Questions FP1 (1491 questions)

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CAIE FP1 2019 June Q8
10 marks Standard +0.8
8 Find the particular solution of the differential equation $$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 50 \sin t$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).
CAIE FP1 2019 June Q9
11 marks Challenging +1.2
9 A cubic equation \(x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) has real roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - \frac { 5 } { 12 } \\ \alpha \beta \gamma & = - 12 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 90 \end{aligned}$$
  1. Find the values of \(c\) and \(d\).
  2. Express \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) in terms of \(b\).
  3. Show that \(b ^ { 3 } - 15 b + 126 = 0\).
  4. Given that \(3 + \mathrm { i } \sqrt { } ( 12 )\) is a root of \(y ^ { 3 } - 15 y + 126 = 0\), deduce the value of \(b\).
CAIE FP1 2019 June Q10
11 marks Challenging +1.8
10 Let \(I _ { n } = \int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { n } x \mathrm {~d} x\), where \(n \geqslant 0\).
  1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cot ^ { n + 1 } x \right)\), or otherwise, show that $$I _ { n + 2 } = \frac { 1 } { n + 1 } - I _ { n }$$ The curve \(C\) has equation \(y = \cot x\), for \(\frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  2. Find, in an exact form, the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the line \(x = \frac { 1 } { 4 } \pi\) and the \(x\)-axis.
CAIE FP1 2019 June Q11 EITHER
Challenging +1.3
A \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) , \quad \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ respectively, where \(b\) is a positive constant.
  1. Find \(\mathbf { A }\) in terms of \(b\).
  2. Find \(\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0 \\ 2 \\ - 2 \end{array} \right)\).
  3. It is given that $$\mathbf { A } ^ { n } \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) = \left( \begin{array} { l } 4 \\ 4 \\ 0 \end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) = \left( \begin{array} { c } - 1 \\ 0 \\ b ^ { - 1 } \end{array} \right) .$$ Find the values of \(n\) and \(b\).
CAIE FP1 2019 June Q11 OR
Challenging +1.2
The positive variables \(y\) and \(t\) are related by $$y = a ^ { t }$$ where \(a\) is a positive constant.
  1. (a) By differentiating \(\ln y\) with respect to \(t\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a\).
    (b) Write down \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
  2. Determine the set of values of \(a\) for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
    A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(a\) and \(t\), and show that, when \(t = 2\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP1 2002 November Q1
5 marks Standard +0.3
1 Given that $$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$ find \(\sum _ { n = 1 } ^ { N } \| _ { n }\) in terms of \(N\) and \(x\). Hence determine the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity for cases where this exists.
CAIE FP1 2002 November Q2
5 marks Standard +0.8
2 The equation $$x ^ { 4 } + x ^ { 3 } + A x ^ { 2 } + 4 x - 2 = 0$$ where \(A\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\). Find a polynomial equation whose roots are $$\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$$ Given that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$$ find the value of \(A\).
CAIE FP1 2002 November Q3
6 marks Standard +0.8
3 It is given that, for \(n = 0,1,2,3 , \ldots\), $$a _ { n } = 17 ^ { 2 n } + 3 ( 9 ) ^ { n } + 20$$ Simplify \(a _ { n + 1 } - a _ { n }\), and hence prove by induction that \(a _ { n }\) is divisible by 24 for all \(n \geqslant 0\).
CAIE FP1 2002 November Q4
7 marks Challenging +1.2
4 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } e ^ { - x ^ { 2 } } d x$$
  1. Find \(I _ { 1 }\) in terms of c .
  2. Show that $$I _ { n + 2 } = \frac { n + 1 } { 2 } I _ { n } - \frac { 1 } { 2 \mathrm { e } }$$
  3. Find \(I _ { 5 }\) in terms of \(e\).
CAIE FP1 2002 November Q5
8 marks Challenging +1.2
5 The curve \(C\) has polar equation \(r \theta = 1\), for \(0 < \theta \leqslant 2 \pi\).
  1. Use the fact that \(\frac { \sin \theta } { \theta }\) tends to 1 as \(\theta\) tends to 0 to show that the line with carresian equation \(y = 1\) is an asymptote to \(C\).
