This is the Multiple Choice Questions Part 5 of the Series in Calculus topic in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

#### MCQ Topic Outline included in Mathematics Board Exam Syllabi

- MCQ in Complex Variables | MCQ in Derivatives and Applications | MCQ in Integration and Applications | MCQ in Transcendental Functions | MCQ in Partial Derivatives | MCQ in Indeterminate forms | MCQ in Multiple Integrals | MCQ in Differential Equations | MCQ in Maxima/Minima and Time Rates

#### Continue Practice Exam Test Questions Part 5 of the Series

⇐ MCQ in Calculus Part 4 | Mathematics Board Exam

**Choose the letter of the best answer in each questions.**

201. Evaluate: Lim (tan^{3 }3x) / x^{3} as x approaches 0.

a. 0

b. 31

c. 27

d. Infinity

Answer: **Option C**

Solution:

202. Evaluate the integral xcosx dx.

a. xsinx + cosx + C

b. x2sinx + C

c. xcosx + sinx + C

d. 2xsinx + cosx + C

Answer: **Option A**

Solution:

203. Find the limit: sin2x/sin3x as x approaches to 0.

a. 1/3

b. 3/4

c. 2/3

d. 0

Answer: **Option C**

Solution:

204. A snowball is being made so that its volume is increasing at the rate of 8 ft^{3}/min. Find the rate at which the radius is increasing when the snowball is 4 ft in diameter.

a. 0.159 ft/min

b. 0.015 ft/min

c. 0.259 ft/min

d. 0.325 ft/min

Answer: **Option A**

Solution:

205. A stone is dropped into a still pond. Concentric circular ripples spread out, and the radius of the disturbed region increases at the rate of 16 cm/s. At what rate does the area of the disturbed increase when its radius is 4 cm?

a. 304.12 cm^{2}/s

b. 503.33 cm^{2}/s

c. 402.12 cm^{2}/s

d. 413.13 cm^{2}/s

Answer: **Option C**

Solution:

206. Find the limit (x + 2)/(x – 3) as x approaches 3.

a. 0

b. infinity

c. indeterminate

d. 3

Answer: **Option B**

Solution:

207. A man 1.8 m. tall is walking at the rate of 1.2 m/s away from a lamp post 6.7 m high. At what rate is the tip of his shadow receding from the lamp post?

a. 2.16 m/s

b. 1.64 m/s

c. 1.83 m/s

d. 1.78 m/s

Answer: **Option B**

Solution:

208. A man on a wharf is pulling a rope tied to a raft at a rate of 0.6 m/s. If the hands of the man pulling the rope are 3.66 m above the water, how fast is the raft approaching the wharf when there are 6.1 m of rope out?

a. -1.75 m/s

b. -0.25 m/s

c. -0.75 m/s

d. -0.54 m/s

Answer: **Option C**

Solution:

209. Evaluate the limit: tanx / x as x approaches 0.

a. 0

b. undefined

c. 1

d. infinity

Answer: **Option C**

Solution:

210. A man is riding his car at the rate of 30 km/hr toward the foot of a pole 10 m high. At what rate is he approaching the top when he is 40 m from the foot of the pole?

a. -5.60 m/s

b. -6.78 m/s

c. -8.08 m/s

d. -4.86 m/s

Answer: **Option C**

Solution:

211. Find the point on the curve y = x3 at which the tangent line is perpendicular to the line 3x + 9y = 4.

a. (1, 1)

b. (1, -1)

c. (-1, 2)

d. (-2, -1)

Answer: **Option A**

Solution:

212. A boy wishes to use 100 feet of fencing to enclose a rectangular garden. Determine the maximum possible area of his garden.

a. 625 ft^{2}

b. 524 ft^{2}

c. 345 ft^{2}

d. 725 ft^{2}

Answer: **Option A**

Solution:

213. Find the equation of the tangent line to the curve x^{3} + y^{3} = 9 at the given point (1, 2).

a. x + 4y = 9

b. 2x + 4y = 5

c. 4x – y = 9

d. 4x – 2y = 10

Answer: **Option A**

Solution:

214. Find the area of the largest rectangle whose base is on the x axis and whose upper two vertices lie on the curve y = 12 – x^{2}.

a. 24

b. 32

c. 16

d. 36

Answer: **Option B**

Solution:

215. Find the radius of the largest right circular cylinder inscribed in a sphere of radius 5.

a. 4.08 units

b. 1.25 units

c. 5.14 units

d. 8.12 units

Answer: **Option A**

Solution:

216. A rectangular box open at the top is to be constructed from a 12 x 12-inch piece of cardboard by cutting away equal squares from the four corners and folding up the sides. Determine the size of the cutout that maximizes the volume of the box.

a. 6 inches

b. 1.5 inches

c. 2 inches

d. 3 inches

Answer: **Option C**

Solution:

