the differential equation $$\frac{dx}{dt} = k(1 - 2x)(1 - 4x), \text{ where } k \text{ is a positive constant.}$$
  1. Solve this differential equation and hence show that $$\ln \left| \frac{1 - 2x}{1 - 4x} \right| = 2kt + c, \text{ where } c \text{ is an arbitrary constant.}$$ [7]
  2. Given that \(x = 0\) when \(t = 0\), find an expression for \(x\) in terms of \(k\) and \(t\). [4]
  3. Find the limiting value of the concentration \(x\) as \(t\) becomes very large. [2]
Edexcel C4 Q1
8 marks Moderate -0.3
  1. Express \(1.5 \sin 2x + 2 \cos 2x\) in the form \(R \sin (2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate. [4]
  2. Express \(3 \sin x \cos x + 4 \cos^2 x\) in the form \(a \cos 2x + b \sin 2x + c\), where \(a\), \(b\) and \(c\) are constants to be found. [2]
  3. Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos^2 x\). [2]
Edexcel C4 Q2
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 + 1}{(1 + x)(3 - x)}, \quad 0 \leq x < 3.$$
  1. Given that \(f(x) = A + \frac{B}{1 + x} + \frac{C}{3 - x}\), find the values of the constants \(A\), \(B\) and \(C\). [4]
The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
  1. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p\), \(q\) and \(r\) are rational constants to be found. [5]
Edexcel C4 Q3
12 marks Moderate -0.8
A student tests the accuracy of the trapezium rule by evaluating \(I\), where $$I = \int_{0.5}^{1.5} \left(\frac{3}{x} + x^4\right) dx.$$
  1. Complete the student's table, giving values to 2 decimal places where appropriate.
    \(x\)0.50.7511.251.5
    \(\frac{3}{x} + x^4\)6.064.32
    [2]
  2. Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of \(I\). [4]
  3. Use integration to calculate the exact value of \(I\). [4]
  4. Verify that the answer obtained by the trapezium rule is within 3\% of the exact value. [2]
Edexcel C4 Q4
10 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a cross-section \(R\) of a dam. The line \(AC\) is the vertical face of the dam, \(AB\) is the horizontal base and the curve \(BC\) is the profile. Taking \(x\) and \(y\) to be the horizontal and vertical axes, then \(A\), \(B\) and \(C\) have coordinates \((0, 0)\), \((3\pi^2, 0)\) and \((0, 30)\) respectively. The area of the cross-section is to be calculated. Initially the profile \(BC\) is approximated by a straight line.
  1. Find an estimate for the area of the cross-section \(R\) using this approximation. [1]
The profile \(BC\) is actually described by the parametric equations. $$x = 16t^2 - \pi^2, \quad y = 30 \sin 2t, \quad \frac{\pi}{4} \leq t \leq \frac{\pi}{2}.$$
  1. Find the exact area of the cross-section \(R\). [7]
  2. Calculate the percentage error in the estimate of the area of the cross-section \(R\) that you found in part (a). [2]
Edexcel C4 Q5
10 marks Moderate -0.3
  1. Prove that, when \(x = \frac{1}{15}\), the value of \((1 + 5x)^{-\frac{1}{3}}\) is exactly equal to \(\sin 60°\). [3]
  2. Expand \((1 + 5x)^{-\frac{1}{3}}\), \(|x| < 0.2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  3. Use your answer to part (b) to find an approximation for \(\sin 60°\). [2]
  4. Find the difference between the exact value of \(\sin 60°\) and the approximation in part (c). [1]
Edexcel C4 Q6
11 marks Standard +0.3
  1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2} \ln 2.$$ [6]
\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Find the volume of the solid of revolution generated. [2]
  2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]
Edexcel C4 Q7
12 marks Standard +0.3
\includegraphics{figure_3} The curve \(C\) with equation \(y = 2e^x + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3.
  1. Find the equation of the normal to \(C\) at \(M\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]
This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N(n, 0)\).
  1. Show that \(n = 14\). [1]
The point \(P(\ln 4, 13)\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(PN\), as shown in Fig. 3.
  1. Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found. [7]
Edexcel C4 Q8
13 marks Standard +0.3
Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((\mathbf{9i} - \mathbf{2j} + \mathbf{k})\), \((\mathbf{6i} + \mathbf{2j} + \mathbf{6k})\) and \((\mathbf{3i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.
  1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]
Given that \(C\) lies on \(l\),
  1. find the value of \(p\) and the value of \(q\), [2]
  2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]
The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).
