This is the Multiple Choice Questions Part 3 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.

### Multiple Choice Questions Topic Outline

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

### Online Questions and Answers in Calculus Series

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

**Calculus MCQs**

**MCQs from Number 1 – 50**Answer key:

**PART I**

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

**PART II**

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

**PART III**

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

**PART IV**

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

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

101. What is the area (in square units) bounded by the curve y^2 = x and the line x – 4 = 0?

- A. 30/3
- B. 31/3
- C. 32/3
- D. 29/3

102. Find the area bounded by the curve y = x^2 + 2, and the lines x = 0 and y = 0 and x = 4.

- A. 88/3
- B. 64/3
- C. 54/3
- D. 64/5

103. Find the area bounded by the parabolas y = 6x – x^2 and y = x^2 – 2x. Note: The parabolas intersect at points (0,0) and (4,8).

- A. 44/3 square units
- B. 64/3 square units
- C. 74/3 square units
- D. 54/2 square units

104. Find the area bounded by the parabolas x^2 = 4y and y = 4.

- A. 21.33
- B. 33.21
- C. 31.32
- D. 13.23

105. Find the area bounded by the line x – 2y + 10 = 0, the x-axis, the y-axis and x = 10.

- A. 75
- B. 50
- C. 100
- D. 25

106. What is the area (in square units) bounded by the curve y^2 = 4x and x^2 = 4y?

- A. 5.33
- B. 6.67
- C. 7.33
- D. 8.67

107. Find the area enclosed by the curve x^2 + 8y + 16 = 0, the x-axis, the y-axis and the line x – 4 =0.

- A. 7.67 sq. units
- B. 8.67 sq. units
- C. 9.67 sq. units
- D. 10.67 sq. units

108. What is the area bounded by the curve y = x^3, the x-axis and the line x = -2 and x = 1?

- A. 4.25
- B. 2.45
- C. 5.24
- D. 5.42

109. Find the area in the first quadrant bounded by the parabola y^2 = 4x, x = 1 & x = 3.

- A. 9.555
- B. 9.955
- C. 5.955
- D. 5.595

110. Find the area (in sq. units) bounded by the parabolas x^2 – 2y = 0 and x^2 + 2y – 8 = 0.

- A. 11.7
- B. 4.7
- C. 9.7
- D. 10.7

111. What is the area between y = 0, y = 3x^2, x = 0 and x = 2?

- A. 8
- B. 24
- C. 12
- D. 6

112. What is the area bounded by the curve y^2 = x and the line x – 4 = 0?

- A. 11
- B. 31/3
- C. 10
- D. 32/3

113. Find the area of the curve r^2 = a^2 cos 2θ.

- A. a
- B. 2a
- C. a^2
- D. a^3

114. Locate the centroid of the plane area bounded by y = x^2 and y = x.

- A. 0.4 from the x-axis and 0.5 from the y-axis
- B. 0.5 from the x-axis and 0.4 from the y-axis
- C. 0.5 from the x-axis and 0.5 from the y-axis
- D. 0.4 from the x-axis and 0.4 from the y-axis

115. Find the coordinates of the centroid of the plane area bounded by the parabola y = 4 – x^2 and the x-axis.

- A. (0,1)
- B. (0,1.6)
- C. (0,2)
- D. (1,0)

116. Locate the centroid of the plane area bounded by the equation y^2 = 4x, x = 1 and the x-axis on the first quadrant.

- A. (3/4, 3/5)
- B. (3/5, 3/4)
- C. (3/5, 3/5)
- D. (3/5, 2/3)

117. Find the length of arc of the parabola x^2 = 4y from x = -2 to x = 2.

- A. 4.2 units
- B. 4.6 units
- C. 4.9 units
- D. 5.2 units

118. Find the surface area (in square units) generated by rotating the parabola arc y = x^2 about the x-axis from x = 0 to x = 1.

- A. 5.33
- B. 4.98
- C. 5.73
- D. 4.73

119. The area enclosed by the ellipse (x^2)/9 + (y^2)/4 = 1 is revolved about the line x = 3. What is the volume generated?

- A. 355.3
- B. 360.1
- C. 370.3
- D. 365.1

120. The area in the second quadrant of the circle x^2 + y^2 = 36 is revolved about the line y + 10 = 0. What is the volume generated?

- A. 2218.33
- B. 2228.83
- C. 2233.43
- D. 2208.53

121. The area bounded by the curve y^2 = 12x and the line x = 3. What is the volume generated?

- A. 179
- B. 181
- C. 183
- D. 185

122. Given the area in the first quadrant bounded by x^2 = 8y, the line y – 2 = 0 and the y-axis. What is the volume generated when the area is revolved about the line y – 2 = 0?

- A. 28.41
- B. 27.32
- C. 25.83
- D. 26.81

123. Find the volume (in cubic units) generated by rotating a circle x^2 + y^2 + 6x + 4y + 12 = 0 about the y-axis.

- A. 39.48
- B. 47.23
- C. 59.22
- D. 62.11

124. Given the area in the first quadrant by x^2 = 8y, the line x = 4 and the x-axis. What is the volume generated by revolving this area about the y-axis?

