the spaces pro OBAFEMI AWOLOWO UNIVERSITY, LED DEPARTMENT OF MATHEMATI Semester Examination), 2022/2023 Academic Session Harmattan Semester (Mi IE, NIGERIA MTH 101-Elementary Mathematics 1 All notations have their usual meanings, 1 <3. |. The non-empty sets Suppose sets X,¥,2.C £, and X’ 3 ¥" > Z/, Where ¢ is the universal set. Find (urynz. leg! xe we © ¥ wler'y vez @®)x ©) z Given that the sets X = {x |? 524.6 < x62} and¥ = {941 |2—-4=0, yen Find XAY, (aout ig BQ} Bred py i ®) 2,0} 4205 - G0) hd WA = {0,2,0,3} and B= fx|xeR and? — 58°46 =0}. Then which of the following is not true? Hua) (A)A=B ®)Ace a () BCA (D) Ais equivalentto B H and T are such that HT CC, and HOT =0, where £ is the unis versal set. Find (H — ry wr OH @r OH Let £7 NN J f(x) = 2x for all EN, Which of the following statement is corres about f. (A) F is an injection only (B) Fis.a surjection only © fsa bijection () Fis neither an injection nor a surjec- tion X22,3 ‘Time Allowed - 1 Hour V6 Let FRR Fe) = 24341. Find F'8). y A) 2,35 8) (~2,23 © (414 @) {1,43 7, Given that n(A~B) (ANB) = 18, deten 24, n(AUB) =65 and ine the value of n(B). (ay*s9 @®) 41 Oz @) 42 ; ‘<8: In the scenario where there are 9S students in the Art ¢lass/and 97 in the Dance elass, with 12 students enrolied in oth classes, deter. imine the total number of students attending either the Art or Dance classes (when both classes meet simultaneously) and at differ. ent Hours, given that a student must atleast attend 4 class, (A) 92 and 80 respectively (B) 100 and 92 respectively x 9, Solve the equation X42 472 xt at 6 a8 8 gt A-35 ©]-ss (B) ~45 @) -6.5 410. Solve the equation 1 t t We eet HGS A) 2ors » B) 207 © -20r-s ©) -20r-7 (©) 80 and 92 respectively Sc (©) 80 and 68 respectively c (a) = Page) of 2, &11. Solve for nonzero x for which 1 1 (#s3)+4(o+4)s0-0 Zz +42+e < > cat (A) -2 twice sed tein (B) -24 Vitwice oe (©) 2st VBI twice Abn ©) ~1 four times 12, LetkeN. Then i? equals 41> 0 O-1 Ms ®)1 ©) x 13. If 2x? — 5x7 is divided by (ax-+b), the quotient is (x~3) and the remainder is ¢ Finda-+b-+be. (ata e-4> he (2 ©) 49 24s roo. B® @) =1- AR RAL) = 24: ‘ 14. If PG) = By = + Tape then +-B+C gives a7 @)1 15. Given that an + in + 8 (-1 | ony 'b 4 Yaa bas] logs(y-1)-+iog, ) 1 and Joga (y + 1) + logyx then (x,) become (4-5/9) © (-3,5/3) 16. Find the range of value of 2 for which the roots of the equation (2 ~ 1)? — 24 (A 1) =Oare real. be Aosas2 4 (B)A>2 O2zso DAN M sdejuticasr eke 17. If @ and Bare the roots of the equation ax? —bx—1 = 0,44 0, then 25+ de, when expressed as a function ofa and bis (A) 6+ 3ab (B) ~8° —3ab © @/(8 +3ab) D) «°/(6° ~3ab) 18. Determine the values of K far which the equation (x4 INGe+2) = x47)k has equal roots, A) 1/91 q © -1,1/9 = ®) 1/91 ®) =1,-1/9 a 19, Iatb+c=8 and ab-+-be-+ac=14, then wha sthe value of (ot 408 +03 — abe)? (5) (a) 176 (©) 400 x fe (e) e2a ©) 536 20, The product of two numbers is 12. The sum of the larger number and twice the smaller number is 11. Find the two num. bers. B) (3,5/3) @) (3-5/3) oh = Ce PK Cnt ROU C oe Hee lees ons AtBh a \ sayy, Vee 