Questions FP3 (539 questions)

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AQA FP3 2016 June Q8
17 marks Challenging +1.2
8 The diagram shows the sketch of part of a curve, the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is \(r = 1 + \tan \theta\).
The point \(A\) is the point on the curve at which \(\theta = \frac { \pi } { 3 }\).
The perpendicular, \(A N\), from \(A\) to the initial line intersects the curve at the point \(B\).
  1. Find the exact length of \(O A\).
    1. Given that, at the point \(B , \theta = \alpha\), show that \(( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }\).
    2. Hence, or otherwise, find \(\alpha\) in terms of \(\pi\).
  3. Show that the area of triangle \(O A B\) is \(\frac { 3 + 2 \sqrt { 3 } } { 6 }\).
  4. Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment \(A B\).
    [0pt] [7 marks]
    \includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
    \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
OCR FP3 2007 June Q7
10 marks Standard +0.3
  1. Show that \(\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1\).
  2. Write down the seven roots of the equation \(z ^ { 7 } = 1\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) and show their positions in an Argand diagram.
  3. Hence express \(z ^ { 7 } - 1\) as the product of one real linear factor and three real quadratic factors.
OCR FP3 2013 June Q2
9 marks Challenging +1.2
  1. Write down the operation table and, assuming associativity, show that \(G\) is a group.
  2. State the order of each element.
  3. Find all the proper subgroups of \(G\). The group \(H\) consists of the set \(\{ 1,3,7,9 \}\) with the operation of multiplication modulo 10 .
  4. Explaining your reasoning, determine whether \(H\) is isomorphic to \(G\).
OCR FP3 2016 June Q7
12 marks Challenging +1.2
  1. Use de Moivre's theorem to show that $$\sin 6 \theta \equiv \cos \theta \left( 6 \sin \theta - 32 \sin ^ { 3 } \theta + 32 \sin ^ { 5 } \theta \right)$$
  2. Hence show that, for \(\sin 2 \theta \neq 0\), $$- 1 \leqslant \frac { \sin 6 \theta } { \sin 2 \theta } < 3$$
Edexcel FP3 Q1
6 marks Standard +0.8
  1. Solve the equation
$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\), where \(a\) is a rational number.
Edexcel FP3 Q2
7 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{045545c7-06d9-40b6-9d01-fc792ab0aa07-01_222_241_525_2042} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$ Calculate
  1. \(\mathbf { b } \times \mathbf { c }\),
  2. \(\mathbf { a . } ( \mathbf { b } \times \mathbf { c } )\),
  3. the area of triangle \(O B C\),
  4. the volume of the tetrahedron \(O A B C\).
Edexcel FP3 Q4
8 marks Challenging +1.2
4. Given that \(y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer as a simplified fraction.
  2. Hence, or otherwise, find $$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$ giving your answer in the form \(\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)\), where \(a\) and \(b\) are integers.
Edexcel FP3 Q5
4 marks Challenging +1.3
5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$
  1. Find an expression for \(\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5\).
  2. Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
  3. Find \(I _ { 4 }\) in the form \(k \pi\), where \(k\) is a fraction.
Edexcel FP3 Q6
10 marks Challenging +1.2
6. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants. The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
  1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
  2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
  3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).
Edexcel FP3 Q7
9 marks Standard +0.3
7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
  1. the value of \(\alpha\),
  2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
    Given that \(\alpha = 2\),
  3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
Edexcel FP3 Q8
8 marks Challenging +1.8
8. A curve, which is part of an ellipse, has parametric equations $$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { a } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \text { where } c = \cos \theta$$ and where \(k\) and \(\alpha\) are constants to be found.
  2. Using the substitution \(c = \frac { 3 } { 4 } \sinh u\), or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.
Edexcel FP3 Q9
8 marks Challenging +1.8
9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$
  1. Show that, for \(n > 0\), $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
  2. Find \(I _ { 2 }\).
AQA FP3 Q5
18 marks Standard +0.8
5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$
  1. Sketch, on an Argand diagram, the locus of \(z\).
  2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
  3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).
AQA FP3 Q6
17 marks Challenging +1.2
6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).
    1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$
    2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
    3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
  2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).