  2. Sketch \(C\). The points \(P\) and \(Q\) on \(C\) correspond to \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\) respectively.
  3. Find the area of the sector \(O P Q\), where \(O\) is the origin.
  4. Show that the length of the are \(P Q\) is $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \sqrt { } \left( 1 + \theta ^ { 2 } \right) } { \theta ^ { 2 } } \mathrm {~d} \theta$$
CAIE FP1 2002 November Q6
9 marks Standard +0.8
6 A curve has equation \(x ^ { 3 } + x y ^ { 2 } - y ^ { 3 } = 3\).
  1. Show that there is no point of the curve at which \(\frac { d y } { d x } = 0\).
  2. Find the values of \(\frac { d y } { d x }\) and \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 1 , - 1 )\).
CAIE FP1 2002 November Q7
9 marks Challenging +1.2
7 Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), show that
  1. \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\).
  2. \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\). Hence show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta )$$ Find a similar expression for \(\cos ^ { 6 } \theta\), and hence express \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in the fom \(a \cos 2 \theta + b \cos 6 \theta\).
CAIE FP1 2002 November Q8
12 marks Challenging +1.2
8 The value of the assets of a large commercial organisation at time \(t\), measured in years, is \(\\) \left( 10 ^ { 8 } y + 10 ^ { 9 } \right)\(. The variables \)y\( and \)t$ are related by the differential equation $$\frac { d ^ { 2 } y } { d t ^ { 2 } } + 5 \frac { d y } { d t } + 6 y = 15 \cos 3 t - 3 \sin 3 t$$ Find \(y\) in terms of \(t\), given that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 2\) when \(t = 0\). Show that, for large values of \(t\), the value of the assets is less than \(\\) 9.5 \times 10 ^ { 8 }$ for about a third of the time.
CAIE FP1 2002 November Q9
12 marks Challenging +1.2
9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), which meet in the line \(/\), have vector equations $$\begin{aligned} & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) , \\ & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) , \end{aligned}$$ respectively. Find a vector equation of the line \(l\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\). Find a vector equation of the plane \(\Pi _ { 3 }\) which contains \(l\) and which passes through the point with position vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find also the equation of \(\Pi _ { 3 }\) in the form \(a x + b y + c z = d\). Deduce, or prove otherwise, that the system of equations $$\begin{aligned} & 6 x - 5 y - 4 z = - 32 \\ & 5 x - y + 3 z = 24 \\ & 9 x - 2 y + 5 z = 40 \end{aligned}$$ has an infinite number of solutions.
CAIE FP1 2002 November Q10
13 marks Challenging +1.8
10 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { H }\), where $$\mathbf { H } = \left( \begin{array} { r r r r } 1 & 2 & - 3 & - 5 \\ - 1 & 4 & 5 & 1 \\ 2 & 3 & 0 & - 3 \\ - 3 & 5 & 7 & 2 \end{array} \right)$$
  1. Find the dimension of the range space of T .
  2. Find a basis for the null space of \(T\).
  3. It is given that \(\mathbf { x }\) satisfies the equation $$\mathbf { H } \mathbf { x } = \left( \begin{array} { r } 2 \\ - 10 \\ - 1 \\ - 15 \end{array} \right)$$ Using the fact that $$\mathbf { H } \left( \begin{array} { r } 1 \\ - 3 \\ 1 \\ - 2 \end{array} \right) = \left( \begin{array} { r } 2 \\ - 10 \\ - 1 \\ - 15 \end{array} \right) ,$$ find the least possible value of \(| \mathbf { x } |\).
    [0pt] [For the vector \(\mathbf { x } = \left( \begin{array} { c } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \\ x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)\).]
