UNIVERSITY OF LAGOS ay SCHOOL OF POUNDATIGN STUDIES 7am FIRST INCOURSE EXAMIRATIOR 202472025 SESSION ae Tee. mnvsics-sc1-3155 a OPTION A sth Instruction: Answer ALL, quottions in this Section Time Atiowed: 2 ¥4 bes 1. Heat flows from one body to another when they have A. Different specific heat B, Different atomic structure “c.Ditferont temperatures D. Different heat content Temperature is. ‘A. 2 form of heat transfer whore energy is radiated in the form of rays. B. a form of hoattranafer. C. a form of energy contained in an object associated with the motion of molecules. ~D._ a moasuro of the hotness or coldness of an object. 3, Determine tho absolute temperature at which the Fahrenheit temperature is twice the Celsius temperature A. 433.15k B. 342.15K C, 444.15k D. 160.15k 4, A constant-volume gas thermometer is used to measure the temperature of an object. When the ‘thormometer is in contact with wator at its triplo point (273.16 K) the pressure in the thermometer is 8.500x 10 Pa, Whon it is in contact withthe object the pressure is 9.650x 10° Pa, The ‘temporaturo of the object is? A 310K Be o241K Ce y3I10K4 Dd. 314k 5. Which of the following properties is not considered in choosing mercury as a thermometic tigi? aa A. Density B. Expansivity C. Conductivity D._Opacity+40 6. What will be the power of a concave lens of focal length Sm? A. 5D, BF-5D. C.02D. D. -02D. 4 “7. A thermoestic thermometer read 2.1y¥ at triple point of water. What will be the indicated ‘voltage when placed in a theatre of temperature 26°C. A. 2347 2B 43m Cc. 33uv D. 83uV ("8 Abhelium gas thermometer record a pressure of 70.0 torr atthe triple poin of water and pressure of 90,20 torr at the system of interest . What i2 the system temperature in Kelvin A. 267.88K. B. 346.72K C. 349,99K Be“450,79K \ | 9. Which of the following statement is true about the anomalous expansion of water are correct? I There is contraction between 0°C and 4 °C II, There is expansion between between 0°C and 100 °C Ml. The volume is minimum at 4°C IV, The volume is maximum at 4°C. Mand only B.Wanditonly —C.Tand only. Hand 1V only \Q 10. The coefficient of linear expansion of iron is 1.0 10° per°C. ‘The surface area of an iron cube, with an edge length of 5.0 cm, will increase by what amount ifit is heated from 10°C to 60°C? (0.0125 cm? B, 0,025 cm? €.0.075 cm? + D015 em? 11. An aluminum ruler is calibrated at 25°C. Calculate the percentage error in the ruler on a cold day ‘when the temperature is -5 °C, Take the linear expansion coefficient for aluminum as 2.3 x 10° K* b A. 0.046% B. -0.046% C. 0.069% D. -0.069% 12. An iron rod ball at 40°c is d amg containing water at 40°c.theheatwill | A. flow from iron ball to water BB. not flow from iron ball to water or from water to iron ballC. flow from water to iron ball ( D. increase the temperature of both 13, Bimetallic strips are employed in ‘A. Vapor-pressure B. Liquid-expansion C. Metal-expansion D. Resistance 14, A metal of mass 200 g at a temperature of 100 °C is placed in 100 g of water at 25 °C ina container of negligible heat capacity. Ifthe final steady temperature is 30 °C. Calculate the specific heat capacity of the metal. (Specific heat capacity of water is 4200 J/kgk) A. 150KkgK B.3000KgK CC. 320IkgK iD, 1960IkgK. 15. Appiece of copper of mass 0.55k, internal energy of the copper. (sp* ‘heated from 57°c to 100°c. What is the increase in the heat capacity of copper =380/Kg™K"* ) A.8990J B. 7990) C. 7000) ~—-D. 87903 16, A fixed mass of gas of volume 600cm? at temperature of 27°C is cooled at constant pressure to a ‘a temperature of 0°C . What is the change in volume. ri a s ONS BG On” A. Sdom? VeV C.300cm? ‘D. 600cm? 17, 5000 J of heat is added to a system, and 2000 J of work is done by the system. Employing the First law of thermodynamics, what is the change in internal energy of the system? ‘A. 7000J B, 30003 C. S000J D. 2000 & 18. Coefficient of Performance (COP) of a refrigeration cycle is given by........ A.COP=WIQ. B.COP=W-Q C.COP=W+Q — ~D.COP=QUW 19, Which of the following is not a mode of heat transfer Charging B. Conduction C, Convection D. Radiation | © 20. When an object is placed in front of a plane mirror the image is, ‘A. Upright, magnified and real By Upright, the same size and virtual . Inverted, demagnified and real D. Inverted, magnified and virtual 2° 21. Using dimensional analysis, simplify these units spintere Tis joule s _p6M'L BML oc, MP p, M'L!2. nthe exrenion F = 0A, Fis fre, ie ae, and # is velocity gradient, Which ofthe following gives tho dimensions of 1 A MIL'T! p/MLT? —¢, ML'T! op, ML"T* 23. Photon is quantum of radiation with energy = hy whoro vis frequency and hs Planck's constant. Tho dimensions ofh are the same as that of {sy Linear impulse (B) Angular impulse (C)Lineas momentum —(D) Angular momentum 24, The equation for the magnotic force F on a current-carrying conductor of length ! placed in a ‘magnetic field B is F = Blisin0, where is the current, Bis the magnetic field strength, and @ is the angle between I and B. The dimension of B is (A)MLTAT —(B)MLT*A (@) MTA" (D) MLA 25. Given the vectors A =2i+3} + 4k, V=A.(BxC), (A) 182 cubic units (B) 80 cubic units (C)32 cubic units (D) 12 cubic units, j + 3k, and C = 2i + jk, find the magnitude of V if 26, Given the foreos Fi resultant force. (A)61+3 (B)3i-4j+6k — (C)4L+2)-6k (0) 4j-2k i+ 4j-2k, Fa = i-2}+ 3k, and Fs =2i + j -k acting on a body, find the 3, 21. Adeatify vector quantities among the following: 1. Displacement./ 4 oo Il. Eloctric field intensity. IIL. Acceleration 1, Magnetic induction V. Work Wt, 1, Il, and TV only B.1, Nv, andVonly C. Lv and V only D. I, and Ill only 28, Whatis th total displacement ofa girl who travels 12.m northwards, 5 m eastwards, and 7 m southwards? , A.Sim,noth B.Sm,east C.7.07m,S45°W 7.07 m, N4S°E 29. Aman walks $km south and then 3 km inthe direction 60° west of south. His distance from the starting point is A700km = -B.750km C.800km ——D. 10.72km 30. A stone is thrown horizontally from the top of a cliff with a speed of 10 mvs. If t takes 2 seconds to reach the ground, whats the height ofthe cliff? (A)20m = (B)40m (C)60m =D)80m He 31. A ball is thrown vertically upward with an initial velocity of 30 m/s. Wh after 4 seconds? (A) 10 m/supward (B}10 m/s downward —{C}20 m/s upward —{D)20-m/s downward — G, 32: An object moves ina straight line with a constant velocity of 10s. How far does ittavel in 15 seconds?| A)S0m —(B)100m_—(eF150m_—(D) 200m 83 33. A car with an initial velocity of 20.0 mvs is subjected to a uniform retardation of 2.0 ml/s?, Calculate the time required to completely stop the car. (A)Ss @Bl0s (20s (D)40s. 22 34. Two blocks of masses m and 2m are placed on a smooth horizontal surface as shown. In the first case only a force is applied from the left Later only a force Fis applied from the right. Ifthe force acting at the interface ofthe two blocks in the two cases is same, then Fi:F> is, A fel AL B12 Oz Mis IP) 35. When an objects at rest on the inclined rough surface, (A) static and kinetic frietions acting on the object is zero (B) static fiction is zero but kinetic friction is not zero (©) state friction isnot zero and kinetic friction is zero (D) static and frictions are not zero Fh 2» 36. How