Questions FP1 (1491 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
CAIE FP1 2017 Specimen Q4
7 marks Challenging +1.2
4 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that, for all positive integers \(n\), $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum \(\sum _ { n = 1 } ^ { N } a _ { n }\) is denoted by \(S _ { N }\).
  1. Find the value of \(S _ { 30 }\) correct to 3 decimal places.
  2. Find the least value of \(N\) for which \(S _ { N } > 4.9\).
CAIE FP1 2017 Specimen Q5
8 marks Standard +0.8
5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$
  1. Write down the value of \(p\) and find the value of \(q\).
  2. Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]
CAIE FP1 2017 Specimen Q6
10 marks Standard +0.3
6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3 .
  1. Find corresponding eigenvectors.
    It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\).
  2. Find the corresponding eigenvalue.
  3. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).
CAIE FP1 2017 Specimen Q7
10 marks Challenging +1.2
7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & - 3 & 1 \\ 3 & - 5 & - 7 & 7 \\ 5 & - 9 & - 13 & 9 \\ 7 & - 13 & - 19 & 11 \end{array} \right)$$
  1. Find the rank of \(\mathbf { M }\) and a basis for the null space of T .
  2. The vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 4 \end{array} \right)\) is denoted by \(\mathbf { e }\). Show that there is a solution of the equation \(\mathbf { M x } = \mathbf { M e }\) of the form \(\mathbf { x } = \left( \begin{array} { c } a \\ b \\ - 1 \\ - 1 \end{array} \right)\), where the constants \(a\) and \(b\) are to be found.
CAIE FP1 2017 Specimen Q8
11 marks Standard +0.8
8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which \(C\) has no stationary points.
  2. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.
CAIE FP1 2017 Specimen Q9
12 marks Challenging +1.3
9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2 .$$
  2. Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\). [6]
CAIE FP1 2017 Specimen Q10
12 marks Challenging +1.3
10
  1. Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
  2. Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
  3. Deduce a quadratic equation, with integer coefficients, having roots \(\sec ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\sec ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\). [3]
CAIE FP1 2017 Specimen Q11 EITHER
Challenging +1.8
The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\).
  1. Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\).
  2. Find the position vector of the point \(Q\).
  3. Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).
CAIE FP1 2017 Specimen Q11 OR
Standard +0.8
The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\).
  1. Sketch \(C\).
  2. Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\).
  3. Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\).
CAIE FP1 2015 June Q1
6 marks Standard +0.8
1 The quartic equation \(x ^ { 4 } - p x ^ { 2 } + q x - r = 0\), where \(p , q\) and \(r\) are real constants, has two pairs of equal roots. Show that \(p ^ { 2 } + 4 r = 0\) and state the value of \(q\).
CAIE FP1 2015 June Q2
6 marks Challenging +1.2
2 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { 4 \theta }\) for \(0 \leqslant \theta \leqslant \alpha\), where \(\alpha\) is measured in radians. The length of \(C\) is 2015 . Find the value of \(\alpha\).
CAIE FP1 2015 June Q3
7 marks Standard +0.8
3 Prove by mathematical induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }\). State the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\).
CAIE FP1 2015 June Q4
7 marks Challenging +1.2
4 Use the formula for \(\tan ( A - B )\) in the List of Formulae (MF10) to show that $$\tan ^ { - 1 } ( x + 1 ) - \tan ^ { - 1 } ( x - 1 ) = \tan ^ { - 1 } \left( \frac { 2 } { x ^ { 2 } } \right)$$ Deduce the sum to \(n\) terms of the series $$\tan ^ { - 1 } \left( \frac { 2 } { 1 ^ { 2 } } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 2 ^ { 2 } } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 3 ^ { 2 } } \right) + \ldots .$$
CAIE FP1 2015 June Q5
9 marks Challenging +1.8
5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 n \theta } { \cos \theta } \mathrm {~d} \theta\), where \(n\) is a non-negative integer.
  1. Use the identity \(\sin P + \sin Q \equiv 2 \sin \frac { 1 } { 2 } ( P + Q ) \cos \frac { 1 } { 2 } ( P - Q )\) to show that $$I _ { n } + I _ { n - 1 } = \frac { 2 } { 2 n - 1 } , \text { for all positive integers } n$$
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 8 \theta } { \cos \theta } d \theta\).
CAIE FP1 2015 June Q6
9 marks Challenging +1.8
6 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Use the binomial expansion of \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, to show that $$\binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \ldots + \binom { n } { n } \cos n \theta = 2 ^ { n } \cos ^ { n } \left( \frac { 1 } { 2 } \theta \right) \cos \left( \frac { 1 } { 2 } n \theta \right) - 1$$ Find $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$
CAIE FP1 2015 June Q7
10 marks Standard +0.8
7 The curve \(C\) has equation \(x ^ { 2 } + 2 x y - 4 y ^ { 2 } + 20 = 0\). Show that if the tangent to \(C\) at the point \(( x , y )\) is parallel to the \(x\)-axis then \(x + y = 0\). Hence find the coordinates of the stationary points on \(C\), and determine their nature.
