M3 MCQ FOR DBATU | diplomachakhazana

 

Unit 1 :- Laplace transform

 

1. If f(t) = 1, then its Laplace Transform is given by?
a)s
b) ¹⁄_s
c)1
d)Does not exist

2. If f(t) = tⁿ where, ‘n’ is an integer greater than zero, then its Laplace Transform is given by?
a) n!
b) tⁿ⁺¹
c) n! ⁄ sⁿ⁺¹
d) Does not exist

3. If f(t)=√t, then its Laplace Transform is given by?
a) ¹⁄₂
b) ¹⁄_s
c) √π ⁄ 2√s
d) Does not exist

4. If f(t) = sin(at), then its Laplace Transform is given by?
a) cos(at)
b) 1 ⁄ a^sin(at)
c) Indeterminate
d) a ⁄ s²+a²

5. If f(t) = tsin(at) then its Laplace Transform is given by?
a) 2as ⁄ (s²+a²)²
b) a ⁄ s²+a²
c) Indeterminate
d) √π ⁄ 2√s

6. If f(t) = e^at, its Laplace Transform is given by?
a) a ⁄ s²+a²
b) √π ⁄ 2√s
c) 1 ⁄ s-a
d) Does not exist

7. If f(t) = t^p where p > – 1, its Laplace Transform is given by?
a) √π ⁄ 2√s
b) f(t) = tsin(at)
c) γ(p+1) ⁄ s^p+1
d) Does not exist

8. If f(t) = cos(at), its Laplace transform is given by?
a) s ⁄ s²+a²
b) a ⁄ s²+a²
c) √π ⁄ 2√s
d) Does not exist

9. If f(t) = tcos(at), its Laplace transform is given by?
a) 1 ⁄ s-a
b) s² – a² ⁄ (s²+a²)²
c) Indeterminate
d) s²at

10. If f(t) = sin(at) – atcos(at), then its Laplace transform is given by?
a) Indeterminate form is encountered
b) a³ ⁄ (s² + a²)²
c) 2a³ ⁄ (s² – a²)²
d) 2a³ ⁄ (s² + a²)²

11. If f(t) = sin(at) – atcos(at), then its Laplace transform is given by?
a) s(s2−a2)(s2+a2)2
b) s(s2−3a2)(s2+a2)2
c) Indeterminate
d) 2as2 / (s2+a2)2

 

12. If f(t) = cos(at) – atsin(at), then its Laplace transform is given by?
a) sinat²
b) s(s2−a2)/(s2+a2)2
c) Γ(p+1)sp+1
d) Does not exist

13. If f(t) = cos(at) + atsin(at), its Laplace transform is given by?
a) s+as−a
b) a3(s2+a2)2
c) s(s2+3a2) / (s2+a2)2
d) Does not exist

14. If f(t) = sin(at + b), its Laplace transform is given by?
a) Indeterminate
b) (s)sin(b)+acos(b) / s2+a2
c) s2−a2(s−a)2
d) 2a3(s2+a2)

15. If f(t) = cos(at + b), its Laplace transform is given by?
a) as2+a2
b) 2as(s2+a2)2
c) scos(b)−asin(b) / s2+a2
d) Does not exist

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1. If f(t) = sinhat, then its Laplace transform is?
a) e^at
b) s ⁄ s²-a²
c) a ⁄ s²-a²
d) Exists only if ‘t’ is complex

2. If f(t) = coshat, its Laplace transform is given by?
a) s ⁄ s²-a²
b) s+a ⁄ s-a
c) Indeterminate
d) (sinh(at))²

3. If f(t) = e^at sin(bt), then its Laplace transform is given by?
a) s²-a² ⁄ (s – a)²
b) b ⁄ (s + a)² + b²
c) b ⁄ (s – a)² + b²
d) Indeterminate

4. If f(t) = e^at cos(bt), then its Laplace transform is?
a) 2a³ ⁄ (s² + a²)
b) s+a ⁄ s-a
c) Indeterminate
d) s-a ⁄ (s – a)² + b²

