Optional Mathematics: Unit-wise Objective Question Bank

Optional Mathematics in grade 8 is examined by your own school, in a written paper of 50 marks and two hours. Group A of that paper is ten one-mark objective items. They are the cheapest marks in the paper and the easiest to throw away: a sign slipped in a rotation rule, a degree confused with a grade, a median found before the data were put in order. This chapter drills those items unit by unit for all ten units of the Optional Mathematics 8 book, in the same order as the book.

Each section opens with a facts line, the few results that the items of that unit actually use. Then come twelve items, and then a key that gives the answer with a one-line reason. Work an item in under a minute, check it in ten seconds, and move on. The reasons matter more than the letters: a reason you can repeat is a mark you will not lose next time.

20.1 Relation and Function

Facts. (a,b)=(c,d) exactly when a=c and b=d. A\times B=\{(a,b): a\in A, b\in B\} and n(A\times B)=n(A)\times n(B). A relation from A to B is a subset of A\times B, so there are 2^{n(A)\times n(B)} relations. The domain is the set of first components and the range the set of second components; R^{-1} swaps every pair. A function gives every member of the domain exactly one image; a graph is a function if no vertical line meets it twice.

1. If (a+3,\ 2b-1)=(5,\ 7), then a+b is [1]a6b2c4d8
2. If (x,\ y)=(y,\ 4), then x is [1]a0b4c2d-4
3. If A=\{1,2,3\} and B=\{4,5\}, the number of members of A\times B is [1]a5b6c8d9
4. If n(A\times B)=12 and n(A)=4, then n(B) is [1]a3b8c16d48
5. For A=\{p,q\} and B=\{1,2\}, which pair belongs to A\times B? [1]a(1,p)b(q,2)c(2,q)d(p,q)
6. If n(A)=2 and n(B)=2, the number of relations from A to B is [1]a4b8c16d2
7. The range of R=\{(1,2),(2,4),(3,6)\} is [1]a\{1,2,3\}b\{2,4,6\}c\{1,2,3,4,6\}d\{2\}
8. On A=\{1,2,3,4\}, the relation R=\{(x,y): y=x+1\} is [1]a\{(1,2),(2,3),(3,4)\}b\{(1,2),(2,3),(3,4),(4,5)\}c\{(2,1),(3,2),(4,3)\}d\{(1,1),(2,2)\}
9. The inverse of R=\{(a,1),(b,2)\} is [1]a\{(1,a),(2,b)\}b\{(a,2),(b,1)\}c\{(1,b),(2,a)\}d\{(a,b),(1,2)\}
10. Which kind of arrow diagram is not a function? [1]aone-to-onebmany-to-onecone-to-manydconstant
11. If f(x)=5-2x, then f(-3) is [1]a-1b11c-11d1
12. If f(x)=x^2 with domain \{-2,-1,0,1,2\}, the range is [1]a\{-4,-1,0,1,4\}b\{0,1,4\}c\{0,1,2,4\}d\{1,4\}

Answers to 20.1

1. (a) — a+3=5 gives a=2; 2b-1=7 gives b=4; 2+4=6.

2. (b) — the second components give y=4, and the first give x=y=4.

3. (b) — 3\times2=6.

4. (a) — 4\times n(B)=12.

5. (b) — the first component must come from A and the second from B.

6. (c) — A\times B has 4 pairs, and a set of 4 has 2^4=16 subsets.

7. (b) — the range is the set of second components.

8. (a) — (4,5) is not allowed because 5\notin A.

9. (a) — swap the components of each pair.

10. (c) — one member with two images breaks the rule of a function.

11. (b) — 5-2(-3)=5+6=11.

12. (b) — squares are never negative, and repeated images are listed once.

20.2 Polynomials

Facts. A polynomial has only whole-number powers of the variable. The degree is the highest power (for a term in several variables, the sum of the powers). By degree: constant 0, linear 1, quadratic 2, cubic 3, biquadratic 4; by terms: monomial, binomial, trinomial. Standard form lists the powers in descending order; the first coefficient is the leading coefficient. Equal polynomials have equal coefficients of like powers. The degree of a product is the sum of the degrees.

