SECTION A STUDENT MATERIAL Prt Wht nd Wh.?
Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are either of the sttements true? Bk up our nswers with proofs. Prolem 3 If nd re non-zero rel numers, omment on the following sttements: () If > then < () If < then > Prolem 4 Is it ever true tht = + +? Prolem 5 Whih of these identities re true nd whih re flse? () log log = log () log = log log () (e) log log + log = log (d) = log ( ) log log = log (f) log = log = log (g) ( ) (h) ( ) = The Centre for Tehing Mthemtis = log
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 6 Is it ever true tht + = +? Prolem 7 Eplin the fll in the following rgument: + 5 4= 5 3 3 7 + 3 4( 5) = 5 3 3 7 Hene 5 = 7 3 + 3 = 5 3 3 = 5 3 3 7 3 3 7 Prolem 8 Eplin the fll in the following rgument: We know tht 3 > so 3 log > log log 3 > log 3 > > 8 4 > The Centre for Tehing Mthemtis 3
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 9 Eplin the fll in the following rgument: Suppose tht = Then = Hene = ( ) ( + ) = ( ) ( + ) = Now sustituting = =, we otin =. Prolem 0 For wht vlues of k n we solve the following equtions? sin = k, sin = k, sin = k Prolem Given tht sin os = k, wht hppens if k > ½? Prolem Whih of the following re identities, nd whih re equtions tht n e solved? () sin + os = (e) tn = tn () os = os (f) + = + () ( + ) = + + (g) sin = sin (d) sin + os = (h) = 3 Prolem 3 Suggest vlues for in rdins for whih sin, os, tn. Prolem 4 Are there n vlues of for whih the following equtions re true sttements? () sin = sin () os = os () tn = tn 4 The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt 8 6 4 Prolem 5 The figure on the left shows grph of = sint + π / 3 t Wht re the vlues of, nd Prolem 6 Wht does the word 'liner' men? Disuss it in the ontet of the following equtions: () 3 + 4 = 7 () + 3z = 6 Prolem 7 f() = + + with > 0 Under wht irumstnes will the eqution f() = 0 hve () () () no roots? one root? two rel roots? Sketh grphs of f() to illustrte the three ses. Prolem 8 Ftorise + + if the roots of + + = 0 re α nd β. Prolem 9 Wht n e sid out if () < () > for > 0 () >? The Centre for Tehing Mthemtis 5
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 0 Give n emple of () () () n eqution n epression n identit. Prolem Wh n ( )( + ) not e rewritten s A B + ( ) ( + )? Prolem Wh n ( )( + ) not e rewritten s A B + ( ) +? Prolem 3 Wht is k if ln = klog? Wht does this tell ou out the grphs of = ln nd = log? Prolem 4 How is the grph of = relted to the grph of = where, > 0? Cn ou prove this lgerill? Prolem 5 t r The formul I = I + 0 gives the vlue of n investment fter 00 period of t ers. () Eplin wht I 0 nd r represent. () Sketh the grph of I ginst t. () How does the role of the letters I nd t differ from tht of I 0 nd r in the eqution? 6 The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 6 If n! = n (n ) (n )... whih hs n ftors, nd! = whih hs ftors, nd! = whih hs ftor, how n we justif the onvention tht 0! =? Prolem 7 Eplin wh () ( 4) = ( 4) = or = = 4 () < < () < > Prolem 8 In plne n two lines must either e prllel or interset t some point. () () () (d) Wht hppens to n three lines in plne? Wht n hppen to n two lines in three dimensionl spe? Wht n hppen to n two plnes in three dimensionl spe? Wht n hppen to n three plnes in three dimensionl spe? Prolem 9 Write down funtion with modulus sign in it, the grph of whih never goes ove the is. Prolem 30 If trnsformtion A mps the grph of = f() into the grph of = f() nd trnsformtion B mps the grph of = f() onto tht of = f( + π), wh does the grph of = sin( + π) not look like = sin under the trnsformtion BA? Prolem 3 Wh re the grphs of = nd = different? The Centre for Tehing Mthemtis 7
