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

Prove that cosh-1x = ln(x + √x2 – 1).
a. Show, by using exponential form, that the curve with equation y = cosh 2x + sinhx has exactly one stationary point. b. Determine, in exact logarithmic form, the x-coordinate of the turning
Solve 7 = 17 tanh + 28 sechx.Write all solutions in exact form unless otherwise stated.
Prove the following hyperbolic identities.a. cosh(A – B) = cosh A cosh B – sinh A sinh Bb. sinh3x = 3 sinhx + 4 sinh3xc. sinhA – sinh B = 2cosh(A + B/2) sinh(A – B/2)
Prove that tanh-1 ½ ln(1 + x/1 – x).
Solve cosechx - 2cothx = 2.Write all solutions in exact form unless otherwise stated.
Given that tanh(x/2) = t, prove that sinhx = 2t/1 – t2.
Prove that sech-1 x = ln(1 + √1 – x2/x).
Sketch the graph of y = |3 cosh x - 5|.Write all solutions in exact form unless otherwise stated.
Prove that tanh-1 A + tanh-1 B = tanh-1(A + B/1 + AB).
Sketch the graph of y = 4 - cosech(x - 2).Write all solutions in exact form unless otherwise stated.
a. Prove that 4cosh3x – 3coshx = cosh 3x.b. Hence, solve cosh3x = 5 coshx.
Sketch the graph of y = |2 + 1/2 coth(x + 1)|.Write all solutions in exact form unless otherwise stated.
Solve 4cosh x + sinhx = 8. Write your answers in the form b ln a, where a and b are integers.Write all solutions in exact form unless otherwise stated.
Solve 10cosh x + 2 sinhx = 11, giving your answers in the form In a, where a is a rational number.Write all solutions in exact form unless otherwise stated.
Given that sech-1 x = In (1 + √1 – x2/x, solve 4sechx - 3 tanh2x = 1.Write all solutions in exact form unless otherwise stated.
In each case, state whether or not the given vectors are eigenvectors of the matrix. If they are eigenvectors, write down their corresponding eigenvalues. a.b.c. 1 2 4 3 A = and ez =
Given that the matrix A has eigenvalue λ and corresponding eigenvector e, find the eigenvalue for A2.
State whether or not the following matrices are diagonalizable.a.b.c. 3 -2 2 -1
Using matrix algebra, determine whether each of the following has unique solutions and, if so, state those solutions.a. 3x – y = 6 -6x + 2y = -13b. 2x + y = 4, 3x – y = 7c. 5x – 4y = 2, 10x –
Find the eigenvalues and corresponding eigenvectors of the matrixFind a non-singular matrix M and a diagonal matrix D such that (A – 21)3 = MDM-1, where I the 3 × 3 identity matrix. 3 A = 3
Given thatShow that the following vectors are eigenvectors, and determine their eigenvalues. 1 2 4 01 0 3 1 2, A =
The matrix A has eigenvalue A and corresponding eigenvector e, and the matrix B has eigenvalue is and corresponding eigenvector e. Show that the matrices AB and BA have the same eigenvalues and
Given that the matrixis not diagonalizable over reals, find the values of k. -1 k -7 -3
Matrix and matrix by considering AX = B, where find:a. α and β such that there are no solutionsb. α and β such that there are an infinite number of solutionsc. α and β such that
You are given the matrix Find the characteristic equation of A.Hence, or otherwise, determine the unknown constants for A3 + αA2 + βA + γI = 0.Hence, or otherwise, find A-1. 1 0 A = (0 2
Given thathas eigenvectors determine the corresponding eigenvalues.By considering M2e1 and Me1, find the eigenvalues for M2.Hence determine the eigenvalues and corresponding eigenvectors of
The matrix A has eigenvalue λ and corresponding eigenvector e, and the matrix B has eigenvalue µ and corresponding eigenvector e. Find the eigenvalue and corresponding eigenvector of A - 2B.
Given that the matrixis diagonalizable, find the smallest positive value of k, is an integer, that gives integer eigenvalues. -7 -10 k 4-
Determine if the following systems of equations have a unique solution, an infinite number of solutions or no solution. If there is a unique solution or an infinite number of solutions, calculate the
A 3 × 3 matrix A has eigenvalues -1, 1, 2, with corresponding eigenvectors,respectively.Findi. The matrix A,ii. A2n, where n is a positive integer. 1
Find the eigenvalues and corresponding eigenvectors for the following matrices. a.b. 3 4
Find the eigenvalues and eigenvectors for each of the following matrices.a. b. 4 -2) -6 5.