1. Which of these is not a polynomial? [1]a3x^2+2b\sqrt{3}\,x+1c\frac{2}{x}+1d7
2. The degree of 3x^2y^2-5x^3+y is [1]a2b3c4d5
3. The polynomial 4x^3-2x+1 is a [1]acubic trinomialbcubic binomialcquadratic trinomialdbiquadratic
4. The leading coefficient of 2-7x+3x^2-x^3 is [1]a2b-7c3d-1
5. A polynomial of degree 1 is called [1]aconstantblinearcquadraticdcubic
6. If f(x)=x^3-2x+1, then f(2) is [1]a5b3c13d1
7. If (p-2)x^2+5x=3x^2+qx, then (p,q) is [1]a(5,5)b(1,5)c(5,3)d(3,5)
8. (2x^2-3x+1)+(x^2+3x-4) equals [1]a3x^2-3b3x^2+6x-3cx^2-3d3x^2-6x-3
9. (x^2+2x)-(3x^2-x) equals [1]a-2x^2+3xb-2x^2+xc4x^2+xd-2x^2-3x
10. (x-3)^2 equals [1]ax^2-9bx^2-6x+9cx^2+9dx^2-3x+9
11. (2x+1)(x-4) equals [1]a2x^2-7x-4b2x^2-9x-4c2x^2+7x-4d2x^2-8x-4
12. If f(x) has degree 4 and g(x) has degree 3, then f(x)\cdot g(x) has degree [1]a1b7c12d4

Answers to 20.2

1. (c) — \frac{2}{x}=2x^{-1} has a negative power; a surd coefficient such as \sqrt{3} is allowed.

2. (c) — the term 3x^2y^2 has degree 2+2=4, more than 3 or 1.

3. (a) — highest power 3 and three terms.

4. (d) — in standard form -x^3+3x^2-7x+2, the first coefficient is -1.

5. (b) — degree 1 is linear.

6. (a) — 8-4+1=5.

7. (a) — p-2=3 gives p=5, and q=5.

8. (a) — the x terms cancel: -3x+3x=0.

9. (a) — x^2-3x^2=-2x^2 and 2x-(-x)=3x.

10. (b) — (a-b)^2=a^2-2ab+b^2; the middle term -6x is not optional.

11. (a) — 2x^2-8x+x-4=2x^2-7x-4.

12. (b) — degrees add: 4+3=7.

20.3 Sequence and Series

Facts. The general term t_n gives the term in position n. Equal first differences d: t_n=a+(n-1)d; equal ratios: t_n=ar^{n-1}; equal second differences: the rule contains n^2. A number is a term only if solving t_n= number gives a natural number n. \sum_{k=1}^{n}t_k adds t_k for k=1 to n; the number of terms is upper limit - lower limit +1. 1+2+\cdots+n=\frac{n(n+1)}{2}; 1+3+\cdots+(2n-1)=n^2; t_n=S_n-S_{n-1}.

1. The 6th term of the sequence with t_n=n^2-1 is [1]a35b36c11d25
2. The general term of 5, 8, 11, 14, \ldots is [1]a3n+2b5ncn+3d3n+5
3. The next term of 1, 3, 6, 10, 15, \ldots is [1]a20b21c25d18
4. The next term of 64, 32, 16, 8, \ldots is [1]a0b2c4d6
5. Which term of 3, 7, 11, \ldots is 43? [1]a9thb10thc11thd12th
6. The general term of 1, -1, 1, -1, \ldots is [1]a(-1)^nb(-1)^{n+1}c-1dn
7. \sum_{k=1}^{3}k^2 equals [1]a6b9c14d36
8. \sum_{k=1}^{4}3 equals [1]a3b7c12d81
9. 1+3+5+7+9 in sigma notation is [1]a\sum_{k=1}^{5}(2k-1)b\sum_{k=1}^{9}kc\sum_{k=1}^{5}2kd\sum_{k=0}^{5}(2k+1)
10. The sum of the first 20 natural numbers is [1]a200b210c400d420
11. The number of terms in \sum_{k=5}^{12}t_k is [1]a7b8c12d17
12. If S_n=n^2 for every n, then t_3 is [1]a9b5c3d4

Answers to 20.3

1. (a) — 6^2-1=35.

2. (a) — the difference is 3 and each term is 2 more than 3n.

3. (b) — the differences are 2, 3, 4, 5, so the next is 15+6=21.

4. (c) — each term is half the one before.

5. (c) — t_n=4n-1=43 gives n=11.

6. (b) — it must be +1 when n=1; (-1)^{n+1} does that.

7. (c) — 1+4+9=14.

8. (c) — the constant 3 is added 4 times.

