Category: Engineering Mathematics Objective Questions

250+ TOP MCQs on Using Properties of Divergence and Curl and Answers

Engineering Mathematics Problems focuses on “Using Properties of Divergence and Curl”.

1. Find the divergence of this given vector (vec{F}=x^3 yvec{i}+3xy^2 zvec{j}+3zxvec{k}).
a) 3x2 y+6xyz+x
b) 2x2 y+6xyz+3x
c) 3x2 y+3xyz+3x
d) 3x2 y+6xyz+3x
View Answer

Answer: d
Explanation: We know that divergence of a vector is given by
(bigtriangledown.vec{F}=frac{partial(x^3 y)}{partial x}+frac{partial(3xy^2 z)}{partial y}+frac{partial(3xz)}{partial z})
(bigtriangledown.vec{F}=3x^2 y+6xyz+3x).

2. Find the divergence of this given vector (vec{r}=12x^6 y^6 vec{i}+3x^3 y^3 zvec{j}+3x^2 yz^2 vec{k}).
a) (12x^5 y^6+2x^3 yz+6x^2 yz)
b) (72x^5 y^6+2x^3 yz+3x^2 yz)
c) (72x^5 y^6+2x^3 yz+6x^2 yz)
d) (6x^5 y^6+2x^3 yz+6x^2 yz)
View Answer

Answer: c
Explanation: We know that divergence of a vector is given by
(bigtriangledown.vec{r}=frac{partial(12x^6 y^6)}{partial x}+frac{partial(3x^3 y^3 z)}{partial y}+frac{partial(3x^2 yz^2)}{partial z})
(bigtriangledown.vec{r}=12×6x^5 y^6+x^3×2y×z+3x^2×y×2z)
(bigtriangledown.vec{r}=72x^5 y^6+2x^3 yz+6x^2 yz).

3. Find the curl for (vec{r}=x^2 yzvec{i}+(3x+2y)zvec{j}+21z^2 xvec{k}).
a) (vec{i}(3x+2y)-vec{j}(11z^2-x^2 y)+vec{k}(3z-x^2 z))
b) (vec{i}(x+2y)-vec{j}(21z^2-x^2 y)+vec{k}(3z-x^2 z))
c) (-vec{i}(3x+2y)-vec{j}(21z^2-x^2 y)+vec{k}(3z-x^2 z))
d) (vec{i}(3x+2y)-vec{j}(21z^2-x^2 y)+vec{k}(3z-x^2 z))
View Answer

Answer: c
Explanation: We know that the curl for any vector quantity is given by
(bigtriangledown.vec{r}=begin{bmatrix}vec{i}&vec{j}&vec{k}\frac{partial}{partial x}&frac{partial}{partial y}&frac{partial}{partial z}\x^2 yz&(3x+2y)z&21z^2 xend{bmatrix})
(bigtriangledown.vec{r}=vec{i}left (frac{partial(21z^2 x)}{partial y}-frac{partial((3x+2y)z)}{partial z}right)-vec{j}left (frac{partial(21z^2 x)}{partial x}-frac{partial(x^2 yz)}{partial z}right))
(+vec{k}left (frac{partial((3x+2y)z)}{partial x}-frac{partial(x^2 yz)}{partial y}right))
(bigtriangledown.vec{r}=-vec{i}(3x+2y)-vec{j}(21z^2-x^2 y)+vec{k}(3z-x^2 z)).

4. Find the curl for ((vec{r})=y^2 z^3 vec{i}+x^2 z^2 vec{j}+(x-2y)vec{k}).
a) (-2vec{i}(1+x^2 z)-vec{j}(1-3y^2 z^2)+vec{2k}(xz^2-yz^3))
b) (-2vec{i}(1+x^2 z)-vec{j}(1-3y^2 z^2)+vec{k}(xz^2-yz^3))
c) (-2vec{i}(1+x^2 z)-vec{j}(1-32z^2)+vec{2k}(xz^2-yz^3))
d) (vec{i}(1+x^2 z)-vec{j}(1-3y^2 z^2)+vec{2k}(xz^2-yz^3))
View Answer

Answer: a
Explanation: We know that the curl for any vector quantity is given by
(bigtriangledown.vec{r}=begin{bmatrix}vec{i}&vec{j}&vec{k}\partial/partial x&partial/partial y&partial/partial z\y^2 z^3&x^2 z^2&(x-2y)end{bmatrix})
(bigtriangledown.vec{r}=vec{i}left (frac{partial(x-2y)}{partial y}-frac{partial(x^2 z^2)}{/partial z}right )-vec{j}left (frac{partial(x-2y)}{partial x}-frac{partial(y^2 z^3)}{partial z}right )+vec{k}left(frac{partial(x^2 z^2)}{partial x}-frac{partial(y^2 z^3)}{partial y}right ))
(bigtriangledown.vec{r}=-2vec{i}(1+x^2 z)-vec{j}(1-3y^2 z^2)+vec{2k}(xz^2-yz^3)).