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 3 Sketh the grph of = 4, nd omment on our nswer. Prolem 33 Cn urve with n eqution of the form = f() where f is funtion of ever e smmetril in the is? Prolem 34 Find possile equtions for these grphs. () 5 () () π -3 - π/6 7π/6-5 - (d) (3,3) (e) (f) (,3) (g) (h) (i) (ii) -3π -π π 3π 5π - 5 8 The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 35 Wht does the r represent in the polr eqution r = osθ? Should we llow r to e negtive? How does this ffet the urve r = osθ? Prolem 36 The funtion f() hs the propert tht f(α) = 0. Write down ftor of f(). If, in ddition, f '(α) = 0 wht else n e dedued? Prolem 37 If P() is divided ( ) the reminder is P(). Wht is the reminder if P() is divided ( )? Prolem 38 Wht hppens to pprohes? s pprohes, nd to s Prolem 39 Funn things n hppen to funtion f() when gets lrge. As gets lrge, wht hppens to 3 + () 5 4 e () 00 () 00 +? Prolem 40 Use our lultor to disover wht hppens to pprohes 0? sin θ θ s θ Does it mtter whether θ is mesured in rdins or degrees? sin Cn ou dedue wht hppens to s pprohes 0 ( in rdins)? The Centre for Tehing Mthemtis 9
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 4 Investigte s h 0. + h Dedue lim + h for n 0. h 0 h Prolem 4 'If = f() hs mimum vlue t = then f '() = 0'. Is the ove sttement true? Is the onverse of the sttement true? If the sttement or its onverse is flse provide ounter emple. Prolem 43 Wht is point of infletion? How is one found? Prolem 44 Sketh =. d Wht is if d () > ½ () < ½ () = ½? Prolem 45 Differentite the following with respet to : e, π, 5, ln3, e, ln. Prolem 46 Wh, in lulus, re ngles mesured in rdins nd e used s the se of logrithms? 0 The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 47 Integrte the following with respet to : e, π,, ln3. 3 Prolem 48 Sen wrote: sin sin os d = + Jne wrote: os sin os d = + Cn oth e orret? Prolem 49 Find the rte of hnge of the funtion f() = ln(ln(sin)). Eplore the grph of the funtion. Prolem 50 Show wh the sttement d = is flse. Prolem 5 = f ( ) 4 6 8 0 - From the figure ove find: () 0 f ( ) d () 4 f ( ) d 4 () 0 0 f ( ) d The Centre for Tehing Mthemtis 0
Student Mteril Prt Prolem 5 Wht nd Wh? Eploring Pure Mths A =f() B If f ( ) d is the re under the urve AB in the figure on the left, stte the re under the urve AB whih is ove the line =. Prolem 53 d N B Write down the integrl whih gives the re etween the urve AB nd the is in the figure on the left, i.e. ABNM. M A = f() Prolem 54 When is the re A under the urve = f() from to equl to f ( ) d? Wht is the onnetion etween re nd integrl in other ses?.. Prolem 55 Look t the grphs numer to 5 elow. To find the re etween two positive urves = f() nd = g() where f() > g() (s in grph )we would s: Are A = [re under = f()] [re under = g()] = f ( ) d g( ) d = [ f ( ) g( )] d Wht is the equivlent result when one, or oth, urves re elow the is, or ross it t some point? Wht hppens if the urves ross eh other? 3. 4. 5. = f ( ) A =f() = f ( ) = f ( ) = f ( = g ( = g ( ) =g() Prolem 56 = g ( ) = g ( ) Compre the following urves. (i) = 4 (ii) + = 4 In eh se, when is the grdient equl to? (iii) = osθ = sinθ The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 57 Wht must e the form of first order differentil eqution so tht it n e solved (i) diret integrtion (ii) seprting the vriles, (iii) using n pproimte numeril method? Whih method(s) n e used on the following equtions? () d = 3 + + d () d = +, d given = when = () d sin =, d π given = when = (d) (e) (f) d = d d = + d d = os given = when = 0 d Prolem 58 Wht hppens to the integrls () M d () M d s M? Prolem 59 If the ontinuous funtion f(, ) hs two sttionr points whih re oth mim then there must e minimum etween them. True or flse? Prolem 60 Given tht the series f ( ) onverges, wht hppens to the integrl n= M f ( ) d sm? The Centre for Tehing Mthemtis 3
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 6 For wht vlues of does the series onverge? k=0sin k Prolem 6 Find ( + 3 d I I, I ( + ) d = 3 3 4 d 0 = ) 0 = 0 3 nd using Simpson's rule with n (even) numer of strips. How urte is our nswer in eh se? Would the trpezium rule give the sme nswers? Prolem 63 Wh is there no error in using Simpson's rule to evlute 3 ( + 3 ) 5 + d? Prolem 64 The trpezium rule n e used to give n et vlue for the integrl of liner funtion. Suppose tht f ( ) = 3 0 3 Wht hppens if the trpezium rule is used, with four intervls, to evlute 3 0 f ( ) d? 4 The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 65 The ontinuous funtion f() hs the following properties: f( ) =, f() = nd f '() < 0 for. How mn roots of the eqution f() = 0 re there in the intervl [, ]? Prolem 66 For the funtion in Prolem 65, how mn pplitions of the isetion method re neessr to gurntee tht the root will e lulted with mimum error of less thn 0.05? Prolem 67 Show tht n vetor 0 nd. 0 n e written in terms of the unit vetors Prolem 68 Drw on grph pper the line r = + t 4 3 Inlude on our digrm two emples of the vetor r. Prolem 69 If, nd re vetors wht n ou dedue from the equtions. = 0 nd. = 0? Prolem 70 A tringle is formed the three vetors, nd s shown on the left. Wht n ou s out the tringle if: (). = 0 (). < 0 (). > 0 (d). =. The Centre for Tehing Mthemtis 5