9. (a) — the kth odd number is 2k-1; option (d) starts at k=0 and has six terms, ending with 11.

10. (b) — \frac{20\times21}{2}=210.

11. (b) — 12-5+1=8.

12. (b) — t_3=S_3-S_2=9-4=5.

20.4 Limit

Facts. A sequence has limit l if its terms come as close to l as we wish and stay that close; the limit need not be a term. \lim_{n\to\infty}\frac{1}{n}=0; \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots approaches 1. x\rightarrow a means x approaches a but never equals it. For a polynomial, substitute; for a \frac{0}{0} form, factorise and cancel first. The limit exists only if the left-hand and right-hand values agree.

1. The sequence 2, 1.5, 1.25, 1.125, \ldots tends to [1]a0b1c2d1.1
2. \lim_{n\to\infty}\frac{5}{n} is [1]a5b0c\inftyd1
3. \lim_{n\to\infty}\frac{n+1}{n} is [1]a0b1c2d\infty
4. The sum \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots approaches [1]a\frac{1}{2}b1c2d\infty
5. \lim_{x\to3}(2x-1) is [1]a5b6c7d3
6. \lim_{x\to0}(x^2+4) is [1]a0b2c4dnot defined
7. \lim_{x\to2}\frac{x^2-4}{x-2} is [1]a0b2c4dnot defined
8. \lim_{x\to1}\frac{x^2-1}{x-1} is [1]a0b1c2dnot defined
9. For f(x)=\frac{x^2-9}{x-3}, the value f(3) is [1]a6b0c9dnot defined
10. "x\rightarrow5^-" means that x approaches 5 [1]athrough values greater than 5bthrough values less than 5cand becomes 5dfrom both sides
11. If \lim_{x\to a^-}f(x)=2 and \lim_{x\to a^+}f(x)=3, then \lim_{x\to a}f(x) [1]ais 2bis 3cis 2.5ddoes not exist
12. As the number of sides increases, the perimeter of a regular polygon inscribed in a circle of radius 7 cm approaches (take \pi=\frac{22}{7}) [1]a44 cmb154 cmc42 cmd49 cm

Answers to 20.4

1. (b) — the terms are 1+1, 1+0.5, 1+0.25, …; the extra part halves towards 0.

2. (b) — \frac{5}{n} shrinks without end.

3. (b) — \frac{n+1}{n}=1+\frac{1}{n}\rightarrow1.

4. (b) — the part still missing after n terms is \frac{1}{2^n}\rightarrow0.

5. (a) — substitute: 6-1=5.

6. (c) — substitute: 0+4=4.

7. (c) — \frac{(x-2)(x+2)}{x-2}=x+2\rightarrow4.

8. (c) — \frac{(x-1)(x+1)}{x-1}=x+1\rightarrow2.

9. (d) — \frac{0}{0} has no value, although the limit is 6.

10. (b) — the minus sign means "from the left", through smaller values.

11. (d) — the two sides disagree, so there is no single limit.

12. (a) — the polygons tend to the circle: 2\pi r=2\times\frac{22}{7}\times7=44 cm.

20.5 Matrix

Facts. A matrix with m rows and n columns has order m\times n and mn elements; a_{ij} is in row i, column j. In this bank a matrix written [(a,b),(c,d)] has first row a, b and second row c, d. Row, column, zero, rectangular and square matrices are named by shape; diagonal, scalar, unit, symmetric (a_{ij}=a_{ji}) and triangular matrices are square. Equal matrices have the same order and equal corresponding elements. Only matrices of the same order can be added or subtracted; kA multiplies every element by k.

