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mathematics
cambridge international as & a level mathematics probability & statistics

Cambridge International AS & A Level Mathematics Pure Mathematics 2 & 3 Coursebook 1st Edition Sue Pemberton, Julianne Hughes, Julian Gilbey - Solutions

Leti. Express f(x) in the formii. Show that f(x) = 2 12 + 8x - x² (2-x)(4+x²)
Leti. Express f(x) in partial fractions.ii Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2. f(x) = x² - 8x +9 (1-x)(2-x)²
The diagram shows the curve y = x2 ln x and its minimum point M.i. Find the exact values of the coordinates of M.ii. Find the exact value of the area of the shaded region bounded by the curve, the x-axis and the line x = e. M C X
Find which of the following pairs of position vectors are perpendicular to one another.For any position vectors that are not perpendicular, find the acute angle between them.a.b.c. a = 28 -2 3 b = 7 -1 3
Relative to the origin O, the position vectors of the points A and B are given byFind angle BOA. OA = -5 0 3 and OB = 7 2
The diagram shows the vectors DE and DF on 1 centimetre squared paper.Using 1cm as 1 unit of displacement:a. Find the column vector EFb. Show that DF - DE = EF. F E
The position vectors of the points A, B,C and D are given by a, b, c and d, respectively, wherea. i. Findii. Deduce that ABCD is a parallelogramb. Find the coordinates ofi. The point M, the midpoint of the line ABii. the point P such that 2 (13 5 --0-0-- 2 5 c= -3 and d 4 4 -6 -6 0
Write down the vector equation for the line through the point A that is parallel to vector b when:a. A(0, −1, 5) and b = 2i + 6j – kb. A( 0, 0, 0 ) and b = 7i – j – kc. A(7, 2, −3) and b = 3i – 4k.
Relative to an origin O, the position vectors of the points A and B are given by OAvector = 5i +3j and OBvecoter = 5i + kj + μk.a. In the case where λ = 0 and μ = 7, find the unit vector in the direction of ABvector.b. Given that triangle OAB is right-angled with area 5√34 and μ = 10, find
Given that y = 2 when x = 0, solve the differential equation.obtaining an expression fory4 in terms of x. 3 y-=1+y+ 山
Write each of your answers to question 1 in parametric form.
Given that the lines with vector equationsr = 2i + 9j + k + (i – 4j + 5k) and r = 11i + 9j + pk + μ(−i − 2j + 16k)intersect at the point P, find the value of p and the position vector of the point P.
Relative to the origin O, the position vectors of the points P and Q are given bywhere k is a constant. OP 5k -3 00 7k+9 9)' k k+2 -1
Given that the vectors 6i − 2j + 5k and ai + 4j − 2k are perpendicular, find the value of the constant a.
Show that the line through the points with position vectors 9i + 2j – 5k and i + 7j + k is parallel to the line r = λ(16i − 10j − 12k).
Relative to an origin O, the position vector of A is i − 2j + 5k and the position vector of B is 3i + 4j + k.a. Find the magnitude of ABvector.b. Use Pythagoras’ theorem to show that OAB is a right-angled triangle.c. Find the exact area of triangle OAB.
a. Given that x < 5, findb. A chemical reaction takes place between two substances A and B. When this happens, a third substance, C, is produced. After t hours there are 5 − x grams of A, there are 10 − x grams of B and there are x grams of C present. The rate of increase of x is
Find the parametric equations of each of the lines.a. r = 2i + 13j + k + t(i + j − k)b. r = 10j + t(2i + 5j)c. r = i − 3j + t(2i + 3j + 4k)
The vector equation of the line L1 is given bya. Find the vector equation of the line L2 that is parallel to L1 and which passes through the point A(5, −3, 2).b. Show that A(5, −3, 2) is the foot of the perpendicular from the point B(4, −7, 2) to the line L2. 0-0-0 = -1 8 -3
The points A, B and C are position vectors relative to the origin O, given bya. Find a vector equation for the line L passing through A and B.b. The line through C, perpendicular to L, meets L at the point N. Find the exact coordinates of N. OA= 5 OB -1 and OC 2 =
The gradient of a curve is such that, at the point (x, y), the gradient of the curve is proportional to x√y. At the point (3, 4) the gradient of this curve is -5.a. Form and solve a differential equation to fi nd the equation of this curve.b. Find the gradient of the curve at the point (−3, 4) .
The position vectors of the points A, B and C are given by a = 5i + j + qk.Given that the length of AB is equal to the length of AC, find the exact possible values of the constant q.
