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Edexcel AEA 2017 Specimen Q2
11 marks Challenging +1.8
2.(a)On separate diagrams,sketch the curves with the following equations.On each sketch you should label the exact coordinates of the points where the curve meets the coordinate axes.
  1. \(y = 8 + 2 x - x ^ { 2 }\)
  2. \(y = 8 + 2 | x | - x ^ { 2 }\)
  3. \(y = 8 + x + | x | - x ^ { 2 }\) (b)Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$
Edexcel AEA 2017 Specimen Q3
12 marks Challenging +1.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_609_631_264_724} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_172_168_781_1548}
Figure 1 shows a regular pentagon \(O A B C D\). The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O A }\) and \(\mathbf { q } = \overrightarrow { O D }\) respectively. Let \(k\) be the number such that \(\overrightarrow { D B } = k \overrightarrow { O A }\).
  1. Write down \(\overrightarrow { A C }\) in terms of \(\mathbf { p } , \mathbf { q }\) and \(k\) as appropriate.
  2. Show that \(\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }\)
  3. Hence find the value of \(k\) By considering triangle \(D B C\), or otherwise,
  4. find the exact value of \(\sin 54 ^ { \circ }\)
Edexcel AEA 2017 Specimen Q4
13 marks Challenging +1.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-12_428_897_251_593} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of weight \(W\) lies on a rough plane.The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) .The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\) The particle is held in equilibrium by a force of magnitude 1.2 W .The force makes an angle \(\theta\) with the plane,where \(0 < \theta < \pi\) ,and acts in a vertical plane containing a line of greatest slope of the plane,as shown in Figure 2.
  1. Find the value of \(\theta\) for which there is no frictional force acting on the particle. The minimum value of \(\theta\) for the particle to remain in equilibrium is \(\beta\)
  2. Show that $$\beta = \arccos \left( \frac { \sqrt { 5 } } { 3 } \right) - \arctan \left( \frac { 1 } { 2 } \right)$$
  3. Find the range of values of \(\theta\) for which the particle remains in equilibrium with the frictional force acting up the plane.
Edexcel AEA 2017 Specimen Q5
13 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-16_745_862_258_667} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Show that the area of the finite region between the curves \(y = \tan ^ { 2 } x\) and \(y = 4 \cos 2 x - 1\) in the interval \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), shown shaded in Figure 3, is given by $$2 \sqrt { 2 \sqrt { 3 } } - 2 \sqrt { 2 \sqrt { 3 } - 3 }$$
\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-16_2255_51_315_1987}
Edexcel AEA 2017 Specimen Q6
18 marks Challenging +1.2
6.(i)Eden,who is confused about the laws of logarithms,states that $$\left( \log _ { 5 } p \right) ^ { 2 } = \log _ { 5 } \left( p ^ { 2 } \right)$$ and \(\log _ { 5 } ( q - p ) = \log _ { 5 } q - \log _ { 5 } p\) However,there is a value of \(p\) and a value of \(q\) for which both statements are correct.
Determine these values.
(ii)(a)Let \(r \in \mathbb { R } ^ { + } , r \neq 1\) .Prove that $$\log _ { r } A = \log _ { r ^ { 2 } } B \Rightarrow A ^ { 2 } = B$$ (b)Solve $$\log _ { 4 } \left( 3 x ^ { 3 } + 26 x ^ { 2 } + 40 x \right) = 2 + \log _ { 2 } ( x + 2 )$$
\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-20_2261_53_317_1977}
Edexcel AEA 2017 Specimen Q7
25 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-25_670_682_301_694} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A circular tower of radius 1 metre stands in a large horizontal field of grass.A goat is attached to one end of a rope and the other end of the rope is attached to a fixed point \(O\) at the base of the tower.The goat cannot enter the tower. Taking the point \(O\) as the origin( 0,0 ),the centre of the base of the tower is at the point \(T ( 0,1 )\) ,where the unit of length is the metre. The rope has length \(\pi\) metres and you may ignore the size of the goat.
The curve \(C\) shown in Figure 4 represents the edge of the region that the goat can reach.