CAIE FP1 2002 November Q11 EITHER
Standard +0.8
The vector \(\mathbf { e }\) is an eigenvector of the square matrix \(\mathbf { G }\). Show that
  1. \(\mathbf { e }\) is an eigenvector of \(\mathbf { G } + k \mathbf { I }\), where \(k\) is a scalar and \(\mathbf { I }\) is an identity matrix,
  2. \(\mathbf { e }\) is an cigenvector of \(\mathbf { G } ^ { 2 }\). Find the eigenvalues, and corresponding eigenvectors, of the matrices \(\mathbf { A }\) and \(\mathbf { B } ^ { 2 }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 3 & 0 \\ 1 & 0 & 1 \\ - 1 & 3 & 2 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r } - 5 & - 3 & 0 \\ 1 & - 8 & 1 \\ - 1 & 3 & - 6 \end{array} \right)$$
CAIE FP1 2002 November Q11 OR
Standard +0.8
The curve \(C\) has equation $$y = \frac { ( x - a ) ( x - b ) } { x - c }$$ where \(a , b , c\) are constants, and it is given that \(0 < a < b < c\).
  1. Express \(y\) in the form $$x + P + \frac { Q } { x - c }$$ giving the constants \(P\) and \(Q\) in terms of \(a , b\) and \(c\).
  2. Find the equations of the asymptotes of \(C\).
  3. Show that \(C\) has two stationary points.
  4. Given also that \(a + b > c\), sketch \(C\), showing the asymptotes and the coordinates of the points of intersection of \(C\) with the axes.
CAIE FP1 2004 November Q1
4 marks Challenging +1.2
1 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 5 & 2 & 6 \\ 2 & 0 & - 1 & 7 \\ 3 & - 1 & - 2 & 10 \\ 4 & 10 & 13 & 29 \end{array} \right)$$ Find the dimension of the null space of T .
CAIE FP1 2004 November Q2
5 marks Challenging +1.8
2 The curve \(C\) is defined parametrically by $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ where \(a\) is a positive constant. Find the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis.
CAIE FP1 2004 November Q3
6 marks Standard +0.8
3 Given that $$\alpha + \beta + \gamma = 0 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 14 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 18$$ find a cubic equation whose roots are \(\alpha , \beta , \gamma\). Hence find possible values for \(\alpha , \beta , \gamma\).
CAIE FP1 2004 November Q4
6 marks Standard +0.8
4 The curve \(C\) has polar equation $$r = \mathrm { e } ^ { \frac { 1 } { 5 } \theta } , \quad 0 \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$$
  1. Draw a sketch of \(C\).
  2. Find the length of \(C\), correct to 3 significant figures.
CAIE FP1 2004 November Q5
7 marks Challenging +1.2
5 Let $$S _ { N } = \sum _ { n = 1 } ^ { N } ( - 1 ) ^ { n - 1 } n ^ { 3 }$$ Find \(S _ { 2 N }\) in terms of \(N\), simplifying your answer as far as possible. Hence write down an expression for \(S _ { 2 N + 1 }\) and find the limit, as \(N \rightarrow \infty\), of \(\frac { S _ { 2 N + 1 } } { N ^ { 3 } }\).
CAIE FP1 2004 November Q6
8 marks Standard +0.3
6 Write down all the 8th roots of unity. Verify that $$\left( z - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 } - ( 2 \cos \theta ) z + 1$$ Hence express \(z ^ { 8 } - 1\) as the product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form.
CAIE FP1 2004 November Q7
8 marks Standard +0.8
7 The curve \(C\) has equation $$x y + ( x + y ) ^ { 5 } = 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 5 } { 6 }\) at the point \(A ( 1,0 )\) on \(C\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
CAIE FP1 2004 November Q8
9 marks Challenging +1.8
8 The sequence of real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 1\) and $$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { \lambda }$$ where \(\lambda\) is a constant greater than 1 . Prove by mathematical induction that, for \(n \geqslant 2\), $$a _ { n } \geqslant 2 ^ { \mathrm { g } ( n ) }$$ where \(g ( n ) = \lambda ^ { n - 1 }\). Prove also that, for \(n \geqslant 2 , \frac { a _ { n + 1 } } { a _ { n } } > 2 ^ { ( \lambda - 1 ) \mathrm { g } ( n ) }\).