Questions and Answers of Cambridge International AS & A Level Further Mathematics

A company sends its employees to a psychologist to try to improve their sales productivity. The following table shows the sales figures, in thousands of dollars, of six employees before and after
The contents of jars of beans may be assumed to be normally distributed. The contents, in grams, of a random sample of nine jars are as follows.460, 449,458, 455, 461, 456, 459, 457, 453a. Calculate
In a large school, a sample of 50 boys and 60 girls complete a 100m race. The estimators of the data are given.Find the 95% confidence interval for the difference in times between the boys and girls
At a petrol station, the manager thinks that one of the pumps is not working properly and is giving out more petrol than it should. She decides to test this claim by filling up ten buckets with 5
The waiting time for a particular train that runs daily is measured over 36 days. The average waiting time is found to be 37.2 minutes, with a standard deviation of 3.2 minutes. Find a 99% confidence
Samples are taken from two different types of honey and the viscosity (i.e. how 'runny' the honey is) is measured.Assuming normal distributions, test at the 5% significance level whether there is a
An economist believes that a typical basket of weekly food, bought by a family of four, costs more in Eastville than in Weston. Seven stores are randomly selected in each of these two towns and the
Find the line of intersection of the planes ∏1: 3x - y + z = 4 and ∏2: 5x + y - 3z = 6. Hence, find the equation of the plane that contains this line of intersection and is perpendicular to ∏1.
Prove, by mathematical induction, that 8n – 3n is divisible by 5 for all n ≥ 1.
The curve r = √(ln θ) is defined for π/2 ≤ θ ≤ π. Find the area bounded by the curve over this interval, giving 2 your answer correct to 3 significant figures.
Sketch the curve r = sec2 θ for the interval 0 ≤ 9 ≤ 2π.
Given that andconfirm that det(AB) = det(B) and det(BA) = det(A) × det(B). 3 4) A = 5
Find the matrix represented by each of the following transformations.a. A reflection in the x-axis followed by an enlargement of scale factor 2 centered at the origin.b. A rotation of 90°
Given that a. Find C such that BC = A + A2b. Determine D, where ADB = I. 9. A = and B = 1 2 3 1 -4
Verify that AB ≠ BA, where 32) 3 -5 4 12 A = and B = %3D -9 4
Given thatandfind C such that ACB = 1. 3 1 A = 5
The matrix A is such that Find the values of x such that A is singular. 4 A = 1 х-3
a. You are given the matrix Show thatand state the value of k.b. Given also thatfind the values of b and m. A = 2. 4.
You are given the matrixa. If the matrix is singular, find the value of the constant a.b. If a = 4, find A-1. 1 2 0 2 0 a+4, a A = 2 a-3
Given thatevaluate the following:a. A2 + 2ABb. B2Ac. A3 - BA A = 1 and B = %3D -4 2.
a. Determine the value of k such thathas no inverse.b. Given that k = 8, determine B where BA2 = I. (2 3 A = 5 k
Show that can always be inverted for all values of x. - 2 B = 5 х+1
For the matrixdetermine the transformation matrix that produces the following image.a.b.c. 15 6 2 2 4
The following transformations are given.T1 is a rotation of 90° clockwise about the origin.T2 is a reflection in the line y = —x.T3 is an enlargement of factor 2 centered at the origin.Given
The matrix A is given asa. Find the invariant lines for this matrix.b. The matrix A is applied to the vertices of the triangle PQR, and the resulting image has vertices at the points (11, 12), (19,
Given that and Bfind:a. AB2b. A2 - AB (2 -5 1 A = 6. 1 3 -2, 0 -2,
Find the inverse of the matrix 1 3 -2 A = |0 4 1 1 1 2,
State which of the following matrices are invertible.a.b.c. nす
Find the inverse of the matrix of the following matrices.a.b.c.d. 7 -2 -5
Find the determinant of each of the following matrices.a.b.c. 3 5 3 1 4 3 -5 3/
Determine the area of the image produced in each case.a. Triangle A(1, 3), B(1, 7), C(4, 3) transformed byb. Square A(1, 1), B(7, 2), C(6, 8), D(0, 7) transformed byc. Trapezium represented
Given thatand calculate the matrices:a. 2A + 5Bb. AB2c. BA2d. 2A + 3A2 1 2 -5 9 -1 -5 10, A = 3
Given that find the matrix A2 and, hence, show that A(A – 5I) = 21, where I is the identity matrix. From this equation show that the inverse matrix (1 2) A = 3 4
You are given the matrixShow that the row operation r1 → cr1 – ar3 changes the determinant by a factor of c. ad g be h A = le f i fi)
Find the invariant lines for each of the following shearing matrices.a.b.c. 3 1
Ifshow that A = 1
Determine if the following matrices are singular or non-singular.a.b. 1 2 3 2 8 7 1 10 11,
Given that find the value of the constant a. a a 2a За -а = 2 4а 2а а
Find the single matrix that is a combination of 90° anticlockwise rotation about the origin, followed by an enlargement of factor 2 with centre of enlargement origin, followed by a reflection in the
Given that andfind the determinant of the matrix AB. 1 2 3 A = ( 0 1 4 1 2 0/
You are given the matricesEvaluate BICIA, where I is the identity matrix. 0 -3 0 -1 andC = 5 9 -3 -2 1 1 8 3 6 0 0 -1 0, A = 4 B = 1 7 3 4.
