 25.1: A geometric sequence has S1 = 2 and S2 = 8. Find: a the common rati...
 25.2: Show that the sum of the first forty terms of the series ln 2 + ln ...
 25.3: Suppose f(x) = bex and g(x) = ln(bx). Find: a (f g)(x) b (g f)(x) c...
 25.4: Consider f(x) = 2(x b)2 + 2. a State the coordinates of the vertex....
 25.5: a Expand (x 2)3. b Hence, find the coefficient of x3 in (3x2 7)(x 2)3.
 25.6: Consider f(x) = p1 2x. Find: a f(0) b f(4) c the domain of f d the ...
 25.7: Let a = sin 20 and b = tan 50. In terms of a and b, write expressio...
 25.8: Suppose f(x) = cos x and g(x)=2x. Solve the following equations on ...
 25.9: Consider the graphs illustrated. Copy and complete the following ta...
 25.10: In the given figure, the perimeter of sector AB is (12 + 2) cm. a F...
 25.11: Let f(x) = x2 + 6. a Can you solve f(x)=3? b What does this tell us...
 25.12: Consider u = i 2j + k and v = 3i + pj k. a Find p if u is perpendic...
 25.13: Consider the parallelogram ABCD illustrated. a Find !BA and !BC. b ...
 25.14: a, b, c, d, e, f, g, h, i, j, k, l, and m are 13 data values which ...
 25.15: For the data set fa, b, cg, the mean is 17:5 and the standard devia...
 25.16: A dart is thrown at the dartboard shown. It is equally likely to la...
 25.17: For the function y = f(x) with graph shown, sketch the graphs of: a...
 25.18: Consider g(x)=3 2 cos(2x). a Find g0 (x). b Sketch y = g0 (x) for 6...
 25.19: A and B are mutually exclusive events where P(A) = x and P(B0 )=0:4...
 25.20: For the function g(x), the sign diagrams for g0 (x) and g00(x) are ...
 25.21: Consider f(x) = xe12x. a Show that f0 (x) = e12x(1 2x). b Find the ...
 25.22: A particle moves in a straight line so that its position s at time ...
 25.23: Solve for x: a log3 27 = x b e52x = 8 c ln(x2 3) ln(2x)=0
 25.24: The graph shows the velocity v m s1 of an object at time t seconds,...
 25.25: Suppose Z 2 1 f(x) dx = 10. Find the value of: a Z 2 1 (f(x) 6) dx ...
 25.26: a b Hence calculate Z 4 0 (sin cos ) 2 d.  + x + + x g'(x) g (x)...
 25.27: The following table shows the probability distribution for a discre...
 25.28: Consider the infinite geometric sequence: e, 1 e , 1 e3 , 1 e5 , .....
 25.29: a Use the formula n r = n! r! (n r)! to evaluate 6 2 . b Hence stat...
 25.30: The random variable X is normally distributed with mean and standar...
 25.31: The examination marks for 200 students are displayed on the cumulat...
 25.32: The line L has equation y = tan 3 x. a Find p given that the point ...
 25.33: The acceleration of an object moving in a straight line is given by...
 25.34: Consider f(x) = e3x4 + 1. a Show that f1(x) = ln(x 1) + 4 3 . b Cal...
 25.35: Given that sin A = 2 5 and 2 6 A 6 , find: a cos A b tan A c sin 2A
 25.36: Consider the infinite geometric sequence 160, 80p 2, 80, 40p2, .......
 25.37: A particle is initially located at P(3, 1, 2). It moves with fixed ...
 25.38: A journalist is investigating the consistency of online reviews for...
 25.39: Suppose g(x) = e x 4 where 0 6 x 6 4. a Sketch y = g(x) on the give...
 25.40: [QS] is a diagonal of quadrilateral PQRS where PQ = 3 cm, QR = 7 cm...
 25.41: Suppose f(x) = 1 4x2 + 3x + 4. a Find f0 (x) in simplest form. b i ...
 25.42: a Consider the geometric sequence: 4, 12, 36, 108, .... i Write dow...
 25.43: Suppose A has position vector 3i + 2j k, B has position vector 2i j...
 25.44: Consider the functions f(x) = 1 2x 1 and g(x) = p3x. Find exactly t...
 25.45: a Consider the quadratic function y = x2 + 12x 20. i Explain why th...
 25.46: Suppose f(x)=4x 3 and g(x) = x + 2. a Find f1(x) and g1(x), the inv...
 25.47: Hannah, Heidi, and Holly have different sets of cards, but each set...
 25.48: Bag C contains 4 blue and 1 yellow ticket. Bag D contains 2 blue an...
