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Questions

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120

Marks total

Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 2 2014

2014

13 questions worth 120 marks in total. Each part is annotated: check the hint if you are stuck, or reveal the worked solution once you have committed to an approach.

In the triangle $A B C$ , the base $A B$ is of length 1 unit and the angles at $A$ and $B$ are $α$ and $β$ respectively, where $0 < α ⩽ β$ . The points $P$ and $Q$ lie on the sides $A C$ and $B C$ respectively, with $A P = P Q = QB = x$ . The line $P Q$ makes an angle of $θ$ with the line through $P$ parallel to $A B$ .
Show that $x cos θ = 1 - x cos α - x cos β$ , and obtain an expression for $x sin θ$ in terms of $x$ , $α$ and $β$ . Hence show that $(1 + 2 cos (α + β)) x^{2} - 2 (cos α + cos β) x + 1 = 0 .$ Show that $(*)$ is also satisfied if $P$ and $Q$ lie on $A C$ produced and $B C$ produced, respectively. [By definition, $P$ lies on $A C$ produced if $P$ lies on the line through $A$ and $C$ and the points are in the order $A$ , $C$ , $P$ .]
State the condition on $α$ and $β$ for $(*)$ to be linear in $x$ . If this condition does not hold (but the condition $0 < α ⩽ β$ still holds), show that $(*)$ has distinct real roots.
Find the possible values of $x$ in the two cases (a) $α = β = 4 5^{\circ}$ and (b) $α = 3 0^{\circ}$ , $β = 9 0^{\circ}$ , and illustrate each case with a sketch.
Trigonometry
This question concerns the inequality $\int_{0}^{π} (f (x))^{2} d x ⩽ \int_{0}^{π} (f^{'} (x))^{2} d x .$
Show that $(*)$ is satisfied in the case $f (x) = sin n x$ , where $n$ is a positive integer. Show by means of counterexamples that $(*)$ is not necessarily satisfied if either $f (0) \neq = 0$ or $f (π) \neq = 0$ .
You may now assume that $(*)$ is satisfied for any (differentiable) function $f$ for which $f (0) = f (π) = 0$ . By setting $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ and $c$ are suitably chosen, show that $π^{2} ⩽ 10$ . By setting $f (x) = p sin \frac{1}{2} x + q cos \frac{1}{2} x + r$ , where $p$ , $q$ and $r$ are suitably chosen, obtain another inequality for $π$ . Which of these inequalities leads to a better estimate for $π^{2}$ ?
Integration
Show, geometrically or otherwise, that the shortest distance between the origin and the line $y = m x + c$ , where $c ⩾ 0$ , is $c (m^{2} + 1)^{- \frac{1}{2}}$ .
The curve $C$ lies in the $x$ - $y$ plane. Let the line $L$ be tangent to $C$ at a point $P$ on $C$ , and let $a$ be the shortest distance between the origin and $L$ . The curve $C$ has the property that the distance $a$ is the same for all points $P$ on $C$ . Let $P$ be the point on $C$ with coordinates $(x, y (x))$ . Given that the tangent to $C$ at $P$ is not vertical, show that $(y - x y^{'})^{2} = a^{2} (1 + (y^{'})^{2}) .$ By first differentiating $(*)$ with respect to $x$ , show that either $y = m x \pm a (1 + m^{2})^{\frac{1}{2}}$ for some $m$ or $x^{2} + y^{2} = a^{2}$ .
Now suppose that $C$ (as defined above) is a continuous curve for $- \infty < x < \infty$ , consisting of the arc of a circle and two straight lines. Sketch an example of such a curve which has a non-vertical tangent at each point.
Coordinate Geometry & Conics
By using the substitution $u = 1/ x$ , show that for $b > 0$ $\int_{1/ b}^{b} \frac{x ln x}{( a ^{2} + x ^{2} ) ( a ^{2} x ^{2} + 1 )} d x = 0 .$
By using the substitution $u = 1/ x$ , show that for $b > 0$ , $\int_{1/ b}^{b} \frac{arctan x}{x} d x = \frac{π ln b}{2} .$
By using the result $\int_{0}^{\infty} \frac{1}{a ^{2} + x ^{2}} d x = \frac{π}{2 a}$ (where $a > 0$ ), and a substitution of the form $u = k / x$ , for suitable $k$ , show that $\int_{0}^{\infty} \frac{1}{( a ^{2} + x ^{2} ) ^{2}} d x = \frac{π}{4 a ^{3}} (a > 0) .$
Integration
Given that $y = xu$ , where $u$ is a function of $x$ , write down an expression for $\frac{d y}{d x}$ .
Use the substitution $y = xu$ to solve $\frac{d y}{d x} = \frac{2 y + x}{y - 2 x}$ given that the solution curve passes through the point $(1, 1)$ . Give your answer in the form of a quadratic in $x$ and $y$ .
Using the substitutions $x = X + a$ and $y = Y + b$ for appropriate values of $a$ and $b$ , or otherwise, solve $\frac{d y}{d x} = \frac{x - 2 y - 4}{2 x + y - 3},$ given that the solution curve passes through the point $(1, 1)$ .
Differential Equations
By simplifying $sin (r + \frac{1}{2}) x - sin (r - \frac{1}{2}) x$ or otherwise show that, for $sin \frac{1}{2} x \neq = 0$ , $cos x + cos 2 x + \dots + cos n x = \frac{sin ( n + \frac{1}{2} ) x - sin \frac{1}{2} x}{2 sin \frac{1}{2} x} .$ The functions $S_{n}$ , for $n = 1$ , $2$ , …, are defined by $S_{n} (x) = r = 1 \sum n \frac{1}{r} sin r x (0 ⩽ x ⩽ π) .$
Find the stationary points of $S_{2} (x)$ for $0 ⩽ x ⩽ π$ , and sketch this function.
Show that if $S_{n} (x)$ has a stationary point at $x = x_{0}$ , where $0 < x_{0} < π$ , then $sin n x_{0} = (1 - cos n x_{0}) tan \frac{1}{2} x_{0}$ and hence that $S_{n} (x_{0}) ⩾ S_{n - 1} (x_{0})$ . Deduce that if $S_{n - 1} (x) > 0$ for all $x$ in the interval $0 < x < π$ , then $S_{n} (x) > 0$ for all $x$ in this interval.
Prove that $S_{n} (x) ⩾ 0$ for $n ⩾ 1$ and $0 ⩽ x ⩽ π$ .
Sequences & Series
The function $f$ is defined by $f (x) = ∣ x - a ∣ + ∣ x - b ∣$ , where $a < b$ . Sketch the graph of $f (x)$ , giving the gradient in each of the regions $x < a$ , $a < x < b$ and $x > b$ . Sketch on the same diagram the graph of $g (x)$ , where $g (x) = ∣2 x - a - b ∣$ . What shape is the quadrilateral with vertices $(a, 0)$ , $(b, 0)$ , $(b, f (b))$ and $(a, f (a))$ ?
Show graphically that the equation $∣ x - a ∣ + ∣ x - b ∣ = ∣ x - c ∣,$ where $a < b$ , has $0$ , $1$ or $2$ solutions, stating the relationship of $c$ to $a$ and $b$ in each case.
For the equation $∣ x - a ∣ + ∣ x - b ∣ = ∣ x - c ∣ + ∣ x - d ∣,$ where $a < b$ , $c < d$ and $d - c < b - a$ , determine the number of solutions in the various cases that arise, stating the relationship between $a$ , $b$ , $c$ and $d$ in each case.
Functions & Curve Sketching
For positive integers $n$ , $a$ and $b$ , the integer $c_{r}$ ( $0 ⩽ r ⩽ n$ ) is defined to be the coefficient of $x^{r}$ in the expansion in powers of $x$ of $(a + b x)^{n}$ . Write down an expression for $c_{r}$ in terms of $r$ , $n$ , $a$ and $b$ .