5. If f(t) = e^at sinh(bt) then its Laplace transform is?
a) e^-as ⁄ s
b) s+a ⁄ (s – a)² + b²
c) b ⁄ (s – a)² – b²
d) Does not exist

6. If f(t) = ¹⁄_a sinh(at), then its Laplace transform is?
a) 1⁄s²-a²
b) 2a ⁄ (s – b)² + b²
c) n! ⁄ (s – a)^n-1
d) Does not exist

7. If f(t) = tⁿ ⁄ n, then its Laplace transform is?
a) s+a(s−a)(s−a)2+b2
b) b2(s−a)(s−a)2+b2
c) 2a3(s2+a2)
d) (n−1)! / sn+1

8. If f(t) = ¹ ⁄ _√Πt, then its Laplace transform is?
a) s2−a2 /(s−a)2
b) S^-1/2
c) n! /(s−a)^n−1
d) n! /(s−a)^n−1

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9. If f(t) = ^t⁄₂ a sinat, then its Laplace transform is?
a) b ⁄ (s + a)² + b²
b) 2a ⁄ (s – b)² + b²

c) Indeterminate
d) s ⁄ (s² + a²)²

10. If f(t) = δ(t), then its Laplace transform is?
a) s + a ⁄ (s – a)² + b²
b) a³ ⁄ (s² + a²)²
c) 1
d) Does not exist

11. If f(t) = te^-at, then its Laplace transform is?
a) 1 /(s+a)2
b) 2a /(s−b)2+b2
c) a3 / s2+a2)2
d) Indeterminate

12. If f(t) = u(t), then its Laplace transform is?
a) scos(b)−asin(b)s2+a2
b) 1/2
c) s/s2−a2
d) b /(s−a)2+b2

13. f(t) = t, then its Laplace transform is?
a) (s)sin(b)+acos(b)s2+a2
b) 2as2(s2+a2)2
c) Γ(p+1) / sp+1
d) 1/s²

14. If f(t)=1beatsinh(bt), then its Laplace transform is?
a) 1/s

b) Indeterminate
c) b(s−a)2−b2
d) f(t)=1 / (s−a)2−b2

15. If L { f(t) } = F(s), then L {kf(t)} = ?
a) F(s)
b) k F(s)
c) Does not exist
d) F(^s⁄_k)

 

1. Laplace of function f(t) is given by?
a) F(s)=∫∞−∞f(t)e−stdt
b) F(t)=∫∞−∞f(t)e−tdt
c) f(s)=∫∞−∞f(t)e−stdt
d) f(t)=∫∞−∞f(t)e−tdt

2. Laplace transform any function changes it domain to s-domain.
a) True
b) False

3. Laplace transform if sin⁡(at)u(t) is?
a) s ⁄ a²+s²
b) a ⁄ a²+s²
c) s² ⁄ a²+s²
d) a² ⁄ a²+s²

4. Laplace transform if cos⁡(at)u(t) is?
a) s ⁄ a²+s²
b) a ⁄ a²+s²
c) s² ⁄ a²+s²
d) a² ⁄ a²+s²

5. Find the laplace transform of e^t Sin(t).
a) a / a2+(s+1)2
b) a / a2+(s−1)2
c) s+1 / a2+(s+1)2
d) s+1 / a2+(s+1)2

6. Laplace transform of t² sin⁡(2t).
a) [12s2−16 / (s2+4)4]
b) [3s2−4 / (s2+4)3]
c) [12s2−16 / (s2+4)6]
d) [12s2−16 / (s2+4)3]

7. Find the laplace transform of t⁵⁄₂.
a) 158√πs5/2
b) 15/8   /  √πs7/2
c) 94√πs7/2
d) 154√πs7/2

8. Value of ∫∞−∞etSin(t)Cos(t)dt = ?
a) 0.5
b) 0.75
c) 0.2
d) 0.71

 

 

 

9. Value of ∫∞−∞etSin(t)dt = ?
a) 0.50
b) 0.25
c) 0.17
d) 0.12

 

 

10. Value of ∫∞−∞etlog(1+t)dt = ?
a) Sum of infinite integers
b) Sum of infinite factorials
c) Sum of squares of Integers
d) Sum of square of factorials

.

