Questions FP1 (1385 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE FP1 2011 June Q11 EITHER
Challenging +1.3
Use de Moivre's theorem to prove that $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ State the exact values of \(\theta\), between 0 and \(\pi\), that satisfy \(\tan 3 \theta = 1\). Express each root of the equation \(t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0\) in the form \(\tan ( k \pi )\), where \(k\) is a positive rational number. For each of these values of \(k\), find the exact value of \(\tan ( k \pi )\).
CAIE FP1 2011 June Q11 OR
Challenging +1.2
The curve \(C\) has equation $$y = \frac { x ^ { 2 } + \lambda x - 6 \lambda ^ { 2 } } { x + 3 }$$ where \(\lambda\) is a constant such that \(\lambda \neq 1\) and \(\lambda \neq - \frac { 3 } { 2 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and deduce that if \(C\) has two stationary points then \(- \frac { 3 } { 2 } < \lambda < 1\).
  2. Find the equations of the asymptotes of \(C\).
  3. Draw a sketch of \(C\) for the case \(0 < \lambda < 1\).
  4. Draw a sketch of \(C\) for the case \(\lambda > 3\).
CAIE FP1 2011 June Q1
Standard +0.3
1 Find \(2 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }\). Hence find \(1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 n ) ^ { 2 }\), simplifying your answer.
CAIE FP1 2011 June Q2
Standard +0.3
2 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 1 \end{array} \right)\). Prove by mathematical induction that, for every positive integer \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$
CAIE FP1 2011 June Q3
Standard +0.3
3 Find a cubic equation with roots \(\alpha , \beta\) and \(\gamma\), given that $$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$ Hence find the numerical values of the roots.
CAIE FP1 2011 June Q4
Standard +0.8
4 The curve \(C\) has equation $$2 x y ^ { 2 } + 3 x ^ { 2 } y = 1$$ Show that, at the point \(A ( - 1,1 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - 4\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
CAIE FP1 2011 June Q5
Standard +0.8
5 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$ where \(n \geqslant 0\). Use the fact that \(\tan ^ { 2 } x = \sec ^ { 2 } x - 1\) to show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 1 } { n - 1 } - I _ { n - 2 }$$ Show that \(I _ { 8 } = \frac { 1 } { 7 } - \frac { 1 } { 5 } + \frac { 1 } { 3 } - 1 + \frac { 1 } { 4 } \pi\).
CAIE FP1 2011 June Q6
Challenging +1.2
6 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$\begin{array} { l l } C _ { 1 } : & r = a \\ C _ { 2 } : & r = 2 a \cos 2 \theta , \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi \end{array}$$ where \(a\) is a positive constant. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram. The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point with polar coordinates ( \(a , \beta\) ). State the value of \(\beta\). Show that the area of the region bounded by the initial line, the arc of \(C _ { 1 }\) from \(\theta = 0\) to \(\theta = \beta\), and the arc of \(C _ { 2 }\) from \(\theta = \beta\) to \(\theta = \frac { 1 } { 4 } \pi\) is $$a ^ { 2 } \left( \frac { 1 } { 6 } \pi - \frac { 1 } { 8 } \sqrt { } 3 \right)$$
CAIE FP1 2011 June Q7
Challenging +1.2
7 A curve \(C\) has parametric equations \(x = \mathrm { e } ^ { t } \cos t , y = \mathrm { e } ^ { t } \sin t\), for \(0 \leqslant t \leqslant \pi\). Find the arc length of \(C\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2011 June Q8
Standard +0.8
8 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t$$ Find the particular solution, given that \(x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\) when \(t = 0\). State an approximate solution for large positive values of \(t\).
CAIE FP1 2011 June Q9
Standard +0.8
9 The curve \(C\) with equation $$y = \frac { a x ^ { 2 } + b x + c } { x - 1 }$$ where \(a , b\) and \(c\) are constants, has two asymptotes. It is given that \(y = 2 x - 5\) is one of these asymptotes.
  1. State the equation of the other asymptote.
  2. Find the value of \(a\) and show that \(b = - 7\).
  3. Given also that \(C\) has a turning point when \(x = 2\), find the value of \(c\).
  4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\).
CAIE FP1 2011 June Q10
Standard +0.8
10 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$l _ { 1 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \mu ( 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) .$$ Find a cartesian equation of the plane \(\Pi\) containing \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vector of the foot of the perpendicular from the point with position vector \(\mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k }\) to \(\Pi\). The line \(l _ { 3 }\) has equation \(\mathbf { r } = \mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k } + v ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )\). Find the shortest distance between \(l _ { 1 }\) and \(l _ { 3 }\).
CAIE FP1 2011 June Q11 EITHER
Challenging +1.2
A \(3 \times 3\) matrix \(\mathbf { A }\) has eigenvalues \(- 1,1,2\), with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right) , \quad \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) ,$$ respectively. Find
  1. the matrix \(\mathbf { A }\),
  2. \(\mathbf { A } ^ { 2 n }\), where \(n\) is a positive integer.