You are given the matrixa. Find A2 and A3.b. Find the value of k such that A3 – kA2 + 2A – 41 = 0c. Hence, determine A-1. 1 0 2) A = 2 1 1 \0 1 1,
Determine the single matrix required for each of the following transformation combinations.a. Rotation about the z-axis by 90° clockwise, followed by a stretch of factor 2 in the x-direction,
The matrix A is given asa. Find A2 and A3.b. Determine An. (you do not need to prove your result. 2. 0 A = -1 0 -1 0 1,
Given thatand that determine C, where ACB = I. 0 2) 2 1 -1 -1 0/ 1 A =
If find B such that AB = I. A = 1 2 6 10 3 3 7 00 N6600 323 2.
Find the Cartesian form for the polar equation r = sin θ.
Find the area contained in the curve r = θ between θ = π/4 and θ = π/2.
The curve C has polar equation r = a(1 — e-θ), where a is a positive constant and 0 ≤ θ < 2π.i. Draw a sketch of C.ii. Show that the area of the region bounded by C and the lines θ = ln 2
Find the coordinates of all the points where the curve r = cos θ + 2 meets the coordinates axes.
The curve C has polar equation r= 3 + 2 cos θ, for -π < θ ≤ π. The straight line I has polar equation r cos θ = 2.Sketch both C and I on a single diagram.Find the polar coordinates of the
Find the maximum distance of the curve r = cos θ - sin θ from the origin, and the coordinates at which this happens.
A curve, C, has polar equation r =2 - sin θ, for 0 ≤ θ ≤ 2π. Sketch C.
Given that r = 1 + cos θ, show that the maximum value of y is 3√3/4.
The curve C has polar equation r = 2 sin θ(1 - cost)), for θ ≤ 9 ≤ π.Find dr/dθ and hence find the polar coordinates of the point of C that is furthest from the pole.Sketch C.Find the exact
A Cartesian equation is given as x3 - y + y2x = 0. Find the polar equivalent.
Without using a calculator, find the area bounded by the curve r = eθ and the lines θ = 1 and θ = 2.
Sketch the curve r = 1 - 2sin θ for the interval [0, 2π]. Sketch also the inner loop and state its domain.
Find the maximum distance of the curve r = eθ cos (θ – π/6) from the origin for the interval 0 ≤ θ ≤ π/2.
A polar curve is given as r = cos 2θ. Show that the Cartesian form is (x2 + y2)3/2 = x2 - y2.
The diagram shows the curves r = 1 + cos θ and r = 1 + sin θ. Without using a calculator, find the area of the shaded region, giving your answer in an exact form. r =1+ sin 6 - 20 r=1+ cos 0 1.5-
Sketch the curve r = cos2 θ, for -π/2 ≤ θ ≤ π/2.
A polar curve is given as r = cos 2θ + cos θ.a. Show that x = cos θ + cos2 θ - 2cos3 θ.b. Differentiate the result in part a and show that for stationary points sinθ(6 cos2 + cos θ - 1) =
Sketch the curve r = sec (θ – π/4), showing the coordinates of all points of intersection with the coordinate axes.
Without using a calculator, find the area enclosed by the curve r = θ eθ and the lines θ = 1 and θ = 2.
The curve, C, is defined as r = cos 3θ, for 0 ≤ θ ≤ 2π. Sketch C.
In each case, find the value of p such that the vectors given are perpendicular.a. pi + 2j + 3k and 4i + j + 2kb. 3i + 5j + 2k and i + pj - 4k
For each case, find the vector equation of the line through the points given. Give your answer in the form r = a + bt.a. (2, 3, 5) and (-7, 1, 6)b. (4, 1, 1) and (-5, 6, 1)c. (0, 2, 4) and (1, 1, -1)
Find the equation of the plane passing through the points given in each case.a. A(2, 4, 1), B(3, 0, -I), C(8, 1, 1) b. A(3, -2, 3), B(1, -8, 9), C(1, 0, 2) c. A(1, -3, 0), B(5, 2, -4), C(3,
The lines and 12 have equations r = 8i + 2j + 3k + 2(1 - 2j) and r = 5i + 3j - 14k + 12(2j - 3k) respectively. The point P on It and the point Q on I2 are such that PQ is perpendicular to both I1 and
Determine the equation of the line that passes through (1, 7, -2) and is perpendicular to both 2i - 3j + 5k and 41 - j + 2k.
The cross product of i + aj + k and 3i + 4k is 8i - j - 6k. What is the value of α?
