STEP 2 2019

Let $f(x)=(x-p)g(x)$, where $g$ is a polynomial. Show that the tangent to the curve $y=f(x)$ at the point with $x=a$, where $a\ne p$, passes through the point $(p,0)$ if and only if $g^{\prime }(a)=0$.

The curve $C$ has equation$$y=A(x-p)(x-q)(x-r),$$where $p$, $q$ and $r$ are constants with $p<q<r$, and $A$ is a non-zero constant.

(i) The tangent to $C$ at the point with $x=a$, where $a\ne p$, passes through the point $(p,0)$. Show that $2a=q+r$ and find an expression for the gradient of this tangent in terms of $A$, $q$ and $r$.

(ii) The tangent to $C$ at the point with $x=c$, where $c\ne r$, passes through the point $(r,0)$. Show that this tangent is parallel to the tangent in part (i) if and only if the tangent to $C$ at the point with $x=q$ does not meet the curve again.

The function f satisfies $f(0)=0$ and $f^{\prime }(t)>0$ for $t>0$. Show by means of a sketch that, for $x>0$,$${\int }_0^{x}f(t)dt+{\int }_0^{f(x)}{f}^{-1}(y)dy=xf(x).$$(i) The (real) function g is defined, for all $t$, by$$g(t{)}^3+g(t)=t.$$Prove that $g(0)=0$, and that $g^{\prime }(t)>0$ for all $t$. Evaluate ${\int }_0^{2}g(t)dt$.

(ii) The (real) function h is defined, for all $t$, by$$h(t{)}^3+h(t)=t+2.$$Evaluate ${\int }_0^{8}h(t)dt$.

For any two real numbers $x_1$ and $x_2$, show that$$|x_1+x_2|\le |x_1|+|x_2|.$$Show further that, for any real numbers $x_1,x_2,\dots ,x_n$,$$|x_1+x_2+\cdots +x_n|\le |x_1|+|x_2|+\cdots +|x_n|.$$(i) The polynomial $f$ is defined by$$f(x)=1+a_{1}x+a_2{x}^2+\cdots +a_{n-1}{x}^{n-1}+x^n$$where the coefficients are real and satisfy $|a_i|\le A$ for $i=1,2,\dots ,n-1$, where $A\ge 1$.

(a) If $|x|<1$, show that$$|f(x)-1|\le \frac{A|x|}{1-|x|}.$$(b) Let $\omega$ be a real root of $f$, so that $f(\omega )=0$. In the case $|\omega |<1$, show that$$\frac{1}{1+A}\le |\omega |\le 1+A.\text{\hspace{1em}(*)}$$(c) Show further that the inequalities $(\ast )$ also hold if $|\omega |\ge 1$.

(ii) Find the integer root or roots of the quintic equation$$135x^5-135x^4-100x^3-91x^2-126x+135=0.$$

You are not required to consider issues of convergence in this question.

For any sequence of numbers $a_1,a_2,\dots ,a_m,\dots ,a_n$, the notation $\prod _{i=m}^n{a}_i$ denotes the product $a_m{a}_{m+1}\cdots a_n$.

(i) Use the identity $2\cos x\sin x=\sin (2x)$ to evaluate the product$$\cos \left(\frac{\pi }{9}\right)\cos \left(\frac{2\pi }{9}\right)\cos \left(\frac{4\pi }{9}\right).$$(ii) Simplify the expression$$\prod _{k=0}^n\cos \left(\frac{x}{2^k}\right)\left(0<x<\frac{1}{2}\pi \right).$$Using differentiation, or otherwise, show that, for $0<x<\frac{1}{2}\pi$,$$\sum _{k=0}^n\frac{1}{2^k}\tan \left(\frac{x}{2^k}\right)=\frac{1}{2^n}\cot \left(\frac{x}{2^n}\right)-2\cot (2x).$$(iii) Using the results $\underset{\theta \to 0}{\lim }\frac{\sin \theta }{\theta }=1$ and $\underset{\theta \to 0}{\lim }\frac{\tan \theta }{\theta }=1$, show that$$\prod _{k=1}^{\infty }\cos \left(\frac{x}{2^k}\right)=\frac{\sin x}{x}$$and evaluate$$\sum _{j=2}^{\infty }\frac{1}{2^{j-2}}\tan \left(\frac{\pi }{2^j}\right).$$

The sequence $u_0,u_1,\dots$ is said to be a constant sequence if $u_n=u_{n+1}$ for $n=0,1,2,\dots$. The sequence is said to be a sequence of period 2 if $u_n=u_{n+2}$ for $n=0,1,2,\dots$ and the sequence is not constant.