24408 +] att ae Cas AGU) AGE C wel =) 248 [= SA 498 1 -6 2% 6 on SAKE Aa, OD Cais vaseba «okt Ik (A) x58, ¥=3/2 oes B) x=4, yaar oe Boe ~ 3k +2 ~ Ik Ox=4y=3 oe H(B- Ae 42 " D) x=8 y=3 ine-* Remark o& Cat G40OBAFEMI AWOLOWO UNIVERSITY, ILE-IFE, NIGERIA, : DEPARTMENT OF MATHEMATICS Rain Semester (Mid-Semester Examination), 2022/2023 Academie Session ‘MTH 102-Elementary Mathematics 11 am apes ‘Time Allowed - | Hour Instructions: Attempt all questions. Use HB pencil ONLY, Write your Names and Registration Number in the spaces provided on your OMR sheet. Shade Option E i {Al totations have thelr usual menage een Ei the comes answerisnocamong the options provided. 2 0 1, Find im (2+3)"+1, 6. Forall n € Z, which of the followin; 2 general solution of the equation Nor (A) -1 (C) -28 Saat GereG.gsg (B) +1 @) +e @ — £08(30) +2c0s(@) = 0. (A) 360° 120° (C) 1360" +90" ©. 20 yn * 13609 30° P 60° 2. HF) = LE, the 2x) —1 is equal B16 E50 a eel} ION EOE cue sont \ : a 7 to 7. Suppose cos(@) —2sin(@) = Reos(@ — a) kk ey and R > 0. In which quadrant is a? @) 1+x (A) First © Third 4 (B) Second @) Fourth Ss ay . — ye (ays x > 0, then 7 8. Given that cos?(@) + (a-+b)c0s(8) ab = \cequals . 0, where a > 1 and b> 1. Which on of the following are real values of e Cat) LS lasyy? Here me log(x) at (A. = ‘ 7 ( 2 ) He (A) @ = cos" (=a) and 8 J Ix y y(‘22h +4) ¥ 1 (B) @ =cos”!(a) and 6 = (©) 6 =sin-(—a) and @ (D) 6 =sin“"(a) and @ = sin!(b) a 3 | 9, cos(A+B)cos(A—B) equals, = cae oe oe (A) cos*(A) —sin*(a) (B) cos*(A) + c0s°(8) aPifi-i. Ww > Cat na)” (©) 4(cos(2A) +e08(28)) wv. © ave HL (By av yall m i 1 Ls mi ® 10, Given that, 62 find a such thay” AL [1's oiven wa oi teGadired in Faian, Differ ‘tua aie Gy Jag Chtiate cos™'(sin(36")) with respect to 6. % a iy = cos(@)cos(20) c0s(48) cos(80).aATo | fie \- 198 (c) 962 (A) 16 «C) 64 (B) -90 Sintt”s aye Sie (B) 32 (D) 128 eas ig YE x GstiCy -© Gas CA4 8) CmiBD4 SA ag é Pave 1 of? Casn-© )Wy b nena [°( b>O. (A) 1+ sin(b) —c0s(6) (B) 1+ sin(6) + c05(b) (©) 1=sin(b) —cos(6) (D) 1 +5in(b) +eos(B) "| A LT cohkn T=sin(26)) d@, where id (asneg int Tent 12, Evaluate f (e'+2¢"4)* de (A) 344102 © 3-40pe tae B) 4432 ©) 4102 13. A particle moves in a straight line with speed v(¢) = sin(w#) cos?(wr). Find its posi- tion function x= f(), if f(0) =0. 3 my ste c03°(wt) 1 . Be ww 3 in'(we) 1 cos*(wt) | 1 ©) Ge aw © ww 14, Find the area bounded by the curve y = xe™*, "the x-axis and the ordinates x=0 and (A) Se 1 © et+1 B) ~Se#+1 @) eet ae ' 15, Bvaluate f sin*(eos*(a)dx. J = fac, ne (Ce (ay —Heos*(a) + $sin*(a) +6 ®) sind (a) + $sin8) +6 (grat 1 © Joos (x)— peos*(s) + Lid xo (x) +e _® —Esin%(a) we ee, ie to fem & we Som Fue 0 2¢"(ur) 16. Given that the slope of the normal to the curve 22/3 +y%/ = 2 at the point (a1, B) is 5 ‘Then (ct, B) is not equal to (@) (11) © (1-1) *¢ ®) (4-1) © C1) ay xia 17. Find dela) te? xan oa AZ oF (B) 4a (@) 4x? 18, If the radius of @ spherical balloon is de- creased by.0.10%, by what percentage does the volume change? ae (A) 0.3% decrease (C) 0.38 increase {71 (B) 3% increase (D) 3% decrease 