5. What is the divergence and curl of the vector (vec{F}=x^2 yvec{i}+(3x+y) vec{j}+y^3 zvec{k}).
a) (y^3+2xy+1,vec{i}(3y^2 z)+vec{j}(3-x^2))
b) (y^3+2xy+1,vec{i}(3y^2 z)+vec{k}(3-x^2))
c) (3y^3+2xy+1,vec{i}(3y^2 z)+vec{k}(3-x^2))
d) (y^3+xy+1,vec{i}(3y^2 z)+vec{k}(3-x^2))
View Answer

Answer: b
Explanation: We know that divergence of a vector is given by
(bigtriangledown.vec{F}=frac{partial(x^2 y)}{partial x}+frac{partial(3x+y)}{partial y}+frac{partial(y^3 z)}{partial z})
(bigtriangledown.vec{F}=y^3+2xy+1)
We know that the curl for any vector quantity is given by
(bigtriangledown.vec{r}=begin{bmatrix}vec{i}&vec{j}&vec{k}\frac{partial}{partial x}&frac{partial}{partial y}&frac{partial}{partial z}\x^2 y&3x+y&(y^3 z)end{bmatrix})
(bigtriangledown.vec{r}=vec{i}left (frac{partial(y^3 z)}{partial y}-frac{partial(3x+y)}{partial z}right )-vec{j}left (frac{partial(y^3 z)}{partial x}-frac{partial(x^2 y)}{partial z}right )+vec{k}left (frac{partial(3x+y)}{partial x}-frac{partial(x^2 y)}{partial y}right ))
(bigtriangledown.vec{r}=vec{i}(3y^2 z)+vec{k}(3-x^2)).

 

To practice all areas of Engineering Mathematics Problems, here is complete set of 1000+ Multiple Choice Questions and Answers.

 

250+ TOP MCQs on Derivation and Solution of Two-dimensional Wave Equation and Answers

Fourier Analysis and Partial Differential Equations Multiple Choice Questions on “Derivation and Solution of Two-dimensional Wave Equation”.

1. Who discovered the one-dimensional wave equation?
a) Jean d’Alembert
b) Joseph Fourier
c) Robert Boyle
d) Isaac Newton
View Answer

Answer: a
Explanation: Jean-Baptiste le Rond d’Alembert (16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was co-editor with Denis Diderot of the Encyclopédie. He was the person responsible for the discovery of wave equation.

2. Wave equation is a third-order linear partial differential equation.
a) True
b) False
View Answer

Answer: b
Explanation: The wave equation is a second-order linear partial differential equation which is developed for the description of waves (water waves, sound waves, seismic waves, light waves), acoustics, electromagnetics, and fluid dynamics.

3. In which of the following fields, does the wave equation not appear?
a) Acoustics
b) Electromagnetics
c) Pedology
d) Fluid Dynamics
View Answer

Answer: c
Explanation: The wave equation arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

4. The wave equation is known as d’Alembert’s equation.
a) True
b) False
View Answer

Answer: a
Explanation: D’Alembert’s formula for obtaining solutions to the wave equation is named after him. The wave equation is sometimes referred to as d’Alembert’s equation.

5. Which of the following statements is false?
a) Equations that describe waves as they occur in nature are called wave equations
b) The problem of having to describe waves arises in fields like acoustics, electromagnetics, and fluid dynamics
c) Jean d’Alembert discovered the three-dimensional wave equation
d) Jean d’Alembert discovered the one-dimensional wave equation
View Answer

Answer: c
Explanation: The one-dimensional wave equation was discovered by d’Alembert in 1746 and by 1756, the three-dimensional wave equation was discovered by Euler.

6. What is the order of the partial differential equation, (frac{∂z}{∂x}-(frac{∂z}{∂y})^3=0)?
a) Order-5
b) Order-1
c) Order-4
d) Order-2
View Answer

Answer: b
Explanation: The order of an equation is defined as the highest derivative present in the equation. Hence, in the given equation, (frac{∂z}{∂x}-(frac{∂z}{∂y})^5=0,) the order is 1.

7. The half-interval method in numerical analysis is also known as __________
a) Newton-Raphson method
b) Regula Falsi method
c) Taylor’s method
d) Bisection method
View Answer

Answer: d
Explanation: The Bisection method, also known as binary chopping or half-interval method, is a starting method which is used, where applicable, for few iterations, to obtain a good initial value.

8. Wave equation is a linear elliptical partial differential equation.
a) False
b) True
View Answer

Answer: a
Explanation: Wave equation is a linear second order hyperbolic partial differential equation whereas a Laplace equation is a linear elliptical partial differential equation.

9. Which of the following is the condition for a second order partial differential equation to be hyperbolic?
a) b2-ac < 0
b) b2-ac=0
c) b2-ac>0
d) b2-ac= < 0
View Answer

Answer: c
Explanation: For a second order partial differential equation to be hyperbolic, the equation should satisfy the condition, b2-ac>0.