  1. Write down the equation of \(C\) for \(y < 0\) When the goat is at the point \(G ( x , y )\) ,with \(x > 0\) and \(y > 0\) ,as shown in Figure 4 ,the rope lies along \(O A G\) where \(O A\) is an arc of the circle with angle \(O T A = \theta\) radians and \(A G\) is a tangent to the circle at \(A\) .
  2. With the aid of a suitable diagram show that $$\begin{aligned} & x = \sin \theta + ( \pi - \theta ) \cos \theta \\ & y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta \end{aligned}$$
  3. By considering \(\int y \frac { \mathrm {~d} x } { \mathrm {~d} \theta } \mathrm {~d} \theta\), show that the area, in the first quadrant, between \(C\), the positive \(x\)-axis and the positive \(y\)-axis can be expressed in the form $$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$
  4. Show that \(\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u\)
  5. Hence find the area of grass that can be reached by the goat.
OCR FP3 2009 January Q1
5 marks Standard +0.8
1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).
  1. In each case, write down the smallest possible value of \(n\) :
    1. if \(G\) is cyclic,
    2. if \(G\) has a proper subgroup of order 3,
    3. if \(G\) has at least two elements of order 2 .
    4. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).
OCR FP3 2009 January Q2
5 marks Standard +0.3
2
  1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
  2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.
OCR FP3 2009 January Q3
6 marks Challenging +1.2
3 Two skew lines have equations $$\frac { x } { 2 } = \frac { y + 3 } { 1 } = \frac { z - 6 } { 3 } \quad \text { and } \quad \frac { x - 5 } { 3 } = \frac { y + 1 } { 1 } = \frac { z - 7 } { 5 } .$$
  1. Find the direction of the common perpendicular to the lines.
  2. Find the shortest distance between the lines.
OCR FP3 2009 January Q4
9 marks Standard +0.8
4 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$
OCR FP3 2009 January Q5
9 marks Standard +0.8
5 The variables \(x\) and \(y\) are related by the differential equation $$x ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x y + x + 1 .$$
  1. Use the substitution \(y = u - \frac { 1 } { x }\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x ^ { 2 } \frac { \mathrm {~d} u } { \mathrm {~d} x } = u .$$
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR FP3 2009 January Q6
13 marks Standard +0.8
6
[diagram]
The cuboid \(O A B C D E F G\) shown in the diagram has \(\overrightarrow { O A } = 4 \mathbf { i } , \overrightarrow { O C } = 2 \mathbf { j } , \overrightarrow { O D } = 3 \mathbf { k }\), and \(M\) is the mid-point of \(G F\).
  1. Find the equation of the plane \(A C G E\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\).
  2. The plane \(O E F C\) has equation \(\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { k } ) = 0\). Find the acute angle between the planes \(O E F C\) and \(A C G E\).
  3. The line \(A M\) meets the plane \(O E F C\) at the point \(W\). Find the ratio \(A W : W M\).
OCR FP3 2009 January Q7
13 marks Standard +0.3
7
  1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
    1. Prove that the set of real numbers, together with the operation \(*\), forms a group.
    2. State, with a reason, whether the group is commutative.
    3. Prove that there are no elements of order 2.
    4. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied.
OCR FP3 2009 January Q8
12 marks Challenging +1.3
8
  1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 6 } \theta \equiv - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )$$
  2. Replace \(\theta\) by ( \(\frac { 1 } { 2 } \pi - \theta\) ) in the identity in part (i) to obtain a similar identity for \(\cos ^ { 6 } \theta\).
  3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sin ^ { 6 } \theta - \cos ^ { 6 } \theta \right) \mathrm { d } \theta\).
OCR FP3 2012 January Q1
7 marks Standard +0.3
1 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y } .$$
  1. Use the substitution \(y = u x\), where \(u\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } = \frac { 2 } { u } .$$
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR FP3 2012 January Q2
7 marks Standard +0.8
2
  1. Show that \(\left( z ^ { n } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { n } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 n } - ( 2 \cos \theta ) z ^ { n } + 1\).
  2. Express \(z ^ { 4 } - z ^ { 2 } + 1\) as the product of four factors of the form \(\left( z - e ^ { \mathrm { i } \alpha } \right)\), where \(0 \leqslant \alpha < 2 \pi\).