19, Ify2(04+3) =20, find the ewo values of when x=2. y 1 1 A) 85 Oa 1 — ®) +3 (D) +2 “The slope of the tangent line to the cirele Pi yP= 25 at the point (3,—4) is 3 4 no) —e(ay 2 4 eG OF 4 ©) -5 Pare ? af?OBAFEMI AWOLOWO UNIVERSITY, ILE-IFE, NIGERIA DEPARTMENT OF MATHEMATICS f Rain Semester Examination, 2022/2023 Academic Session MTTH 102-Elementary Mathematies IT ‘Type 4 ‘Time Allowed - 2 Hours Instructions: Attempt all questions. Use HB pencil ONLY. Write your Names and Registration Number in the spaces provided on your OMR sheet. Shade Option I if the correct answer is not among the options provided. All notations have their usual meanings. | 1A patticle moves in a straight line with 5. Given that cos*(@) — (S$4) cos(8) + # = 0, speed v(t) = sin(wt) cos*(wt). Find its posi- ‘where |a| < 1 and |b| < 1. Which of the fol- tion function x = f(t), if f(0) =0. lowing is/are real values of 0? cost(wt) 1 cos'(wt) 1 - Ow Bw of - aw Sw sin?(wt) 1 sit) | 8 ay OS tae 2. Find the equation of the normal to the ellipse (iw) @ sin £4 % atthe point (-3.2) @ y-x=1 © y=x @) y=—xt1 » (A) Only ie 2 (B) Only ii. S (C) Both if. and iv. (D) Both i, and ii Goat 102 43. Evaluate A (e+267)ax. (A) 3—4(1n2) (C) 3441n2 46, 1 7G2+1)=20, ind the vatne(s of when x=2, 4 1 ee (@) 4in2 (B) 44312 cst oat staf Lyi 3 Pp aetcts ney eck! 1 oy Lay A FQL-k 4 2¢ ¥ sr dy HOB YY). y Ws iat eyes | Beep Potts gay BN seats Agel TEES 2 0 Waid) Mepakity wad the tae | Ap TARY Sas EE = ordinates of the foci of the ellipse — | 7. A circle Af has the Centre at W(3,4) and rae TO0y? + bax? = 6400 dus of r= 5. This ciclo interscot y-asts 0 fone intercept and the x-axis at two intercepts. (A) 2/3 and (42,0) 2 ‘What is the’ area of the triangle formed by a these intercepts? | (C) 5/3 and (46,0) (A) 128q.unit (©) 164, unit } | (©) Vand (42,1) (B) 18Sq. unit DF 24 Sq, unit | ! * Pues8, Find the area bounded by the curve y* = 4x, the x-axis and the ordinates x=4 and x=9 4 16 AZ OF 10 @? OF xc 9. Ifthe dfference between mean and mode of a distribution is 48 and median is 12. The means 1G 2 Maen pm (A) 36 © 36 @) 42 O38 E = 3¢45 10. cos(A +B)cos(A—B) equals (A) c0s*(4) + 008%(B) ~1 ®) lc0s(2A) +-e0s(2B)) (© cos*(A) —sin?(B) ®) All of the above M1. Suppose x>0 and y= \/x—4/%. Find 4 P. 5 4 ee (ay 2ve-1 (B) 2ve+1 ©) ve 12. Consider the following distribution ‘Marks Obtained Ramof 5 tadents | More than or equal 0 More than or equal to 10 se ‘More than or equal to 20 More than or equal t0 30 | 51 ‘More than or equal to 40 [48 [More than orequal to 50} 42] “The Frequency of the marks more than 30 bout less than 40, (30. 40), is A4 4 C51 M3 13. Given that, @ Ann, ne%, find «+B and psahaba Sint) _ sa? sin() Bcos(a°@) cos(a@) cos(a?@) cos( a’ (A) IOand16 (©) dana (B) 9and 16 (Dy 18 and 32 14, Find tim +4" +1 4 alg (A) +e (c) +1 * = oat 9,8 Cop VCs CU when y =x sin“!