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Questions

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120

Marks total

Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 3 1991

1991

16 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.

Evaluate $r = 1 \sum n \frac{6}{r ( r + 1 ) ( r + 3 )} .$
Expand $ln (1 + x + x^{2} + x^{3})$ as a series in powers of $x$ , where $∣ x ∣ < 1$ , giving the first five non-zero terms and the general term.
Expand $e^{x l n (1 + x)}$ as a series in powers of $x$ , where $- 1 < x ⩽ 1$ , as far as the term in $x^{4}$ .
Sequences & Series
The distinct points $P_{1}, P_{2}, P_{3}, Q_{1}, Q_{2}$ and $Q_{3}$ in the Argand diagram are represented by the complex numbers $z_{1}, z_{2}, z_{3}, w_{1}, w_{2}$ and $w_{3}$ respectively. Show that the triangles $P_{1} P_{2} P_{3}$ and $Q_{1} Q_{2} Q_{3}$ are similar, with $P_{i}$ corresponding to $Q_{i}$ ( $i = 1, 2, 3$ ) and the rotation from $1$ to $2$ to $3$ being in the same sense for both triangles, if and only if $\frac{z _{1} - z _{2}}{z _{2} - z _{3}} = \frac{w _{1} - w _{2}}{w _{1} - w _{3}} .$ Verify that this condition may be written $det z_{1} w_{1} 1 z_{2} w_{2} 1 z_{3} w_{3} 1 = 0.$
Show that if $w_{i} = z_{i}^{2}$ ( $i = 1, 2, 3$ ) then triangle $P_{1} P_{2} P_{3}$ is not similar to triangle $Q_{1} Q_{2} Q_{3} .$
Show that if $w_{i} = z_{i}^{3}$ ( $i = 1, 2, 3$ ) then triangle $P_{1} P_{2} P_{3}$ is similar to triangle $Q_{1} Q_{2} Q_{3}$ if and only if the centroid of triangle $P_{1} P_{2} P_{3}$ is the origin. {[}The centroid of triangle $P_{1} P_{2} P_{3}$ is represented by the complex number $\frac{1}{3} (z_{1} + z_{2} + z_{3})$ .{]}
Show that the triangle $P_{1} P_{2} P_{3}$ is equilateral if and only if $z_{2} z_{3} + z_{3} z_{1} + z_{1} z_{2} = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} .$
Complex Numbers
The function $f$ is defined for $x < 2$ by $f (x) = 2 x^{2} - x + x^{2} - 1 - 2 x^{2} + x .$ Find the maximum and minimum points and the points of inflection of the graph of $f$ and sketch this graph. Is $f$ continuous everywhere? Is $f$ differentiable everywhere?

Find the inverse of the function $f$ , i.e. expressions for $f^{- 1} (x),$ defined in the various appropriate intervals.
Differentiation
The point $P$ moves on a straight line in three-dimensional space. The position of $P$ is observed from the points $O_{1} (0, 0, 0)$ and $O_{2} (8 a, 0, 0) .$ At times $t = t_{1}$ and $t = t_{1}^{'}$ , the lines of sight from $O_{1}$ are along the lines $\frac{x}{2} = \frac{z}{3}, y = 0 and x = 0, \frac{y}{3} = \frac{z}{4}$ respectively. At times $t = t_{2}$ and $t = t_{2}^{'}$ , the lines of sight from $O_{2}$ are $\frac{x - 8 a}{- 3} = \frac{y}{1} = \frac{z}{3} and \frac{x - 8 a}{- 4} = \frac{y}{2} = \frac{z}{5}$ respectively. Find an equation or equations for the path of $P$ .
Coordinate Geometry & Conics
The curve $C$ has the differential equation in polar coordinates $\frac{d ^{2} r}{d θ ^{2}} + 4 r = 5 sin 3 θ, for \frac{π}{5} ⩽ θ ⩽ \frac{3 π}{5},$ and, when $θ = \frac{π}{2},$ $r = 1$ and $\frac{d r}{d θ} = - 2.$

