Questions FP1 - A-Level Maths

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CAIE FP1 2003 November Q3

6 marks Challenging +1.2

Three $n \times 1$ column vectors are denoted by $\mathbf{x}_1$, $\mathbf{x}_2$, $\mathbf{x}_3$, and $\mathbf{M}$ is an $n \times n$ matrix. Show that if $\mathbf{x}_1$, $\mathbf{x}_2$, $\mathbf{x}_3$ are linearly dependent then the vectors $\mathbf{Mx}_1$, $\mathbf{Mx}_2$, $\mathbf{Mx}_3$ are also linearly dependent. [2] The vectors $\mathbf{y}_1$, $\mathbf{y}_2$, $\mathbf{y}_3$ and the matrix $\mathbf{P}$ are defined as follows: $$\mathbf{y}_1 = \begin{pmatrix} 1 \\ 5 \\ 7 \end{pmatrix}, \quad \mathbf{y}_2 = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}, \quad \mathbf{y}_3 = \begin{pmatrix} 5 \\ 51 \\ 55 \end{pmatrix},$$ $$\mathbf{P} = \begin{pmatrix} 1 & -4 & 3 \\ 0 & 2 & 5 \\ 0 & 0 & -7 \end{pmatrix}$$

Show that $\mathbf{y}_1$, $\mathbf{y}_2$, $\mathbf{y}_3$ are linearly dependent. [2]
Find a basis for the linear space spanned by the vectors $\mathbf{Py}_1$, $\mathbf{Py}_2$, $\mathbf{Py}_3$. [2]

Find the length of $PQ$ in terms of $t$. [4]
Hence show that the lines $l_1$ and $l_2$ do not intersect, and find the maximum length of $PQ$ as $t$ varies. [3]
The plane $\Pi_1$ contains $l_1$ and $PQ$; the plane $\Pi_2$ contains $l_2$ and $PQ$. Find the angle between the planes $\Pi_1$ and $\Pi_2$, correct to the nearest tenth of a degree. [4]

CAIE FP1 2003 November Q10

12 marks Standard +0.8

Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf{A}$, where $$\mathbf{A} = \begin{pmatrix} 6 & 4 & 1 \\ -6 & -1 & 3 \\ 8 & 8 & 4 \end{pmatrix}.$$ [8] Hence find a non-singular matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{A} + \mathbf{A}^2 + \mathbf{A}^3 = \mathbf{PDP}^{-1}$. [4]

CAIE FP1 2003 November Q11

28 marks Challenging +1.2

Answer only one of the following two alternatives. EITHER The curve $C$ has equation $y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}$.

Express $y$ in the form $P + \frac{Q}{x-2} + \frac{R}{x+3}$. [3]
Show that $\frac{dy}{dx} = 0$ for exactly one value of $x$ and find the corresponding value of $y$. [4]
Write down the equations of all the asymptotes of $C$. [3]
Find the set of values of $k$ for which the line $y = k$ does not intersect $C$. [4]

OR A curve has equation $y = \frac{5}{3}x^{\frac{3}{2}}$, for $x \geq 0$. The arc of the curve joining the origin to the point where $x = 3$ is denoted by $R$.

Find the length of $R$. [4]
Find the $y$-coordinate of the centroid of the region bounded by the $x$-axis, the line $x = 3$ and $R$. [5]
Show that the area of the surface generated when $R$ is rotated through one revolution about the $y$-axis is $\frac{232\pi}{15}$. [5]

CAIE FP1 2005 November Q1

4 marks Standard +0.8

Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z^5 = -16 + (16\sqrt{3})i,$$ giving each root in the form $re^{i\theta}$. [4]

CAIE FP1 2005 November Q2

6 marks Challenging +1.2

The sequence $u_1, u_2, u_3, \ldots$ is such that $u_1 = 1$ and $$u_{n+1} = -1 + \sqrt{(u_n + 7)}.$$

Prove by induction that $u_n < 2$ for all $n \geqslant 1$. [4]
Show that if $u_n = 2 - \varepsilon$, where $\varepsilon$ is small, then $$u_{n+1} \approx 2 - \frac{1}{6}\varepsilon.$$ [2]

CAIE FP1 2005 November Q3

7 marks Standard +0.8

The curve $C$ has equation $$y = \frac{x^2}{x + \lambda},$$ where $\lambda$ is a non-zero constant. Obtain the equations of the asymptotes of $C$. [3] In separate diagrams, sketch $C$ for the cases where

$\lambda > 0$,
$\lambda < 0$.

[4]

CAIE FP1 2005 November Q4

7 marks Standard +0.3

Solve the differential equation $$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 24e^{2x},$$ given that $y = 1$ and $\frac{dy}{dx} = 9$ when $x = 0$. [7]

CAIE FP1 2005 November Q5

7 marks Challenging +1.8

In the equation $$x^3 + ax^2 + bx + c = 0,$$ the coefficients $a$, $b$ and $c$ are real. It is given that all the roots are real and greater than $1$.