OCR FP3 2012 January Q3
7 marks Challenging +1.2
3 A multiplicative group contains the distinct elements \(e , x\) and \(y\), where \(e\) is the identity.
  1. Prove that \(x ^ { - 1 } y ^ { - 1 } = ( y x ) ^ { - 1 }\).
  2. Given that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) for some integer \(n \geqslant 2\), prove that \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\).
  3. If \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\), does it follow that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) ? Give a reason for your answer.
OCR FP3 2012 January Q4
10 marks Standard +0.3
4 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { 2 }\) and the point \(A\) is ( \(7,3,7\) ). \(M\) is the point where the perpendicular from \(A\) meets \(l\).
  1. Find, in either order, the coordinates of \(M\) and the perpendicular distance from \(A\) to \(l\).
  2. Find the coordinates of the point \(B\) on \(A M\) such that \(\overrightarrow { A B } = 3 \overrightarrow { B M }\).
OCR FP3 2012 January Q5
11 marks Challenging +1.3
5 The variables \(x\) and \(y\) satisfy the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 5 \mathrm { e } ^ { - 2 x }$$
  1. Find the complementary function of the differential equation.
  2. Given that there is a particular integral of the form \(y = p x \mathrm { e } ^ { - 2 x }\), find the constant \(p\).
  3. Find the solution of the equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\).
OCR FP3 2012 January Q6
9 marks Standard +0.3
6 The plane \(\Pi\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 3 \\ - 5 \end{array} \right)\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 3 \\ 0 \\ - 1 \end{array} \right)\).
  1. Express the equation of \(\Pi\) in the form r.n \(= p\).
  2. Find the point of intersection of \(l\) and \(\Pi\).
  3. The equation of \(\Pi\) may be expressed in the form \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \mathbf { c }\), where \(\mathbf { c }\) is perpendicular to \(\left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right)\). Find \(\mathbf { c }\).
OCR FP3 2012 January Q7
9 marks Challenging +1.8
7 The set \(M\) consists of the six matrices \(\left( \begin{array} { l l } 1 & 0 \\ n & 1 \end{array} \right)\), where \(n \in \{ 0,1,2,3,4,5 \}\). It is given that \(M\) forms a group ( \(M , \times\) ) under matrix multiplication, with numerical addition and multiplication both being carried out modulo 6 .
  1. Determine whether ( \(M , \times\) ) is a commutative group, justifying your answer.
  2. Write down the identity element of the group and find the inverse of \(\left( \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right)\).
  3. State the order of \(\left( \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right)\) and give a reason why \(( M , \times )\) has no subgroup of order 4.
  4. The multiplicative group \(G\) has order 6. All the elements of \(G\), apart from the identity, have order 2 or 3 . Determine whether \(G\) is isomorphic to ( \(M , \times\) ), justifying your answer.
OCR FP3 2012 January Q8
12 marks Challenging +1.3
8
  1. Use de Moivre's theorem to prove that $$\tan 5 \theta \equiv \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
  2. Solve the equation \(\tan 5 \theta = 1\), for \(0 \leqslant \theta < \pi\).
  3. Show that the roots of the equation $$t ^ { 4 } - 4 t ^ { 3 } - 14 t ^ { 2 } - 4 t + 1 = 0$$ may be expressed in the form \(\tan \alpha\), stating the exact values of \(\alpha\), where \(0 \leqslant \alpha < \pi\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR FP3 2013 January Q1
7 marks Standard +0.3
1 Two planes have equations $$x + 2 y + 5 z = 12 \text { and } 2 x - y + 3 z = 5$$
  1. Find the acute angle between the planes.
  2. Find a vector equation of the line of intersection of the planes.
OCR FP3 2013 January Q2
6 marks Standard +0.8
2 The elements of a group \(G\) are the complex numbers \(a + b \mathrm { i }\) where \(a , b \in \{ 0,1,2,3,4 \}\). These elements are combined under the operation of addition modulo 5 .
  1. State the identity element and the order of \(G\).
  2. Write down the inverse of \(2 + 4 \mathrm { i }\).
  3. Show that every non-zero element of \(G\) has order 5 .
OCR FP3 2013 January Q3
8 marks Standard +0.8
3 Solve the differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = x ^ { 4 } \mathrm { e } ^ { 2 x }\) for \(y\) in terms of \(x\), given that \(y = 0\) when \(x = 1\).