(y) Va ae 96 © Winn 36 | ri PY uy ©) Fn=36 ©) Fn 36 Le 16. Find the equation of the tangent to the parabola y* = 8y+4x— 32 at point (-1,2); A) y= Oyax-1 4 (Dy yt2=1 dy a1 -£@) y-x=1 as REE «4 18. Which of the following is not real solution afticoution sacar Sea . sin(20) +3sin() 9 C88 14) (Ayeor(-32) © =F FH (B) 2% © ( (29-"") x 4g & “ \0)19. Consider the frequency distribution table foes ta ee eS re Px ax [x where, 5 EN. Find bale Leper that the cumulative frequency of the fifth observation is 25: 7 D2 Snob Po Sey * )l3 Ol2 (D) 2 1) (A) i. i. and iii, only (B) ii i and iv. WS 4 (C)1. and ii. only 3 (©) ii. ii. and iv. only 1% spe ($10) sin (3,5) and (x,y) externally in the ratio 5:2. Find (x,y). 1) © (6,11) @) (4,7) 29, Evaluate Sf (vizameay 40, where b>O, (A) 1 =sin(b) —cos(s) (B) 1 +sin(b) —cos(s) © 1 +sin(b) +cos(b) (DY 1 +sin(b) +c08(6) (3,6), (B) (11,8) 30. If the radius of a spherical balloon is de- creased by 0.10%, by what perwentage does the volume change? (AY 0.34% decrease (C) 39% increase (B) 0.3% ine se (D) 3% decrease Pace 3h31. Suppose cos(6) —2sin(@) = Reos(@ +a) and R > 0. In which quadrant is ? © Fourth (D) Third (A) Second (B) First 32. Find the area of the triangle having vertices at (4,5), (5,6), and (—3, 2) ' 4s 2 (35 w4 33. Given that @ is measured in radian. Differ- entiate cos~'(sin(3°)) with respect to 8 (A) 96 (©) -98 (B) 907 By -967 ( st yey x > 0, then 2 ; , 2 equals / Weta " 3 las) Y (2 =A) Haclettre 3 ay MO ®) +(22 44) fg.) ae uy filled © +(S82-4) eal y I og Br) id 35. Vertices of quacrilateral ABCD _ are (0,0), B(4,5), C(9,9) and _D(5,4).. What ‘is the shape of the quadrila- eral? (A) Rhombus (BY Rectangle but not a square (©) Paraltelogram bit not a Rhombus (D) Square 36 37. 38, 39. . Let In a coordinate system, P= (2,7) and —3). Which of the following could be coordinate of R if POR is an isosceles tr- angle? () (12,3) Gi.) (+6, —9) Git) (-117,2) (A) i-and iii. only (C) i. and ii. oniy (B) i only (®) ii. and iii, only Given that the slope of the normal to the curve 4/3 45/3 =2 at the point (ct, B) is 3 ‘Then (cr, B) is not equal to (1) © (.-1) @®) (-1,-1) @) (11) ~ Find the probability of occurrence of atleast one of A and B, if A and B are two indepen- dept events? (A) P(A) — PB) 41 yt PA)PE) (CO) P(A’) —P(B’) 1 (D) 14+ P(A')P(B) Find the equation of the parabola with focus (—4,3)_ and directrix x-yt2=0. £0" ‘ (A) 2412049? -8y +46 =0 (B) 12x47 + 8y—46 =0 (©) 2 12x—2xy +? + 8y—46 (DY 2? + 12x+2ay +)? -8y +46 , @eR and f(x) £0, if oe slog(f(x)+C, then what is F(x) equal to ? (A) xtor ©) 5+a (B) x-a@ (D) 2x+a@ see , : ’ Prow 4 of 4OBAFEMI AWOLOWO UNIVERSITY, ILE-IFE, NIGERIA DEPARTMENT OF MATHEMATICS 2022/2023 RAIN SEMESTER EXAMINATIONS MTH 104 - VECTORS Tuesday, 16th July, 2024 Duration Allowed: 2 Hours QUESTION TYPE: 3 instructions: Fill your OMR Sheet correctly and shade the correct option on the OMR Sheet. If the correct answer is not inary option “Eon the OMR Sheet. Make sure your same with your Question Type, es SE ed 1. The magnitude of the resultant 26. Pid the coordinates of the of two forces with magnitudes 27° joint where the line through and 4F, inclined at an angle 300° to each other, is 7 (A) 7eV3 B) TPVE 44a) (0,-1.8) (B) (—3,0,5) (ro =e dag ow Bie 7 (0-81) (D) (0,6) 2. Determine the displacement # of 7. Evaluate S(@?