SL Paper 1
Let \(f\,{\text{:}}\,\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) be defined by \(f\left( {x,\,y} \right) = \left( {x + 3y,\,2x - y} \right)\).
Given that A is the interval \(\left\{ {x\,{\text{:}}\,0 \leqslant x \leqslant 3} \right\}\) and B is the interval \(\left\{ {y\,{\text{:}}\,0 \leqslant x \leqslant 4} \right\}\) then describe A × B in geometric form.
Show that the function \(f\) is a bijection.
Hence find the inverse function \({f^{ - 1}}\).
The group \(\left\{ {G,\left. * \right\}} \right.\) is defined on the set \(G = \left\{ {1,2,3,4,5,\left. 6 \right\}} \right.\) where \( * \) denotes multiplication modulo \(7\).
Draw the Cayley table for \(\left\{ {G,\left. * \right\}} \right.\) .
(i) Determine the order of each element of \(\left\{ {G,\left. * \right\}} \right.\) .
(ii) Find all the proper subgroups of \(\left\{ {G,\left. * \right\}} \right.\) .
Solve the equation \(x * 6 * x = 3\) where \(x \in G\) .
Let \(G\) denote the set of \(2 \times 2\) matrices whose elements belong to \(\mathbb{R}\) and whose determinant is equal to 1. Let \( * \) denote matrix multiplication which may be assumed to be associative.
Let \(H\) denote the set of \(2 \times 2\) matrices whose elements belong to \(\mathbb{Z}\) and whose determinant is equal to 1.
Show that \(\{ G,{\text{ }} * \} \) is a group.
Determine whether or not \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).
The relations \({\rho _1}\) and \({\rho _2}\) are defined on the Cartesian plane as follows
\(({x_1},{\text{ }}{y_1}){\rho _1}({x_2},{\text{ }}{y_2}) \Leftrightarrow x_1^2 - x_2^2 = y_1^2 - y_2^2\)
\(({x_1},{\text{ }}{y_1}){\rho _2}({x_2},{\text{ }}{y_2}) \Leftrightarrow \sqrt {x_1^2 + x_2^2} \leqslant \sqrt {y_1^2 + y_2^2} \).
For \({\rho _1}\) and \({\rho _2}\) determine whether or not each is reflexive, symmetric and transitive.
For each of \({\rho _1}\) and \({\rho _2}\) which is an equivalence relation, describe the equivalence classes.
The permutation \(P\) is given by
\[P = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&4&5&6&2&1 \end{array}} \right).\]
Determine the order of \(P\), justifying your answer.
Find \({P^2}\).
The permutation group \(G\) is generated by \(P\). Determine the element of \(G\) that is of order 2, giving your answer in cycle notation.
The set \(P\) contains all prime numbers less than 2500.
The set \(Q\) is the set of all subsets of \(P\).
The set \(S\) contains all positive integers less than 2500.
The function \(f:{\text{ }}S \to Q\) is defined by \(f(s)\) as the set of primes exactly dividing \(s\), for \(s \in S\).
For example \(f(4) = \{ 2\} ,{\text{ }}f(45) = \{ 3,{\text{ }}5\} \).
Explain why only one of the following statements is true
(i) \(17 \subset P\);
(ii) \(\{ 7,{\text{ }}17,{\text{ }}37,{\text{ }}47,{\text{ }}57\} \in Q\);
(iii) \(\phi \subset Q\) and \(\phi \in Q\), where \(\phi \) is the empty set.
(i) State the value of \(f(1)\), giving a reason for your answer.
(ii) Find \(n\left( {f(2310)} \right)\).
Determine whether or not \(f\) is
(i) injective;
(ii) surjective.
The group \(\{ G,{\text{ }} * \} \) has a subgroup \(\{ H,{\text{ }} * \} \). The relation \(R\) is defined, for \(x,{\text{ }}y \in G\), by \(xRy\) if and only if \({x^{ - 1}} * y \in H\).
