R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of


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1 Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of the function f? Which digrm shows sketch of f ( )? omin Rnge f()   f() 8 f ( ) f() f ( ) 8 f ( ) The digrm shows prt of the grph of f ( ). Q(, ) f ( ) P(, 8) The curve psses through the points P(, 8) nd Q(, ). Which of the following represents the eqution of the curve? 8 f ( ) 7 log 7 Pge
2 The digrm elow shows sketch of f ( ). f ( ) Higher Mthemtics Ojective Test Prctice ook 6 The digrm shows sketch of f ( ). f ( ) Which eqution elow represents f ( )? sin sin( ) Which digrm elow shows sketch of f ( )? sin sin The digrm elow shows sketch of g( ). g( ) Which eqution elow represents g( )? cos cos cos cos Pge
3 Higher Mthemtics Ojective Test Prctice ook 7 The digrm shows sketch of f ( ). f ( ) 8 Functions f nd g re defined on suitle domins f ( ) nd g( ) Find n epression for f ( g( )). ( ) Which digrm elow shows sketch of f ( )? 9 When 8 is written in the form ( p) q, wht is the vlue of q? 9 7 Here re two sttements out the grph of the curve with eqution ( ) ( ) () There is turning point on the is. () The minimum vlue of is t. Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. Wht is the minimum vlue of f( ) ( )( )? 7 Pge
4 Higher Mthemtics Ojective Test Prctice ook qudrtic function f is given f ( ) c, where nd c Which grph elow is possile sketch of f ( )? The digrm shows prt of the grph of f ( ). f ( ) The curve cuts the is t (, ) nd the is t (,), (, ) nd (, ). Which of the following represents the eqution of the curve? cos cos Here re two sttements out the circle with eqution ( ) ( ) nd the line with eqution. () The line intersects the circle t (6, 6) () Prt of the line is dimeter of the circle. Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. sequence is defined the recurrence reltion u u 9, u. n n Wht is the vlue of u? Pge
5 Higher Mthemtics Ojective Test Prctice ook 6 sequence is defined the recurrence reltion u u, u n n Wht is the vlue of u? 7 sequence is defined the recurrence reltion n n u u, u Wht is the vlue of u? Here re two sttements out the limits of sequences. () The sequence generted un 7u n 6 hs limit 8 s n. () The sequence generted u n hs no limit. Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. 9 sequence is generted the recurrence reltion un u n 6. Wht is the limit of this sequence s n? u n Wht re the roots of the qudrtic eqution ( )( )? nd nd nd nd Solve the inequlit Solve ( )( ). or or Solve ( 8)( ). 8 or 8 or 8 8 Wht is the nture of the roots of the qudrtic eqution Two rel equl roots Two rel distinct roots No rel roots 6 7 6? Three rel distinct roots Pge
6 Higher Mthemtics Ojective Test Prctice ook For wht vlues of p does p (p ) p hve equl roots? 6 Wht is the eqution of the prol with roots nd, pssing through the point (, ) qudrtic eqution is given 8( )( ) It hs roots t nd. Wht is the intercept? 6 9 Wht is the reminder when is divided ( )? 7 7 The digrm shows sketch of the curve with eqution k( )( )( ) Evlute log 9 log 6 log 8. Wht re the vlues of nd k? (, ) k Epress log log 8 log in form without using logrithms. 8 8 Pge 6
7 Higher Mthemtics Ojective Test Prctice ook The epression log ( e) log ( e ) cn e written in the form e P loge Q. Wht re the vlues of P nd Q? P Q e Which digrm elow shows the grph of log? e The digrm shows the grph with eqution log ( ). (, ) log ( ) Wht re the vlues of nd? 6 Wht is the solution of log? If log 6, wht is the vlue of? Which of the following equtions is equivlent to log log log? 8 Pge 7