=-y)de+ (y? +2)dy OC y>@.8,—4,-1) and (1,-2.5) crosses ‘the ye—plane. » @ particle at t = Qn if its ve- aloug the line Joining the two tet, locity and dsp at = points A(0, 1) and B(1-2) 44 0 are both 0 and if its acceler- 7 gg % ation at any time ¢ is given by AFB. @2 més za i ae“, (t+ 1), 3sinz) Sess | 8 y-! me ej -2_ayt 23 gap 8. Determine the centre of mass of I~! ‘i (A) (e* — 1)i + 20n9 — an 4 system of masses spread over a 3c (B) 24 3a8\} 4iteome te ; Cin, Ae en 44 (A+ 3n8)7 + (oF % space as Ske on 31-2) +68, 10ky 4h sia yon 2 — 10k, 12kg on 7} — ke and #4 a ee et 2m — 1k + (Br3 + ekg on 101 ~ 193. 5K? -t, mit or . “ (A) 2:(105, -10, — orto ToL atte, (D) e-44(O4.50)} — seine (A) $(105, 10, —101) aoe et BH age PD 34 (201,10, —205) 9) “res tase S'Given that A(1,0,5) and (C) £(10, 101, ~108) a > ne “MAGI, 2,9) are points, find, cor- 'D) 4,(105, 10, 101 a 73) gn 1 decimal place, the acute () ail ) doy, angle between the line AB and Rog the line with .yector equation 9. Find ‘the acute angle between the oe r= (1+3}+4k) +425 438). two vectors —13i + 5} ~ 12% and Fey M64, (A) 56.9° —(B) 65.92 81+ 57 + 4k wadéar, MOBI (D) a1 (A) 208-2) (B) oos-¥( 8) 4 4. Find the value of the scalar for 5 gs which the vectors (A—2)i-+3-4.26 (C) eos) (D) cas-1¢4) and ~12i ~ 4] — 8k are parallel e Be i i Ea 3 (D) 8 10. Let P, @ and R denote three (A) ~2 (B) 4(C) -3 (D) 5 points’ relative to. an origin O 5. Given that a x B+ a for Which of the following is NOT a the vectors i423 —& and correct expression for PQ? b= 21-74, find the value of 4. (A) PQ = PR~@R ‘ (A) -2 (B) £(D) 0 (B) PQ = GR - Pk i et ey la ny a Let o15. Which two of the following rela- tionships (/) to (IV) hold if AB is perpendicular to CD? (1) |AB x CD| =|AB\\CD| (1) AB = MCD), for 4 € R\{0} (11) AB-CD=~1 UV) AB- Cb = 0. (A) (2) and (11) (B) (7) and (IV) dua. de (sont 6ruy ! SRE ALOE G 19. + respectively, compute the scalar 20. dyys atde, fe oct & (C) (7) and (IV) % (D) (12) and (112) Xx. 11. The position vectors of t ts r $3) i 2, PSR eine Sig 16 Gin tt tro = Qare Bi—8(1+0)}+6ak and thar, find tho cornet valleo of a % 7 as Gah — (20 Aa 13)k. Suppose 1 2 is.a straight ee = Gs, esata ot (A) m=) mas 2 ‘ee (ay 1 eat) ge Se, S Az B-3 ©1 wi (C) m=45 D) m=33 ee Oe in ie A= 50 4sint}+ethang 2” TRE Ct cion of the parabola og OB Ee TERT) a Vortex o(2,—4, 1), latus teo- he, @°F B= costi+1%h, evaluate 14-8) sum units, in the plane contain- 7%, ae dé ing the two perpendicular vectors " oats tno ra a(i,4,—3) and 0(3,0,1), given te See “fore (A) (0,0,0) (B) S145 thal the axis of the parabola is @, Yep %e(C) -5t?sint + tem, pent isgivenas Fe of}y 2 yy eee i (A) = (24+ 3+ Bae rc ° af - Wompute toe projection of toe Sa Sat er 33 ~ GE on ‘the plane (B) 7 = @- 3+ Bt4+ 1 es 1 @ so 1 — oe _ Ay yy Aeorseepe sae 2 i iL a) qq(208 — 525 ~ sak) le (QC) F = 2+3y-— Feet ‘iia ae 2 (B) s(—"w0i + 599 + 338) 3 IR aS) een (©), +o — 59} 338) O07 ape erie Ae pe “ag e “ ae E+) Jaeocg (D) = (202 + 59) + 33%) ‘ As 7 a es iW 18. If A = P — tj + (2t + Yk and 14. What is the projection of the vec- Bs 3)i + j — th, then! GE tor 5i— 12} — 18k on the vector 45 a, dp Bit 4j — 3k? gax Qatar if: (A) 1672 (B) 3.1v3 (A) (1,7,2) ® (1,6,2) (C) 13V5 (D) 6.572 (C) (0,7, -3) (D) (0.3.7) by bee Given that |@| and || denote the magnitudes of vectors @ and 6, product of the vectors ([b|d-+ |alb) and (Ba — {a[). (A) -1 (B) 0 (C)1 (D) -2 Find the obtuse angle BAO if it is known that OA that OB = Pare ge a seh pa oe Oj) tee 3 2 Satephlats 4521. 