Find the particular solution for each of the following differential equations, using the values given.a.b. cos²x = y given y = 5 when x = 0 dx
A curve has equation 2x2 + 3y2 − 2x + 4y = 4. Find the equation of the tangent to the curve at the point (1, −2).
a. Find the parametric equations of the line L through the points A(0, 4, −2) and B(1, 1, 6).b. Hence find the coordinates of the point where the line L crosses the Oxy plane.
The variables w and t satisfy the differential equationGiven that when t = 0 w = 0, show that the solution to the differential equation can be written in the form dw dt 0.001(100-w)².
a. Expressin the formb. The variables x and t satisfy the differential equationIt is given that t = 0 when x = 0.i. Solve this differential equation and obtain an expression for x in terms of t.ii. State what happens to the value of x when t becomes large. 1 (2-x)(x + 1)
Find the equation of the curve that the equation.and passes through the point (0, 1). Give your answer in the form y = f(x). dy 1 + y² dx 4+x² =
Given that dy/dx = ex-y and y = ln 2 when x = 0, obtain an expression for y in terms of x.
The cost of designing an aircraft, $A per kilogram, at time t years after 1950 can be modelled as a continuous variable. The rate increase of A is directly proportional to A.a. Write down a differential equation that is satisfied by A.b. In 1950, the cost of designing an aircraft was $12 per
The gradient of a curve at the point (x, y) is 5y3x.a. Write down the differential equation satisfi ed by y.b. Given that the curve passes through the point (1, 1), fi nd y2 as a function of x.
A liquid is heated so that its temperature is x°C after t seconds. It is given that the rate of increase of x is proportional to (100 − x). The initial temperature of the liquid is 25°C.a. Form a differential equation relating x, t and a constant of proportionality, k, to model this
Find the general solution of the equation 3 dy - + 2y = 1. d.x
The size of a population, P, at time t minutes is to be modelled as a continuous variable such that the rate of increase of P is directly proportional to P.a. Write down a differential equation that is satisfied by P.b. The initial size of the population is P0. Show that P = P0ekt, where k is a
Maria makes a large dish of curry on Monday, ready for her family to eat on Tuesday. She needs to put the dish of curry in the refrigerator but must let it cool to room temperature, 18°C, first. The temperature of the curry is 94°C when it has finished cooking.At 6pm, Maria places the hot dish in
The half-life of a radioactive isotope is the amount of time it takes for half of the isotope in a sample to decay to its stable form.Carbon-14 is a radioactive isotope that has a half-life of 5700 years.It is given that the rate of decrease of the mass, m, of the carbon-14 in a sample is
Solve the differential equation dx/dt = 4xcos2 t given that x = 1 when t = 0.Give your answer as an expression for x in terms of t.
Anya carried out an experiment and discovered that the rate of growth of her hair was constant. At the start of her experiment, her hair was 20 cm long. After 20 weeks, her hair was 26cm long. Form and solve a differential equation to find a direct relationship between time, t, and the length of
a. Use the substitution u = ln x to findb. Given that x > 0 > y 0, find the general solution of the differential equation In xdy/dx = y/x. 1 x ln - dx. x
The variables x and t satisfy the differential equationfor x . 0 where k is a constant. When t = 0, x = 1 and when t = 0.5, x = 2. Solve the differential equation, finding the exact value of k , and show that (3x + 2x³) dx = k(3x² + x4) di
A bottle of water is taken out of a refrigerator. The temperature of the water in the bottle is 4°C. The bottle water is taken outside to drink. The air temperature outside is constant at 24°C.It is given that the rate at which the water in the bottle warms up is proportional to the difference in
The number of customers, n, of a food shop t months after it opens for the first time can be modelled as a continuous variable. It is suggested that the number of customers increasing at a rate that is proportional to the square root of n.a. Form and solve a differential equation relating n
Doubling time is the length of time it takes for a quantity to double in size or value.The number of bacteria in a liquid culture can be modelled as a continuous variable and grows at a rate proportional to the number of bacteria present. Initially, there are 5000 bacteria in the culture. After 3.5
Show that for small values of x2, (1 − 2x2)−2 − (1 + 6x2)2/3 ≈ kx4, where the value of constant k to be determined.
Leti. Express f(x) in partial fractions.ii. Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2. f(x) = 4x² + 12 (x + 1)(x - 3)²
Leti. Express f(x) in partial fractions.ii. Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x3 f(x) = 3x (1+x)(1+2x²)
Find the following integrals.a.b.c.d.e.f. S 1 x² +9 dx
Expand (1 − 3x)−5 in ascending powers of x, up to and including the term in x3.