Prove that $a < -3$. [1]
By considering the sum of the squares of the roots, prove that $a^2 > 2b + 3$. [2]
By considering the sum of the cubes of the roots, prove that $a^3 < -9b - 3c - 3$. [4]

CAIE FP1 2005 November Q6

8 marks Challenging +1.2

Let $$I_n = \int_0^1 (1 + x^2)^{-n} dx,$$ where $n \geqslant 1$. By considering $\frac{d}{dx}(x(1 + x^2)^{-n})$, or otherwise, prove that $$2nI_{n+1} = (2n - 1)I_n + 2^{-n}.$$ [5] Deduce that $I_3 = \frac{3}{32}\pi + \frac{1}{4}$. [3]

CAIE FP1 2005 November Q7

8 marks Challenging +1.2

Write down an expression in terms of $z$ and $N$ for the sum of the series $$\sum_{n=1}^N 2^{-n}z^n.$$ [2] Use de Moivre's theorem to deduce that $$\sum_{n=1}^{10} 2^{-n}\sin\left(\frac{1}{10}n\pi\right) = \frac{1025\sin\left(\frac{1}{10}\pi\right)}{2560 - 2048\cos\left(\frac{1}{10}\pi\right)}.$$ [6]

CAIE FP1 2005 November Q8

9 marks Standard +0.8

Find the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is $$y = x^2(1 - x).$$ [7] Deduce the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is $$y = x(1 - x)^2.$$ [2]

CAIE FP1 2005 November Q9

10 marks Challenging +1.2

The planes $\Pi_1$ and $\Pi_2$ have vector equations $$\mathbf{r} = \lambda_1(\mathbf{i} + \mathbf{j} - \mathbf{k}) + \mu_1(2\mathbf{i} - \mathbf{j} + \mathbf{k}) \quad \text{and} \quad \mathbf{r} = \lambda_2(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + \mu_2(3\mathbf{i} + \mathbf{j} - \mathbf{k})$$ respectively. The line $l$ passes through the point with position vector $4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}$ and is parallel to both $\Pi_1$ and $\Pi_2$. Find a vector equation for $l$. [6] Find also the shortest distance between $l$ and the line of intersection of $\Pi_1$ and $\Pi_2$. [4]

CAIE FP1 2005 November Q10

11 marks Standard +0.8

It is given that the eigenvalues of the matrix $\mathbf{M}$, where $$\mathbf{M} = \begin{pmatrix} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{pmatrix},$$ are $1, 3, 4$. Find a set of corresponding eigenvectors. [4] Write down a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1},$$ where $n$ is a positive integer. [2] Find $\mathbf{P}^{-1}$ and deduce that $$\lim_{n \to \infty} 4^{-n}\mathbf{M}^n = \begin{pmatrix} -\frac{1}{3} & 0 & -\frac{1}{3} \\ \frac{4}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix}.$$ [5]

CAIE FP1 2005 November Q11

11 marks Challenging +1.8

Find the rank of the matrix $\mathbf{A}$, where $$\mathbf{A} = \begin{pmatrix} 1 & 1 & 2 & 3 \\ 4 & 3 & 5 & 10 \\ 6 & 6 & 13 & 13 \\ 14 & 12 & 23 & 45 \end{pmatrix}.$$ [3] Find vectors $\mathbf{x_0}$ and $\mathbf{e}$ such that any solution of the equation $$\mathbf{A}\mathbf{x} = \begin{pmatrix} 0 \\ 2 \\ -1 \\ -3 \end{pmatrix} \quad (*)$$ can be expressed in the form $\mathbf{x_0} + \lambda\mathbf{e}$, where $\lambda \in \mathbb{R}$. [5] Hence show that there is no vector which satisfies $(*)$ and has all its elements positive. [3]

CAIE FP1 2005 November Q12

24 marks Challenging +1.3

Answer only one of the following two alternatives. **EITHER** Show that $\left(n + \frac{1}{2}\right)^3 - \left(n - \frac{1}{2}\right)^3 \equiv 3n^2 + \frac{1}{4}$. [1] Use this result to prove that $\sum_{n=1}^N n^2 = \frac{1}{6}N(N + 1)(2N + 1)$. [2] The sums $S$, $T$ and $U$ are defined as follows: \begin{align} S &= 1^2 + 2^2 + 3^2 + 4^2 + \ldots + (2N)^2 + (2N + 1)^2,
T &= 1^2 + 3^2 + 5^2 + 7^2 + \ldots + (2N - 1)^2 + (2N + 1)^2,
U &= 1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2N)^2 + (2N + 1)^2. \end{align} Find and simplify expressions in terms of $N$ for each of $S$, $T$ and $U$. [5] Hence

describe the behaviour of $\frac{S}{T}$ as $N \to \infty$, [1]
prove that if $\frac{S}{U}$ is an integer then $\frac{T}{U}$ is an integer. [3]

**OR** The curves $C_1$ and $C_2$ have polar equations $$r = 4\cos\theta \quad \text{and} \quad r = 1 + \cos\theta$$ respectively, where $-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi$.

Show that $C_1$ and $C_2$ meet at the points $A\left(\frac{4}{3}, \alpha\right)$ and $B\left(\frac{4}{3}, -\alpha\right)$, where $\alpha$ is the acute angle such that $\cos\alpha = \frac{1}{3}$. [2]
In a single diagram, draw sketch graphs of $C_1$ and $C_2$. [3]
Show that the area of the region bounded by the arcs $OA$ and $OB$ of $C_1$, and the arc $AB$ of $C_2$, is $$4\pi - \frac{1}{3}\sqrt{2} - \frac{13}{2}\alpha.$$ [7]