Show that $C$ forms a closed loop and that the area of the region enclosed by $C$ is $\frac{π}{5} + \frac{25}{48} [sin (\frac{π}{5}) - sin (\frac{2 π}{5})] .$
Differential Equations
The transformation $T$ from $(x y)$ to $(x^{'} y^{'})$ in two-dimensional space is given by $(x^{'} y^{'}) = (cosh u sinh u sinh u cosh u) (x y),$ where $u$ is a positive real constant. Show that the curve with equation $x^{2} - y^{2} = 1$ is transformed into itself. Find the equations of two straight lines through the origin which transform into themselves.

A line, not necessary through the origin, which has gradient $tanh v$ transforms under $T$ into a line with gradient $tanh v^{'}$ . Show that $v^{'} = v + u$ .

The lines $ℓ_{1}$ and $ℓ_{2}$ with gradients $tanh v_{1}$ and $tanh v_{2}$ transform under $T$ into lines with gradients $tanh v_{1}^{'}$ and $tanh v_{2}^{'}$ respectively. Find the relation satisfied by $v_{1}$ and $v_{2}$ that is the necessary and sufficient for $ℓ_{1}$ and $ℓ_{2}$ to intersect at the same angle as their transforms.

In the case when $ℓ_{1}$ and $ℓ_{2}$ meet at the origin, illustrate in a diagram the relation between $ℓ_{1}$ , $ℓ_{2}$ and their transforms.
Trigonometry
Prove that $\int_{0}^{\frac{1}{2} π} ln (sin x) d x = \int_{0}^{\frac{1}{2} π} ln (cos x) d x = \frac{1}{2} \int_{0}^{\frac{1}{2} π} ln (sin 2 x) d x - \frac{1}{4} π ln 2$ and $\int_{0}^{\frac{1}{2} π} ln (sin 2 x) d x = \frac{1}{2} \int_{0}^{π} ln (sin x) d x = \int_{0}^{\frac{1}{2} π} ln (sin x) d x .$ Hence, or otherwise, evaluate $\int_{0}^{\frac{1}{2} π} ln (sin x) d x .$ {[}You may assume that all the integrals converge.{]}
Given that $ln u < u$ for $u ⩾ 1$ deduce that $\frac{1}{2} ln x < x for x ⩾ 1.$ Deduce that $\frac{ln x}{x} \to 0$ as $x \to \infty$ and that $x ln x \to 0$ as $x \to 0$ through positive values.
Using the results of parts (i) and (ii), or otherwise, evaluate $\int_{0}^{\frac{1}{2} π} x cot x d x .$
Integration
The integral $I_{k}$ is defined by $I_{k} = \int_{0}^{θ} cos^{k} x cos k x d x .$ Prove that $2 k I_{k} = k I_{k - 1} + cos^{k} θ sin k θ .$
Prove that $1 + m cos 2 θ + (2 m) cos 4 θ + \dots + (r m) cos 2 r θ + \dots + cos 2 m θ = 2^{m} cos^{m} θ cos m θ .$
Using the results of (i) and (ii), show that $m \frac{sin 2 θ}{2} + (2 m) \frac{sin 4 θ}{4} + \dots + (r m) \frac{sin 2 r θ}{2 r} + \dots + \frac{sin 2 m θ}{2 m}$ is equal to $cos θ sin θ + cos^{2} θ sin 2 θ + \dots + \frac{1}{r} 2^{r - 1} cos^{r} θ sin r θ + \dots + \frac{1}{m} 2^{m - 1} cos^{m} θ sin m θ .$
Sequences & Series
The parametric equations $E_{1}$ and $E_{2}$ define the same ellipse, in terms of the parameters $θ_{1}$ and $θ_{2}$ , (though not referred to the same coordinate axes). $E_{1} : E_{2} : x = a cos θ_{1}, x = \frac{k cos θ _{2}}{1 + e cos θ _{2}}, y = b sin θ_{1}, y = \frac{k sin θ _{2}}{1 + e cos θ _{2}},$ where $0 < b < a,$ $0 < e < 1$ and $0 < k$ . Find the position of the axes for $E_{2}$ relative to the axes for $E_{1}$ and show that $k = a (1 - e^{2})$ and $b^{2} = a^{2} (1 - e^{2}) .$