(a) Show that \(R\) is an equivalence relation.
(b) Given that \(G = \{ 0,{\text{ }} \pm 1,{\text{ }} \pm 2,{\text{ }} \ldots \} \), \(H = \{ 0,{\text{ }} \pm 4,{\text{ }} \pm 8,{\text{ }} \ldots \} \) and \( * \) denotes addition, find the equivalence class containing the number \(3\).
\(G\) is a group. The elements \(a,b \in G\) , satisfy \({a^3} = {b^2} = e\) and \(ba = {a^2}b\) , where \(e\) is the identity element of \(G\) .
Show that \({(ba)^2} = e\) .
Express \({(bab)^{ - 1}}\) in its simplest form.
Given that \(a \ne e\) ,
(i) show that \(b \ne e\) ;
(ii) show that \(G\) is not Abelian.
The set \({{\rm{S}}_1} = \left\{ {2,4,6,8} \right\}\) and \({ \times _{10}}\) denotes multiplication modulo \(10\).
(i) Write down the Cayley table for \(\left\{ {{{\rm{S}}_1},{ \times _{10}}} \right\}\) .
(ii) Show that \(\left\{ {{{\rm{S}}_1},{ \times _{10}}} \right\}\) is a group.
(iii) Show that this group is cyclic.
Now consider the group \(\left\{ {{{\rm{S}}_1},{ \times _{20}}} \right\}\) where \({{\rm{S}}_2} = \left\{ {1,9,11,19} \right\}\) and \({{ \times _{20}}}\) denotes multiplication modulo \(20\). Giving a reason, state whether or not \(\left\{ {{{\rm{S}}_1},{ \times _{10}}} \right\}\) and \(\left\{ {{{\rm{S}}_1},{ \times _{20}}} \right\}\) are isomorphic.
Let S be the set of matrices given by
\(\left[ \begin{array}{l}
a\\
c
\end{array} \right.\left. \begin{array}{l}
b\\
d
\end{array} \right]\) ; \(a,b,c,d \in \mathbb{R}\), \(ad - bc = 1\)
The relation \(R\) is defined on \(S\) as follows. Given \(\boldsymbol{A}\) , \(\boldsymbol{B} \in S\) , \(\boldsymbol{ARB}\) if and only if there exists \(\boldsymbol{X} \in S\) such that \(\boldsymbol{A} = \boldsymbol{BX}\) .
Show that \(R\) is an equivalence relation.
The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State, with a reason, whether or not there are any non-zero values of \(n\) , other than \(1\), for which \(R\) is an equivalence relation.
Consider the set \(S = \{ 0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5\} \) under the operation of addition modulo \(6\), denoted by \({ + _6}\).
Construct the Cayley table for \(\{ S,{\text{ }}{ + _6}\} \).
Show that \(\{ S,{\text{ }}{ + _6}\} \) forms an Abelian group.
State the order of each element.
Explain whether or not the group is cyclic.
Prove that the function \(f:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}\) defined by \(f(x,{\text{ }}y) = (2x + y,{\text{ }}x + y)\) is a bijection.
Show that the set \(S\) of numbers of the form \({2^m} \times {3^n}\) , where \(m,n \in \mathbb{Z}\) , forms a group \(\left\{ {S, \times } \right\}\) under multiplication.
Show that \(\left\{ {S, \times } \right\}\) is isomorphic to the group of complex numbers \(m + n{\rm{i}}\) under addition, where \(m\), \(n \in \mathbb{Z}\) .
The relation \(R\) is defined on the set \(\mathbb{Z}\) by \(aRb\) if and only if \(4a + b = 5n\) , where \(a,b,n \in \mathbb{Z}\).
Show that \(R\) is an equivalence relation.
State the equivalence classes of \(R\) .
The function \(f:{\mathbb{R}^ + } \times {\mathbb{R}^ + } \to {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) is defined by \(f(x,{\text{ }}y) = \left( {xy,{\text{ }}\frac{x}{y}} \right)\).