Find the exact value of each of these integrals.a.b.c. S8² 1 x² +9 d.x
Use the substitution u = 1 + 3 tan x to find the exact value of 0 (1 + 3 tan x) COS X - dx
Expand 32/(x + 2)3 in ascending powers of x, up to and including the term in x3.
i. Prove that cot θ + tan θ = 2 cosec 2θ.ii. Hence show that czt fr cosec 20 de = In In 3.
Expand 1/√4 - 2x in ascending powers of x, up to and including the term in x2.
The integral I is defined byi. Use the substitution x = t2 + 1 to show thatii. Hence find the exact value of I. I= ² 47³ In(1² + 1) dr. Jo
i. Express 2/(x + 1)(x + 3) in partial fractions.ii. Using your answer to part i, show thatiii. Hence show that 2 1 1 (es) (x + ! (x + 3) (x+1) +1 x+3 + + 1 (x+3) ².
a. Findb. Find the exact value of [(4 + tan² 2x) dx.
Can the binomial expansion be used to expand each of these expressions? (a)(b)Give reasons for your answers. (3x-1)-2
Expand in ascending powers of x up to and including the term in x2, simplifying the coefficients. 1+ 3x √(1+2x)
a. Expand (2 − x)−1 and (1 + 3x)−2 giving the first 3 terms in each expansion.b. Use your answers to part a to find the first 3 terms in the expansion of (2 − x)−1(1 + 3x)−2, stating the range of values of x for which this expansion is valid.
a. Express 21/(x - 4)(x + 3)in partial fractions.b. Hence obtain the expansion of 24/(X - 4)(X + 3) in ascending powers of x, up to and including the term in x2.
Find the values of A, B,C and D such that 4x3 - 9x² + 11x-4 x2(2x-1) = B A+-+ с D 2x-1
Expand 1/√1 + 3x in ascending powers of x, up to and including the term in x3, simplifying the coefficients.
Expand (2x − 1)−3 in ascending powers of x, up to and including the term in x3.
The first 3 terms in the expansion of (a − 5x)−2 are 1/4 + 5/4x + bx2. Find the value of a, the value of b and the term in x3.
a. Factorise 2x3 − 3x2 − 3x + 2 completely.b. Hence expressin partial fractions. x²-13x-5 2x³ 3x²-3x + 2
Expand 10/(2 - x)2 in ascending powers of x, up to and including the term in x2, simplifying the coefficients.
Show that the expansion of (1 + x)1/2 is not valid when x = 3.
In the expansion of √3 + ax where a ≠ 0, the coefficient of the term in x2 is 3 times the coefficient of the term in x3. Find the value of a.
a. Factorise 2x3 − 11x2 + 12x + 9 completelyb. Hence expressin partial fractions. 24 - x 2x3 11x² + 12x +9
Expand 1/√4 - 5x in ascending powers of x, up to and including the term in x2, simplifying the coefficients.
Given that (1 − 3x)-4 - (1 + 2x)3/2 ≈ 9x + kx2 for small values of x, find the value of the constant k.
The polynomial p(x) = 2x3 + 5x2 + ax + b is exactly divisible by 2x + 1 and leaves a remainder of 9 when divided by x + 2.a. Find the value of a and the value of b.b. Factorise p(x) completely.c. Express 120/p(x) in partial fractions.
Expressin the form 12 x²(2x - 3)
Given that α/1 - x + b/1 + 2x ≈ -3 + 12x for small values of x, find the value of a and the value of b.
a. Express 2/x(x+2) in partial fractions.b. Find an expression for the sum of the first n terms of the series:c. Find the sum to infinity of this series. 2 2 2 2 183 284 385 486 57 + + + + x + '
Expressin the form A/x + 2 + Bx + C/x2 + 5. 8x² + 4x + 21 (x + 2)(x² + 5)
When (1 + ax)−3, where a is a positive constant, is expanded the coefficients of x and x2 are equal.a. Find the value of a.b. When a has this value, obtain the first 5 terms in the expansion.
a. By sketching graphs of y = ex and y = x + 6, determine number of roots of the equation ex − x − 6 = 0.b. Show, by calculation, that ex − x − 6 = 0 has a root that lies between x = 2.0 and x = 2.1.
i. Use the trapezium rule with two intervals to estimate the value ofgiving your answer correct to 2 decimal places.ii. Find 0 1 6 + 2e.* -dx, giving
The terms of a sequence with first term x1 = 1 are defined by the iterative formula:The terms converge to the value α.a. Use this formula to find the value of α correct to 2 decimal places. Give the value of each term you calculate to 4 decimal places.b. The value α is the root of an equation of
The equation of a curve is xy(x − 6y) = 9a3, where a is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the x-axis, and find the coordinates of this point.