For given $n$ , $a$ and $b$ , let $m$ denote a value of $r$ for which $c_{r}$ is greatest (that is, $c_{m} ⩾ c_{r}$ for $0 ⩽ r ⩽ n$ ).

Show that $\frac{b ( n + 1 )}{a + b} - 1 ⩽ m ⩽ \frac{b ( n + 1 )}{a + b} .$ Deduce that $m$ is either a unique integer or one of two consecutive integers.

Let $G (n, a, b)$ denote the unique value of $m$ (if there is one) or the larger of the two possible values of $m$ .
Evaluate $G (9, 1, 3)$ and $G (9, 2, 3)$ .
For any positive integer $k$ , find $G (2 k, a, a)$ and $G (2 k - 1, a, a)$ in terms of $k$ .
For fixed $n$ and $b$ , determine a value of $a$ for which $G (n, a, b)$ is greatest.
For fixed $n$ , find the greatest possible value of $G (n, 1, b)$ . For which values of $b$ is this greatest value achieved?
Combinatorics & Binomial
A uniform rectangular lamina $A B C D$ rests in equilibrium in a vertical plane with the corner $A$ in contact with a rough vertical wall. The plane of the lamina is perpendicular to the wall. It is supported by a light inextensible string attached to the side $A B$ at a distance $d$ from $A$ . The other end of the string is attached to a point on the wall above $A$ where it makes an acute angle $θ$ with the downwards vertical. The side $A B$ makes an acute angle $ϕ$ with the upwards vertical at $A$ . The sides $B C$ and $A B$ have lengths $2 a$ and $2 b$ respectively. The coefficient of friction between the lamina and the wall is $μ$ .
Show that, when the lamina is in limiting equilibrium with the frictional force acting upwards, $d sin (θ + ϕ) = (cos θ + μ sin θ) (a cos ϕ + b sin ϕ) .$
How should $(*)$ be modified if the lamina is in limiting equilibrium with the frictional force acting downwards?
Find a condition on $d$ , in terms of $a$ , $b$ , $tan θ$ and $tan ϕ$ , which is necessary and sufficient for the frictional force to act upwards. Show that this condition cannot be satisfied if $b (2 tan θ + tan ϕ) < a$ .
Mechanics
A particle is projected from a point $O$ on horizontal ground with initial speed $u$ and at an angle of $θ$ above the ground. The motion takes place in the $x$ - $y$ plane, where the $x$ -axis is horizontal, the $y$ -axis is vertical and the origin is $O$ . Obtain the Cartesian equation of the particle's trajectory in terms of $u$ , $g$ and $λ$ , where $λ = tan θ$ .