11. Find the laplace transform of y(t)=e^t.t.Sin(t)Cos(t).
a) 4(s−1)  / [(s−1)2+4]2
b) 2(s+1)  /  [(s+1)2+4]2
c) 4(s+1) / [(s+1)2+4]2
d) 2(s−1) /  [(s−1)2+4]2

12. Find the value of ∫∞0tsin(t)cos(t).
a) s ⁄ s²+2²
b) a ⁄ a²+s⁴
c) 1
d) 0

13. Find the laplace transform of y(t)=e^|t-1| u(t).
a) 2s / 1−s2es
b) 2s / 1+s2e−s
c) 2s / 1+s2es
d) 2s / 1−s2e−s

 

 

1. Transfer function may be defined as ____________
a) Ratio of out to input
b) Ratio of laplace transform of output to input
c) Ratio of laplace transform of output to input with zero initial conditions
d) None of the mentioned

2. Poles of any transfer function is define as the roots of equation of denominator of transfer function.
a) True
b) False

3. Zeros of any transfer function is define as the roots of equation of numerator of transfer function.
a) True
b) False

4. Find the poles of transfer function which is defined by input x(t)=5Sin(t)-u(t) and output y(t)=Cos(t)-u(t).
a) 4.79, 0.208
b) 5.73, 0.31
c) 5.89, 0.208
d) 5.49, 0.308

5. Find the equation of transfer function which is defined by y(t)-∫₀^t y(t)dt + ^d⁄_dt x(t) – 5Sin(t) = 0.
a) s(e−as−1) / s−1
b) (e−as−s) / s−1
c) s(e−as−s) / s−1
d) s(e−as−s2) / s−1

6. Find the poles of transfer function given by system ^d2⁄_dt² y(t) – ^d⁄_dt y(t) + y(t) – ∫₀^t x(t)dt = x(t).
a) 0, 0.7 ± 0.466
b) 0, 2.5 ± 0.866
c) 0, 0 .5 ± 0.866
d) 0, 1.5 ± 0.876

7. Find the transfer function of a system given by equation ^d2⁄_dt² y(t-a) + x(t) + 5 ^d⁄_dt y(t) = x(t-a).
a) (e^-as-s)/(1+e^-as s²)
b) (e^-as-5s)/(e^-as s²)
c) (e^-as-s)/(2+e^-as s²)
d) (e^-as-5s)/(1+e^-as s²)

8. Any system is said to be stable if and only if ____________
a) It poles lies at the left of imaginary axis
b) It zeros lies at the left of imaginary axis
c) It poles lies at the right of imaginary axis
d) It zeros lies at the right of imaginary axis

 

9. The system given by equation 5 ^d3⁄_dt³ y(t) + 10 ^d⁄_dt y(t) – 5y(t) = x(t) + ∫₀^t x(t)dt, is?
a) Stable
b) Unstable
c) Has poles 0, 0.455, -0.236±1.567
d) Has zeros 0, 0.455, -0.226±1.467

10. Find the laplace transform of input x(t) if the system given by ^d3⁄_dt³ y(t) – 2 ^d2⁄_dt² y(t) –^d⁄_dt y(t) + 2y(t) = x(t), is stable.
a) s + 1
b) s – 1
c) s + 2
d) s – 2

11. The system given by equation y(t – 2a) – 3y(t – a) + 2y(t) = x(t – a) is?
a) Stable
b) Unstable
c) Marginally stable
d) 0

 

 

1. Time domain function of sa2+s2 is given by?
a) Cos(at)
b) Sin(at)
c) Cos(at)Sin(at)
d) Sin(t)

2. Inverse Laplace transform of 1(s+1)(s−1)(s+2) is?
a) –¹⁄₂ e^t + ¹⁄₆ e^-t + ¹⁄₃ e^2t
b) –¹⁄₂ e^-t + ¹⁄₆ e^t + ¹⁄₃ e^-2t
c) ¹⁄₂ e^-t – ¹⁄₆ e^t – ¹⁄₃ e^-2
d) –¹⁄₂ e^-t + ¹⁄₆ e^-t + ¹⁄₃ e^-2

