Questions — CAIE (7646 questions)

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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.
CAIE FP1 2007 November Q5
7 marks Standard +0.8
5 The positive variables \(x\) and \(y\) are related by $$y = x ^ { 2 } + 2 \ln ( x y )$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when both \(x\) and \(y\) are equal to 1 .
CAIE FP1 2007 November Q6
8 marks Standard +0.3
6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } , 3 \mathbf { j }\) and \(4 \mathbf { k }\) respectively. Find a vector which is perpendicular to the plane \(\Pi _ { 1 }\) containing \(A , B\) and \(C\). The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } ) + \mu ( \mathbf { j } - \mathbf { k } ) .$$ Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE FP1 2007 November Q7
9 marks Standard +0.8
7 The curve \(C\) has polar equation $$r = \theta \sin \theta ,$$ where \(0 \leqslant \theta \leqslant \pi\). Draw a sketch of \(C\). Find the area of the region enclosed by \(C\), leaving your answer in terms of \(\pi\).
CAIE FP1 2007 November Q8
10 marks Challenging +1.8
8 Let \(I _ { n } = \int _ { 0 } ^ { \ln 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } \mathrm {~d} x\).
  1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 1 } \right] = n \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } - 4 ( n - 1 ) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 2 } .$$
  2. Hence show that $$n I _ { n } = 4 ( n - 1 ) I _ { n - 2 } + \frac { 3 } { 2 } \left( \frac { 5 } { 2 } \right) ^ { n - 1 } .$$
  3. Use the result in part (ii) to find the \(y\)-coordinate of the centroid of the region bounded by the axes, the line \(x = \ln 2\) and the curve $$y = \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 } .$$ Give your answer correct to 3 decimal places.
CAIE FP1 2007 November Q9
10 marks Challenging +1.2
9 Write down, in any form, all the roots of the equation \(z ^ { 5 } - 1 = 0\). Hence find all the roots of the equation $$( w - 1 ) ^ { 4 } + ( w - 1 ) ^ { 3 } + ( w - 1 ) ^ { 2 } + w = 0$$ and deduce that none of them is real. Find the arguments of the two roots which have the smaller modulus.
CAIE FP1 2007 November Q10
10 marks Challenging +1.2
10 The vectors \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 } , \mathbf { b } _ { 4 }\) are defined as follows: $$\mathbf { b } _ { 1 } = \left( \begin{array} { c } 1 \\ 0 \\ 0 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 2 } = \left( \begin{array} { c } 1 \\ 1 \\ 0 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 3 } = \left( \begin{array} { c } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 4 } = \left( \begin{array} { c } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) .$$ The linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 }\) is denoted by \(V _ { 1 }\) and the linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 4 }\) is denoted by \(V _ { 2 }\).
  1. Give a reason why \(V _ { 1 } \cup V _ { 2 }\) is not a linear space.
  2. State the dimension of the linear space \(V _ { 1 } \cap V _ { 2 }\) and write down a basis. Consider now the set \(V _ { 3 }\) of all vectors of the form \(q \mathbf { b } _ { 2 } + r \mathbf { b } _ { 3 } + s \mathbf { b } _ { 4 }\), where \(q , r , s\) are real numbers. Show that \(V _ { 3 }\) is a linear space, and show also that it has dimension 3 . Determine whether each of the vectors $$\left( \begin{array} { l } 4 \\ 4 \\ 2 \\ 5 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { l } 5 \\ 4 \\ 2 \\ 5 \end{array} \right)$$ belongs to \(V _ { 3 }\) and justify your conclusions.
CAIE FP1 2007 November Q11
11 marks Challenging +1.2
11 Find the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 1 & 4 \\ 1 & 1 & - 1 \\ 2 & 1 & 1 \end{array} \right)$$ and corresponding eigenvectors. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \mathbf { A } - k \mathbf { I } ,$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(k\) is a real number. Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { B } ^ { 3 } = \mathbf { P D } \mathbf { P } ^ { - 1 } .$$
CAIE FP1 2007 November Q12 EITHER
Challenging +1.2
The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + 4 }$$ where \(a\), \(b\) and \(c\) are constants. It is given that \(y = 2 x - 5\) is an asymptote of \(C\).