Prove that \(f\) is a bijection.
The transformations T1, T2, T3, T4, in the plane are defined as follows:
T1 : A rotation of 360° about the origin
T2 : An anticlockwise rotation of 270° about the origin
T3 : A rotation of 180° about the origin
T4 : An anticlockwise rotation of 90° about the origin.
The transformation T5 is defined as a reflection in the \(x\)-axis.
The transformation T is defined as the composition of T3 followed by T5 followed by T4.
Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the operation of composition of transformations.
Show that T1, T2, T3, T4 under the operation of composition of transformations form a group. Associativity may be assumed.
Show that this group is cyclic.
Write down the 2 × 2 matrices representing T3, T4 and T5.
Find the 2 × 2 matrix representing T.
Give a geometric description of the transformation T.
\(\{ G,{\text{ }} * \} \) is a group of order \(N\) and \(\{ H,{\text{ }} * \} \) is a proper subgroup of \(\{ G,{\text{ }} * \} \) of order \(n\).
(a) Define the right coset of \(\{ H,{\text{ }} * \} \) containing the element \(a \in G\).
(b) Show that each right coset of \(\{ H,{\text{ }} * \} \) contains \(n\) elements.
(c) Show that the union of the right cosets of \(\{ H,{\text{ }} * \} \) is equal to \(G\).
(d) Show that any two right cosets of \(\{ H,{\text{ }} * \} \) are either equal or disjoint.
(e) Give a reason why the above results can be used to prove that \(N\) is a multiple of \(n\).
The set \(S\) contains the eight matrices of the form\[\left( {\begin{array}{*{20}{c}}
a&0&0\\
0&b&0\\
0&0&c
\end{array}} \right)\]where \(a\), \(b\), \(c\) can each take one of the values \( + 1\) or \( - 1\) .
Show that any matrix of this form is its own inverse.
Show that \(S\) forms an Abelian group under matrix multiplication.
Giving a reason, state whether or not this group is cyclic.
Prove that the number \(14 641\) is the fourth power of an integer in any base greater than \(6\).
For \(a,b \in \mathbb{Z}\) the relation \(aRb\) is defined if and only if \(\frac{a}{b} = {2^k}\) , \(k \in \mathbb{Z}\) .
(i) Prove that \(R\) is an equivalence relation.
(ii) List the equivalence classes of \(R\) on the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
The group \(\left\{ {G, + } \right\}\) is defined by the operation of addition on the set \(G = \left\{ {2n|n \in \mathbb{Z}} \right\}\) .
The group \(\left\{ {H, + } \right\}\) is defined by the operation of addition on the set \(H = \left\{ {4n|n \in \mathbb{Z}} \right\}\)
Prove that \(\left\{ {G, + } \right\}\) and \(\left\{ {H, + } \right\}\) are isomorphic.
Use the Euclidean algorithm to find \(\gcd (162,{\text{ }}5982)\).
The relation \(R\) is defined on \({\mathbb{Z}^ + }\) by \(nRm\) if and only if \(\gcd (n,{\text{ }}m) = 2\).
(i) By finding counterexamples show that \(R\) is neither reflexive nor transitive.
(ii) Write down the set of solutions of \(nR6\).
A sample of size 100 is taken from a normal population with unknown mean μ and known variance 36.
Another investigator decides to use the same data to test the hypotheses H0 : μ = 65 , H1 : μ = 67.9.
An investigator wishes to test the hypotheses H0 : μ = 65, H1 : μ > 65.
He decides on the following acceptance criteria:
Accept H0 if the sample mean \(\bar x\) ≤ 66.5
Accept H1 if \(\bar x\) > 66.5
Find the probability of a Type I error.
She decides to use the same acceptance criteria as the previous investigator. Find the probability of a Type II error.
Find the critical value for \({\bar x}\) if she wants the probabilities of a Type I error and a Type II error to be equal.