1. A matrix of order 2\times3 has [1]a2 columnsb3 rowsc6 elementsd5 elements
2. In the matrix [(5,2,7),(1,9,4)], the element a_{21} is [1]a2b1c9d7
3. A matrix with only one column is a [1]arow matrixbcolumn matrixcsquare matrixdunit matrix
4. The matrix [(3,0),(0,3)] is a [1]aunit matrixbscalar matrixcrow matrixdzero matrix
5. A square matrix with a_{ij}=a_{ji} for all i and j is [1]adiagonalbsymmetricctriangulardzero
6. The unit matrix of order 2 is [1]a[(1,1),(1,1)]b[(1,0),(0,1)]c[(0,1),(1,0)]d[(0,0),(0,0)]
7. If [(x,3),(4,y)]=[(2,3),(4,5)], then x+y is [1]a5b7c9d10
8. Two matrices can be added only when they have [1]athe same number of rowsbthe same ordercthe same first elementda zero element
9. [(1,2),(3,4)]+[(4,3),(2,1)] equals [1]a[(5,5),(5,5)]b[(4,6),(6,4)]c[(3,1),(-1,-3)]d[(5,5),(5,6)]
10. [(5,6),(7,8)]-[(1,2),(3,4)] equals [1]a[(4,4),(4,4)]b[(6,8),(10,12)]c[(4,3),(2,1)]d[(5,4),(4,4)]
11. If A=[(2,-1),(3,0)], then -2A is [1]a[(-4,2),(-6,0)]b[(4,-2),(6,0)]c[(-4,-2),(-6,0)]d[(0,-3),(1,-2)]
12. For matrices of the same order, A+B=B+A is the [1]aassociative propertybcommutative propertycidentity propertydinverse property

Answers to 20.5

1. (c) — 2 rows and 3 columns make 2\times3=6 elements.

2. (b) — row 2, column 1.

3. (b) — by definition.

4. (b) — diagonal with equal diagonal elements; it would be the unit matrix only if they were 1.

5. (b) — the leading diagonal acts as a mirror.

6. (b) — 1s on the leading diagonal, 0s elsewhere.

7. (b) — x=2 and y=5.

8. (b) — only then do the elements have matching positions.

9. (a) — every pair of matching elements adds to 5.

10. (a) — every difference is 4.

11. (a) — multiply every element by -2; -2\times-1=2.

12. (b) — the order of adding does not matter.

20.6 Coordinate Geometry

Facts. Distance =\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}; from the origin \sqrt{x^2+y^2}. Internal division in the ratio m_1:m_2: \left(\frac{m_1x_2+m_2x_1}{m_1+m_2},\frac{m_1y_2+m_2y_1}{m_1+m_2}\right); mid-point \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). Slope m=\frac{y_2-y_1}{x_2-x_1}; horizontal lines have slope 0 and vertical lines no slope; y=mx+c has slope m and y-intercept c. Area of a triangle =\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|; zero area means collinear.

1. The distance between (1,2) and (4,6) is [1]a5b7c25d\sqrt{7}
2. The distance of (-5,12) from the origin is [1]a7b13c17d169
3. The mid-point of (-2,5) and (6,-1) is [1]a(2,2)b(4,4)c(-4,3)d(2,3)
4. The point dividing the join of (2,1) and (8,4) internally in the ratio 2:1 is [1]a(6,3)b(4,2)c(5,2.5)d(10,5)
5. If (3,4) is the mid-point of (1,2) and (x,y), then (x,y) is [1]a(5,6)b(2,3)c(4,6)d(7,8)
6. The slope of the line joining (-1,3) and (2,9) is [1]a2b3c\frac{1}{2}d6
7. The slope of a line parallel to the y-axis is [1]a0b1cnot definedd-1
8. The equation of the x-axis is [1]ax=0by=0cy=xdx+y=0
9. The line y=3x-2 cuts the y-axis at [1]a(0,3)b(0,-2)c(-2,0)d(3,0)
10. The slope of the line 4x-2y=5 is [1]a4b-2c2d-4
11. The area of the triangle with vertices (0,0), (6,0) and (0,4) is [1]a24b12c10d6
12. The "triangle" with vertices (1,1), (2,2) and (3,3) has area [1]a0b1c3d6

Answers to 20.6

1. (a) — \sqrt{9+16}=5.

2. (b) — \sqrt{25+144}=13.

3. (a) — \left(\frac{-2+6}{2},\frac{5-1}{2}\right).

4. (a) — \left(\frac{2(8)+1(2)}{3},\frac{2(4)+1(1)}{3}\right)=(6,3).

5. (a) — x=2(3)-1=5 and y=2(4)-2=6.

6. (a) — \frac{9-3}{2+1}=2.

7. (c) — its run is 0, and division by 0 is not defined.

8. (b) — every point on the x-axis has y=0.

9. (b) — put x=0: y=-2.

10. (c) — y=2x-\frac{5}{2}.

11. (b) — a right triangle with legs 6 and 4: \frac{1}{2}\times6\times4.

12. (a) — the points lie on the line y=x, so they are collinear.