(i) A sequence of real numbers is defined by $u_0=a$ and $u_{n+1}=f(u_n)$ for $n=0,1,2,\dots$, where$$f(x)=p+(x-p)x,$$and $p$ is a given real number.

Find the values of $a$ for which the sequence is constant.

Show that the sequence has period 2 for some value of $a$ if and only if $p>3$ or $p<-1$.

(ii) A sequence of real numbers is defined by $u_0=a$ and $u_{n+1}=f(u_n)$ for $n=0,1,2,\dots$, where$$f(x)=q+(x-p)x,$$and $p$ and $q$ are given real numbers.

Show that there is no value of $a$ for which the sequence is constant if and only if $f(x)>x$ for all $x$.

Deduce that, if there is no value of $a$ for which the sequence is constant, then there is no value of $a$ for which the sequence has period 2.

Is it true that, if there is no value of $a$ for which the sequence has period 2, then there is no value of $a$ for which the sequence is constant?

Note: You may assume that if the functions $y_1(x)$ and $y_2(x)$ both satisfy one of the differential equations in this question, then the curves $y=y_1(x)$ and $y=y_2(x)$ do not intersect.

(i) Find the solution of the differential equation$$\frac{dy}{dx}=y+x+1$$that has the form $y=mx+c$, where $m$ and $c$ are constants.

Let $y_3(x)$ be the solution of this differential equation with $y_3(0)=k$. Show that any stationary point on the curve $y=y_3(x)$ lies on the line $y=-x-1$. Deduce that solution curves with $k<-2$ cannot have any stationary points.

Show further that any stationary point on the solution curve is a local minimum.

Use the substitution $Y=y+x$ to solve the differential equation, and sketch, on the same axes, the solutions with $k=0$, $k=-2$ and $k=-3$.

(ii) Find the two solutions of the differential equation$$\frac{dy}{dx}=x^2+y^2-2xy-4x+4y+3$$that have the form $y=mx+c$.

Let $y_4(x)$ be the solution of this differential equation with $y_4(0)=-2$. (Do not attempt to find this solution.)

Show that any stationary point on the curve $y=y_4(x)$ lies on one of two lines that you should identify. What can be said about the gradient of the curve at points between these lines?

Sketch the curve $y=y_4(x)$. You should include on your sketch the two straight line solutions and the two lines of stationary points.

(i) The points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$, respectively. Each of these vectors is a unit vector (so $\mathbf{a}\cdot \mathbf{a}=1$, for example) and$$\mathbf{a}+\mathbf{b}+\mathbf{c}=0.$$Show that $\mathbf{a}\cdot \mathbf{b}=-\frac{1}{2}$. What can be said about the triangle $ABC$? You should justify your answer.

(ii) The four distinct points $A_i$ ($i=1,2,3,4$) have unit position vectors ${\mathbf{a}}_i$ and$$\sum _{i=1}^4{\mathbf{a}}_i=0.$$Show that ${\mathbf{a}}_1\cdot {\mathbf{a}}_2={\mathbf{a}}_3\cdot {\mathbf{a}}_4$.

(a) Given that the four points lie in a plane, determine the shape of the quadrilateral with vertices $A_1$, $A_2$, $A_3$ and $A_4$.

(b) Given instead that the four points are the vertices of a regular tetrahedron, find the length of the sides of this tetrahedron.

The domain of the function $f$ is the set of all $2\times 2$ matrices and its range is the set of real numbers. Thus, if $M$ is a $2\times 2$ matrix, then $f(M)\in \mathbb{R}$. The function $f$ has the property that $f(MN)=f(M)f(N)$ for any $2\times 2$ matrices $M$ and $N$.

(i) You are given that there is a matrix $M$ such that $f(M)\ne 0$. Let $I$ be the $2\times 2$ identity matrix. By considering $f(MI)$, show that $f(I)=1$.

(ii) Let $J=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. You are given that $f(J)\ne 1$. By considering $J^2$, evaluate $f(J)$.

Using $J$, show that, for any real numbers $a$, $b$, $c$ and $d$,$$f\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=-f\left(\begin{array}{cc}c& d\\ a& b\end{array}\right)=f\left(\begin{array}{cc}d& c\\ b& a\end{array}\right).$$(iii) Let $K=\left(\begin{array}{cc}1& 0\\ 0& k\end{array}\right)$ where $k\in \mathbb{R}$. Use $K$ to show that, if the second row of the matrix $A$ is a multiple of the first row, then $f(A)=0$.