10. Which of the following statements is true?
a) Hyperbolic equations have three families of characteristic curves
b) Hyperbolic equations have one family of characteristic curves
c) Hyperbolic equations have no families of characteristic curves
d) Hyperbolic equations have two families of characteristic curves
View Answer

Answer: d
Explanation: The canonical variables ξ and η for a hyperbolic pde satisfy the equations,
(aζ_x+(b+sqrt{b^2-ac}) ζ_y=0 , and , aη_x+(b+sqrt{b^2-ac}) η_y=0 )
The families of curves ξ = constant and η = constant are the characteristic curves. Hence, hyperbolic equations have two families of characteristic curves.

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here is complete set of 1000+ Multiple Choice Questions and Answers.

 

250+ TOP MCQs on Taylor Mclaurin Series and Answers

Engineering Mathematics Multiple Choice Questions on “Taylor Mclaurin Series – 1”.

1. The Mclaurin Series expansion of sin(ex) is?
a) sin(1)+(frac{x.cos(1)}{1!}+sum_{n=2}^{infty}sum_{a=0}^{infty}frac{x^n.(-1)^a}{n!}timesfrac{(2a+1)^n}{(2a+1)!})
b) (frac{e^x}{1!}+frac{e^{3x}}{3!}+frac{e^{5x}}{5!}…infty)
c) (-frac{e^x}{1!}+frac{e^{3x}}{3!}-frac{e^{5x}}{5!}…infty)
d) (sum_{n=2}^{infty}sum_{a=0}^{infty}frac{x^n.(-1)^a}{n!}times frac{(2a+1)^n}{(2a+1)!})
Answer: a
Explanation: We know the series expansion for sin(x) is
sin(t)=(frac{t}{1!}-frac{t^3}{3!}+frac{t^5}{5!}…infty)
Substituting t=ex we have
sin(ex)=(frac{e^x}{1!}-frac{e^3x}{3!}+frac{e^5x}{5!}…infty)
Now using
(e^x=1+frac{x}{1!}+frac{x^2}{2!}+frac{x^3}{3!}+…infty)
We have
sin(ex)=(frac{1+frac{x}{1!}+frac{x^2}{2!}+frac{x^3}{3!}+…infty}{1!}-frac{1+frac{3x}{1!}+frac{(3x)^2}{2!}+frac{(3x)^3}{3!}+…infty}{3!}+frac{1+frac{5x}{1!}+frac{(5x)^2}{2!}+frac{(5x)^3}{3!}+…infty}{5!})
Grouping terms with same power we have
(=(frac{1}{1!}-frac{1}{3!}+frac{1}{5!}…infty)+x(frac{1}{1!}-frac{3}{3!}+frac{5}{5!}…infty))
+(frac{x^2}{2!}(frac{1}{1!}-frac{3^2}{3!}+frac{5^2}{5!}…infty)+…infty)
We can rewrite the last terms of the series as a double series we have
sin(1)+(frac{x.cos(1)}{1!}+sum_{n=2}^{infty}sum_{a=0}^{infty}frac{x^n.(-1)^a}{n!}timesfrac{(2a+1)^n}{(2a+1)!})

2. What is the coefficient of x101729 in the series expansion of cos(sin(x))?
a) 0
b) 1⁄101729!
c) -1⁄101729!
d) 1
Answer: a
Explanation: We know that the series expansion of cos(x) is
cos(t)=1-(frac{t^2}{2!}+frac{t^4}{4!}….infty)
Now substituting t=sin(x) we have
cos(sin(x))=(1-frac{1}{2!}times (frac{x}{1!}-frac{x^3}{3!}+frac{x^5}{5!}+…infty)^2+frac{1}{4!}times (frac{x}{1!}-frac{x^3}{3!}+frac{x^5}{5!}+…infty)^4+..infty)
Observe that every term has odd powered series raised to an even term.
Thus, we must have only even powered terms in the above series expansion. The coefficient of any odd powered term is zero.

3. Let τ(X) be the Taylor Series expansion of f(x) = x3 + x2 + 1019 centered at a = 1019, then what is the value of the expression 2(τ(1729))2 + τ(1729) * f(1729) – 3(f(1729))2 + 1770?
a) 1770
b) 1729
c) 0
d) 1
Answer: a
Explanation: Observe first off that the given function is a polynomial and so any other representation (Taylor Series here) which is continuous and differentiable has to be the
same polynomial. This gives us
τ(x) = f(x)
We now evaluate the expression as follows
= 2(f(1729))2 + (f(1729))2 – 3(f(1729))2 + 1770
= 3(f(1729))2 – 3(f(1729))2 +1770
= 1770