Find the equation of the plane that contains the line r = 2i - j + 3k + (-i + 4j - 5k)t and the point (2, 6, 3).
Find a Cartesian equation of the plane H passing through the points with coordinates (2, —1, 3), (4, 2, - 5) and (-1, 3, -2).The plane ∏2 has Cartesian equation 3x - y + 2z = 5. Find the acute
Given that the cross product of 2i + αj - k and βi + 4j + 2k is 10i - 5j + 5k, find the values of α and β.
Determine the equation of the line that passes through (3, -5, I) and is perpendicular to both -i + 2j - 4k and 3i + 2j.
Find the equation of the plane that is perpendicular to the planes 2x - 4y + z = 10 and x - 4z = 2 and that contains the point (2, 3, 1).
The points A, B, C have position vectors 4i + 5j + 6k, 51 + 7j + 8k, 2i + 6j + 4k, respectively, relative to the origin O. Find a Cartesian equation of the plane ABC.The point D has position vector
Find the common perpendicular to the two vectors given in each case.a. a = 5i - 8j + 2k, b = 3i - 4j - k,b. a = 2i - 10j + 3k, b = 5i - 7j + 4kc. a = 12j + 7k, b = -6i + j + 9kd. a = 21 + 17j, b = 41
Three points are given as A(2, 2, 1), B(1, 0,4) and C(2,-3, 1). Find the shortest distance between the point C and the line through A and B. Give your answer in the form a√b/c.
Find, in Cartesian form, the equation of the plane containing the points A(1, -1, 1), B(0, 0, 5) and C(2, 3, 7).
Find the area of the triangle in each of the following cases.a. Triangle OAB with A(6, -7, 21) and B(3, 2, -5).b. Triangle ABC with A(0, 0, 2), B(-4, 9, 3) and C(2, 0, 7).c. Triangle ABQ with A(5, 1,
The points P and Q are on the lines r = k + (2i + 3j + k)s and r = i + j + 4k + 4jt, respectively. Given that P̅Q̅(vector) is perpendicular to both lines, find the position vectors O̅P̅(vector)
Find the shortest distance between the point (5, 2, 6) and the plane that passes through the point (2,-3, 2) and is perpendicular to the direction i + j.
Two points are given as A(2, 1, 3) and B(5, k, 7). Given that the triangle OAB has area √3/2, determine the possible values of k.
The lines r = i - 2j + 3k + (4i - j)s and r = 3i – 5k + (i + 4j + 3k)t are skew.a. Show that the lines do not intersect.b. Find the angle between the lines.c. Find the shortest distance between the
Find the acute angle between the planes ∏1: 2x - y + 4z = 13 and ∏2: 3y - 2z = 4.
In each of the following cases, determine the volume of the tetrahedron.a. Tetrahedron OABC with A(-2, 4, 1), B(8, 1, 0) and C(5, 6, -7).b. Tetrahedron ABCD with A(1, 1, -1), .8(0, 4,0), C(0, 6, 7)
The plane P is given as x + 2y - 5z = 7 and the line l as r = -3i + 4j + 6k + (11i + j + 8k)t.a. Find the angle between the line and the plane.b. Find the equation of the plane that contains the line
The lines Li r = - k + (1 - j - 2k)t and L2 r = 2i + j + (3i + cos θj + k)s, where 0 ≤ θ < 2π, are skew.a. Show that these lines do not intersect regardless of the value of θ.b. Determine
The tetrahedron A(2, 5, m), B(1, 7, 2), C(1, 8, 3) and D(9, 8, -1) has volume 2. Determine the value of m.
In both of the following cases, determine the value(s) of the unknown constants for which the two lines intersect.a. r = ai + 4j + 2k + (i - j - 3k)s and r = 4i + 3j - k + (2i + 2j - k)tb. r = 2j +
Find the distance of the point (p, q, r) from the plane ax + by + cz = d, giving your answer in a simplified form.
Four points are given as A(-1, -2, 3), B(0, 0, 9), C(-2, 4, -1) and D(2,7, 1).Find the shortest distance between the plane through ABD and the point C, giving your answer to 3 significant figures.
Prove, by mathematical induction, that n3 + 2n is divisible by 3 for all n ≥ 1.
Prove, by mathematical induction, that Ere + 3) =n(n- + 1) (n + 5) for all n 1. r=1
Prove by mathematical induction that, for all non-negative integers n, 112n + 25n + 22 is divisible by 24.
Prove, by mathematical induction, that the sum of the first n terms of an arithmetic sequence iswhere a is the first term and d is the common difference. [2a + (n- 1)d], 2.
It is given that ur = r x r! for r = 1, 2, 3, .... Let Sn = u1 + u2 + u3 + … + un. Write down the values of 2! – S1, 3! - S2, 4! - S3, 5! - S4.Conjecture a formula for Sn.Prove, by mathematical
Given that un+1 = 2un + 1 and that u1 = 1, show, by mathematical induction, that un = 2n – 1 for all n ≥ 1.

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