CAIE FP1 2011 June Q11 OR
Challenging +1.8
Determine the rank of the matrix $$\mathbf { A } = \left( \begin{array} { l l l l } 1 & - 1 & - 1 & 1 \\ 2 & - 1 & - 4 & 3 \\ 3 & - 3 & - 2 & 2 \\ 5 & - 4 & - 6 & 5 \end{array} \right)$$ Show that if $$\mathbf { A x } = p \left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 5 \end{array} \right) + q \left( \begin{array} { l } - 1 \\ - 1 \\ - 3 \\ - 4 \end{array} \right) + r \left( \begin{array} { l } - 1 \\ - 4 \\ - 2 \\ - 6 \end{array} \right)$$ where \(p , q\) and \(r\) are given real numbers, then $$\mathbf { x } = \left( \begin{array} { c } p + \lambda \\ q + \lambda \\ r + \lambda \\ \lambda \end{array} \right) ,$$ where \(\lambda\) is real. Find the values of \(p , q\) and \(r\) such that $$p \left( \begin{array} { l } 1
2
3
5 \end{array} \right) + q \left( \begin{array} { l } - 1
- 1
- 3
- 4 \end{array} \right) + r \left( \begin{array} { l } - 1
- 4
- 2
- 6 \end{array} \right) = \left( \begin{array} { r } 3
7
8
CAIE FP1 2011 June Q15
Challenging +1.2
15 \end{array} \right) .$$ Find the solution \(\mathbf { x } = \left( \begin{array} { l } \alpha \\ \beta \\ \gamma \\ \delta \end{array} \right)\) of the equation \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } 3 \\ 7 \\ 8 \\ 15 \end{array} \right)\) for which \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 11 } { 4 }\).
CAIE FP1 2012 June Q1
Standard +0.8
1 The roots of the cubic equation \(x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0\) are \(\alpha , \beta , \gamma\). Find the values of
  1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\),
  2. \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
CAIE FP1 2012 June Q2
Standard +0.3
2 Prove, by mathematical induction, that, for integers \(n \geqslant 2\), $$4 ^ { n } > 2 ^ { n } + 3 ^ { n }$$
CAIE FP1 2012 June Q3
Standard +0.3
3 Given that \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\), show that $$\mathrm { f } ( r - 1 ) - \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$ Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
CAIE FP1 2012 June Q4
Standard +0.8
4 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\). Sketch the graph of \(C\). Find the area of the region \(R\) enclosed by \(C\) and the initial line. The half-line \(\theta = \frac { 1 } { 5 } \pi\) divides \(R\) into two parts. Find the area of each part, correct to 3 decimal places.
CAIE FP1 2012 June Q5
Standard +0.3
5 A matrix \(\mathbf { A }\) has eigenvalues \(- 1,1\) and 2 , with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) ,$$ respectively. Find \(\mathbf { A }\).
CAIE FP1 2012 June Q6
Challenging +1.2
6 Write down the values of \(\theta\), in the interval \(0 \leqslant \theta < 2 \pi\), for which \(\cos \theta + \mathrm { i } \sin \theta\) is a fifth root of unity. By writing the equation \(( z + 1 ) ^ { 5 } = z ^ { 5 }\) in the form $$\left( \frac { z + 1 } { z } \right) ^ { 5 } = 1$$ show that its roots are $$- \frac { 1 } { 2 } \left\{ 1 + \mathrm { i } \cot \left( \frac { k \pi } { 5 } \right) \right\} , \quad k = 1,2,3,4$$
CAIE FP1 2012 June Q7
Challenging +1.2
7 The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 4 \\ 2 & 1 & 4 & 11 \\ 3 & 4 & 1 & 9 \\ 4 & - 3 & 18 & 37 \end{array} \right) \quad \text { and } \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & - 1 \\ 2 & 3 & 0 & 1 \\ 3 & 4 & 1 & 0 \\ 4 & 5 & 2 & 0 \end{array} \right)$$ respectively. The null space of \(\mathrm { T } _ { 1 }\) is denoted by \(K _ { 1 }\) and the null space of \(\mathrm { T } _ { 2 }\) is denoted by \(K _ { 2 }\). Show that the dimension of \(K _ { 1 }\) is 2 and that the dimension of \(K _ { 2 }\) is 1 . Find the basis of \(K _ { 1 }\) which has the form \(\left\{ \left( \begin{array} { c } p \\ q \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { c } r \\ s \\ 0 \\ 1 \end{array} \right) \right\}\) and show that \(K _ { 2 }\) is a subspace of \(K _ { 1 }\).
CAIE FP1 2012 June Q8
Standard +0.8
8 Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 10 \mathrm { e } ^ { - 2 x }$$ given that \(y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\).
CAIE FP1 2012 June Q9
Standard +0.3
9 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 2 x + 3 } { x ^ { 2 } + 2 }$$ Show that, for all \(x , 1 \leqslant y \leqslant \frac { 5 } { 2 }\). Find the coordinates of the turning points on \(C\). Find the equation of the asymptote of \(C\). Sketch the graph of \(C\), stating the coordinates of any intersections with the \(y\)-axis and the asymptote.
CAIE FP1 2012 June Q10
Challenging +1.2
10 The curve \(C\) has equation $$y = 2 \left( \frac { x } { 3 } \right) ^ { \frac { 3 } { 2 } }$$ where \(0 \leqslant x \leqslant 3\). Show that the arc length of \(C\) is \(2 ( 2 \sqrt { 2 } - 1 )\). Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 3\).