CAIE FP1 2015 June Q8
10 marks Standard +0.3
8 A line, passing through the point \(A ( 3,0,2 )\), has vector equation \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\). It meets the plane \(\Pi\), which has equation \(\mathbf { r } \cdot ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) = 3\), at the point \(P\). Find the coordinates of \(P\). Write down a vector \(\mathbf { n }\) which is perpendicular to \(\Pi\), and calculate the vector \(\mathbf { w }\), where $$\mathbf { w } = \mathbf { n } \times ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The point \(Q\) lies in \(\Pi\) and is the foot of the perpendicular from \(A\) to \(\Pi\). Use the vector \(\mathbf { w }\) to determine an equation of the line \(P Q\) in the form \(\mathbf { r } = \mathbf { u } + \mu \mathbf { v }\).
CAIE FP1 2015 June Q9
11 marks Standard +0.8
9 Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 3 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 10 x = 2 \sin t - 3 \cos t$$ given that, when \(t = 0 , x = 3.3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.9\).
CAIE FP1 2015 June Q10
11 marks Standard +0.8
10 The curve \(C\) has equation \(y = \frac { 4 x ^ { 2 } - 3 x } { x ^ { 2 } + 1 }\). Verify that the equation of \(C\) may be written in the form \(y = - \frac { 1 } { 2 } + \frac { ( 3 x - 1 ) ^ { 2 } } { 2 \left( x ^ { 2 } + 1 \right) }\) and also in the form \(y = \frac { 9 } { 2 } - \frac { ( x + 3 ) ^ { 2 } } { 2 \left( x ^ { 2 } + 1 \right) }\). Hence show that \(- \frac { 1 } { 2 } \leqslant y \leqslant \frac { 9 } { 2 }\). Without differentiating, write down the coordinates of the turning points of \(C\). State the equation of the asymptote of \(C\). Sketch the graph of \(C\), stating the coordinates of the intersections with the coordinate axes and the asymptote.
CAIE FP1 2015 June Q11 EITHER
Challenging +1.8
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & 2 & 3 & 4 \\ 1 & - 1 & 2 & 3 \\ 1 & - 3 & 3 & 5 \\ 1 & 4 & 2 & 2 \end{array} \right)$$ The range space of T is denoted by \(V\).
  1. Determine the dimension of \(V\).
  2. Show that the vectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 2 \\ - 1 \\ - 3 \\ 4 \end{array} \right) , \left( \begin{array} { l } 3 \\ 2 \\ 3 \\ 2 \end{array} \right)\) are a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
  3. State, with a reason, whether \(W\) is a vector space.
  4. Show that if the vector \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(W\) then \(x + y \neq z + t\).
CAIE FP1 2015 June Q11 OR
Standard +0.3
One of the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 4 & 2 \\ - 4 & \alpha & 6 \\ 2 & 6 & - 2 \end{array} \right)$$ is - 9 . Find the value of \(\alpha\). Find
  1. the other two eigenvalues, \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\), of \(\mathbf { M }\), where \(\lambda _ { 1 } > \lambda _ { 2 }\),
  2. corresponding eigenvectors for all three eigenvalues of \(\mathbf { M }\). It is given that \(\mathbf { x } = a \mathbf { e } _ { 1 } + b \mathbf { e } _ { 2 }\), where \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) are eigenvectors of \(\mathbf { M }\) corresponding to the eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) respectively, and \(a\) and \(b\) are scalar constants. Show that \(\mathbf { M x } = p \mathbf { e } _ { 1 } + q \mathbf { e } _ { 2 }\), expressing \(p\) and \(q\) in terms of \(a\) and \(b\). {www.cie.org.uk} after the live examination series. }
CAIE FP1 2007 November Q1
4 marks Challenging +1.2
1 A curve is defined parametrically by $$x = a t ^ { 2 } , \quad y = a t$$ where \(a\) is a positive constant. The part of the curve joining the point where \(t = 0\) to the point where \(t = \sqrt { } 2\) is rotated through one complete revolution about the \(x\)-axis. Show that the area of the surface obtained is \(\frac { 13 } { 3 } \pi a ^ { 2 }\).
CAIE FP1 2007 November Q2
5 marks Standard +0.8
2 Express $$\frac { 2 n + 3 } { n ( n + 1 ) }$$ in partial fractions and hence use the method of differences to find $$\sum _ { n = 1 } ^ { N } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$ in terms of \(N\). Deduce the value of $$\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$
CAIE FP1 2007 November Q3
6 marks Challenging +1.2
3 Prove by induction that, for all \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x ^ { 2 } } \right) = \mathrm { P } _ { n } ( x ) \mathrm { e } ^ { x ^ { 2 } } ,$$ where \(\mathrm { P } _ { n } ( x )\) is a polynomial in \(x\) of degree \(n\) with the coefficient of \(x ^ { n }\) equal to \(2 ^ { n }\).
CAIE FP1 2007 November Q4
7 marks Standard +0.8
4 The roots of the equation $$x ^ { 3 } - 8 x ^ { 2 } + 5 = 0$$ are \(\alpha , \beta , \gamma\). Show that $$\alpha ^ { 2 } = \frac { 5 } { \beta + \gamma } .$$ It is given that the roots are all real. Without reference to a graph, show that one of the roots is negative and the other two roots are positive.