Questions FP1 - A-Level Maths

OCR MEI FP1 2006 June Q8

10 marks Moderate -0.3

Verify that $2 + \mathrm{j}$ is a root of the equation $2x^3 - 11x^2 + 22x - 15 = 0$. [5]
Write down the other complex root. [1]
Find the third root of the equation. [4]

OCR MEI FP1 2006 June Q9

13 marks Standard +0.3

Show that $r(r+1)(r+2) - (r-1)r(r+1) \equiv 3r(r+1)$. [2]
Hence use the method of differences to find an expression for $\sum_{r=1}^{n} r(r+1)$. [6]
Show that you can obtain the same expression for $\sum_{r=1}^{n} r(r+1)$ using the standard formulae for $\sum_{r=1}^{n} r$ and $\sum_{r=1}^{n} r^2$. [5]

OCR MEI FP1 2007 June Q1

3 marks Moderate -0.8

You are given the matrix $\mathbf{M} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}$.

Find the inverse of $\mathbf{M}$. [2]
A triangle of area 2 square units undergoes the transformation represented by the matrix $\mathbf{M}$. Find the area of the image of the triangle following this transformation. [1]

OCR MEI FP1 2007 June Q2

3 marks Easy -1.2

Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}

OCR MEI FP1 2007 June Q3

5 marks Easy -1.2

Find the values of the constants $A$, $B$, $C$ and $D$ in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]

OCR MEI FP1 2007 June Q4

7 marks Moderate -0.8

Two complex numbers, $\alpha$ and $\beta$, are given by $\alpha = 1 - 2\mathrm{j}$ and $\beta = -2 - \mathrm{j}$.

Represent $\beta$ and its complex conjugate $\beta^*$ on an Argand diagram. [2]
Express $\alpha\beta$ in the form $a + b\mathrm{j}$. [2]
Express $\frac{\alpha + \beta}{\beta}$ in the form $a + b\mathrm{j}$. [3]

OCR MEI FP1 2007 June Q5

6 marks Standard +0.3

The roots of the cubic equation $x^3 + 3x^2 - 7x + 1 = 0$ are $\alpha$, $\beta$ and $\gamma$. Find the cubic equation whose roots are $3\alpha$, $3\beta$ and $3\gamma$, expressing your answer in a form with integer coefficients. [6]

OCR MEI FP1 2007 June Q6

6 marks Moderate -0.8

Show that $\frac{1}{r+2} - \frac{1}{r+3} = \frac{1}{(r+2)(r+3)}$. [2]
Hence use the method of differences to find $\frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \frac{1}{5 \times 6} + \ldots + \frac{1}{52 \times 53}$. [4]

OCR MEI FP1 2007 June Q7

6 marks Moderate -0.3

Prove by induction that $\sum_{r=1}^{n} 3^{r-1} = \frac{3^n - 1}{2}$. [6]

OCR MEI FP1 2007 June Q8

14 marks Standard +0.3

A curve has equation $y = \frac{x^2 - 4}{(x-3)(x+1)(x-1)}$.

Write down the coordinates of the points where the curve crosses the axes. [3]
Write down the equations of the three vertical asymptotes and the one horizontal asymptote. [4]
Determine whether the curve approaches the horizontal asymptote from above or below for
1. large positive values of $x$,
2. large negative values of $x$. [3]
Sketch the curve. [4]

OCR MEI FP1 2007 June Q9

11 marks Standard +0.8

The cubic equation $x^3 + Ax^2 + Bx + 15 = 0$, where $A$ and $B$ are real numbers, has a root $x = 1 + 2\mathrm{j}$.

Write down the other complex root. [1]
Explain why the equation must have a real root. [1]
Find the value of the real root and the values of $A$ and $B$. [9]

OCR MEI FP1 2007 June Q10

11 marks Standard +0.8

You are given that $\mathbf{A} = \begin{pmatrix} 1 & -2 & k \\ 2 & 1 & 2 \\ 3 & 2 & -1 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} -5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5 \end{pmatrix}$ and that $\mathbf{AB}$ is of the form $\mathbf{AB} = \begin{pmatrix} k-n & 0 & 0 \\ 0 & k-n & 0 \\ 0 & 0 & k-n \end{pmatrix}$.

Find the value of $n$. [2]
Write down the inverse matrix $\mathbf{A}^{-1}$ and state the condition on $k$ for this inverse to exist. [4]
Using the result from part (ii), or otherwise, solve the following simultaneous equations. \begin{align} x - 2y + z &= 1
2x + y + 2z &= 12
3x + 2y - z &= 3 \end{align} [5]

CAIE FP1 2013 November Q22

Moderate -0.5

22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r }

CAIE FP1 2013 November Q7

Standard +0.3

7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l }

CAIE FP1 2013 November Q9

Challenging +1.8

9 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis. Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$.

CAIE FP1 2013 November Q11

Challenging +1.3

11 Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form $\cos \theta + \mathrm { i } \sin \theta$, where $- \pi < \theta \leqslant \pi$. Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right) .$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form $\cos \theta \pm \mathrm { i } \sin \theta$. Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that $v = y ^ { 3 }$, show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for $y$ in terms of $x$, given that when $x = 0 , y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$. \end{document}

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