217. Find dy / dx if y = 5^{(2x + 1)}.

a. (5^{(2x + 1)}) ln25

b. (5^{(2x + 1)}) ln(2x + 1)

c. (5^{(2x + 1)}) ln5

d. (5^{(2x + 1) }ln15

Answer: **Option A**

Solution:

218. An athlete at point A on the shore of a circular lake of radius 1 km wants to reach point B on the shore diametrically opposite A. If he can row a boat 3 km/hr and jog 6 km/hr, at what angle with the diameter should he row in order to reach B in the shortest possible time?

a. 30°

b. 50°

c. 45°

d. 60°

Answer: **Option A**

Solution:

219. Find the area of the region above the x axis bounded by the curve y = -x^{2} + 4x – 3.

a. 1.333 square units

b. 3.243 square units

c. 2.122 square units

d. 1.544 square units

Answer: **Option A**

Solution:

220. Find the volume of the solid of revolution formed by rotating the region bounded by the parabola y = x^{2} and the lines y = 0 and x = 2 about the x axis.

a. 25.01 cu. units

b. 15.50 cu. units

c. 20.11 cu. units

d. 30.14 cu. units

Answer: **Option C**

Solution:

221. A publisher estimates that in t months after he introduces a new magazine, the circulation will be C(t) = 150t^{2} + 400t + 7000 copies. If this prediction is correct, how fast will the circulation increase 6 months after the magazine is introduced?

a. 1200 copies/month

b. 2202 copies/month

c. 2000 copies/month

d. 2200 copies/month

Answer: **Option D**

Solution:

222. What is the order and degree of the differential equation y’’’ + xy’’ + 2y(y’)^{2} +xy = 0.

a. first order, second degree

b. second order, third degree

c. third order, first degree

d. third order, second degree

Answer: **Option C**

Solution:

223. A curve is defined by the condition that at each of its points (x, y), its slope is equal to twice the sum of the coordinates of the point. Express the condition by means of a differential equation.

a. dy / dx = 2x + 2y

b. dy / dx = 2x + 2ydy

c. dy = 2xdx + 2y

d. x + y = 2y’

Answer: **Option A**

Solution:

224. Find the first derivative of ln (cosx).

a. cscx

b. –tanx

c. secx

d. cotx

Answer: **Option B**

Solution:

225. Find the number of equal parts into which a given number N must be divided as that their product will be a maximum.

a. N/2e

b. N/e

c. 2N/e^{2}

d. 2N/e

Answer: **Option B**

Solution:

226. An object moves along the x – axis so that its x-coordinate obeys the law x = 3t^{2} + 8t + 1. Find the time when its velocity and acceleration are the same.

a. 2/3

b. 3/5

c. 3/4

d. 4/5

Answer: **Option A**

Solution:

227. Assuming that the earth is a perfect sphere, with radius 4000 miles. The volume of ice at the north and south poles is estimated to be 8,000,000 cubic miles. If this ice were melted and if the resulting water were distributed uniformly over the globe, approximately what should be the depth of the added water at any point on the earth?

a. 120 ft.

b. 320 ft.

c. 210 ft.

d. 230 ft.

Answer: **Option C**

Solution:

228. Find the equation of the curve passing through the point (3, 2) and having slope 2x^{2} – 5 at any point (x, y).

a. 2x^{3} – 15x – 3y + 2 = 0

b. 3x^{3} – 5x – 2y – 1 = 0

c. 2x^{3} + 5x – 3y – 21 = 0

d. 5x^{3} – 3x – 3y + 1 = 0

Answer: **Option A**

Solution:

229. Find the centroid of the region bounded by y = x^{2}, y = 0, and x = 1.

a. (1/4, 2/3)

b. (2/3, 5/4)

c. (3/4, 3/10)

d. (3/5, 5/10)

Answer: **Option C**

Solution:

230. Find the point of inflection of the curve x^{3} – 3x^{2} – x + 7.

a. 2, 3

b. 2, 6

c. 1, 5

d. 1, 4

Answer: **Option D**

Solution:

231. Find two numbers whose sum is 36 if the product of one by the square of the other is a maximum.

a. 12, 23

b. 25, 11

c. 16, 20

d. 12, 24

Answer: **Option D**

Solution:

232. Find the minimum distance from the curve y = 2 square root of 2x to the

a. 3.56

b. 4.66

c. 5.66

d. 2.66

Answer: **Option C**

Solution:

233. Divide 60 into 3 parts so that the product of the three parts will be the maximum. Find the product.

a. 6,000

b. 8,000

c. 4,000

d. 12,000

Answer: **Option B**

Solution:

234. A particle moves along a path whose parametric equations are x = t^{3} and y = 2t^{2}. What is the acceleration of that particle when t = 5 seconds?

a. 30.26 m/s^{2}

b. 18.56 m/s^{2}

c. 21.62 m/s^{2}

d. 23.37 m/s^{2}

Answer: **Option A**

Solution:

235. Find the area bounded by the curve 5y^{2} = 16x and the curve y^{2} = 8x – 24.

a. 36

b. 25

c. 16

d. 14

Answer: **Option C**

Solution:

236. Find the area in the first quadrant bounded by the parabola y^{2} = 4x and the line x = 3 and x = 1

a. 5.595

b. 4.254

c. 6.567

d. 7.667

Answer: **Option A**

Solution:

237. Find the area enclosed by the curve x^{2} + 8y + 16 = 0, the line x = 4 and the coordinate axes.

a. 8.97

b. 10.67

c. 9.10

d. 12.72

Answer: **Option B**

Solution:

238. Find the volume of the solid formed by rotating the curve 4x^{2} + 9y^{2} = 36 about the line 4x + 3y – 20 = 0

a. 356.79

b. 138.54

c. 473.74

d. 228.56

Answer: **Option C**

Solution:

239. Determine the moment of inertia of a rectangle 100cm by 300cm with respect to a line through its center of gravity and parallel to the shorter side.

a. 225 x 106 cm^{4}

b. 125 x 106 cm^{4}

c. 325 x 106 cm^{4}

d. 235 x 106 cm^{4}

Answer: **Option A**

Solution:

240. Find the area of the region bounded by y^{2} = 8x and y = 2x.

a. 3/4

b. 5/4

c. 4/3

d. 5/6

Answer: **Option C**

Solution:

241. Two posts, one 8 ft. high and the other 12 ft. high, stand 15 ft. apart from each other. They are to be stayed by wires attached to a single stake at ground level, the wires running to the tops of the posts. How far from the shorter post should the stake be placed to use the least amount of wire?

a. 6 ft.

b. 5 ft.

c. 9 ft.

d. 8 ft.

Answer: **Option A**

Solution:

242. At the maximum point, the second derivative of the curve is

a. 0

b. Negative

c. Undefined

d. Positive

Answer: **Option B**

Solution:

243. Determine the curvature of the curve y^{2} = 16x at the point (4, 8).

a. -0.0442

b. -0.1043

c. -0.0544

d. -0.0254

Answer: **Option A**

Solution:

244. Determine the value of the integral of sin53xdx from 0 to pi over 6.

a. 0.457

b. 1.053

c. 0.0178

d. 0.178

Answer: **Option D**

Solution:

245. A body moves such that its acceleration as a function of time is a = 2 + 12t, where “a” is in m/s^{2}. If its velocity after 1 s is 11 m/s. find the distance traveled after 5 seconds.

a. 256 m

b. 340 m

c. 290 m

d. 420 m

Answer: **Option C**

Solution:

246. A runner and his coach are standing together on a circular track of radius 100 meters. When the coach gives a signal, the runner starts to run around the track at a speed of 10 m/s. How fast is the distance between the runners has run ¼ of the way around the track?

a. 5.04 m/s

b. 6.78 m/s

c. 5.67 m/s

d. 7.07 m/s

Answer: **Option D**

Solution:

247. A telephone company has to run a line from a point A on one side of a river to another point B that is on the other side, 30 km down from the point opposite A. the river is uniformly 10 km wide. The company can run the line along the shoreline to a point C then run the line under the river to b. the cost of laying the line along the shore is P5000 per km, and the cost of laying it under water is P12, 000 per km. Where the point C should be located to minimize the cost?

a. 5.167 km

b. 6.435 km

c. 4.583 km

d. 3.567 km

Answer: **Option C**

Solution:

248. The height of a projectile thrown vertically at any given time is define by the equation h(t) = -16t^{2} + 256t. What is the maximum height reach by the projectile?

a. 1567 ft

b. 1920 ft

c. 1247 ft

d. 1024 ft

Answer: **Option D**

Solution:

249. The density of the rod is the rate of change of its mass with respect to its given length. A certain rod has length of 9 feet and a total mass of 24 slugs. If the mass of a section of the rod of length x from its left end is proportional to the square root of this length, calculate the density of the rod 4 ft from its left end.

a. 1 slug/ft

b. 2 slugs/ft

c. 3 slugs/ft

d. 4 slugs/ft

Answer: **Option B**

Solution:

250. It costs 0.05 x^{2} + 6x + 100 dollars to produce x pounds of soap. Because of quantity discounts, each pound sells for 12 – 0.15x dollars. Calculate the marginal profit when 10 pounds of soap is produced.

a. $ 9

b. $ 2

c. $ 12

d. $ 7

Answer: **Option B**

Solution:

### Online Questions and Answers in Calculus Series

Following is the list of multiple choice questions in this brand new series:

**MCQ in Calculus**

**MCQ from Number 1 – 50**Answer key:

**PART 1**

**MCQ from Number 51 – 100**Answer key:

**PART 2**

**MCQ from Number 101 – 150**Answer key:

**PART 3**

**MCQ from Number 151 – 200**Answer key:

**PART 4**

**MCQ from Number 201 – 250**Answer key:

**PART 5**

**MCQ from Number 251 – 300**Answer key:

**PART 6**

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