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 7 For eh of the following prolems, in whih, nd re non-zero vetors, whih of the options must neessril e true? (). = 0 ().. = 0 () + + = 0 (d). () = 0 (e). =. Options (i) The vetor is prllel to the vetor. (ii) The vetor is perpendiulr to the vetor. (iii) The vetors, nd re oplnr. (iv) Both the vetor nd the vetor re unit vetors. (v) The reltionship is not possile (vi) The reltionship is true for ll non-zero vetors. Prolem 7 In the following epressions,, nd d re ritrr vetors. (i). (. ) (vi) ( + + d) (ii) ( ) + (vii) (. ) (. d) (iii) (. ) (viii) (. )( d) (iv) ( ) + ( d) (i) (. )(. d) (v) ( ) + (. d) () (. ) (. d) () () () Whih of the epressions represent undefined opertions? For whih of the llowle epressions is the result slr quntit? For whih of the llowle epressions is the result vetor quntit? Prolem 73 If = + i is solution of f() = 3 3 + 4 = 0, write down the other solutions of f() = 0. Prolem 74 If Re(z z ) = Re(z )Re(z ) nd Im(z z ) = Im(z )Im(z ), must oth z nd z e rel? 6 The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 75 If f(θ) = osθ + isinθ, whih of the following must lws e true? (i) (ii) (iii) (iv) f(θ + Φ) = f(θ) + f(φ) f(θ + Φ) = f(θ) f(φ) f( θ) = f(θ) [f(θ)] n = f(nθ) nεz. Prolem 76 z is omple numer nd is rel, suh tht z =. For the two ses () = nd () = 3, wh n rg(z) in one se tke ll vlues nd in the other se tke onl finite rnge of vlues? Prolem 77 The omple numer z is suh tht z =. Whih of the following must e true for ll possile z? (i) Re(z) < 3 () Re(z) > 0 Prolem 78 z = k Wht does this men in words? Prolem 79 rg(z ) = k Wht does this men in words? Prolem 80 Sketh the points represented z = in n Argnd digrm. Wht hppens to this sketh if z is repled z? The Centre for Tehing Mthemtis 7
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 8 If z = + i; re the sttements z = + nd rg(z) = tn true for ll nd? Prolem 8 Wht hppens to () () () the line from the origin to the point (3, 4) when the oordintes re multiplied i? the squre with verties (0, 0), (, 0), (, ), (0, ) when the oordintes re multiplied i? the retngle with verties (, ), (5, ), (, 3), (5, 3) when the oordintes re multiplied? i Prolem 83 Wht does z 3 mesure? Wht n ou s out z if () z 3 = () z + i 4 3 + i 5 5 () z 3 = z? Prolem 84 Wht does rg(z i) mesure? Wht n ou s out z if π () rg(z i) = 4 () () rg(z ) 3 π rg(z + ) = rg(z + i)? 8 The Centre for Tehing Mthemtis
Wht nd Wh? Student Mteril Eploring Pure Mths Prt Prolem 85 A rel numer hs zero or two rel squre roots, one rel ue root, zero or two rel fourth roots et. How mn omple roots does omple numer hve? How do the differ? Prolem 86 D A C (,) B Does ever geometri trnsformtion hve n inverse? The trnsformtion represented the mtri M = is pplied to the unit squre ABCD shown on the left. The result is tht the squre eomes A'B'C'D'. M does not eist s the mtri M is singulr. Desrie in words wh there n e no geometri trnsformtion whih would trnsform A'B'C'D' into ABCD. Prolem 87 How ould the mtries P = ( ) Q = e omined to form 0 3 R = 4 3 0 () () () 3 3 mtri 3 mtri mtri? Prolem 88 Whih geometril trnsformtion is represented the mtri 0 J =? 0 Whih of the mtries 0 0 0 (i) (ii) (iii) 0 0 0 0 (iv) 0 n e epressed in the form J n when n is n integer? The Centre for Tehing Mthemtis 9
Student Mteril Prt Wht nd Wh? Eploring Pure Mths Prolem 89 Whih of the following mtries hve determinnt? (i) () (iv) ( ) (ii) d (v) 0 0 0 (iii) d 0 0 Prolem 90 If A, B nd C re three squre mtries of the sme size, whih of the following sttements re not neessril true? (i) (A ) = A (ii) (A ) T = (A T ) (iii) det(ab) = det(ba) (iv) (A B) (A + B) = A B (v) det(a ) = (deta) (vi) If AB = AC, then B = C (vii) If A = B, then B = A 0 The Centre for Tehing Mthemtis