4. Find the Taylor series expansion of the function cosh(x) centered at x = 0.
a) (1-frac{x^2}{2!}+frac{x^4}{4!}+….infty)
b) (frac{x}{1!}+frac{x^3}{3!}+frac{x^5}{5!}….infty)
c) (1+frac{x^2}{2!}+frac{x^4}{4!}+….infty)
d) (1+frac{x}{1!}+frac{x^2}{2!}+….infty)
Answer: c
Explanation: We know the general expression for the expansion of the Taylor series
(tau[f(x)]=f(a)+frac{x.f^{(1)}(a)}{1!}+frac{x^2.f^{(2)}(a)}{2!}+…..infty)
Given a=0 we substitute in the equation to get
(tau[f(x)]=f(0)+f^{(1)}(0)times frac{x}{1!}+f^{(2)}(0)times frac{x^2}{2!}+….infty)
Now the nth derivatives can be calculated as
(f^{(n)}(x)=(frac{e^x+e^{-x}}{2})^{(n)})
(=frac{e^x+(-1)^ne^{-x}}{2})
Substituting x=0 yields the final expansion
(f^{(n)}(x)=frac{1+(-1)^n}{2})
We get
(tau[f(x)]=1+(0)times frac{x}{1!}+(1)times frac{x^2}{2!}+(0)times frac{x^3}{3!}+…..infty)
(tau[f(x)]=1+frac{x^2}{2!}+frac{x^4}{4!}+…infty)

5. To find the value of sin(9) the Taylor Series expansion should be expanded with center as ___________
a) 9
b) 8
c) 7
d) Some delta (small) interval around 9
Answer: d
Explanation: The Taylor series gives accurate results around some point taken as center. As we need the value of 9 the center nearer to the point should be taken.

6. f(1) (n) = g(n) (0) holds good for some functions f(x) and g(x). Now let the coordinate axes containing graph g(x) be rotated by 30 degrees clockwise, then the corresponding Taylor series for the transformed g(x) is?
a) g(0)+(frac{e^x – 1}{sqrt{3}}+frac{sum_{n=1}^infty f^{(1)}(n)x^n}{n!})
b) (g(0) + frac{g^{(1)}.x}{1!} + frac{g^{(2)}(1).x^2}{2!}+…infty)
c) No unique answer exist
d) Such function is not continuous
Answer: a
Explanation: We have f(1) (n) = g(n) (0)
As the coordinate axes containing f(x) is rotated the tan(30) term gets added to the derivative of f(1)new(n)=g(n)(0)-tan(30)
We have
g(n)(0)=f(1)(n)+tan(30)
The Taylor expansion centered at 0 for g(x) is given by
g(x)=(g(0)+frac{g^{(1)}(0).x}{1!}+frac{g^{(2)}(0).x^2}{2!}+…..infty )
Now substituting g(n)(0)=f(1)(n)+tan(30) we have
g(x)=(g(0)+frac{(f^{(1)}(1)+tan(30)).x}{1!}+frac{(f^{(1)}(2)+tan(30)).x^2}{2!}+….infty)
g(x)=(g(0)+(frac{f^{(1)}(1).x}{1!}+frac{f^{(1)}(2).x}{2!}+…infty)+tan(30)times(frac{x}{1!}+frac{x^2}{2!}+….infty))
=(g(0)+frac{sum_{n=1}^{infty}f^{(1)}(n).x^n}{n!}+frac{1}{sqrt{3}}times(e^x-1))

7. Let τ(f(x)) denote the Taylor series for some function f(x). Then the value of τ(τ(τ(f(1729)))) – 2τ(τ(f(1729))) + τ(f(1729)) is?
a) 1729
b) -1
c) 1
d) 0
Answer: d
Explanation: We know that the Mclaurin Series for any given function always yields a polynomial (finite OR infinite).
Further the Mclaurin series of this polynomial (i.e.τ(τ(f(x)))) is also a polynomial. Due to uniqueness of this polynomial, no matter how many nested Mclaurin series we might find, they are all equal. Thus, we have
τ(τ…….(f(x))….)) = f(x)
Substituting this into our required expression we have
= f(1729) – 2f(1729) + f(1729)
= 0.

8. Let Mclaurin series of some f(x) be given recursively, where an denotes the coefficient of xn in the expansion. Also given an = an-1 / n and a0 = 1, which of the
following functions could be f(x)?
a) ex
b) e2x
c) c + ex
d) No closed form exists
Answer: a
Explanation: Observing the recurrence relation we have
an=(frac{a_{n-1}}{n}=frac{a_{n-2}}{n(n-1)})
an=(frac{a_0}{n(n-1)(n-2)…3times 2 times 1})
Thus one could deduce that
an=(frac{1}{n!})
Putting this into the Mclaurin expansion we have
f(x)=a0+a1x+a2x2+a3x3….∞
f(x)=(1+frac{x}{1!}+frac{x^2}{2!}+frac{x^3}{3!}+….infty)
Which is the well known expansion of ex.