3. Inverse laplace transform of 1(s−1)2(s+5) is?
a) ¹⁄₆ e ^{– t} – ¹⁄₃₆ e^t + ¹⁄₃₆ e^-5t
b) ¹⁄₆ e^tt – ¹⁄₃₆ e^t + ¹⁄₃₆ e^-5t
c) ¹⁄₆ e^-tt² – ¹⁄₃₆ e^-t + ¹⁄₃₆ e^5t
d) ¹⁄₆ e^-t t-¹⁄₃₆ e^-t + ¹⁄₃₆ e^5t

4. Find the inverse laplace transform of 1(s2+1)(s–1)(s+5).
a) ¹⁄₁₂ e^t – ¹⁄₁₃ Cos(-t) – ¹⁄₁₂ Sin(-t) – ¹⁄₁₅₆ e^-5t
b) ¹⁄₁₂ e^-t – ¹⁄₁₃ Cos(t) – ¹⁄₁₂ Sin(t) – ¹⁄₁₅₆ e^5t
c) ¹⁄₁₂ e^t – ¹⁄₁₃ Cos(t) – ¹⁄₁₂ Sin(t) – ¹⁄₁₅₆ e^-5t
d) ¹⁄₁₂ e^t + ¹⁄₁₃ Cos(t) + ¹⁄₁₂ Sin(t) + ¹⁄₁₅₆ e^-5t

5. Find the inverse laplace transform of s(s2+4)2.
a) ¹⁄₄ sin(2t)
b) ^t2⁄₄ sin(2t)
c) ^t⁄₄ sin(2t)
d) ^t⁄₄ sin(2t²)

6. Final value theorem states that _________
a) x(0)=limx→∞sX(s)
b) x(∞)=limx→∞sX(s)
c) x(0)=limx→0sX(s)
d) x(∞)=limx→0sX(s)

7. Initial value theorem states that ___________
a) x(0)=limx→∞sX(s)
b) x(∞)=limx→∞sX(s)
c) x(0)=limx→0sX(s)
d) x(∞)=limx→0sX(s)

8. Find the value of x(∞) if X(s)=2s2+5s+12/ss3+4s2+14s+20.
a) 5
b) 4
c) ¹²⁄₂₀
d) 2

9. Find the value of x(0) if X(s)=2s2+5s+12/ss3+4s2+14s+20.
a) 5
b) 4
c) 12
d) 2

10. Find the inverse lapace of (s+1)[(s+1)2+4][(s+1)2+1].
a) ¹⁄₃ e^t [Cos(t) – Cos(2t)].
b) ¹⁄₃ e^-t [Cos(t) + Cos(2t)].
c) ¹⁄₃ e^t [Cos(t) + Cos(2t)].
d) ¹⁄₃ e^-t [Cos(t) – Cos(2t)].

11. Find the inverse laplace transform of Y(s)=\frac{2s}{1-s^2}e^{-s}.
a) -e^{-t + 1} + e^{t – 1}
b) -e^{-t + 1} – e^{t + 1}
c) -e^{-t + 1} + e^{t + 1}
d) -e^{-t + 1} – e^{t – 1}

12. Find the inverse laplace transform of \frac{1}{s(s-1)(s^2+1)}.
a) ¹⁄₂ e^-t + ¹⁄₂ Sin(-t) – ¹⁄₂ Cos(-t)
b) ¹⁄₂ e^t + ¹⁄₂ Sin(t) – ¹⁄₂ Cos(t)
c) ¹⁄₂ e^t + ¹⁄₂ Sin(t) + ¹⁄₂ Cos(t)
d) ¹⁄₂ e^t – ¹⁄₂ Sin(t) – ¹⁄₂ Cos(t)

 

 

 

 

 

1. Find the laplace transform of f(t), where
f(t) = 1 for 0 < t < a
-1 for a < t < 2a
a) 1scoth(as2)
b) 1ssinh(as2)
c) 1se−as
d) 1/stanh(as / 2)