20.7 Trigonometry

Facts. Right angle =90^\circ=100^g=\left(\frac{\pi}{2}\right)^c; 1^g=100^c; \pi^c=180^\circ; \frac{D}{90}=\frac{G}{100}=\frac{2R}{\pi}. \sin\theta=\frac{p}{h}, \cos\theta=\frac{b}{h}, \tan\theta=\frac{p}{b}, with reciprocals cosec, sec, cot. \sin^2\theta+\cos^2\theta=1, 1+\tan^2\theta=\sec^2\theta, 1+\cot^2\theta=\mathrm{cosec}^2\theta. \sin30^\circ=\cos60^\circ=\frac{1}{2}, \tan45^\circ=1, \tan60^\circ=\sqrt{3}. Signs: All, Sin, Tan, Cos. \sin(-\theta)=-\sin\theta, \cos(-\theta)=\cos\theta; 90^\circ\pm\theta changes the ratio to its partner, 180^\circ\pm\theta keeps it.

1. 60^\circ in grades is [1]a54^gb66\frac{2}{3}^gc60^gd75^g
2. 90^g in degrees is [1]a81^\circb100^\circc90^\circd99^\circ
3. \left(\frac{\pi}{6}\right)^c in degrees is [1]a60^\circb30^\circc45^\circd15^\circ
4. 1^g is equal to [1]a60^cb100^cc90'd100'
5. If \sin\theta=\frac{3}{5} and \theta is acute, then \cos\theta is [1]a\frac{4}{5}b\frac{3}{4}c\frac{5}{4}d\frac{5}{3}
6. \sec\theta is equal to [1]a\frac{1}{\sin\theta}b\frac{1}{\cos\theta}c\frac{1}{\tan\theta}d\frac{\sin\theta}{\cos\theta}
7. 1+\tan^2\theta is equal to [1]a\sec^2\thetab\mathrm{cosec}^2\thetac\cos^2\thetad\cot^2\theta
8. \cos60^\circ+\sin30^\circ is equal to [1]a0b\frac{1}{2}c1d\sqrt{3}
9. \tan30^\circ\times\tan60^\circ is equal to [1]a1b\sqrt{3}c\frac{1}{\sqrt{3}}d3
10. \sin150^\circ is equal to [1]a-\frac{1}{2}b\frac{1}{2}c\frac{\sqrt{3}}{2}d-\frac{\sqrt{3}}{2}
11. \cos(-\theta) is equal to [1]a-\cos\thetab\cos\thetac\sin\thetad-\sin\theta
12. From a point 20\sqrt{3} m from the foot of a 20 m pole, the angle of elevation of the top is [1]a30^\circb45^\circc60^\circd90^\circ

Answers to 20.7

1. (b) — 60\times\frac{10}{9}=66\frac{2}{3} grades.

2. (a) — 90\times\frac{9}{10}=81^\circ.

3. (b) — \frac{180^\circ}{6}=30^\circ.

4. (b) — a grade has 100 centesimal minutes.

5. (a) — p=3, h=5, so b=4.

6. (b) — secant is the reciprocal of cosine.

7. (a) — divide p^2+b^2=h^2 by b^2.

8. (c) — \frac{1}{2}+\frac{1}{2}=1.

9. (a) — \frac{1}{\sqrt{3}}\times\sqrt{3}=1.

10. (b) — \sin(180^\circ-30^\circ)=\sin30^\circ, positive in quadrant II.

11. (b) — reflecting in the x-axis does not change x.

12. (a) — \tan\theta=\frac{20}{20\sqrt{3}}=\frac{1}{\sqrt{3}}.

20.8 Vector

Facts. A scalar has magnitude only; a vector has magnitude and direction. \overrightarrow{AB}=\binom{x_2-x_1}{y_2-y_1} (head minus tail); |\vec{a}|=\sqrt{x^2+y^2}; direction \tan\theta=\frac{y}{x}. Unit vector \hat{a}=\frac{\vec{a}}{|\vec{a}|}; null vector \binom{0}{0}; \overrightarrow{BA}=-\overrightarrow{AB}. k\vec{a}=\binom{kx}{ky}; \vec{a}\parallel\vec{b} when x_1y_2-x_2y_1=0 (like if \vec{a}=k\vec{b} with k>0, unlike if k<0). \vec{a}\pm\vec{b} by components; \overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}.