23. 24, . Identify a ved, TA Poy ) () cos V3. ectory traced out by a particle whose position vector on the xz plane is given as asec 6i+ (btan@ + h)k is (A) a hyperbola (B) a parabola {C) a circle (D) an ellipse. ‘The equation of the normal to the parabola at) + 2atj is (A) 7a = atti + 2atj +X'(E+ 9) (B) Fa = ati + 2atj + NG (C) fa = ati + 2atf + G49) (D) Fn = at®l + 2atj + XG - #9) ‘The plane curve described by the equation # = ev/got6i + cvtandj is ec (A) a rectangular hyperbola (B) a parabola (C) an ellipse (D) acirele. Given that the work done (W’) by a force (F*) along a displace- ment (7) on a smooth curve C is given as W = JF di, de termine the work done by a mov ing particle under the force Seld fe = 3071 + (2ay — 2)j + zk along P the space curve whose parametric equations are «= 2, y= ¥ and P= 1, fromt=0 tot=1 22 $82 283 8 22 (py 382 (¢y 288 (0) & (A) 3 8) FC) ay ©) 50 which of the following could be a vector equation of a line. (Aye (B)r (ye (D)r (i-j- +198 iyi) (2i-j-W)-M-1-7-4) (2 j-B) ACE 3- 4) 26, What is the equation of a plane through the point (1,2,3) and perpendicular to ~3i — 5j + 7k? (A) -3 — Sy +72 =2 (B) 32+ 5y—72=6 (C) -32 + 5y - 7: (D) —82 — 5y + Ty 27. Which one of these js not a prop- erty of line integral? (A) foPlaiy)de + Q(e,u)dy = JePlt,y)de + fo 2, v)dy (B) [L2h)) Pla v)de+Q(@, v)dy = — fh) Pla, v)de + QCe. wav (©) fool P(a,v)dz+Q(e, v)dy) = c fo Ple.u)de + ¢ fo Oz y)ay (D) fre? P(e, vide + Q(x, vray = [o0k) Pla, vide + Qe, way — (2309) pea, yd + Ql, dy (re Sesh 28, Which one of the following iden- tities is NOT correct? (Aja G-9=a-5-a-2 (B) dx (6x2) = (@-Ab- (a. De (C) (@x6)-(B x2) = (@xB)-(xd) (D)@- G+ 9=a-b4a-¢ aq a + 29. The position vectors of points A, B and C' relative to an origi 3 are given by 61+3)+2k, 2+nj—k auul 81 +9) respectively. Compute the exact value of the constant n for which [AB] = |CB| Fb negate Com 2 BBB) 7) a7 30. Find, correct to 1 decimal place, the area of a triangle ABC whose sides are represented by vectors +h, %-j-kand -i-3 ye (D) 2 (A) Sl unit? (B) 4.1 unit? (C) 3.1 unit? (D) 2.1 unit? | 3/" ] ’ 1 o-~a/ s 5 iCA_% i [Cohe €OBAFEMI AWOLOWO UNIVERSITY, ILE-IFE, NIGERIA E DEPARTMENT OF MATHEMATICS 2022/2023 RAIN MID-SEMESTER_ EXAMINATION a MTH 104 - VECTORS 18th May, 2024 Duration Allowed: 1 Hour QUESTION TYPE: 1 Instructions: Fill your OMR Sheet correctly and shade the correct option ou he OMR Shes Hon ore nett fo dot in any of the options provided), shade the OMR Sheet. If the correct answer is not in any 0 option ‘B’ on the OMR Sheet. Make sure your shaded OMR Sheet Type is the same with your Question Type. ‘ 2 =2 1, Find the acute angle