9. A function f(x) which is continuous and differentiable over the real domain exists such that f(n) (x) = [f(n + 1) (x)]2, f(0) = a and f(1)(0) = 1.
a) True
b) False
Answer: a
Explanation: Writing out the Mclaurin series we have
f(x)=f(0)+(frac{f^{(1)}(0).x}{1!}+frac{f^{(2)}(0).x^2}{2!}+…infty)
Now sustituting f(n+1)(0) = (sqrt{f^{(n)}(0)})we have
f(x) = f(0) + (frac{f^{(1)}(0).x}{1!}+frac{sqrt{f^{(1)}(0)}.x^2}{2!}+frac{sqrt{sqrt{f^{(1)}(0)}.x^3}}{3!}…infty)
Now f(1)(0)=1 and f(0)=a
Substituting this we have
f(x)=a+(frac{x}{1!}+frac{x^2}{2!}+frac{x^3}{3!}+…infty)
f(x)=a-1+ex
This is a well defined function.

250+ TOP MCQs on Limits and Derivatives of Several Variables and Answers

Engineering Mathematics online test focuses on “Limits and Derivatives of Several Variables – 3”.

1. limx → 1⁡ (x-1)Tan(πx⁄2) is?
a) 0
b) –1⁄π
c) –2⁄π
d) 2⁄π
Answer: c
Explanation:
(lim_{xrightarrow 1}frac{(x-1)sin(frac{pi x}{2})}{cos(frac{pi x}{2})}=frac{0}{0}) (Indeterminate)
By L’Hospital rule
(lim_{xrightarrow 1}frac{(x-1)cos(frac{pi x}{2})frac{pi}{2}+sin(frac{pi x}{2})}{frac{pi}{2}sin(frac{xpi}{2})}=-frac{2}{pi})

2. Value of limit always be in the range of function.
a) True
b) False
Answer: b
Explanation: Because the range of f(x) = {x} is [0,1) and it value at limx → 1⁡ – f(x) is 1 which is not in its range.

3. Which of the following is a necessary Conditions of Sandwich rule?
a) All function must have common domain
b) All function must have common range
c) All function must have common domain and range both
d) Function must not have common domain and range
Answer: a
Explanation: Statement of sandwich theorem is, If Functions f(x), g(x) and h(x)
1. have Common Domain,
2. and, satisfy f(x) ≤ g(x) ≤ h(x) ∀ x ∈ D
Then if f(x) = h(x) = L
=> g(x) = L.

4. The value of limx → 0⁡⁡ [x]Cos(x), [x] denotes the greatest integer function _______
a) lies between 0 and 1
b) lies between -1 and 0
c) lies between 0 and 2
d) lies between -2 and 0
Answer: b
Explanation: limx → 0⁡⁡ [x]Cos(x)
We know that,
x-1 < [x] < x
Multiplying by Cos(x), we get
(x-1)Cos(x) < [x]Cos(x) < xCos(x)
Taking limits, we get
limx → 0 [(x-1)Cos(x)] < limx → 0 [x]Cos(x) < limx → 0[xCos(x)]
=> -1 < limx → 0 [x]Cos(x) < 0.

5. Value of limx → 0[(1+xex)/(1 – Cos(x))].
a) e
b) 1
c) 2
d) Can not be solved
Answer: c
Explanation: =>limx → 0[(1+xex)/(1 – Cos(x))] = 1⁄0 (Indeterminate)
=> By L’Hospital rule
=> limx → 0[(1+xex) / (Sin(x))] = 1⁄0 (Again indeterminate)
=> By L’ Hospital rule
=> limx → 0[((2+x)ex)/ (Cos(x))] = 2.

6. The value of (lim_{xrightarrow 1}[x]cos(frac{pi(1-x)}{2})e^{1/(1-x)}), [x] denotes the greatest integer function.
a) 0
b) 1
c) ∞
d) -∞
Answer: a
Explanation:
(lim_{xrightarrow 1}[x]cos(frac{pi(1-x)}{2})e^{1/(1-x)})
We know that
x-1 ≤ [x] ≤ x
Multiplying by Remaining term of question
((x-1)e^{1/(1-x)}cos(frac{pi(1-x)}{2})≤e^{1/(1-x)}cos(frac{pi(1-x)}{2})≤[x]≤xe^{1/(1-x)}cos(frac{pi(1-x)}{2}))
(lim_{xrightarrow 1}(x-1)e^{1/(1-x)}cos(frac{pi(1-x)}{2})≤lim_{xrightarrow 1}e^{1/(1-x)}cos(frac{pi(1-x)}{2})[x])
(≤lim_{xrightarrow 1}xe^{1/(1-x)}cos(frac{pi(1-x)}{2}))
By rearranging the terms of e1/(1-x) to e-1/(1-x)
(lim_{xrightarrow 1}e^{-1/(x-1)}cos(frac{pi(1-x)}{2})x-1≤lim_{xrightarrow 1}e^{1/(1-x)}cos(frac{pi(1-x)}{2})[x])
(≤lim_{xrightarrow 1}e^{-1/(x-1)}cos(frac{pi(1-x)}{2})x)
(0≤e^{-1/(x-1)}cos(frac{pi(1-x)}{2})[x]≤0)
Hence by sandwich rule
(lim_{xrightarrow 1}e^{1/(1-x)}cos(frac{pi(1-x)}{2})[x]=0)