much centripetal fre is needed to make a body of mas 0.8 kg o move ina circle of aus 40 cm with a speed 2 ms"? (SN @)6N (CIN (BN 37, A'S kg block is placed on a rough inclined plane with an angle of 30° tothe horizontal. The coefficient of kinetic friction between the block and the plane is 0.2. What is the acceleration of the block Gown the plane? (A) 2.5 m/s* (B)3.5 mit (C)45 mis* (D)5.5 mis 38. Ifan object is moving with a constant speed ina circular path, its acceleration is: (A) Zero (B)Constant (C) Increasing —_(D) Centripetal 39. Ifno force is applied to a moving object, it will stop due to (A) gavitational force (B) constant force (C) centripetal force (D) Frictional force )% 40. Two forces with magnitude and directions as shown inthe diagram. What is the magnitude ofthe © resultant force, 4000 N (A)s600N (B) 5492.N (ABN (D) 6500N 5SECTION B-'THEORY Instruction: Answer FOUR ausstions i tls Section, Two 2) auctions from exch Course PHY001 - Mechanics and Properties of Matter any to (2) aves insu Kei In quantum mechanics, the fundamental constant called Planck's constant, h, has dimensions of ‘MLT", Construct a ua with the dimensions of longth from h, a mass m, and c, tho speed of light. (@ marks) Ve xine tel (aii) Differential botweon Vectors and scalar quantities. (1 mark) Give two examples ofeach (2 marks) (b) The two vectors a and b in the Figure below have equal magnitudes of 10.0 m. Find (i) the x component and the y component of their vector sum r, (2 marks) the magnitude of r and (2 marks) i) the angle r makes with the positive direction of the x axis. (marks) (©)() Given that A =2i+5j,B=3i—4), C=21+3k, find the tripe product (AxxB)+C.(3 marks) (ii) What vector M must be added to two vectors P = 2i+3j and Q R=i+2)-3k (2 marks) 4j to give a resultant Ht nord oro ey 2. (aiw State Newton's laws of motion (3 ants aytanal Soe 9 age (aii Define the term is impulse? (2 marks) F = VO eo ()Inatrip, John undertook the following displacements A= 72.4 m, 32° east of north; B= 573 m, 36° south of west and C= 17.8 m due south. (i) Sketch John’s trip. (2 marks) (Gi) What are the magnitude and direction ofits resultant displacement? (4. marks) —____ (©). Show that the path or trajectory taken by a projectile isa parabola ofthe form y= Ax Bx*, where ‘A and B are constants ( 4marks)3. (a State the main difference between fundamental quantities and derived quantities. (2 marks) Give two examples ofeach (2:marks) (0) A particle P of mass 2 kg is subjected to two forces F: and Fs. The forces are given by: Fi=(i+4j) Nand Fs =(Si-2/)N. Find the resultant force F acting on the particloP. (2 marks) i) Determine the: litude of the resultant force F. (2marks) i) If the particle is initially at est, calulate the acceleration ofthe particle P due to the resultant force. Give your answer in vector form, (2:marks) (@)A chunk of clay of mas 0.20 kg is thrown from the ground with an initial speed of 12 m/s tan angle of 30° withthe horizontal a shown below. At the top of its trajectory, te clay strikes a small block of ‘mass 2.3 kg suspended from a 3.0 m long string. The clay sticks tothe block, which then swings freely () Neglecting air resistance, caloulte the horizontal distance D between the launching point of the clay and a point on the floor directly below the inital position of the block. (3 marks) (Gi) Calculate the speed of the block-clay system immediately ater the collision withthe clay. (2 marks) 4. (a) State the laws of solid friction (2 marks) (b) A block m of mass 15 kg rests on the surface ofa rough plane inclined at an angle 30° to the horizontal. The coefficient of