AQA FP3 2006 January Q1
12 marks Standard +0.3
1
  1. Find the roots of the equation \(m ^ { 2 } + 2 m + 2 = 0\) in the form \(a + i b\).
    (2 marks)
    1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x$$
    2. Hence express \(y\) in terms of \(x\), given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) when \(x = 0\).
AQA FP3 2006 January Q2
8 marks Standard +0.3
2
  1. Find \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), where \(a > 0\).
  2. Write down the value of \(\lim _ { a \rightarrow \infty } a ^ { k } \mathrm { e } ^ { - 2 a }\), where \(k\) is a positive constant.
  3. Hence find \(\int _ { 0 } ^ { \infty } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
AQA FP3 2006 January Q3
8 marks Standard +0.3
3
  1. Show that \(y = x ^ { 3 } - x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$
  2. By differentiating \(\left( x ^ { 2 } - 1 \right) y = c\) implicitly, where \(y\) is a function of \(x\) and \(c\) is a constant, show that \(y = \frac { c } { x ^ { 2 } - 1 }\) is a solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 0$$
  3. Hence find the general solution of $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$
AQA FP3 2006 January Q4
14 marks Standard +0.8
4
  1. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 } + \ldots$$ to write down the first four terms in the expansion, in ascending powers of \(x\), of \(\ln ( 1 - x )\).
  2. The function f is defined by $$\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
    1. the first three terms are $$1 + x + \frac { 1 } { 2 } x ^ { 2 }$$
    2. the coefficient of \(x ^ { 3 }\) is zero.
  3. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { \sin x } - 1 + \ln ( 1 - x ) } { x ^ { 2 } \sin x }$$ (4 marks)
AQA FP3 2006 January Q5
17 marks Standard +0.3
5
  1. The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x \ln x + \frac { y } { x }$$ and $$y ( 1 ) = 1$$
    1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\).
    2. Use the formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with your answer to part (a)(i) to obtain an approximation to \(y ( 1.2 )\), giving your answer to three decimal places.
    1. Show that \(\frac { 1 } { x }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 1 } { x } y = x \ln x$$
    2. Solve this differential equation, given that \(y = 1\) when \(x = 1\).
    3. Calculate the value of \(y\) when \(x = 1.2\), giving your answer to three decimal places.
AQA FP3 2006 January Q6
16 marks Challenging +1.2
6
  1. A circle \(C _ { 1 }\) has cartesian equation \(x ^ { 2 } + ( y - 6 ) ^ { 2 } = 36\). Show that the polar equation of \(C _ { 1 }\) is \(r = 12 \sin \theta\).
  2. A curve \(C _ { 2 }\) with polar equation \(r = 2 \sin \theta + 5,0 \leqslant \theta \leqslant 2 \pi\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b572aeb5-bcbb-4d50-964c-7f37e223f51d-5_545_837_559_651} Calculate the area bounded by \(C _ { 2 }\).
  3. The circle \(C _ { 1 }\) intersects the curve \(C _ { 2 }\) at the points \(P\) and \(Q\). Find, in surd form, the area of the quadrilateral \(O P M Q\), where \(M\) is the centre of the circle and \(O\) is the pole.
    (6 marks)
AQA FP3 2007 January Q1
9 marks Standard +0.3
1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln \left( 1 + x ^ { 2 } + y \right)$$ and $$y ( 1 ) = 0.6$$
  1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.
  2. Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.
AQA FP3 2007 January Q2
6 marks Standard +0.8
2 A curve has polar equation \(r ( 1 - \sin \theta ) = 4\). Find its cartesian equation in the form \(y = \mathrm { f } ( x )\).
AQA FP3 2007 January Q3
9 marks Standard +0.3
3
  1. Show that \(x ^ { 2 }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
  2. Solve this differential equation, given that \(y = 1\) when \(x = 2\).
AQA FP3 2007 January Q4
8 marks Challenging +1.2
4
  1. Explain why \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) is an improper integral.
    (1 mark)
  2. Use integration by parts to find \(\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\).
    (3 marks)
  3. Show that \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) exists and find its value.
    (4 marks)
AQA FP3 2007 January Q5
12 marks Challenging +1.2
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 6 + 5 \sin x$$ (12 marks)