1. Which of these is a scalar quantity? [1]aweightbdisplacementcmassdvelocity
2. If A is (2,-1) and B is (5,3), then \overrightarrow{AB} is [1]a\binom{3}{4}b\binom{7}{2}c\binom{-3}{-4}d\binom{3}{2}
3. The magnitude of \binom{-6}{8} is [1]a2b10c14d100
4. The vector \binom{1}{1} makes with the positive x-axis an angle of [1]a30^\circb45^\circc60^\circd90^\circ
5. The unit vector in the direction of \binom{0}{-5} is [1]a\binom{0}{-1}b\binom{0}{1}c\binom{0}{-5}d\binom{1}{0}
6. The magnitude of the null vector is [1]a1b0cnot defineddany number
7. \overrightarrow{BA} is equal to [1]a\overrightarrow{AB}b-\overrightarrow{AB}c2\overrightarrow{AB}d\vec{0}
8. The vectors \binom{2}{6} and \binom{1}{3} are [1]aunlike vectorsblike vectorscunit vectorsdnull vectors
9. If \vec{a}=\binom{4}{-2}, then \frac{1}{2}\vec{a} is [1]a\binom{2}{-1}b\binom{8}{-4}c\binom{4}{-1}d\binom{2}{-2}
10. \binom{k}{2} is parallel to \binom{6}{4} when k is [1]a2b3c4d12
11. \binom{3}{-1}-\binom{1}{2} is equal to [1]a\binom{2}{-3}b\binom{4}{1}c\binom{2}{1}d\binom{-2}{3}
12. By the triangle law, \overrightarrow{PQ}+\overrightarrow{QR} is equal to [1]a\overrightarrow{PR}b\overrightarrow{RP}c\overrightarrow{QP}d\vec{0}

Answers to 20.8

1. (c) — mass has no direction; weight is a force and acts downwards.

2. (a) — \binom{5-2}{3+1}.

3. (b) — \sqrt{36+64}=10.

4. (b) — \tan\theta=1 in quadrant I.

5. (a) — divide by the magnitude 5.

6. (b) — both components are 0.

7. (b) — same length, opposite direction.

8. (b) — \binom{2}{6}=2\binom{1}{3} with a positive scalar.

9. (a) — halve each component.

10. (b) — 4k-6\times2=0 gives k=3.

11. (a) — \binom{3-1}{-1-2}.

12. (a) — from the first tail to the last head.

20.9 Transformation

Facts. Reflection, rotation and translation are isometric (congruent image); enlargement is not. Reflection in the x-axis: (x,y)\rightarrow(x,-y); in the y-axis: (x,y)\rightarrow(-x,y). Rotation about O: +90^\circ\rightarrow(-y,x), 180^\circ\rightarrow(-x,-y), +270^\circ=-90^\circ\rightarrow(y,-x), 360^\circ\rightarrow(x,y). Translation by \binom{a}{b}: (x,y)\rightarrow(x+a,y+b); its inverse is the translation by \binom{-a}{-b}. Points on the mirror line do not move.

1. Which transformation changes the size of a figure? [1]areflectionbtranslationcrotationdenlargement
2. The image of (-4,-1) after reflection in the x-axis is [1]a(-4,1)b(4,-1)c(4,1)d(-1,-4)
3. The image of (6,-2) after reflection in the y-axis is [1]a(-6,-2)b(6,2)c(-6,2)d(-2,6)
4. Under reflection in the y-axis, the point (0,5) goes to [1]a(0,-5)b(0,5)c(5,0)d(-5,0)
5. Rotation through +90^\circ about O sends (3,1) to [1]a(-1,3)b(1,-3)c(-3,-1)d(1,3)
6. Rotation through -90^\circ about O sends (3,1) to [1]a(-1,3)b(1,-3)c(-3,-1)d(3,-1)
7. Rotation through 180^\circ about O sends (-2,7) to [1]a(2,-7)b(-7,2)c(2,7)d(-2,-7)
8. A rotation about O sends (2,3) to (-3,2). The rotation is [1]a+90^\circb-90^\circc180^\circd360^\circ
9. Translation by \binom{-3}{4} sends (5,-1) to [1]a(2,3)b(8,-5)c(2,-5)d(-15,-4)
10. The translation vector that sends (1,1) to (4,-2) is [1]a\binom{3}{-3}b\binom{5}{-1}c\binom{-3}{3}d\binom{3}{3}
11. Rotation through 270^\circ anticlockwise about O sends (x,y) to [1]a(-y,x)b(y,-x)c(-x,-y)d(y,x)
12. The translation that undoes the translation by \binom{2}{-5} is the translation by [1]a\binom{-2}{5}b\binom{2}{5}c\binom{5}{-2}d\binom{-2}{-5}

Answers to 20.9

1. (d) — the others keep every length.