(iv) Let $P=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$. By considering the matrices $P^2$, $P^{-1}$, and $K^{-1}PK$ for suitable values of $k$, evaluate $f(P)$.

A particle $P$ is projected from a point $O$ on horizontal ground with speed $u$ and angle of projection $\alpha$, where $0<\alpha <\frac{1}{2}\pi$.

(i) Show that if $\sin \alpha <\frac{2\sqrt{2}}{3}$, then the distance $OP$ is increasing throughout the flight.

Show also that if $\sin \alpha >\frac{2\sqrt{2}}{3}$, then $OP$ will be decreasing at some time before the particle lands.

(ii) At the same time as $P$ is projected, a particle $Q$ is projected horizontally from $O$ with speed $v$ along the ground in the opposite direction from the trajectory of $P$. The ground is smooth. Show that if$$2\sqrt{2}v>(\sin \alpha -2\sqrt{2}\cos \alpha )u,$$then $QP$ is increasing throughout the flight of $P$.

A small light ring is attached to the end $A$ of a uniform rod $AB$ of weight $W$ and length $2a$. The ring can slide on a rough horizontal rail. One end of a light inextensible string of length $2a$ is attached to the rod at $B$ and the other end is attached to a point $C$ on the rail so that the rod makes an angle of $\theta$ with the rail, where $0<\theta <90^{\circ }$. The rod hangs in the same vertical plane as the rail. A force of $kW$ acts vertically downwards on the rod at $B$ and the rod is in equilibrium.

(i) You are given that the string will break if the tension $T$ is greater than $W$. Show that (assuming that the ring does not slip) the string will break if$$2k+1>4\sin \theta .$$(ii) Show that (assuming that the string does not break) the ring will slip if$$2k+1>(2k+3)\mu \tan \theta ,$$where $\mu$ is the coefficient of friction between the rail and the ring.

(iii) You are now given that $\mu \tan \theta <1$. Show that, when $k$ is increased gradually from zero, the ring will slip before the string breaks if$$\mu <\frac{2\cos \theta }{1+2\sin \theta }.$$

(i) The three integers $n_1$, $n_2$ and $n_3$ satisfy $0<n_1<n_2<n_3$ and $n_1+n_2>n_3$.

Find the number of ways of choosing the pair of numbers $n_1$ and $n_2$ in the cases $n_3=9$ and $n_3=10$.

Given that $n_3=2n+1$, where $n$ is a positive integer, write down an expression (which you need not prove is correct) for the number of ways of choosing the pair of numbers $n_1$ and $n_2$. Simplify your expression.

Write down and simplify the corresponding expression when $n_3=2n$, where $n$ is a positive integer.

(ii) You have $N$ rods, of lengths $1,2,3,\dots ,N$ (one rod of each length). You take the rod of length $N$, and choose two more rods at random from the remainder, each choice of two being equally likely. Show that, in the case $N=2n+1$ where $n$ is a positive integer, the probability that these three rods can form a triangle (of non-zero area) is$$\frac{n-1}{2n-1}.$$Find the corresponding probability in the case $N=2n$, where $n$ is a positive integer.

(iii) You have $2M+1$ rods, of lengths $1,2,3,\dots ,2M+1$ (one rod of each length), where $M$ is a positive integer. You choose three at random, each choice of three being equally likely. Show that the probability that the rods can form a triangle (of non-zero area) is$$\frac{(4M+1)(M-1)}{2(2M+1)(2M-1)}.$$Note: $\sum _{k=1}^K{k}^2=\frac{1}{6}K(K+1)(2K+1)$.

The random variable $X$ has the probability density function on the interval $[0,1]$:$$f(x)=\{\begin{array}{ll}n{x}^{n-1}& 0\le x\le 1,\\ 0& \text{elsewhere,}\end{array}$$where $n$ is an integer greater than 1.

(i) Let $\mu =E(X)$. Find an expression for $\mu$ in terms of $n$, and show that the variance, ${\sigma }^2$, of $X$ is given by$${\sigma }^2=\frac{n}{(n+1{)}^2(n+2)}.$$(ii) In the case $n=2$, show without using decimal approximations that the interquartile range is less than $2\sigma$.

(iii) Write down the first three terms and the $(k+1)$th term (where $0\le k\le n$) of the binomial expansion of $(1+x{)}^n$ in ascending powers of $x$.

By setting $x=\frac{1}{n}$, show that $\mu$ is less than the median and greater than the lower quartile.

Note: You may assume that$$1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots <4.$$