8 Higher Mthemtics Ojective Test Prctice ook 8 Solve log log. 6 9 The digrm shows the grph of log plotted ginst. log The curve with eqution log ( ) cuts the is t P. Wht is the coordinte of P? 9 Given tht f ( ) ( ), find f( ). 8 6 The grph is stright line through the origin with grdient. Find n epression for in terms of. d Given tht, find. d 6 7 n The grph illustrtes the lw p kv. 8 7 log p Wht is the derivtive of? log v Wht is the vlue of k? Pge 8
9 Higher Mthemtics Ojective Test Prctice ook tngent to the curve with eqution hs grdient. Wht is the coordinte of the point of contct of the tngent nd curve? Given tht, for f ( ), for, for Which digrm shows the curve with eqution f ( )? 6 Given tht s t, find the rte of chnge of s with respect to t, when t For wht vlues of is the function with derivtive f ( ) strictl decresing? or 8 For wht vlues of is the function with derivtive f ( ) neither incresing or decresing? nd nd nd nd 9 function is given f ( ). Find the mimum vlue of f in the intervl. 7 Pge 9
10 function f( ) is such tht f ( ) ( ). Which grph could e sketch of f ( )? (, ) Higher Mthemtics Ojective Test Prctice ook Find 6 d 6 c c c 9 c Wht is the integrl of respect to? c Find c c c d, c c c c ( ) with Find d 7 c 7 c 7 c 7 c 7 6 Find 6 d, c c 7 c c Pge
11 7 Find d 6 c 6 c c c Higher Mthemtics Ojective Test Prctice ook 6 The digrm shows the curve with eqution f ( ). c 8 Evlute 8 8 d 9 Evlute 8 d Which of the following gives the re etween the curve nd is? c f ( ) d f ( ) d c f ( ) d f ( ) d c f ( ) d f ( ) d Pge
12 Higher Mthemtics Ojective Test Prctice ook 6 The digrm shows the curve with eqution. 6 The digrm shows the curves with equtions. nd. Which of the following gives the vlue of the shded re? d d d d d Which of the following gives the vlue of the shded re? d d d d Pge
13 Higher Mthemtics Ojective Test Prctice ook 6 6 f ( ) ( cd, ) f ( ) g( ) (, ) Wht is n epression for the shded re in the digrm ove? c ( ) ( ) c f g d ( ) ( ) d f g d ( ) ( ) c f g d ( ) ( ) g f d Find n epression for the totl shded re in the digrm ove. f ( ) d ( ) ( ) f d f d ( ) ( ) f d f d ( ) ( ) f d f d 6 The grdient of the tngent to curve is d given. d If the curve psses through the point (, ) find its eqution. 6 Pge
14 Higher Mthemtics Ojective Test Prctice ook d 66 Given tht d nd when, find n epression for in terms of. = + 67 Given tht p. 8 8 p 6 d, find the vlue of d 68 If 6sin( ), find. d 6 cos( + ) 6 cos( + ) 8 cos( + ) cos( + ) d 69 If cos( ), find. d sin( ) sin( ) sin( ) 6sin( ) 7 Wht is the derivtive of respect to? cos cos sin sin cos 7 Find d ( ) c () c ( ) ( ) c 7 Find d c c ( ) ( ) () c c c 7 Find 6cos d sin() c 6sin() c 8sin() c 8sin( ) c sin with 7 If f find f ( ). ( ) ( ), ( ) 6 ( ) ( ) 7 Find sin( ) d cos( ) c cos( ) c cos() c cos() c Pge
15 Higher Mthemtics Ojective Test Prctice ook 76 Wht is the derivtive of respect to? cos sin sin cos sin with 8 Wht is the distnce etween the points (,, ) nd (,, )? G hs coordintes (,, ) nd H(,, ). Wht is the distnce etween G nd H? 6 78 P is the point (, ), Q (, ), R(,) nd S(, ). Here re two sttements out PQ nd RS. () PQ 9 units () PQ RS Neither sttements is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. 79 The point (, ) lies on the circle with centre (, ). Find the length of the dimeter of the circle. 8 lculte the length of the line joining P(,) to Q(,) Find the grdient of the line in the digrm ove. 8 Find the grdient of the line with eqution. 8 The line with eqution 6 meets the is t the point. Wht is the grdient of the line joining to the point (, )? 9 9 Pge