2. (a) — the sign of y changes.

3. (a) — the sign of x changes.

4. (b) — the point lies on the mirror line.

5. (a) — (-y,x)=(-1,3), into quadrant II.

6. (b) — (y,-x)=(1,-3), into quadrant IV.

7. (a) — both signs change.

8. (a) — (-y,x) for (2,3) is (-3,2).

9. (a) — (5-3,\ -1+4).

10. (a) — \binom{4-1}{-2-1}.

11. (b) — the same as a quarter turn clockwise.

12. (a) — the negative vector brings every point back.

20.10 Statistics

Facts. Mean: \frac{\sum x}{n}, \frac{\sum fx}{N}, or \frac{\sum fm}{N} with mid-values m for grouped data. Median: the \left(\frac{n+1}{2}\right)th item of the ordered data; Q_1 the \left(\frac{n+1}{4}\right)th and Q_3 the \left(\frac{3(n+1)}{4}\right)th item. Mode: the most frequent value; for grouped data M_o=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times i in the modal class. Range =L-S; coefficient of range =\frac{L-S}{L+S}.

1. The mean of 7, 9, 11, 13 is [1]a9b10c11d40
2. If the mean of 6 numbers is 15, their sum is [1]a21b90c2.5d9
3. The mean of the distribution x: 1, 2, 3 with f: 2, 3, 5 is [1]a2b2.3c2.5d3
4. The mid-value of the class 30–40 is [1]a30b35c40d70
5. The median of 8, 3, 6, 1, 9 is [1]a3b6c5.4d9
6. The median of 4, 6, 9, 13, 15, 20 is [1]a9b13c11d11.17
7. For 15 values in ascending order, Q_1 is the [1]a3rd valueb4th valuec8th valued12th value
8. For 15 values in ascending order, Q_3 is the [1]a10th valueb11th valuec12th valued13th value
9. The mode of 3, 5, 5, 6, 8, 8, 8, 10 is [1]a5b6c8d10
10. For the classes 0–10: 5, 10–20: 9, 20–30: 6, the modal class is [1]a0–10b10–20c20–30d0–30
11. The range of 34, 12, 56, 23, 45 is [1]a44b22c56d33
12. If the largest value is 60 and the smallest is 20, the coefficient of range is [1]a40b0.5c2d0.33

Answers to 20.10

1. (b) — \frac{40}{4}=10.

2. (b) — 6\times15=90.

3. (b) — \sum fx=2+6+15=23 and N=10.

4. (b) — \frac{30+40}{2}.

5. (b) — in order 1, 3, 6, 8, 9; the 3rd value is 6.

6. (c) — \frac{9+13}{2}=11.

7. (b) — \frac{15+1}{4}=4.

8. (c) — \frac{3(15+1)}{4}=12.

9. (c) — 8 occurs three times.

10. (b) — it has the highest frequency, 9.

11. (a) — 56-12=44.

12. (b) — \frac{40}{80}=0.5.

Chapter summary

• Relations and functions: n(A\times B)=n(A)n(B); a function gives each member of the domain exactly one image; R^{-1} swaps every pair.
• Polynomials and sequences: whole-number powers only; degree of a product = sum of degrees; t_n=a+(n-1)d; number of terms in \sum_{k=p}^{q} is q-p+1.
• Limit: substitute for polynomials; factorise and cancel a \frac{0}{0} form; no limit when the two sides disagree.
• Matrix: order is rows \times columns; add and subtract only the same order; kA multiplies every element.
• Coordinate geometry: distance, section and mid-point formulas; slope \frac{y_2-y_1}{x_2-x_1}; y=mx+c; area \frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|.
• Trigonometry: 90^\circ=100^g=\frac{\pi}{2} rad; the three identities; standard values; ASTC; allied-angle rules.
• Vector and transformation: head minus tail; parallel when x_1y_2-x_2y_1=0; (x,-y), (-x,y), (-y,x), (-x,-y), (y,-x), (x+a,y+b).
• Statistics: order the data before finding a median or quartile; divide by N=\sum f; range L-S and coefficient \frac{L-S}{L+S}.