7. Evaluate limx → 0(1+Tan(x))Cot(x)
a) 1
b) e
c) ln(2)
d) e2
Answer: b
Explanation:
limx → 0(1+Tan(x))Cot(x) = limtan(x) → 0 (1+Tan(x))1⁄Tan(x) = limt → 0 (1 + t)1⁄t = e.

8. Evaluate limx → 1[(-xx + 1) / (xlog(x))].
a) ee
b) e
c) -1
d) e2
Answer: c
Explanation:
(lim{xrightarrow 1}[(-x^x+1)/(xlog(x))]=(0/0))
By L’Hospital rule,
(-lim_{xrightarrow 1}[x^x(1+xlog(x))/(1+xlog(x))]=-lim_{xrightarrow 1}[x^x]=-1)

9. Find domain of n for which limx → 0enxCot(nx), has non zero value.
a) n ∈ (0,∞) ∩ (1,5)
b) n ∈ (-∞,∞) ∩ (1,5)
c) n ∈ (-∞,∞)
d) n ∈ (-∞,∞) ~ 5
Answer: c
Explanation:
(lim_{xrightarrow 1}frac{e^{nx}cos(nx)}{sin(nx)}=(1/0))
By L’hospital Rule we get
(Rightarrow lim_{xrightarrow 0}frac{ne^{nx}(-sin(nx)+cos(nx))}{ncos(nx)}=n/n=1)
Hence domain of n is n ∈ (-∞,∞).

10. Value of (frac{dSin(x)Cos(x)}{dx}) is
a) Cos(2x)
b) Sin(2x)
c) Cos2(2x)
d) Sin2(2x)
Answer: a
Explanation: (frac{dSin(x)Cos(x)}{dx} = Cos(x) frac{dSin(x)}{dx} + Sin(x) frac{dCos(x)}{dx}) = Cos2(x) – Sin2(x) = Cos(2x).

11. Evaluate (lim_{xrightarrowinfty}(sin(frac{1}{x})+cos(frac{1}{x}))^x)
a) 1
b) e
c) 0
d) e2
Answer: b
Explanation:
(lim_{xrightarrowinfty}(sin(frac{1}{x})+cos(frac{1}{x}))^x)
Putting x=1/y,
(Rightarrow lim_{yrightarrow 0}(sin(y)+cos(y))^{frac{1}{y}})
(Rightarrow lim_{yrightarrow 0}((y-frac{y^3}{3!}+frac{y^5}{5!}-…)+(1-frac{y^2}{2!}+frac{y^4}{4!}-….))^{frac{1}{y}})
Neglecting higher powers of y,(as y is limits to 0 which is very small hence higher power terms can be neglected)
(Rightarrowlim_{yrightarrow 0}(1+y)^{frac{1}{y}})
=>e

12. If (lim_{xrightarrow 0}frac{(x(1+acos(x))-bsin(x))}{x^3}=1), then find the value of a and b.
a) 2.5, -1.5
b) -2.5, -1.5
c) -2.5, 1.5
d) 2.5, 1.5
Answer: b
Explanation:
(lim_{xrightarrow 0}frac{(x(1+acos(x))-bsin(x))}{x^3}=1)
Expanding terms of cos(x) and sin(x) and rearranging we get,
(lim_{xrightarrow 0}frac{(1+a-b)x+(frac{b}{6}-frac{a}{2})x^3+(frac{a}{24}-frac{b}{120})x^5+….}{x^3}=1)
Since, given limit is finite, hence coefficients of powers of x should be zero and x3 should be 1
⇒ 1 + a – b=0
⇒ b⁄6 – a⁄2 = 1
⇒ Solving the above two equations we get, a = -2.5, b = -1.5.

13. (lim_{xrightarrow 0}frac{ax^3+b sin(x)+c cos(x)}{x^5}=1), then find the value of a, b and c.
a) 1.37, -4.13, 4.13
b) 1.37, 4.13, -4.13
c) -1.37, 4.13, 4.13
d) 1.37, 4.13, 4.13
Answer: b
Explanation:
(lim_{xrightarrow 0}frac{ax^3+b sin(x)+c cos(x)}{x^5}=1)
Now expanding the terms of sin(x) and cos(x) and rearranging in powers of x,x3 and x5 and so on,we get
=>(lim_{xrightarrow 0}frac{x(b+c)-x^3(frac{b}{6}+frac{c}{2}-a)+x^5(frac{b}{120}+frac{c}{24})+…}{x^5})
Now, coefficient of x and x3 should be zero and that of x5 should be 1, then
⇒ B + c = 0
⇒ b⁄6 + c⁄2 = a
⇒ b⁄120 + c⁄24 = 1
⇒ By solving these 3 equations, a = 1.37, b = 4.13, c = -4.13.