kineto friction between the plane andthe block is 0.25. A light inextensible string passing over a small smooth pulley at the top ofthe plane connects the block to another block M of sass 13 kg hanging freely, Find the acceleration of the resulting motion (3 marks) and the tension inthe string, (2 marks) Take g=9.8 mis. \4,(¢i) Explain the term acceleration due to gravity (2 marks) (Gil) A projectile is fired into the air from the top of a 200-m tower above the ground level, Its initial velocity is 60 m/s at 60° to the horizontal, Neglecting air resistance, where does the projectile land. What is the speed just before it hits the ground? (3 marks) What isthe maximum height reach? (3 marks) Take 0ms. PHY 002- Heat, Waves and Optics Answer any two (2) questions in this subsection 8/ (© @ State the zeroth law of thermodynamics, (mark) i) What does it meant for two bodies to be in thermal equilibrium? (2-marks) (b) Ina tabular form, distinguish between the following concepts ( Absolute zero and Triple points of water (i) Thermometer and Thermometric substance i) Lower fixed point and Upper fixed point (6 marks) (©) A 60.001 steel gas tank of a cars filled to 98% capacity with gasoline at 22°C. The car sits in the sun and the tank reaches a temperature of 45°C. Ignoring any changes inthe steel tank dimensions and taking B=9.5 x 104, i. Does the the tank overflows wiy?, Gmuks) fi, By what quantity does the tank overflow? (marks) & (#)() Define the linear expansivity of a substance, (2 marks) il) The Volume of a glass vessel is 200 em and its linear expansivity of 11x 10*/°C, if this ‘lass vessel is filled to the rim with mereury (0.18 x 10° *C) How much of the mercury will overflow the vessel if we raise the temperature by 30°C (3 marks) (iii) List three examples of applications of thermal expansion. @ marks) (b) Differentiate between the following: (lea apacity amd Spee et exp ty $$ (ii) Latent heat of fusion and Latent heat of vaporizationii) Isothermal process and Adiabatic process (iv) Isochorie process and Isobaric process __(1 mark each) (©) An iron tea kettle with mass 2.10 kg and containing 2.40 kg of water is placed on a cooker. Ifno heat is lost to the surroundings , how much heat must be added to raise the temperature from 20.0°C to 85.0°C.( specific heat capacity of iron is 470 JKg~1K-1, and that of water is 4190 Kg™*K~) (GB marks) 7. (a) List five postulates of kinetic theory. (5 matks) (b) (i State the equation of state for one mole of an ideal gas and explain each quantity inthe equation. (2 marks) (ii) A cylinder contains oxygen at temperature of 25°C and a pressure of 17 atm in a volume of 160 liters . A piston is lowered into the cylinder, decreasing the volume occupied by the gas to 90 liters and raising the temperature to 30°C. Assuming the oxygen behaves like idea gas under these conditions, what is the gas pressure in its final state? B marks) (©)@ State the first law of thermodynamics. (1 mark) (ii) _A gas ina cylinder expands from a volume of 0.113 to 0.32 m8, Heat flows into the gas just rapidly enough to keep the pressure constant at 1.15 x 105), Calculate the workdone by the gas (2 marks) and the change in internal energy of the gas. (2 marks) 8, (a) State the laws of refractions. (2marks) (b) Explain the following terms (i) Critical angle (ii) Total internal reflections (4marks) (ci) A 2.0 cm high object is placed on the principal axis of a concave mirror at a distance of 12 om from the pole, If the image is inverted, real and 5.0 cm high, find the location of the image and the focal length of the mirror. (4 marks) (ii) A ray of light travels from