  1. Find the position vector of \(D\). [6]
OCR C4 2007 January Q1
3 marks Moderate -0.8
It is given that $$f(x) = \frac{x^2 + 2x - 24}{x^2 - 4x} \quad \text{for } x \neq 0, x \neq 4.$$ Express \(f(x)\) in its simplest form. [3]
OCR C4 2007 January Q2
5 marks Standard +0.3
Find the exact value of \(\int_1^2 x \ln x \, dx\). [5]
OCR C4 2007 January Q3
6 marks Moderate -0.3
The points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to an origin \(O\), where \(\mathbf{a} = 4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{b} = -7\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}\).
  1. Find the length of \(AB\). [3]
  2. Use a scalar product to find angle \(OAB\). [3]
OCR C4 2007 January Q4
5 marks Moderate -0.8
Use the substitution \(u = 2x - 5\) to show that \(\int_2^3 (4x - 8)(2x - 5)^7 \, dx = \frac{17}{72}\). [5]
OCR C4 2007 January Q5
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^3\). [4]
  2. Hence find the coefficient of \(x^3\) in the expansion of \(\left(1 - 3(x + x^3)\right)^{-\frac{1}{2}}\). [3]
OCR C4 2007 January Q6
7 marks Moderate -0.3
  1. Express \(\frac{2x + 1}{(x - 3)^2}\) in the form \(\frac{A}{x - 3} + \frac{B}{(x - 3)^2}\), where \(A\) and \(B\) are constants. [3]
  2. Hence find the exact value of \(\int_4^{10} \frac{2x + 1}{(x - 3)^2} \, dx\), giving your answer in the form \(a + b \ln c\), where \(a\), \(b\) and \(c\) are integers. [4]
OCR C4 2007 January Q7
8 marks Challenging +1.2
The equation of a curve is \(2x^2 + xy + y^2 = 14\). Show that there are two stationary points on the curve and find their coordinates. [8]
OCR C4 2007 January Q8
10 marks Standard +0.3
The parametric equations of a curve are \(x = 2t^2\), \(y = 4t\). Two points on the curve are \(P(2p^2, 4p)\) and \(Q(2q^2, 4q)\).
  1. Show that the gradient of the normal to the curve at \(P\) is \(-p\). [2]
  2. Show that the gradient of the chord joining the points \(P\) and \(Q\) is \(\frac{2}{p + q}\). [2]
  3. The chord \(PQ\) is the normal to the curve at \(P\). Show that \(p^2 + pq + 2 = 0\). [2]
  4. The normal at the point \(R(8, 8)\) meets the curve again at \(S\). The normal at \(S\) meets the curve again at \(T\). Find the coordinates of \(T\). [4]
OCR C4 2007 January Q9
10 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{\sec^2 y}{\cos^2(2x)} \frac{dy}{dx} = 2.$$ [7]
  2. For the particular solution in which \(y = \frac{1}{4}\pi\) when \(x = 0\), find the value of \(y\) when \(x = \frac{1}{8}\pi\). [3]
OCR C4 2007 January Q10
11 marks Standard +0.3
The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5\mathbf{i} + 2\mathbf{j} - 9\mathbf{k}\) and \(4\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) respectively.
  1. Find a vector equation for the line \(PQ\). [2]
The position vector of the point \(T\) is \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).
  1. Write down a vector equation for the line \(OT\) and show that \(OT\) is perpendicular to \(PQ\). [4]
It is given that \(OT\) intersects \(PQ\).
  1. Find the position vector of the point of intersection of \(OT\) and \(PQ\). [3]
  2. Hence find the perpendicular distance from \(O\) to \(PQ\), giving your answer in an exact form. [2]
OCR C4 2005 June Q1
4 marks Moderate -0.8
Find the quotient and the remainder when \(x^4 + 3x^3 + 5x^2 + 4x - 1\) is divided by \(x^2 + x + 1\). [4]
OCR C4 2005 June Q2
5 marks Moderate -0.3
Evaluate \(\int_0^{\frac{\pi}{2}} x \cos x dx\), giving your answer in an exact form. [5]
OCR C4 2005 June Q3
7 marks Standard +0.3
The line \(L_1\) passes through the points \((2, -3, 1)\) and \((-1, -2, -4)\). The line \(L_2\) passes through the point \((3, 2, -9)\) and is parallel to the vector \(\mathbf{4i} - \mathbf{4j} + \mathbf{5k}\).
  1. Find an equation for \(L_1\) in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [2]
  2. Prove that \(L_1\) and \(L_2\) are skew. [5]