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Curve Tracing Questions and Answers – Curves in Cartesian Form and Answers

Differential and Integral Calculus Multiple Choice Questions on “Curves in Cartesian Form”.

1. Which of the following characteristic is not included in the study of general procedure for tracing the algebraic curve?
a) Symmetry
b) Region or Extent
c) Orthogonality
d) Tangents to the Curve at the origin
Answer: c
Explanation: General procedure for tracing the algebraic curve consists of the study of the following characteristics of the curve:

  • Symmetry
  • Region/Extent
  • Asymptotes
  • Origin
  • Tangents to the curve at the origin
  • Maxima and Minima
  • Sign of the first derivative
  • Sign of the second derivative
  • Inflection Point

2. Which of the following is the representation of a plane algebraic curve of nth degree?
a) f(x,y)=ayn+(bx+c) yn-1+(dx2+ex+f) yn-2+⋯+un (x)=0
b) f(x,y)=ayn-1+(bx+c) yn-2+(dx2+ex+f) yn-3+⋯+un (x)=0
c) f(x,y)=ayn+byn-1+cyn-2+⋯+un (x)=0
d) f(x,y)=ayn+(bx+c) yn+(dx2+ex+f) yn+⋯+un (x)=0
Answer: a
Explanation: Plane algebraic curve of nth degree is represented by,
f(x,y)=ayn+(bx+c) yn-1+(dx2+ex+f) yn-2+⋯+un (x)=0
Where a, b, c, d, e, f are all constants and un(x) is a polynomial in x of degree n.

3. If f(x,y)=ayn+(bx+c) yn-1+(dx2+ex+f) yn-2+⋯+un(x)=0…(1), is the algebraic curve of nth order then, what is the condition for the curve to be symmetric about x-axis?
a) Only even powers of x appear in (1)
b) Only odd powers of x appear in (1)
c) Only odd powers of y appear in (1)
d) Only even powers of y appear in (1)
Answer: d
Explanation: If only even powers of y occur in (1), i.e., if y is replaced by -y in (1), the equation (1) remains unaltered or in other words f(x, -y) = f(x, y).

4. Which of the following is not an example for curve symmetric about y axis?
a) x2=4ay
b) x2=ay
c) y2=4ax
d) x2=2ay
Answer: c
Explanation: Out of the given options, y2=4ax is an example of curve symmetric about x axis and not about y axis as shown in the figure below,

5. Which of the following graphs represent symmetric about the origin?
a) y2=4ax
b) x5+y5=5a2x2y
c) x2=4ay
d) x2+y2=a2
Answer: b
Explanation: Out of the given options, x5+y5=5a2 x2 y represents the curve symmetric about the origin and can be observed as,

6. What are the tangents to the curve x3+ y3=3axy at the origin?
a) x=0
b) x=0, y=0
c) y=0
d) x=y
Answer: b
Explanation: Given: x3+ y3=3axy
To find the tangent to the curve at the origin, we need to equate the lowest degree term to 0.
Therefore, 3axy=0, which gives x=0 and y=0 as two tangents to the curve at origin.

7. At the point where (frac{dy}{dx}=0,) the tangent is parallel to y axis.
a) False
b) True
Answer: a
Explanation: At the point where (frac{dy}{dx}=0,) the tangent is parallel to x axis, i.e., horizontal. At the point where (frac{dy}{dx}=∞,) the tangent is vertical, i.e., parallel to y axis.

8. Which of the following is not correct regarding signs of derivatives?
a) If (frac{dy}{dx}>0,) then the curve is increasing in [a, b]
b) If (frac{dy}{dx}<0,) then the curve is increasing in [a, b]
c) If (frac{d^2 x}{dy^2}>0, ) then curve is convex and concave upward (holds water)
d) If (frac{d^2 x}{dy^2}>0, ) then curve is concave downward (spills water)
Answer: b
Explanation: In an interval, a≤x≤b,

  • If (frac{dy}{dx}>0, ) then the curve is increasing in [a, b]
  • If (frac{dy}{dx}< 0, ) then the curve is increasing in [a, b]
  • If (frac{d^2x}{dy^2}>0, ) then curve is convex and concave upward (holds water)
  • If (frac{d^2x}{dy^2}>0, ) then curve is concave downward (spills water)

9. What is the value of the given limit, (lim_{xto 0}⁡frac{25}{x})?
a) 25
b) 0
c) (frac{1}{2} )
d) (frac{3}{2} )
Answer: a
Explanation: Given: (lim_{xto 0}frac{25}{x})
Using L’Hospital’s Rule, by differentiating both the numerator and denominator with respect to x,
(lim_{xto 0}frac{25}{x}= 25.)