Find the values of the following matrices.a.b. 4 0 2 -3
For the system of equation, find the value of k such that there is an infinite number of solutions. x- y = 1 2x - y + 5z = 4 x+ 2y + 15z =k
The matrixhas eigenvectors Find the value of r. -11 3 -6 G = 8 -2 9. 8 4 r -6
For each of the following matrices, find its eigenvalues and eigenvectors. Find also the eigenvalues of B = A2 - 3I.a.b. -2 A =0 1 5, 2 1 1 2
Find which of the following matrices are diagonalizable.a.b.c. 1 0 2 0 3 2 2 01
Three planes are given:Find the unique point where all planes intersect. x - 4y + 4z = 3 x- 7y + 11z = 4 2x - 5y + 3z = 5
Given that has eigenvectorsfind the values of the corresponding eigenvalues, as well as p and q. p -3 0 1 2 1 -1 A = q 4,
The matrix has eigenvectorsa. Find the eigenvalues of A3.b. The matrix B = A - A2. Find the eigenvalues and eigenvectors of B. 1 /4 0 A =0 5 0 3 -6/
The matrix has eigenvectors. Find the matrices P and H such that B4 = PHP-1, where B = A – 3I. 3 3 2 A =|1 5 0 0 0 4
For the three planes, find the point of intersection. x - 6y = 2 2x + 4y - 17z = 4 3x + 12y – 33z = 6
Show that the characteristic equation for the matrix 4 1 -6 -2 3 is 13 – 622 – 92 + 14 0. 8 3 5 4
For the matrix ,find the eigenvalues and eigenvectors for the cases when k = 0 and k = 2. Explain why the matrix A cannot be diagonalized when k = 2. k 1 2) A = 0 2 3 0 0
Given thathas eigenvectorfind the corresponding eigenvalue and the value of a.Hence, find the remaining eigenvalues and eigenvectors. 1 5 7\ A = 1 3 -1 5/
For the system of equations, state the number of solutions when:a. a = 5, b = 5b. a = 10, b = 10c. a = 10, b = 5 x+ y + 3z = 1 x - 2y + 2z = -1 Зх + бу + аг 3b %3D + az:
Find the eigenvalues and corresponding eigenvectors for the matrix a 2) a 2 -26 2 b 0 2a 1, -3
If a matrix, A, has eigenvalue A and corresponding eigenvector e, show the following.a. Ae + A2e = (λ + λ2)eb. Ae + A-1e = (λ + 1/λ)e
You are given the matrix where the eigenvalues are λ1, λ2, λ3, λ4.a. Write down the values of λ1, λ2, λ3, λ4.b. Ifdetermine the value of A6. 1 0 2\ 0 2 A = 0 0 -1 1 1 0 0 0 -2 3.
For each case, find the first derivative dy/dxa. xy = exb. y2x = dyc. tan(x + y) = y
For each case, differentiate the set of parametric equations to obtain the second derivative d2y/dx2.a. x = t2, y = 3t + 1b. x = et, y = t2
By differentiating y = ½ e2x + 1/2e-2x, show that dy/dx = 2sinh 2x.
Find the value of the second derivative d2y/dx2 of x2 = yex + y2 at the point (0, 0).
Find the fourth derivative of each of the following functions.a. y = ex2 y b. y = xcosx
Find the value of the second derivative d2y/dx2 of the parametric equations x = t3, y = 4/t, at the point where t = 2.
Given that y = tanh-1(2x + 3), find dy/dx.
Find the first three non-zero terms of the Maclaurin series for f(x) = 2/(2x2 + 1)3/2.
A curve has equation x2 – 6xy + 25y2 = 16. Show that dy/dx = 0 at the point (3, 1).By finding the value of d2y/dx2 at the point (3, 1), determine the nature of this turning point.
Find the value of the first and second derivatives of In (x + y) = 2y at the point (1, 0).
Find the values of dy/dx and d2y/dx2 of the parametric equations x = e3t, y = et, at the point where t = 1.
Find the first derivative of y = sin-1(x3).
By considering the expansion of y = In(2x + 4), find an approximation, using five terms, to 1n4.02. Give your answer to 7 decimal places.
Given that y = cos-1(1/2x), find dy/dx and d2y/dx2.Hence, determine the Maclaurin expansion of y = cos-1 (1/2x), giving the first three non-zero terms.
Given that xy = sin (x + y), find the first and second derivatives with respect to x.
An implicit curve, C, is defined as y2 = In x + ey - 1. Given that C passes through the point (1, 0), find the values of dy/dx and d2y/dx2 at this point.
Given that x = t3, y = 3t2, find the value of d2y/dx2 when t = 2.