2. Find the laplace transform of f(t), where f(t) = |sin(pt)| and t>0.
a) ps2+p2×cosh(sπ2p)
b) ps2+p2×sinh(sπ2p)
c) p /s2+p2×coth(sπ / 2p)
d) ps2+p2×tanh(sπ2p)

 

Unit 2 :-Inverse laplace transform

1. Find the L−1(s+34s2+9).
a) 14cos(3t2)+12cos(3t2)
b) 14cos(3t4)+12sin(3t2)
c) 12cos(3t2)+12sin(3t2)
d) 14cos(3t / 2)+12sin(3t /2)

2. Find the L−1(1(s+2)4).
a) e−2t×3
b) e−2t×t3 /3
c) e−2t×t^3/6
d) e−2t×t2 / 6

3. Find the L−1(s(s−1)7).
a) e−t(t65!+t56!)
b) et(t65!+t56!)
c) et(t66!+t55!)
d) e−t(t66!+t55!)

4. Find the L−1(s2s+9+s2).
a) e^−t{cos(2√2t)−sin(2t−−√2t)}
b) e^−t{cos(2√2t)−sin(22t−−√2t)}
c) e^−t{cos(2√2t)−cos(2t−−√2t)}
d) e^−2t{cos(2√2t)−sin(22t−−√2t)}

 

 

5. Find the L−1((s+1)(s+2)(s+3)).
a) 2e^-3t-e^-2t
b) 3e^-3t-e^-2t
c) 2e^-3t-3e^-2t
d) 2e^-2t-e^-t

6. Find the L−1((3s+9)(s+1)(s−1)(s−2)).
a) e^-t+6e^t+5e^2t
b) e^-t-e^t+5e^2t
c) e^-3t-6e^t+5e^2t
d) e^-t-6e^t+5e^2t

7. Find the L−1(1(s2+4)(s2+9)).
a) 15(sin(2t)2−sin(t)3)
b) 15(sin(2t)2+sin(3t)3)
c) 15(sin(t)2−sin(3t)3)
d) 1/5(sin(2t)/2−sin(3t)/3)

8. Find the L−1(s(s2+1)(s2+2)(s2+3)).
a) 12cos(t)−cos(√3t)−12cos(√3t)
b) 12cos(t)+cos(√2t)−12cos(√3t)
c) 1/2cos(t)−cos(√2t))−1/2cos(√3t)
d) 12cos(t)+cos(√2t)+12cos(√3t)

9. Find the L−1(s+1(s−1)(s+2)2).
a) 27et−29e−2t+13e−2t×t
b) 2/9et−2/9e−2t+1/3e^−2t×t
c) 29et−29e−3t+13e−2t×t
d) 29et−29e−2t+13e−2t

10. The L−1(3s+8s2+4s+25) is e−st(3cos(21−−√t+2sin(21√t)21√). What is the value of s?

a) 0
b) 1
c) 2
d) 3

 

convolution

1. Find the L−1(1s(s2+4)).
a) 1−sin(t)4
b) 1−cos(t)4
c) 1−sin(2t)4
d) 1−cos(2t) / 4

2. Find the L−1(1s(s+4)12), give the answer in terms of error function.
a) 1/2erf(2t)
b) 1/2erf(√t)
c) 1/2erf(2√t)
d) 1/2erf(4√t)

3. Find the L−1s(s2+4)2.
a) 1/4tcos(2t)
b) 1/4tsin(t)
c) 1/4tsin(2t)
d) 1/2tsin(2t)

 

1. While solving the ordinary differential equation using unilateral laplace transform, we consider the initial conditions of the system.
a) True
b) False

2. With the help of _____________________ Mr.Melin gave inverse laplace transformation formula.
a) Theory of calculus
b) Theory of probability
c) Theory of statistics
d) Theory of residues

3. What is the laplce tranform of the first derivative of a function y(t) with respect to t : y’(t)?
a) sy(0) – Y(s)
b) sY(s) – y(0)
c) s² Y(s)-sy(0)-y'(0)
d) s² Y(s)-sy'(0)-y(0)