10. Oblique asymptotes are parallel to both x and y axes.
a) True
b) False
Answer: b
Explanation: Oblique asymptotes are those which are neither parallel to x-axis nor y-axis and are given by y=mx+c where (m=lim_{xto ∞}⁡(frac{x}{y})=lim_{xto ∞}(y-mx).)

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250+ TOP MCQs on Basic Definitions and Answers

Ordinary Differential Equations Multiple Choice Questions on “”.

1. Which of the following correctly defines ordinary differential equations?
a) A differential equation in which a dependent variable (say ‘y’) depends on only one independent variable (say ’x’)
b) A differential equation in which an independent variable (say ‘y’) depends on only one dependent variable (say ’x’)
c) A differential equation in which a dependent variable (say ‘y’) depends on one or more independent variables (say ’x’, ’t’ etc.)
d) A differential equation in which an independent variable (say ‘y’) depends on one or more dependent variables (say ’x’, ’t’ etc.)
Answer: a
Explanation: A differential equation is an equation involving an unknown function y of one or more independent variables x, t, …… and its derivatives. These are divided into two types, ordinary or partial differential equations.
An ordinary differential equation is a differential equation in which a dependent variable (say ‘y’) is a function of only one independent variable (say ‘x’).

2. A partial differential equation is one in which a dependent variable (say ‘y’) depends on one or more independent variables (say ’x’, ’t’ etc.)
a) False
b) True
Answer: b
Explanation: An ordinary differential equation is divided into two types, ordinary and partial differential equations.
A partial differential equation is one in which a dependent variable depends on one or more independent variables.
Example: (F(x,t,y,frac{∂y}{∂x},frac{∂y}{∂t},……)= 0 )

3. What is the order of the differential equation, y”+y’-x3y=sinx?
a) 2
b) 1
c) 0
d) 3
Answer: a
Explanation: Order of a differential equation is given by the highest order derivative appearing in the differential equation. Hence for the given equation, y”+y’-x3y=sinx, the order is 2.

4. What is the degree of the differential equation, 4x3-6x2 y3+2y=0?
a) 3
b) 5
c) 1
d) 8
Answer: b
Explanation: The degree of an equation that has not more than one variable in each term is the exponent of the highest power to which that variable is raised in the equation. But when more than one variable appears in a term, it is necessary to add the exponents of the variables within a term to get the degree of the equation. Hence, the degree of the equation, 4x3-6x2 y3+2y=0, is 2+3 = 5.

5. Which one of the following is not a criterion for linearity of an ordinary differential equation?
a) The dependent variable y and its derivatives are of first degree
b) The derivatives of the dependent variable y should be of second degree
c) No product terms of y and/or any of its derivatives are present
d) No transcendental functions of y and/or its derivatives occur
Answer: b
Explanation: The criterions for linearity of an ordinary differential equation are:

  • The dependent variable y and its derivatives are of first degree
  • No product terms of y and/or any of its derivatives are present
  • No transcendental functions of y and/or its derivatives occur

6. Which of the following is a type of Iterative method of solving non-linear equations?
a) Graphical method
b) Interpolation method
c) Trial and Error methods
d) Direct Analytical methods
Answer: b
Explanation: There are 2 types of Iterative methods, (i) Interpolation methods (or Bracketing methods) and (ii) Extrapolation methods (or Open-end methods).

7. Which of the following is not an example of linear differential equation?
a) y=mx+c
b) x+x’=0
c) x+x2=0
d) x”+2x=0
Answer: c
Explanation: For a differential equation to be linear the dependent variable should be of first degree. Since in equation x+x2=0, x2 is not a first power, it is not an example of linear differential equation.

8. Which of the following is not a standard method for finding the solutions for differential equations?
a) Variable Separable
b) Homogenous Equation
c) Bernoulli’s Equation
d) Orthogonal Method
Answer: d
Explanation: The following are the different standard methods used in finding the solution of a differential equation:

  • Variable Separable
  • Homogenous Equation
  • Non-homogenous Equation reducible to Homogenous Equation
  • Exact Differential Equation
  • Non-exact Differential Equation that can be made exact with the help of integrating factors
  • Linear First Order Equation
  • Bernoulli’s Equation

9. Solution of a differential equation is any function which satisfies the equation.
a) False
b) True
Answer: b
Explanation: A solution of a differential equation is any function which satisfies the equation, i.e., reduces it to an identity. A solution is also known as integral or primitive.

10. The equation (2frac{dy}{dx} – xy = y^{-2},) is an example for Bernoulli’s equation.
a) False
b) True
Answer: b
Explanation: A first order, first degree differential equation of the form,
(frac{dy}{dx} + P(x). y = Q(x). y^a,) is known as Bernoulli’s equation.

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