Differentiate the following functions with respect to x. Simplify your answer where possible.a. y = ln cosh 4xb. y = x2 sinh(2x2 – 1)c. y = xcosh-15xd. y = tan-1(x + y)
Find the Maclaurin series for the following functions, in each case giving the first four non-zero terms.a. f(x) = secxb. f(x) = tan-1 xc. f(x). sinhx
A parametric curve is defined as x = In (t + 1) and y = et, where t > 0.a. Find dy/dx in terms of t.b. Show that the second derivative with respect to x is always positive.
You are given the function f(x) = cot-1 x.a. Find the Maclaurin series for f(x), giving the first four non-zero terms.b. Use your result from part a to find the series for 1/1 + x2, also giving the
A curve is given as xyα = β, where α, β are constants.a. Find dy/dxb. Show that d2y/dx2 = y(1 + α)/α2x2
The parametric curve x = t2 + t, y= In t is valid for t > 0.a. Show that there are no turning points.b. Show that the second derivative with respect to x is 4t + 1/t2(2t + 1)3
By first finding a four-term Maclaurin series for f(x) = sinh-lx, determine an approximation with respect to x for sink-1 0.2. Give your answer to 8 decimal places.
A curve is defined as y = cosh3x2.a. Find the first derivative with respect to x, stating the number of turning points.b. Find the second derivative with respect to x.
Using standard results, or otherwise, find the Maclaurin series, giving the first four non-zero terms, for the following functions.a. f(x) = cos 3x b. f(x) = x2/1 – 3xc. f(x) = cosh
The curve C is given as (x + y)6 = x. Given that the curve also passes through the point (1, 0), determine the values dy/dx and d2y/dx2 at this point.
If x = sin2 t, y = cos3 t is valid for 0 ≤ t ≤ 2π, find the second derivative with respect to x in terms of t.
A curve, C, is defined as being x = 3 tanh-1 t, y = sin 2t, where 1 ≤ t ≤ 1.a. Find the values of r where the gradient is zero.b. Find the second derivative with respect to x when t = 0.
Establish a result for the first derivative with respect to x for the following functions.a. ay = sin-1 bxb. ay = tan-1 bxc. ay = sinh-1bxd. ay = cosh-1bxe. ay = tanh-1 bx
An implicit curve is defined as sin 2xcos 2y = √3/4. It is known that the curve passes through the point P(π/3, π/6). Find, at the point P:a. dy/dxb. d2y/dx2
Given that x = t2 – 1/g, y = t2 + 1/5, find:a. The value(s) oft for any stationary point(s)b. The nature of any turning point(s).
Consider the function y = tanhx.a. By differentiating twice, show that y" = -2y + 2y3.b. Differentiate with respect to x up to y(5) and use this to determine the Maclaurin series of y = tanhx.
The parametric curve represented by x = e2t + e-2t, y = e2t – e-2t is valid for all t.a. Find dy/dx and state the number of turning points.b. Show that the second derivative with respect to x is -
You are given that Find, without using a reduction formula, a relationship between I5 and I3. tan xdx.
By using the integral test or otherwise, determine whether converges or diverges. 00 Σ 1 n+ 2 n=1
Find the length of the curve y = 4x2/3 from x = 1 to x = 2. Give your answer to 3 decimal places.
A curve has equation y = 1/3x3 + 1.The length of the arc of the curve joining the point where x = 0 to the point where x = 1 is denoted by s.Show that The surface area generated when this arc is
Determine the integral 25 - 9 -2 dx.
Given that find a relationship between I6 and I4. 16 x'e"dx
Find the surface area of revolution formed when the curve represented by x = sin t, y = cost, for t = 0, t = π/2 is rotated about the x-axis by 2π radians.
Show thatDiverges. In n n +4 n=1
LetShow that, for n ≥ 1, (3 + 2n)In = 2nIn-1.Hence find the exact value of I3. x"(1 - x)?dx, for n>0. %3D
Evaluate the integral giving your answer in an exact form. tan tan-xdx Jo
Given thatfind a relationship between I4 and I3. x(1 + x)*dx xp
The curve C is given as y = 1/3(x — 4)2/3. Find the length of the curve from x = 4 to x = 9.
Using the integral test or otherwise, determine whetherconverges or diverges. 1 n=1n" + 1
DetermineHence, determine the value ofgiving the exact solution. 0.5 cosh- 2x dx.
Find the area under the curve y = sinh 2x from x = ln 3 to x = ln 5.
Find a reduction formula forstating the values of n for which it is valid. 4 sin" 2xdx Jo %3D

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