- A. 53.26
- B. 52.26
- C. 51.26
- D. 50.26

125. Find the moment of inertia, with respect to x-axis of the area bounded by the parabola y^2 = 4x and the line x = 1.

- A. 2.03
- B. 2.13
- C. 2.33
- D. 2.53

126. Determine the order and degree of the differential equation (2x)(d^4y)/(dy^4) + (5x^2)(dy/dx)^3 – xy = 0.

- A. Fourth order, first degree
- B. Third order, first degree
- C. First order, fourth degree
- D. First order, third degree

127. Which of the following equations is an exact DE?

- A. (x^2 + 1) dx – xy dy = 0
- B. x dy + (3x -2y) dx = 0
- C. 2xy dx + (2 + x^2) dy = 0
- D. x^2y dy – y dx = 0

128. Which of the following equations is a variable separable DE?

- A. (x + x^2y) dy = (2x + xy^2) dx
- B. (x + y) dx – 2y dy = 0
- C. 2y dx = (x^2 + 1) dy
- D. y^2 dx + (2x – 3y) dy = 0

129. The equation y^2 = cx is the general solution of:

- A. y’ = 2y/x
- B. y’ = 2x/y
- C. y’ = y/(2x)
- D. y’ = x/(2y)

130. Solve the differential equation: x (y – 1) dx + (x + 1) dy = 0. If y = 2 when x = 1, determine y when x = 2.

- A. 1.80
- B. 1.48
- C. 1.55
- D. 1.63

131. If dy = x^2 dx; what is the equation of y in terms of x if the curve passes through (1,1)?

- A. x^2 – 3y + 3 = 0
- B. x^3 – 3y + 2 = 0
- C. x^3 + 3y^2 + 2 = 0
- D. 2y + x^3 + 2 =0

132. Find the equation of the curve at every point of which the tangent line has a slope of 2x.

- A. x = -y^2 + C
- B. y = -x^2 + C
- C. y = y^2 + C
- D. x = y^2 + C

133. Solve (cos x cos y – cot x) dx – sin x sin y dy = 0.

- A. sin x cos y = ln (c cos x)
- B. sin x cos y = ln (c sin x)
- C. sin x cos y = – ln (c sin x)
- D. sin x cos y = – ln (c cos x)

134. Solve the differential equation dy – xdx = 0, if the curve passes through (1,0)?

- A. 3x^2 + 2y – 3 = 0
- B. 2y + x^2 – 1 = 0
- C. x^2 – 2y – 1 = 0
- D. 2x^2 + 2y – 2 = 0

135. What is the solution of the first order differential equation y(k+1) = y(k) + 5.

- A. y(k) = 4 – 5/k
- B. y(k) = 20 + 5k
- C. y(k) = C – k, where C is constant
- D. The solution is non-existent for real values of y

136. Solve (y – sqrt(x^2 + y^2)) dx – xdy = 0

- A. sqrt(x^2 + y^2) + y = C
- B. sqrt(x^2 + y^2 + y) = C
- C. sqrt(x + y) + y = C
- D. sqrt(x^2 – y) + y = C

137. Find the differential equation whose general solution is y = C1x + C2e^x.

- A. (x – 1) y” – xy’ + y = 0
- B. (x + 1) y” – xy’ + y = 0
- C. (x – 1) y” + xy’ + y = 0
- D. (x + 1) y” + xy’ + y = 0

138. Find the general solution of y’ = y sec x.

- A. y = C (sec x + tan x)
- B. y = C (sec x – tan x)
- C. y = C sec x tan x
- D. y = C (sec2 x tan x)

139. Solve xy’ (2y – 1) = y (1 – x)

- A. ln (xy) = 2 (x – y) + C
- B. ln (xy) = x – 2y + C
- C. ln (xy) = 2y – x + C
- D. ln (xy) = x + 2y) + C

140. Solve (x + y) dy = (x – y) dx.

- A. x^2 + y^2 = C
- B. x^2 + 2xy + y^2 = C
- C. x^2 – 2xy – y^2 = C
- D. x^2 – 2xy + y^2 = C

141. Find the differential equation of family of straight lines with slope and y-intercept equal.

- A. xydy = x^3/4
- B. ydx=(x+1)dy
- C. x^2y = x(x+1)dx
- D. y = x^3/4

142. Find the differential equations of the family of lines passing through the origin.

- A. ydx – xdy = 0
- B. xdy – ydx = 0
- C. xdx + ydy = 0
- D. ydx + xdy = 0

143. What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis.

- A. 2xdx – ydy = 0
- B. xdy + ydx = 0
- C. 2ydx – xdy = 0
- D. dy/dx – x = 0

144. Determine the differential equation of the family of lines passing through (h, k).

- A. (y – k)dx – (x – h)dy = 0
- B. (y – h) + (y – k) = dy/dx
- C. (x – h)dx – (y – k)dy = 0
- D. (x + h)dx – (y – k)dy = 0

145. Determine the differential equation of the family of circles with center on the origin.

- A. (y”)^3 – xy + y’ = 0
- B. y” – xyy’ = 0
- C. x + yy’ = 0
- D. (y’)^3 + (y”)^2 + xy = 0

146. Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg of radium decomposes to 96 mg. How many mg will be left after 100 years?

- A. 88.60
- B. 95.32
- C. 92.16
- D. 90.72

147. The population of a country doubles in 50 years. How many years will it be five times as much? Assume that the rate of increase is proportional to the number of inhabitants.

- A. 100 years
- B. 116 years
- C. 120 years
- D. 98 years

148. Radium decomposes at a rate proportional to the amount present. If half of the original amount disappears after 1000 years, what is the percentage lost in 100 years?

- A. 6.70%
- B. 4.50%
- C. 5.36%
- D. 4.30%

149. Find the equation of the family of orthogonal trajectories of the system of parabolas y^2 = 2x + C.

- A. y = Ce^(-x)
- B. y = Ce^(2x)
- C. y = Ce^x
- D. y = Ce^(-2x)

150. According to Newton’s law of cooling, the rate at which a substance cools in air is directly proportional to the difference between the temperature of the substance and that of air. If the temperature of the air is 30° and the substance cools from 100° to 70° in 15 minutes, how long will it take to cool 100° to 50°?

- A. 33.59 min.
- B. 43.50 min.
- C. 35.39 min.
- D. 45.30 min.