This is the Multiple Choice Questions Part 6 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 6 of the Series

โ MCQ in Calculus Part 5 | Mathematics Board Exam

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

251. Find the area of the region bounded by y = x^{2} โ 5x + 6, the axis, and the vertical lines x = 0 and x = 4.

a. 5/7

b. 19/4

c. 17/3

d. 9/2

Answer: **Option C**

Solution:

252. A police car is 20 ft away from a long straight wall. Its beacon, rotating 1 revolution per second, shines a beam of light on the wall. How fast is the beam moving when it is nearest to the police car?

a. 10pi

b. 20pi

c. 30pi

d. 40pi

Answer: **Option D**

Solution:

253. Find area of the largest rectangle that can be inscribed in an equilateral triangle of side 20.

a. 24โ2

b. 39โ3

c. 50โ3

d. 40โ5

Answer: **Option C**

Solution:

254. A hole of 2 radius is drilled through the axis of a sphere of radius 3. Compute the volume of the remaining part.

a. 46.83

b. 59.23

c. 91.23

d. 62.73

Answer: **Option A**

Solution:

255. Find the maximum area of a rectangle circumscribed about a fixed rectangle of length 8 and width 4

a. 67

b. 38

c. 72

d. 81

Answer: **Option C**

Solution:

256. A trough filled with liquid is 2 m long and has a cross section of an isosceles trapezoid 30 cm wide of 50 cm. If the through leaks water at the rate of 2000 cm^{3}/min, how fast is the water level decreasing when the water is 20 cm deep.

a. 13/25

b. 1/46

c. 5/21

d. 11/14

Answer: **Option C**

Solution:

257. Determine the area of the region bounded by the curve y = x^{3} โ 4x^{2} + 3x and the axis, from x = 0 to x =3.

a. 28/13

b. 13/58

c. 29/11

d. 37/12

Answer: **Option D**

Solution:

258. Find the volume of a solid formed by rotating the area bounded by y = x^{2}, y = 8 โ x^{2} and the y axis about the x axis.

a. 268.1

b. 287.5

c. 372.9

d. 332.4

Answer: **Option A**

Solution:

259. The price p of beans, in dollars per basket, and the daily supply x, in thousands of basket, are related by the equation px + 6x + 7p = 5950. If the supply is decreasing at the rate of 2000 baskets per day, what is the rate of change of daily basket price of beans when 100,000 baskets are available?

a. 2.35

b. 1.05

c. 3.15

d. 4.95

Answer: **Option B**

Solution:

260. A flying kite is 100 m above the ground, moving in a horizontal direction at a rate of 10 m/s. How fast is the angle between the string and the horizontal changing when there is 300 m of string out?

a. 1/90 rad/sec

b. 1/30 rad/sec

c. 1/65 rad/sec

d. 1/72 rad/sec

Answer: **Option A**

Solution:

261. If functions f and g have domains D_{f} and D_{g} respectively, then the domain of f/g is given by

a. the union of Df and Dg

b. the intersection of Df and Dg

c. the intersection of Df and Dg without the zeros of function g

d. None of these

Answer: **Option C**

Solution:

262. Let the closed interval [a, b] be the domain of function f. The domain of f(x – 3) is given by

a. the open interval (a , b)

b. the closed interval [a , b]

c. the closed interval [a – 3 , b – 3]

d. the closed interval [a + 3 , b + 3]

Answer: **Option D**

Solution:

263. Let the interval (a , +infinity) be the range of function f. The range of f(x) – 4 is given by

a. the interval (a – 4 , +infinity)

b. the interval (a + 4, +infinity)

c. the interval (a, +infinity)

d. None of these

Answer: **Option A**

Solution:

264. If functions f(x) and g(x) are continuous everywhere then

a. (f/g)(x) is also continuous everywhere.

b. (f /g)(x) is also continuous everywhere except at the zeros of g(x).

c. (f /g)(x) is also continuous everywhere except at the zeros of f(x).

d. more information is needed to answer this question

Answer: **Option B**

Solution:

265. If functions f(x) and g(x) are continuous everywhere and f(1) = 2, f(3) = -4, f(4) = 8, g(0) = 4, g(3) = -6 and g(7) = 0 then lim (f + g)(x) as x approaches 3 is equal to

a. -9

b. -10

c. -11

d. -12

Answer: **Option B**

Solution:

266. If f(x) and g(x) are such that lim f(x) as x –> a = + infinity and lim g(x) as x –> a = 0, then

a. lim [ f(x) . g(x) ] as x –> a is always equal to 0

b. lim [ f(x) . g(x) ] as x –> a is never equal to 0

c. lim [ f(x) . g(x) ] as x –> a may be +infinity or -infinity

d. None of these

Answer: **Option C**

Solution:

267. A critical number c of a function f is a number in the domain of f such that

a. f ‘(c) = 0

b. f ‘(c) is undefined

c. (A) or (B) above

d. None of these

Answer: **Option C**

Solution:

268. The values of parameter a for which function f defined by f(x) = x3 + ax2 + 3x has two distinct critical numbers are in the interval

a. (-infinity , + infinity)

b. (-infinity , -3] U [3 , +infinity)

c. (0 , + infinity)

d. None of these

Answer: **Option D**

Solution:

269. If f(x) has one critical point at x = c, then

a. function f(x – a) has one critical point at x = c + a

b. function – f(x) has a critical point at x = – c

c. f(k x) has a critical point at x = c / k

d. (A) and (C) only

Answer: **Option D**

Solution:

270. The values of parameter a for which function f defined by f(x) = 3x^{3} + ax^{2} + 3 has two distinct critical numbers are in the interval

a. (-infinity , + infinity)

b. (-infinity , -3] U [3 , +infinity)

c. (0 , + infinity)

d. None of these

Answer: **Option B**

Solution:

271. If f(x) = x^{3} -3x^{2} + x and g is the inverse of f, then g ‘(3) is equal to

a. 10

b. 1 / 10

c. 1

d. None of these

Answer: **Option B**

Solution:

272. If functions f and g are such that f(x) = g(x) + k where k is a constant, then

a. f ‘(x) = g ‘(x) + k

b. f ‘(x) = g ‘(x)

c. Both (A) and (B)

d. None of these

Answer: **Option B**

Solution:

273. If f(x) = g(u) and u = u(x) then

a. f ‘(x) = g ‘(u)

b. f ‘(x) = g ‘(u) . u ‘(x)

c. f ‘(x) = u ‘(x)

d. None of these

Answer: **Option B**

Solution:

274. lim [e^{x} – 1] / x as x approaches 0 is equal to

a. 1

b. 0

c. is of the form 0 / 0 and cannot be calculated.

d. None of these

Answer: **Option A**

Solution:

275. If f(x) is a differentiable function such that f ‘(0) = 2, f ‘(2) = -3 and f ‘(5) = 7 then the limit lim [f(x) – f(4)] / (x – 4) as x approaches 4 is equal to

a. 2

b. -3

c. 7

d. 4

Answer: **Option D**

Solution:

276. If f(x) and g(x) are differentiable functions such that f ‘(x) = 3x and g'(x) = 2x^{2} then the limit lim [(f(x) + g(x)) – (f(1) + g(1))] / (x – 1) as x approaches 1 is equal to

a. 5

b. 10

c. 20

d. 15

Answer: **Option A**

Solution:

277. Find the Laplace transform of [ 2/(s+1)] โ [ 4/(s+3)].

A. [ 2 e( exp โt ) โ 4 e( exp โ 3t ) ]

B. [ e( exp โ2t ) + e( exp โ 3t ) ]

C. [ e( exp โ2t ) โ e( exp โ 3t ) ]

D. [ 2 e( exp โt ) ][1โ 2 e( exp โ 3t ) ]

Answer: **Option A**

Solution:

278. Determine the inverse Laplace transform of I(s) = 200 / (s^{2} + 50s + 1625)

A. I(s) = 2 e^(-25t) sin 100t

B. I(s) = 2t e^(-25t) sin 100t

C. I(s) = 2e^(-25t) cos 100t

D. I(s) = 2te^(-25t) cos 100t

Answer: **Option A**

Solution:

279. The inverse Laplace transform of s / [ (s square) + (w square) ] is

A. sin wt

B. w

C. (e exponent wt)

D. cos wt

Answer: **Option D**

Solution:

280. Find the inverse Laplace transform of (2s – 18) / (s^{2} + 9) as a function of x.

A. 2 cos x โ sin 3x

B. 2 cos 3x โ 6 sin 3x

C. 3 cos 2x โ 2 sin 6x

D. 6 cos x โ 3 sin 2x

Answer: **Option B**

Solution:

281. Determine the inverse laplace transform of 1 / (4s^{2} โ 8s).

A. (1/4) e^t sinht

B. (1/2) e^(2t) sinht

C. (1/4) e^t cosht

D. (1/2) e^(2t) cosht

Answer: **Option A**

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 โ 50**Answer key:

**PART 1**

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

**PART 2**

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

**PART 3**

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

**PART 4**

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

**PART 5**

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

**PART 6**

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