{[}The standard polar equation of an ellipse is $r = \frac{ℓ}{1 + e cos θ} .]$

By considering expressions for the length of the perimeter of the ellipse, or otherwise, prove that $\int_{0}^{π} 1 - e^{2} cos^{2} θ d θ = \int_{0}^{π} \frac{1 - e ^{2}}{( 1 + e cos θ ) ^{2}} 1 + e^{2} + 2 e cos θ d θ .$ Given that $e$ is so small that $e^{6}$ may be neglected, show that the value of either integral is $\frac{1}{64} π (64 - 16 e^{2} - 3 e^{4}) .$
Coordinate Geometry & Conics
The equation $x^{n} - q x^{n - 1} + r = 0,$ where $n ⩾ 5$ and $q$ and $r$ are real constants, has roots $α_{1}, α_{2}, \dots, α_{n} .$ The sum of the products of $m$ distinct roots is denoted by $Σ_{m}$ (so that, for example, $Σ_{3} = \sum α_{i} α_{j} α_{k}$ where the sum runs over the values of $i, j$ and $k$ with $n ⩾ i > j > k ⩾ 1$ ). The sum of $m$ th powers of the roots is denoted by $S_{m}$ (so that, for example, $S_{3} = i = 1 \sum n α_{i}^{3}$ ).

Prove that $S_{p} = p^{q}$ for $1 ⩽ p ⩽ n - 1.$ {[}You may assume that for any $n$ th degree equation and $1 ⩽ p ⩽ n$ $S_{p} - S_{p - 1} Σ_{1} + S_{p - 2} Σ_{2} - \dots + (- 1)^{p - 1} S_{1} Σ_{p - 1} + (- 1)^{p} p Σ_{p} = 0.]$ Find expressions for $S_{n},$ $S_{n + 1}$ and $S_{n + 2}$ in terms of $q, r$ and $n$ . Suggest an expression for $S_{n + m},$ where $m < n$ , and prove its validity by induction.
Sequences & Series
A uniform circular cylinder of radius $2 a$ with a groove of radius $a$ cut in its central cross-section has mass $M$ . It rests, as shown in the diagram, on a rough plane inclined at an acute angle $α$ to the horizontal. It is supported by a light inextensible string would round the groove and attached to the cylinder at one end. The other end of the string is attached to the plane at $Q$ , the free part of the string, $P Q,$ making an angle $2 α$ with the inclined plane. The coefficient of friction at the contact between the cylinder and the plane is $μ .$ Show that $μ ⩾ \frac{1}{3} tan α .$