a. By sketching two suitable graphs on the same diagram, show that the cot x = x2 has one root between 0 and π/2 radius.b. Show, by calculation, that this root,α, lies between 0.8 and 1.
a. By sketching each of the graphs of = sec y x andon the same diagram, show that the equationhas exactly two real roots in the intervalb. Show that the equationcan be written in the formc. The two real roots of the equationin the intervalare denoted α and β.Verify by calculation that the
a. By sketching a suitable pair of graphs, deduce the number of roots of the equation x = tan 2x for < x < 2π.b. Verify, by calculation, that one of these roots, α, lies between 2.1 and 2.2.
The diagram shows a container in the shape of a cone with a cylinder on top.The height of the cylinder is 3 times its base radius, r.The volume of the container must be 5500 cm3.The base of the cone has a radius of r cm. a. Write down an expression for the height of the cone in terms of r.b.
a. Show graphically that the equation cosec x = sin x has exactly two roots for 0 < x < 2π.b. Use an algebraic method to find the value of the larger root correct to 3 significant figures.
The parametric equations of a curve are x = e2t, y = 4tet.i. Show that dy/dx = 2(t + 1)/et.ii. Find the equation of the normal to the curve at the point where t = 0.
The diagram shows part of the curve y = cos3 x, where x is in radians. The shaded region between the curve, the axes and the line x = α is denoted by R. The area of R is equal to 0.3.a. Using the substitutionHence show thatb. Use the iterative formulawith α1 = 0.2, to find the value of α correct
The equation x3 − 7x2 + 1 = 0 has two positive roots, α and β, which are such that α lies between 0 and 1 and β lies between 6 and 7.By deriving two suitable iterative formulae from the given equation, carry out suitable iterations to find the value of α and of β, giving each correct
f(x) = 20x3 + 8x2 − 7x − 3a. The equation 20x3 + 8x2 − 7x − 3 = 0 has exactly two roots.Without factorising the cubic equation, show, by calculation, that one of these roots is between 0.5 and 1.b. Show, by sketching the graph of y = f(x) for −1 ≤ x ≤ 1, that the smaller root is
For each of the following curves, find the exact gradient at the point indicated:(i)(ii) at (1,-1). 6 y = 3 cos 2x - 5 sin x at
a. Show graphically that the equation cot x = sin x has a root,α, which is such that 0 < α < π/2.b. Show that the equation in part a can be rearranged as x = sin−1 √cos x.c. Using an iterative formula based on the equation in part b, with an initial value of 0.9, find the value of
A guitar tuning peg is in the shape of a cylinder with a hemisphere at one end. The cylinder is 20mm long and the whole peg is made from 800mm3 of plastic.The base radius of the cylinder is rmm.a. Show that πr3 + 30πr2 − 1200 = 0.b. Show that the value of r is between 3mm and 4mm.
i. By differentiating 1/cos x, show that if y = sec x then dy/dx = sec x tan x.ii. Show thatiii. Deduce thativ. Hence show that 1 secx tan x =sec x + tan x.
The diagram shows the curve = 4e1/2x- 6x + 3 and its minimum point M.i. Show that the x-coordinate of M can be written in the form ln a, where the value of a is to be stated.ii. Find the exact value of the area of the region enclosed by the curve and the x = 0, x = 2 and y = 0. y M
The equation of a curve is x3 − 3x2y + y3 = 3.i. Show that dy/dx = x2 - 2xy/ x2 - y2.'ii. Find the coordinates of the points on the curve where the tangent is parallel to the x-axis.
A curve has parametric equationsThe point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.i. Show thatii. Use an iterative process based on the equation in part i to find the value of p correct to 3 decimal places. Use a starting value of 0.7 and show the
i. Finda.b.ii. Use the trapezium rule with 2 intervals to estimate the value ofgiving your answer correct to 2 decimal places. +6 e2x + 6 e2 dx,
i. By sketching a suitable pair of graphs, show that the equation e2x = 14 − x2 has exactly two real roots.ii. Show by calculation that the positive root lies between 1.2 and 1.3.iii. Show that this root also satisfies the equation x = 1/2 In(14 - x2).iv. Use an iteration process based on the
The diagram shows part of the curve with parametric equationsx = 2 In(t + 2), y = t3 + 2t + 3.i. Find the gradient of the curve at the origin.ii. At the point P on the curve, the value of the parameter is p. It is given that the gradient of the curve at P is 1/2.a. Show that P = 1/3p2 + 2

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