glass to air at an angle of incidence a = 35" as shown below, ‘The ray is partially reflected from the glass-air boundary at an angle a and partially refracted at the angle ay . The index of refraction of the glass is 1.5. The speed of light in air = 3 x 10°m/s . Calculate: i, the speed of light in the glass, (1 mark) angle of reflection a, and (2 marks) iii, angle of refraction as (2 marks)AN UNIVERSITY OF LAGOS § Ue ‘SCHOOL OF FOUNDATION STUDIES FIRST INCOURSE 2024/2025 SESSION See ae MATHEMATICS SCI-J154 Section A-Multiple Choice questions OPTION-B Instruction: Answer ALL questions in this section Time Allowed: 2 Hours 1 nn that x— 2, x—1and 3x5 are three consecutive terms of a geometric progression, Find the possible values of x. Aad B32 an__p Given that wae GD AB B2 Cs D4 The sum of the first m positive integers can be expressed as iin the value of 29 —p. a, met 2 m2 2 m(m +1) 2 snl + 12m +2) 6 The sum 1°+2? +3? +... +4 of the first k natural numbers is e+) pane Kk+I(k+2) 2 R&+)? 4 K(k+DQk+1) 6 ‘ D. A B Gj5. Which of these identities illustrate De-Morgan law. A AU(ANB)=A B. (AnBncy =A uB'UC! C Anas D.An(BuC)=(AnB)U(anc) 6. If P is the set of prime factors of 42 and Q is the set of prime factors of 45, Find Png. A (2) 8. (3) c {3} D7 7. Let Nand z denote the set of natural numbers and positive integers, respectively, Given that A= (re N: x < 4JandB= (xe Z: xz 2},then deduce (AB) x (A\B). A. (042.3) B. ((23),(2.4)) ©. (@1).G.0) D. (3.2), 4.2) 8. Deduce the values of if 97° = 30-2), Bis a ha dD h2 9. Given that logsy 2 = 0.3010, logiy 3 = 0.4771 and logo 7 = 0.8451. Compute Logi» 84, A. 06232 B, 0.9242 C. 16232 b. 19242 10, Find an admissible value of x in 1/, 1og,81 = 2. 1 AG Boa C2 1 DG A.{ sindh H.determine a Sy on A B © dD. 12, Find the derivative of the function /( 13.Find the maximum and minimum values of y=»? 33° +5 AL (0,5)aac> (2.7)ain B, 6.0 na (ae 6, Sans Da db. Ge) + 2,0)eis 14. Given y =e”, obtain the fifth derivative of”. Awe B, 18e" ¢, Sle" b, 2e" —15:For the function y* =1+x", find the derivative y’ at the point (x,y)16, Find the derivative of the function /(x)=sin(e") A. e* case! B. coset C. cos (ete) D. sine" cose* 17. If y=Inx*, find fray A. 12x - 2inx? Bax? 2m? ©. 484° 2Inx? D, -48r*—2y 18.Find the differential coefficient of y=~2x" +4x+7and determine the gradient of the curve at x=-1.5. AL ~ 6x44, -9.4 B. ~6x*+4, -9.5 C. 64441, -90 D. -6x7 +4, -3.5 19.Find the inverse of the function f(x) =2x?+120. Find the first order partial derivative Lot o* sin(x +») A, e cos(x-ty) +sin(x+y) Be cos(x+y)~sin(x+y) Ce cos(x+y) p, 7 c0s(x+y)—e sin(x+ y) 21, Determine lim} 282% 0 Sh? 47h 22.Find the derivative of the function f(x) ~Sxt43x4 (aay 23. Find the maximum and minimum values of y= x*-3x*+5 B. ,5)anc> 257)aia B, 60mu> 21)aia 6. O3)aac> 21a d. (59) . (x) 24..Giveny=e™ , obtain the fifth derivative of ». A, 3eple” ¢, Sle Dp, 243e* 25, For the function y* =1+x", find the derivative y' at the point (x,y) Al a -£ 26, Find the derivative of the function f(x)=sin(e*) A. e* cose B, cose? c, 8 (ee) p, Sine*cose* 27. Tf y=Inx*, find thoy A, 12x 2Inx? B, 12*—2inx* C. 489 —2Ina? D. -48x%-2y 28. Find the differential coefficient of y=-2x' +4x+7and determine the gradient of the curve at x=-15, A. ~6x+4, -94 B. ~6x'+4, -95 © 64441, -90 D, ~6hr=35—29. Find the inverse of the function f(x) =2x? +1 ¥. 