Now consider the trajectories for different values of $θ$ with $u$ fixed. Show that for a given value of $x$ , the coordinate $y$ can take all values up to a maximum value, $Y$ , which you should determine as a function of $x$ , $u$ and $g$ .

Sketch a graph of $Y$ against $x$ and indicate on your graph the set of points that can be reached by a particle projected from $O$ with speed $u$ .

Hence find the furthest distance from $O$ that can be achieved by such a projectile.
Mechanics
A small smooth ring $R$ of mass $m$ is free to slide on a fixed smooth horizontal rail. A light inextensible string of length $L$ is attached to one end, $O$ , of the rail. The string passes through the ring, and a particle $P$ of mass $k m$ (where $k > 0$ ) is attached to its other end; this part of the string hangs at an acute angle $α$ to the vertical and it is given that $α$ is constant in the motion.

Let $x$ be the distance between $O$ and the ring. Taking the $y$ -axis to be vertically upwards, write down the Cartesian coordinates of $P$ relative to $O$ in terms of $x$ , $L$ and $α$ .
By considering the vertical component of the equation of motion of $P$ , show that $k m \overset{x}{¨} cos α = T cos α - k m g,$ where $T$ is the tension in the string. Obtain two similar equations relating to the horizontal components of the equations of motion of $P$ and $R$ .
Show that $\frac{sin α}{( 1 - sin α ) _{∣}^{2}} = k$ , and deduce, by means of a sketch or otherwise, that motion with $α$ constant is possible for all values of $k$ .
Show that $\overset{x}{¨} = - g tan α$ .
Mechanics
The lifetime of a fly (measured in hours) is given by the continuous random variable $T$ with probability density function $f (t)$ and cumulative distribution function $F (t)$ . The hazard function, $h (t)$ , is defined, for $F (t) < 1$ , by $h (t) = \frac{f ( t )}{1 - F ( t )} .$
Given that the fly lives to at least time $t$ , show that the probability of its dying within the following $δ t$ is approximately $h (t) δ t$ for small values of $δ t$ .
Find the hazard function in the case $F (t) = t / a$ for $0 < t < a$ . Sketch $f (t)$ and $h (t)$ in this case.
The random variable $T$ is distributed on the interval $t > a$ , where $a > 0$ , and its hazard function is $t^{- 1}$ . Determine the probability density function for $T$ .
Show that $h (t)$ is constant for $t > b$ and zero otherwise if and only if $f (t) = k e^{- k (t - b)}$ for $t > b$ , where $k$ is a positive constant.
The random variable $T$ is distributed on the interval $t > 0$ and its hazard function is given by $h (t) = (\frac{λ}{θ ^{λ}}) t^{λ - 1},$ where $λ$ and $θ$ are positive constants. Find the probability density function for $T$ .
Probability & Statistics
A random number generator prints out a sequence of integers $I_{1}$ , $I_{2}$ , $I_{3}$ , …. Each integer is independently equally likely to be any one of $1$ , $2$ , …, $n$ , where $n$ is fixed. The random variable $X$ takes the value $r$ , where $I_{r}$ is the first integer which is a repeat of some earlier integer.

Write down an expression for $P (X = 4)$ .
Find an expression for $P (X = r)$ , where $2 ⩽ r ⩽ n + 1$ . Hence show that, for any positive integer $n$ , $\frac{1}{n} + (1 - \frac{1}{n}) \frac{2}{n} + (1 - \frac{1}{n}) (1 - \frac{2}{n}) \frac{3}{n} + \dots = 1 .$
Write down an expression for $E (X)$ . (You do not need to simplify it.)
Write down an expression for $P (X ⩾ k)$ .
Show that, for any discrete random variable $Y$ taking the values $1$ , $2$ , \dots, $N$ , $E (Y) = k = 1 \sum N P (Y ⩾ k) .$ Hence show that, for any positive integer $n$ , $(1 - \frac{1 ^{2}}{n}) + (1 - \frac{1}{n}) (1 - \frac{2 ^{2}}{n}) + (1 - \frac{1}{n}) (1 - \frac{2}{n}) (1 - \frac{3 ^{2}}{n}) + \dots = 0.$
Probability & Statistics