This is the Multiple Choice Questions Part 4 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 4 of the Series

โ MCQ in Calculus Part 3 | Mathematics Board Exam

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

151. An object falls from rest in a medium offering a resistance. The velocity of the object before the object reaches the ground is given by the differential equation dV/dt + V/10 = 32 ft/sec. What is the velocity of the object one second after it falls?

A. 40.54

B. 38.65

C. 30.45

D. 34.12

Answer: **Option C**

Solution:

152. In a tank are 100 liters of brine containing 50 kg. total of dissolved salt. Pure water is allowed to run into the tank at the rate of 3 liters a minute. Brine runs out of the tank at the rate of 2 liters a minute. The instantaneous concentration in the tank is kept uniform by stirring. How much salt is in the tank at the end of one hour?

A. 15.45 kg.

B. 19.53 kg.

C. 12.62 kg.

D. 20.62 kg.

Answer: **Option B**

Solution:

153. A tank initially holds 100 gallons of salt solution in which 50 lbs of salt has been dissolved. A pipe fills the tank with brine at the rate of 3 gpm, containing 2 lbs of dissolved salt per gallon. Assuming that the mixture is kept uniform by stirring, a drain pipe draws out of the tank the mixture at 2 gpm. Find the amount of salt in the tank at the end of 30 minutes.

A. 171.24 lbs

B. 124.11 lbs

C. 143.25 lbs

D. 105.12 lbs

Answer: **Option A**

Solution:

154. If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuously compounded account?

A. P5,750

B. P6,750

C. P7,500

D. P6,350

Answer: **Option B**

Solution:

155. A nominal interest of 3% compounded continuously is given on the account. What is the accumulated amount of P10,000 after 10 years?

A. P13,620.10

B. P13,500.10

C. P13,650.20

D. P13,498.60

Answer: **Option D**

Solution:

156. Simplify the expression i^{1997} + i^{1999}, where i is an imaginary number.

A. 0

B. โi

C. 1 + i

D. 1 โ i

Answer: **Option A**

Solution:

157. Simplify: i^29 + i^21 + i

A. 3i

B. 1 โ i

C. 1 + i

D. 2i

Answer: **Option A**

Solution:

158. Write in the form a + bi the expression i^{3217} โ i^{427} + i^{18}

A. 2i +1

B. โi +1

C. 2i โ 1

D. 1 + i

Answer: **Option C**

Solution:

159. The expression 3 + 4i is a complex number. Compute its absolute value.

A. 4

B. 5

C. 6

D. 7

Answer: **Option D**

Solution:

160. Write the polar form of the vector 3 + j4.

A. 6 cis 53.1ยฐ

B. 10 cis 53.1ยฐ

C. 5 cis 53.1ยฐ

D. 8 cis 53.1ยฐ

Answer: **Option C**

Solution:

161. Evaluate the value of sqrt(-10) x sqrt(-7).

A. i

B. โsqrt(70)

C. sqrt(70)

D. sqrt(17)

Answer: **Option B**

Solution:

162. Simplify (3 โ i)^{2} โ 7(3 โ i) + 10

A. โ (3 + i)

B. 3 + i

C. 3 โ i

D. โ (3 โ i)

Answer: **Option D**

Solution:

163. If A = 40 e^{(j120ยฐ)}, B = 2 cis -40ยฐ, C = 26.46 + j0, solve for A + B + C.

A. 27.7 cis 45ยฐ

B. 35.1 cis 45ยฐ

C. 30.8 cis 45ยฐ

D. 33.4 cis 45ยฐ

Answer: **Option C**

Solution:

164. What is 4i cube times 2i square?

A. -8i

B. 8i

C. -8

D. -8i^{2}

Answer: **Option B**

Solution:

165. What is the simplified complex expression of (4.33 + j2.5) square?

A. 12.5 + j21.65

B. 20 + j20

C. 15 + j20

D. 21.65 + j12.5

Answer: **Option A**

Solution:

166. Find the value of (1 + i)^{5}, where I is an imaginary number.

A. 1 โ i

B. -4(1 + i)

C. 1 + i

D. 4(1 + i)

Answer: **Option B**

Solution:

167. Find the principal 5th root of [ 50(cos 150ยฐ + jsin 150ยฐ) ].

A. 1.9 + j1.1

B. 3.26 โ j2.1

C. 2.87 + j2.1

D. 2.25 โ j1.2

Answer: **Option A**

Solution:

168. What is the quotient when 4 + 8i is divided by i^{3}?

A. 8 โ 4i

B. 8 + 4i

C. -8 + 4i

D. -8 โ 4i

Answer: **Option C**

Solution:

169. If A = -2 โ 3i, and B = 3 + 4i, what is A/B?

A. (18 – i)/25

B. (-18 – i)/25

C. (-18 + i)/25

D. (18 + i)/25

Answer: **Option B**

Solution:

170. Rationalize (4 + 3i) / (2 – i).

A. 1 + 2i

B. (11 + 10i)/5

C. (5 + 2i)/5

D. 2 + 2i

Answer: **Option A**

Solution:

171. Simplify [(2 + 3i)(5 – i)] / [(3 โ 2i)^{2}].

A. (221 โ 91i)/169

B. (21 + 52i)/13

C. (-7 + 17i)/13

D. (-90 + 220i)/169

Answer: **Option C**

Solution:

172. What is the simplified expression of the complex number (6 + j2.5)/(3 + j4)?

A. โ 0.32 + j0.66

B. 1.12 โ j0.66

C. 0.32 โ j0.66

D. โ 1.75 + j1.03

Answer: **Option B**

Solution:

173. Perform the operation: 4 (cos 60ยฐ + i sin 60ยฐ) divided by 2(cos 30ยฐ + i sin 30ยฐ) in rectangular coordinates.

A. square root of 3 โ 2i

B. square root of 3 โ i

C. square root of 3 + i

D. square root of 3 + 2i

Answer: **Option C**

Solution:

174. Find the quotient of (50 + j35)/(8 + j5).

A. 6.47 cis 3ยฐ

B. 4.47 cis 3ยฐ

C. 7.47 cis 30ยฐ

D. 2.47 cis 53ยฐ

Answer: **Option A**

Solution:

175. Three vectors A, B and C are related as follows: A/B = 2 at 180ยฐ, A + C = -5 + j15, C = conjugate of B. Find A.

A. 5 โ j5

B. -10 + j10

C. 10 โ j10

D. 15 + j15

Answer: **Option B**

Solution:

176. Evaluate cosh(jฯ/4).

A. 0.707

B. 1.41 + j0.866

C. 0.5 + j0.707

D. j0.707

Answer: **Option A**

Solution:

177. Evaluate tanh(jฯ/3).

A. 0.5 + j1.732

B. j0.866

C. j1.732

D. 0.5 + j0.866

Answer: **Option C**

Solution:

178. Evaluate in (2 + j3).

A. 1.34 + j0.32

B. 2.54 + j0.866

C. 2.23 + j0.21

D. 1.28 + j0.98

Answer: **Option D**

Solution:

179. Evaluate the terms of a Fourier series 2 e^{(j10ฯt)} + e^{(- j10ฯt)} at t = 1.

A. 2 + j

B. 2

C. 4

D. 2 + j2

Answer: **Option C**

Solution:

180. Given the following series:

sin x = x โ x^{3}/3! + x^{5}/5! + โฆ

cos x 1 โ x^{2}/2! + x^{4}/4! + โฆ

e^x = 1 + x + x^{2}/2! + x^{3}/3! + โฆ

What relation can you draw from these series?

A. e^{x} = cos x + sin x

B. e^{(ix)} = cos x + i sin x

C. e^{(ix)} = I cos x + sin x

D. ie^{x} = i cos x + i sin x

Answer: **Option B**

Solution:

181. One term of a Fourier series in cosine form is 10 cos 40ฯt. Write it in exponential form.

A. 5 e^{(j40ฯt)}

B. 5 e^{(j40ฯt)} + 5 e^{(-j40ฯt)}

C. 10 e^{(-j40ฯt)}

D. 10 e^{(j40ฯt)}

Answer: **Option D**

Solution:

182. Evaluate the determinant:

A. 4

B. 2

C. 5

D. 0

Answer: **Option C**

Solution:

183. Evaluate the determinant:

A. 110

B. -101

C. 101

D. -110

Answer: **Option B**

Solution:

184. Evaluate the determinant:

A. 489

B. 389

C. 326

D. 452

Answer: **Option C**

Solution:

185. Compute the value of x by determinant.

A. -32

B. -28

C. 16

D. 52

Answer: **Option B**

Solution:

186. Given the equations:

**x + y + z = 2**

**3x โ y โ 2z = 4**

**5x โ 2y + 3z = -7**

Solve for y by determinants.

A. 1

B. -2

C. 3

D. 0

Answer: **Option C**

Solution:

187. Solve the equations by Cramerโs Rule:

**2x โ y + 3z = -3**

**3x + 3y โ z = 10**

**-x โ y + z = -4**

A. (2, 1, -1)

B. (2, -1, -1)

C. (1, 2, -1)

D. (-1, -2, 1)

Answer: **Option C**

Solution:

188. If,

What is the cofactor of the second row, third column element?

Answer: **Option B**

Solution:

189. If,

What is the cofactor with the first row, second column element?

Answer: **Option D**

Solution:

190. If a 3 x 3 matrix and its inverse are multiplied together, write the product.

Answer: **Option A**

Solution:

191. If matrix

A. 3

B. 1

C. 0

D. -2

Answer: **Option C**

Solution:

192. Given the matrix equation, solve for x and y.

A. -4, 6

B. -4, 2

C. -41 -2

D. -4, -6

Answer: **Option A**

Solution:

193. If matrix

A. 8

B. 1

C. -4

D. 0

Answer: **Option D**

Solution:

194. If

What is A times B equal to?

Answer: **Option A**

Solution:

195. Matrix

Answer: **Option D**

Solution:

196. Elements of matrix

Find the elements of the product of the two matrices, matrix BC.

Answer: **Option A**

Solution:

197. Transpose the matrix

Answer: **Option B**

Solution:

198. Determine the inverse matrix of

Answer: **Option A**

Solution:

199. k divided by [ (s square) + (k square) ] is the inverse Laplace transform of:

A. cos kt

B. sin kt

C. (e exponent kt)

D. 1.0

Answer: **Option B**

Solution:

200. The Laplace transform of cos wt is

A. s / [(s square) + (w square) ]

B. w / [ (s square) + (w square) ]

C. w / (s + w)

D. s / (s + w)

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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