16 Higher Mthemtics Ojective Test Prctice ook 8 The line through K(, ) nd L(7, ) hs grdient of. Wht is the vlue of? Wht is the eqution of the line through the points (, ) nd (, )? ( ) ( ) ( ) ( ) 89 Wht is the eqution of the line joining the points (, ) nd (6, )? line, pssing through the point (, ), mkes n ngle of with the positive direction of the is. Wht is the eqution of the line? Find the eqution of the line through the points P(, ) nd Q(, 9). 88 line psses through the points (, ) nd (, ). Wht is the grdient nd intercept of this line? m intercept  9 Find the eqution of the line through the points (, ) nd (, ) Wht is the grdient of line prllel to the line with eqution 7? 9 Here re two sttements out the lines PQ : nd RS : 6 () PQ nd RS re perpendiculr () PQ nd RS hve the sme intercept Which of the following is true? Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. Pge 6
17 Higher Mthemtics Ojective Test Prctice ook 9 line L is perpendiculr to the line with eqution 6 Wht is the grdient of the line L? 6 9 line L hs grdient. Wht is the eqution of the line through the point (, ) nd perpendiculr to line L? 98 P is the point (, ) nd Q(, 7). Wht is the grdient nd the midpoint R of the line joining P to Q? m 99 Wht re the coordintes of M, the midpoint of the line joining P(, 7, ) nd Q(,, ). (,, ) (,, ) (,, ) (,, ) R (,) (,) (, ) (, ) 96 line hs eqution. Which of the following lines is perpendiculr to this line? The digrm shows tringle PQR with ltitude PS. Q S R P(, ) 97 line L is perpendiculr to the line with eqution 6 7. Wht is the grdient of line L? P hs coordintes (,) nd the grdient of QR is. Wht is the eqution of ltitude PS? 9 7 Pge 7
18 Higher Mthemtics Ojective Test Prctice ook The digrm shows tringle EFG with medin EH drwn. EH hs grdient. F hs coordintes (,7) nd G is (7,). Wht is the eqution of medin EH? 7 8 The digrm shows tringle PQR with ltitude PS drwn. Q E S R F(, 7) H P(, ) G(7, ) The points P( 9, ), Q(,) nd R(,) re colliner. Wht is the vlue of? 7 circle hs eqution ( ) ( ) 8 Wht re the coordintes of its centre nd length of its rdius? circle hs eqution 6 6 Wht is the rdius of this circle? entre (,) Rdius 8 (, ) 8 (,) 8 (, ) 8 P hs coordintes (,) nd the grdient of QR is. Wht is the eqution of ltitude PS? 7 6 circle hs eqution 7 Wht is the rdius of this circle? 7 Pge 8
19 Higher Mthemtics Ojective Test Prctice ook 7 circle with centre (, ) psses through the point (6, ). Wht is the eqution of the circle? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The point P(,) lies on the circle with eqution ( ) ( ). Wht is the grdient of the tngent t P? 8 circle with centre (, ) psses through the point (, ). Wht is the eqution of the circle? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The point T(9, ) lies on the circle with eqution 6. Wht is the grdient of the tngent t T? 9 9 circle hs centre (, ) nd rdius 6 units. Wht is the eqution of the circle? The point P(, ) lies on the circle with eqution 6 7. Wht is the grdient of the tngent t P? To find the points of intersection of line nd circle the eqution k is formed. Wht is the vlue of k for which the line is tngent to the circle? To find the points of intersection of line nd circle the eqution p is formed. Wht is the vlue of p for which the line tngent to the circle? Pge 9