The string $P Q$ is now detached from the plane and the end $Q$ is fastened to a particle of mass $3 M$ which is placed on the plane, the position of the string remain unchanged. Given that $tan α = \frac{1}{2}$ and that the system remains in equilibrium, find the least value of the coefficient of friction between the particle and the plane.
Mechanics
A smooth tube whose axis is horizontal has an elliptic cross-section in the form of the curve with parametric equations $x = a cos θ y = b sin θ$ where the $x$ -axis is horizontal and the $y$ -axis is vertically upwards. A particle moves freely under gravity on the inside of the tube in the plane of this cross-section. By first finding $\overset{x}{¨}$ and $\overset{y}{¨},$ or otherwise, show that the acceleration along the inward normal at the point with parameter $θ$ is $\frac{ab θ ˙ ^{2}}{a ^{2} sin ^{2} θ + b ^{2} cos ^{2} θ} .$ The particle is projected along the surface in the vertical cross-section plane, with speed $2 b g,$ from the lowest point. Given that $2 a = 3 b,$ show that it will leave the surface at the point with parameter $θ$ where $5 sin^{3} θ + 12 sin θ - 8 = 0.$
Mechanics
A smooth particle $P_{1}$ is projected from a point $O$ on the horizontal floor of a room with has a horizontal ceiling at a height $h$ above the floor. The speed of projection is $8 g h$ and the direction of projection makes an acute angle $α$ with the horizontal. The particle strikes the ceiling and rebounds, the impact being perfectly elastic. Show that for this to happen $α$ must be at least $\frac{1}{6} π$ and that the range on the floor is then $8 h cos α (2 sin α - 4 sin^{2} α - 1) .$ Another particle $P_{2}$ is projected from $O$ with the same velocity as $P_{1}$ but its impact with the ceiling is perfectly inelastic. Find the difference $D$ between the ranges of $P_{1}$ and $P_{2}$ on the floor and show that, as $α$ varies, $D$ has a maximum value when $α = \frac{1}{4} π .$
Mechanics
The end $O$ of a smooth light rod $O A$ of length $2 a$ is a fixed point. The rod $O A$ makes a fixed angle $sin^{- 1} \frac{3}{5}$ with the downward vertical $O N,$ but is free to rotate about $O N .$ A particle of mass $m$ is attached to the rod at $A$ and a small ring $B$ of mass $m$ is free to slide on the rod but is joined to a spring of natural length $a$ and modulus of elasticity $k m g$ . The vertical plane containing the rod $O A$ rotates about $O N$ with constant angular velocity $5 g /2 a$ and $B$ is at rest relative to the rod. Show that the length of $O B$ is $\frac{( 10 k + 8 ) a}{10 k - 9} .$ Given that the reaction of the rod on the particle at $A$ makes an angle $tan^{- 1} \frac{13}{21}$ with the horizontal, find the value of $k$ . Find also the magnitude of the reaction between the rod and the ring $B$ .
Mechanics
A pack of $2 n$ (where $n ⩾ 4$ ) cards consists of two each of $n$ different sorts. If four cards are drawn from the pack without replacement show that the probability that no pairs of identical cards have been drawn is $\frac{4 ( n - 2 ) ( n - 3 )}{( 2 n - 1 ) ( 2 n - 3 )} .$ Find the probability that exactly one pair of identical cards is included in the four.

If $k$ cards are drawn without replacement and $2 < k < 2 n,$ find an expression for the probability that there are exactly $r$ pairs of identical cards included when $r < \frac{1}{2} k .$

For even values of $k$ show that the probability that the drawn cards consist of $\frac{1}{2} k$ pairs is $\frac{1 \times 3 \times 5 \times \dots \times ( k - 1 )}{( 2 n - 1 ) ( 2 n - 3 ) \dots ( 2 n - k + 1 )} .$
Probability & Statistics
The random variables $X$ and $Y$ take integer values $x$ and $y$ respectively which are restricted by $x ⩾ 1,$ $y ⩾ 1$ and $2 x + y ⩽ 2 a$ where $a$ is an integer greater than 1. The joint probability is given by $P (X = x, Y = y) = c (2 x + y),$ where $c$ is a positive constant, within this region and zero elsewhere. Obtain, in terms of $x, c$ and $a,$ the marginal probability $P (X = x)$ and show that $c = \frac{6}{a ( a - 1 ) ( 8 a + 5 )} .$ Show that when $y$ is an even number the marginal probability $P (Y = y)$ is $\frac{3 ( 2 a - y ) ( 2 a + 2 + y )}{2 a ( a - 1 ) ( 8 a + 5 )}$ and find the corresponding expression when $y$ is off.

Evaluate $E (Y)$ in terms of $a$ .
Probability & Statistics