  1. Find the values of \(a\) and \(b\).
  2. Given also that \(C\) has a turning point at \(x = - 1\), find the value of \(c\).
  3. Find the set of values of \(y\) for which there are no points on \(C\).
  4. Draw a sketch of the curve with equation $$y = \frac { 2 ( x - 7 ) ^ { 2 } + 3 ( x - 7 ) - 2 } { x - 3 }$$ [You should state the equations of the asymptotes and the coordinates of the turning points.]
CAIE FP1 2007 November Q12 OR
Challenging +1.8
Show that the substitution \(y = \frac { 1 } { w }\) reduces the differential equation $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 5 y ^ { 2 } = \left( 5 x ^ { 2 } + 4 x + 2 \right) y ^ { 3 }$$ to $$\frac { \mathrm { d } ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} w } { \mathrm {~d} x } + 5 w = - 5 x ^ { 2 } - 4 x - 2$$ Find the general solution for \(w\) in terms of \(x\). Find a function f such that \(\lim _ { x \rightarrow \infty } \left( \frac { y } { \mathrm { f } ( x ) } \right) = 1\).
CAIE FP1 2011 November Q1
5 marks Challenging +1.2
1 The equation \(x ^ { 3 } + p x + q = 0\) has a repeated root. Prove that \(4 p ^ { 3 } + 27 q ^ { 2 } = 0\).
CAIE FP1 2011 November Q2
5 marks Standard +0.3
2 The position vectors of points \(A , B , C\), relative to the origin \(O\), are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\), where $$\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { c } = 3 \mathbf { i } - \mathbf { j } - \mathbf { k }$$ Find \(\mathbf { a } \times \mathbf { b }\) and deduce the area of the triangle \(O A B\). Hence find the volume of the tetrahedron \(O A B C\), given that the volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.
CAIE FP1 2011 November Q3
7 marks Challenging +1.2
3 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = 2 ^ { \frac { 1 } { 2 } n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$
CAIE FP1 2011 November Q4
7 marks Challenging +1.2
4 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 3 & 4 & 2 & 5 \\ 6 & 7 & 5 & 8 \\ 9 & 9 & 9 & 9 \\ 15 & 16 & 14 & 17 \end{array} \right)$$ Find
  1. the rank of \(\mathbf { M }\) and a basis for the range space of T ,
  2. a basis for the null space of T .
CAIE FP1 2011 November Q5
7 marks Standard +0.8
5 The point \(P ( 2,1 )\) lies on the curve with equation $$x ^ { 3 } - 2 y ^ { 3 } = 3 x y$$ Find
  1. the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\),
  2. the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(P\).
CAIE FP1 2011 November Q6
8 marks Challenging +1.2
6 Let \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } ( 1 - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 1\), $$( 3 + 2 n ) I _ { n } = 2 n I _ { n - 1 }$$ Hence find the exact value of \(I _ { 3 }\).
CAIE FP1 2011 November Q7
11 marks Standard +0.8
7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + p x + 1 } { x - 2 }\), where \(p\) is a constant. Given that \(C\) has two asymptotes, find the equation of each asymptote. Find the set of values of \(p\) for which \(C\) has two distinct turning points. Sketch \(C\) in the case \(p = - 1\). Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.
CAIE FP1 2011 November Q8
11 marks Standard +0.3
8 The vector \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf { B }\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\). State the eigenvalues of the matrix \(\mathbf { C }\), where $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & - 1 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 2 \end{array} \right) ,$$ and find corresponding eigenvectors. Show that \(\left( \begin{array} { l } 1 \\ 6 \\ 3 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { D }\), where $$\mathbf { D } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ - 6 & - 3 & 4 \\ - 9 & - 3 & 7 \end{array} \right) ,$$ and state the corresponding eigenvalue. Hence state an eigenvector of the matrix CD and give the corresponding eigenvalue.