20 Higher Mthemtics Ojective Test Prctice ook The circle with eqution 6 intersects the is t the points (, ) nd (, ) (, ) nd (, ) ( 6, ) nd (, ) (, ) nd (6, ) 6 PQ is dimeter of the circle with eqution ( ) ( ) P hs coordintes (, 7), wht re the coordintes of Q? (,) (, ) (, 8) (, 9) 7 The vectors nd cn e epressed in the form i jk nd i j k respectivel. Find? Wht is the mgnitude of vector 7 where hs components? The vector GH cn e epressed s 7i 7jk. The coordintes of H re (,, ). Wht re the coordintes of G? (, 9, ) (,, ) (,, ) (, 9, 7) The points K nd L hve coordintes (6,, ) nd (,, ) respectivel. Epress KL in terms of i, j nd k? 8i + j k 8i + 7j k 8i + 7j k 8i  7j + k 8 The points nd hve coordintes (6,, ) nd (,, ) respectivel. Find If L is the point (7,, ) nd Wht re the coordintes of K? (,, ) (,, ) (9,, ) (9,, ) KL 6 Pge
21 Higher Mthemtics Ojective Test Prctice ook Vector u, where u k, is unit vector. Given tht k, find the vlue of k. Given tht p i j k, find p Which of the following vectors is prllel to vector u, where u? Given tht u i j k nd v i j k, find u v. 7 6 Here re two sttements out 6 u nd v 8 () u is prllel to v () u nd v hve the sme direction Which of the following is true? Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. 8 Given tht u nd v, Find the components of vector u v Pge
22 Higher Mthemtics Ojective Test Prctice ook 9 Given tht u nd v, Wht re the components of u v? Vectors u nd v re shown in the digrm elow. 6 E Find in terms of u nd v. v u v u v u v u v u v E Here re two sttements out 6 nd 8 6 () () is prllel to Which of the following is true? Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. Which of the following is equivlent to c? d e e d c d e Pge
23 Higher Mthemtics Ojective Test Prctice ook Pge is the point (,, ), (8,, ) nd G(8, 7, ). Wht re the components of H? 7 OPQRS is prmid s shown SM is perpendiculr to the plne OPQR. SM Wht re the components of SQ? 8 6 The points (,, ), (, 7, 9) nd (,, 7) re colliner. Wht is the vlue of? The points P(,, ), Q(,, 6) nd R(,, ) re colliner. Find the rtio PQ : QR. : : : : E F G H P Q S R 8 6 M z
24 Higher Mthemtics Ojective Test Prctice ook 7 The points P(,, ), Q(,, ) nd R(9,, ) re colliner. Find the rtio PQ : QR. : : : : Given tht nd, find P is the point (,, ) nd Q is (6,, ). R divides PQ in the rtio :. Wht re the coordintes of R? (,, ) (,, ) (,, ) (,, 9) 9 P is the point (,, ) nd Q is (,, ). R divides PQ in the rtio :. Wht re the coordintes of R? ( 9, 6, ) (, 7, ) (,, ) (,, ) Given tht PQ nd P is (,, ). Wht re the coordintes of R, the midpoint of PQ? (,, ) (,, ) (,, ) (,, ) 6 Given tht p nd q, find p. q q. q Given tht nd, find 7 if is perpendiculr to.   Given tht nd, find if is perpendiculr to. Pge
25 Higher Mthemtics Ojective Test Prctice ook u nd v hve components nd respectivel. Find if u nd v re perpendiculr u nd v re the vectors with components nd respectivel. Wht is the ect vlue of cos where is the ngle etween the vectors? ½ 6 8 In the digrm, r s nd the ngle is. Wht is the vlue of r.( r s )? r s c 7 6 O(,, ) P(,, ) Q(,, ) In the isosceles tringle shown nd c. Find.( c ). Wht is the ngle etween OP nd OQ? 6 9 Pge
26 Higher Mthemtics Ojective Test Prctice ook p r q In the equilterl tringle shown the length of ech side is units. Find q. r. Pge 6
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