 25.49: Two identical tetrahedral dice are rolled. Their four vertices are ...
 25.50: A particle moves in a straight line such that at time t seconds, t ...
 25.51: The graph of f(x) = a sin b(x c) + d is illustrated. A is a local m...
 25.52: a Factorise 4x 2x 20 in the form (2x + a)(2x b) where a, b 2 Z +. b...
 25.53: Suppose f(x) = a cos 2x + b sin2 x where b < 2a, 0 6 x 6 2. a Show ...
 25.54: Suppose S(x) = 1 2 (ex ex) and C(x) = 1 2 (ex + ex). a Show that [C...
 25.55: The size of a population at time t years is given by P(t) = 60 000 ...
 25.56: a Find Z x2e1x3 dx using the substitution u(x)=1 x3. b Hence show t...
 25.57: Suppose f(x) is defined by f : x 7! cos3 x. a State the range of f....
 25.58: [PQ] is the diameter of a semicircle with centre O and radius 5 cm...
 25.59: The graph of the function f(x) = a(x h)2 + k is shown alongside. It...
 25.60: a Find the exact value of x for which: i 212x = 0:5 ii logx 7=5 b S...
 25.61: a Suppose 1 cos 2 sin 2 = p3 where 0 << 2 . i Show that tan = p 3 a...
 25.62: The sum of the first n terms of a series is given by Sn = n3 + 2n 1...
 25.63: Find the exact values of x for which sin2 x + sin x 2=0 and 2 6 x 6 2.
 25.64: If f : x 7! ln x and g : x 7! 3 + x, find: a f1(2) g1(2) b (f g)1(2).
 25.65: The equation of line L is r = 2i 3j + k + t(i + j k), t 2 R . Find ...
 25.66: For the function f(x), f0 (x) > 0 and f00(x) < 0 for all x 2 R , f(...
 25.67: In a team of 30 judo players, 13 have won a match by throwing (T), ...
 25.68: Use the figure alongside to show that cos 36 = 1 + p5 4 .
 25.69: Find a given that the shaded region has area 5 1 6 units2.
 25.70: What can be deduced if A \ B and A [ B are independent events?
 25.71: Solve sin cos = 1 4 on the domain 6 6 .
 25.72: f is defined by x 7! ln(x(x 2)). a State the domain of f(x). b Find...
 25.73: Hat 1 contains three green and four blue tickets. Hat 2 contains fo...
 25.74: A normally distributed random variable X has a mean of 90. The prob...
 25.75: The discrete random variable X has probability function P(X = x) = ...
 25.76: Given x = log3 y2, express logy 81 in terms of x.
 25.77: Matt has noticed that his pet rat Pug does not always eat the same ...
 25.78: The point A(2, 3) lies on the graph of y = f(x). Give the coordinat...
 25.79: Find a trigonometric equation in the form y = a sin(b(xc))+d which ...
 25.80: A and B are events for which P(A)=0:3 + x, P(B)=0:2 + x, and P(A \ ...
 25.81: Simplify: a 9log3 11 b logm n logn m2
 25.82: The graph describes the weight of 40 watermelons. a Estimate the: i...
 25.83: If f : x 7! 2x + 1 and g : x 7! x + 1 x 2 , find: a (f g)(x) b g1(x).
 25.84: A and B are two events such that P(A) = 1 3 and P(B) = 2 7 . Find P...
 25.85: Find x in terms of a if a > 1 and loga(x + 2) = loga x + 2.
 25.86: Consider the expansion (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab...
 25.87: If x + 1 x = a, find in terms of a: a x2 + 1 x2 b x3 + 1 x3
 25.88: The illustrated ellipse has equation x2 16 + y2 4 = 1. a Find the c...
 25.89: Consider f(x) = sin2 x. a Copy and complete the table of values: x ...
 25.90: The graph of f(x) = x + 1 x , x > 0 is shown. a Find f0 (x) and sol...
 25.91: A straight line passes through A(2, 0, 3) and has direction vector ...
 25.92: A cumulative frequency graph for the continuous random variable X i...
Solutions for Chapter 25: Miscellaneous questions
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Solutions for Chapter 25: Miscellaneous questions
Get Full SolutionsThis textbook survival guide was created for the textbook: Mathematics for the International Student: Mathematics SL, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 25: Miscellaneous questions includes 92 full stepbystep solutions. Since 92 problems in chapter 25: Miscellaneous questions have been answered, more than 11398 students have viewed full stepbystep solutions from this chapter. Mathematics for the International Student: Mathematics SL was written by and is associated to the ISBN: 9781921972089.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Solvable system Ax = b.
The right side b is in the column space of A.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.