30. Find the first order partil derivative Sof e*sin+y) A cos(x+ y)+sin(x+ y) Be cos(x+y)—sin(x+ y) C. eFcos(x+y) D. eFcos(x+y)-e*sin(x+y) 31, Find the Identity element of real numbers under the operation * defined by x 1 e yay. A3 8 ‘, D. Same) oes not exist 32, Ina class'of 80 students for first in-course text, 34 students offer Physics, 29 offers Chemistry while 42 offer Mathematics, 12 offer Physics and Chemistry, 14 offer Mathematics and Chemistry, and 8 offer Physics and ‘Mathematics, If 4 students did not offer any of the three subjects, how many students offer all the three subjects? A4 Nae 33, Which of the following numbers is not an irrational number? g9ae 22 9oeregs aise aE“the seventh term, A4 B. 13 a9 Bt 35, The coefficients of the second and third term of the expansion of (1 + cx)* are 6 and 12 respectively. If c is a positive integer, find the value of c. a3 B.2 . a. D4 36, Given that @ and # are the roots of the quadratic equation 3x? + 4x —5 =0. Find the value of a? + g?. a 3 B 2 ae oF 37, What is the remainder when the polynomial x5 — 4x3 + 2x +3 is divided by x— RP aIOP> the value of k if logse 25 = klogs 5. a in wile or 39, What is the value of 427-2, if 16%#8 = 64**3) z 32 c, Bi D. 256 40. The roots of a quadratic equation ax? + bx + c = Oare 1and<. What is +b + ©? (6 4 ac -1 iy “Ob PRETO P> ab~ SECTION B: THEORY Instruction: Answer FOUR in ALL, Two (2) question ch course MAT O01: PURE MATHEMATICS Answer any two (2) a us QUESTION ONE (@) Show clearly that Len 7 i [5 marks] (b) Tf a, b, cand d are in harmonic progression, show that b = 2ac/ (a +c).{5 marks] (©) A binary operation * is defined on the set of real number by m+n = m-+mn-+n, ‘Show that the operation * is commutative. Solve the equation x + (x +2) = -2. {5 marks} QUESTION TWO (©) With the aid of mathematical induction, show that £(x")=nx"#, [5 marks] (©) Given functions f(x) =3x7 -x4+5 and g(x) =6x + 7x? +7x-+10. Compute fog [5 marks] (6) Using Mathematical induction, show that m* ~ m is divisible by 3 for all natural numbers m. [ marks} QUESTION THREE + (@) Let A and B be set in the same universal set such that 4AB = (4~B)U(B~ 4). Then illustrate AABin the Venn diagram [5 marks} (b) Find the possible values of the constant h for which the equation h(x—4) = x —4x+1has equal roots. B marks] (©) The roots of the equation? + 2px + q = O differ by 2. Show that p? = 1 +9. G marks] QUESTION FOUR () When the polynomial H(x)=dk? + fe? +72-+10is divided by x-1, the remainder is -10 and when itis divided by x-3, then remainder is 36, Find the values of the constants d and f. 1 marks] (b) Expand (-3) * binomially up tothe third order and state the range of lidity of the expansion, 5 marks}(€) The first three terms of the expansion (1 +r2)" in ascending powers of x are 1+ 20x-+ 160x?, Find the values of m and r. {5 marks} MAT 002: CALCULUS. ‘Answer ony TWO (2) question subsecti QUESTION FIVE 3x42, x<2 Ci sa jo oad if x2-2, Then, find: @lim f() Gi) lim f() [5 marks] (b) Calculate & for the following parametrically defined plane curve & x(D=Scost, y()=Ssinz*. [5 marks] (d) Determine the equation of the tangent and normal to the curve = at the point ( 1-4). [5 marks] QUESTION SIX (a) Determine the maximum area of a rectangular piece of land that can be enclosed by 1200m of fencing. {5 marks] (b) Obtain the expression of cos4x in the ascending powers of x as far as the term.x* mks] (©) Evaluate the integrals | [5 marks] 10QUESTION seven (©) Evaluate lim, !=sinx [5 marks) osx (b) Find the derivative of the function f(x) =2x? +4x+1 from the first principle [5 marks} (OIF y? =6xy- =I, prove that 2 [5 marks] QUESTION EIGHT Pen (@) Evaluate (E—VG*= 28-2) 4, (5 marks) 1 (b) Assuming y is a differentiable function of x, find the derivative of yey Sx" —xsinx=0 {8 merks} (©) Acurvey has sredient 2. 3x? ~6x+2. Find the equation of the curve if it posses ‘through the Origin, [5 morks} n