between the ‘A) cos'(—22) (B) cos) two vectors 13) + 5) ~ 12k and 4 ay3 avs —31 +5) + 4h. 3) cos“"(—=)_(D) cos“ "(3 7 (©) cos (57) (D) e085) 8 6 (AS cost S 18, coe 5) (yee ‘p 6. Let P, Q and R denote three a3 ad points’ relative to an origin (©) cos '(5)— (D) eas) Which of the following is NOT a 2. Which of the following identities correct expression for PQ? is NOT correct? (A) FQ = PR-GR (Aa G-O=a-5-a-2 | (8) PQ=GR-PR (B) éx bx a =(2-ab-@-He (C) PQ = PR+ RQ (©) (x1) Gx} = (@xB)-(Exd)——(D) PQ = RQ RP (D)a-(64+0 =a-b+a-2 tors Zand 6, eval- ) 4b Ox a). m4 7. For any two ve 3. Given thet |] and [2] denote the uate S2- (2x magnitudes cL pute tye seaar (A) Sb— 4 product of the vectors (|b|a-+|ald) (B) 7(a—2) ‘and ({Bja — [@)8) % a (A) -1 (B)0 (1 (D) 2 i _ 8 What is a unit vector that would bis be perpendicular to both of the 4. If the vectors @ = 2 b= i+ 3h and ; *) evaluate the triple product i-j + 3k, vectors 31 - j — 2k and i—j— k? ee, ANG 9 + 28 (alse J + 2h) f ax (bx d. (A) i959 +38 (B) 36 (c 5 Be ane BAO if it is ene 9, Compute the projection of the hat = i-j-k and that vector 7 sé on the plane oe Oa ree soi te) i on the planei — 59) Oye 20% + 59} + 33k) (Orgt- 20 — 59} - 33h) D)p_(208 +59) + 33k) 10. What is the projection of the vec- tor i — 12) ~ 13k on the vector Bit 4j — 3k? (A) 1.6v3(B) 3.1v3 (C) 13V8 (D) 65v2 The position vectors of two points P and Q relative to an origin O are 21 — (1 +.a)j + Gak and 85 + 3205 — (2a + 19k. Suppose that OP@ isa straight line, find the value of the constant a. nL 1 5 2 The magnitude of the resultant of two forces with magnitudes 27 and 4F, inclined at an angle 300° to each other, is (A) TFV¥3 (B) 7PV3 ) BF VT (C) 2FVv7 (D) (13) Find, correct to 1 decimal place. the area of a triangle ABC whose sides are represented by vectors i-j-+k, %—j—k and -i~3j-k (A) 5.1 unit? (B) 4.1 unit? (C) 3.1 unit? (D) 2.1 unit? What is the equation of a plane through the point (1,2,3) and perpendicular to —3i — 57 + 7h? {A) - Sy +72 (B) 32 + 5y — ~3z + Sy —7z Sy+Ty @®) (C1 M1 12, fa U 15. Which two of the following rela- tionships (I) to (IV) hold if AB is perpendicular to CD (1) [AB x CD| = |ABIICD| (11) AB = MCD). for \ € B\{0} U1) AB. GD = -1 ie 4B-CD ) (2) and uy ) ZT) and IV) ‘o (2) and (IV) (D) (2) and (171) 16. Identify which of the following could be @ vector equation of a line. (a) (B) (Cz -j-h) (D) 2 = (23-j—-A)-A(--j—Ak) 17. Given that A(1,0,5) and B(-1,2,9) are points, find, cox- rect to 1 decimal place, the acute angle between the line AB and the line with vector equation (4-39+48) +\(-i-2) +38). {A) 56.9% (B) 65.9° (C) 331° (D) 241° 18. The position vectors of points A, B and C relative to an origin O are given by 6i-+33+2h, 24+nj—k and 8% respectively. Compute the exact value of the constant nm for which |AB] = |CBI. (A) 2 (B)7 ( 19. Find the coordinates of the int where the line through '3,—4,—1) and (1,—2,5) crosses the yz—plane. (A) (0,-1.8) (C) @-8,1) 20. Given that vectors 3i ~ 3) ~ Smk and 3i7i+3)+mk are perpendic- ular, find the exact values of m (B) (~3.0.5) (D) (0.0, 6) (A) n= (C) m