4. Solve the Ordinary Differential Equation by Laplace Transformation y’’ – 2y’ – 8y = 0 if y(0) = 3 and y’(0) = 6.
a) 3e^tcos(3t)+tsint(3t)
b) 3e^tcos(3t)+te−tsint(3t)
c) 2e^−tcos(3t)−2t3sint(3t)
d) 2e^−tcos(3t)−2te−t3sint(3t)

5. Solve the Ordinary Differential Equation y’’ + 2y’ + 5y = e^-t sin(t) when y(0) = 0 and y’(0) = 1.(Without solving for the constants we get in the partial fractions).
a) et[Acost+A1sint+Bcos(2t)+(B1)2sin(2t)]
b) e−t[Acost+A1sint+Bcos(2t)+B1sin(2t)]
c) e−t[Acost+A1sint+Bcos(2t)+(B1)2sin(2t)]
d) et[Acost+A1sint+Bcos(2t)+(B1)sin(2t)]

6. Solve the Ordinary Diferential Equation using Laplace Transformation y’’’ – 3y’’ + 3y’ – y = t² e^t when y(0) = 1, y’(0) = 0 and y’’(0) = 2.
a) 2e^tt^5 / 720+2e^tt/6+4e^tt^2 /24
b) ett5720+2e−t+2ett6+4ett224
c) e−tt5720+e−t+2e−tt6+4e−tt224
d) 2e−tt5720+e−t+2e−tt6+4e−tt224

7. Take Laplace Transformation on the Ordinary Differential Equation if y’’’ – 3y’’ + 3y’ – y = t² e^t if y(0) = 1, y’(0) = b and y’’(0) = c.
a) (s3−3s2+3s−1)Y(s)+(−as2+(3a−b)s+(−3a−c))=2(s−1)3
b) (s3−3s2+3s−1)Y(s)+(−as2+(3a−b)+(−3a−c)s)=2(s−1)3
c) (s3−3s2+3s)Y(s)+(−as+(3a−b)s+(−3a−c))=2(s−1)3
d) (s3−3s2+3s−1)Y(s)+(−as2+(3a−b)s+(−3a−c))=2(s−1)3

8. What is the inverse Laplace Transform of a function y(t) if after solving the Ordinary Differential Equation Y(s) comes out to be Y(s)=s2−s+3(s+1)(s+2)(s+3) ?
a) 1/2e−t+9/2e^−3t−3e−2t
b) −1/2e−t+9/2e^−2t−3e−3t
c) 1/2e−t−3/2e^−2t−3e−3t
d) −1/2et+9/2e^2t−3e3t

9. For the Transient analysis of a circuit with capacitors, inductors, resistors, we use bilateral Laplace Transformation to solve the equation obtained from the Kirchoff’s current/voltage law.
a) True
b) False

10. While solving an Ordinary Differential Equation using the unilateral Laplace Transform, it is possible to solve if there is no function in the right hand side of the equation in standard form and if the initial conditions are zero.
a) True
b) False

 

1. Find the L(sin³ t).
a) 3/ 4(s2+1)−1/ 4(s2+9)
b) 3/4(s2+1)−3/4(s2+9)
c) 3/4(s2+1)−9/4(s2+9)
d) 3/4(s2−1)−3/4(s2+9)

2. Find the L(e2t(1+t)2).
a) 1 /s−2+2/(s−2)3+2/(s−2)2
b) 3/s−2+2/(s−2)3+2/(s−2)2
c) 1//s−2+2/(s+2)3+2/(s−2)2
d) 1/s−2+2/(s−2)3

3. Find the Laplace Transform of g(t) which has value (t-1)³ for t>1 and 0 for t<1.
a) e^−2as×6/ s4
b) e^−as×2/s5
c) e^−as×6/s4
d) e^−as×2/4s4

4. Find the L(t e^-2t sinh⁡(4t)).
a) 8s+16 / (s2+2s−12)2
b) 2s+16 / (s2+2s−12)2
c) 8s+16 / (s2+21s−12)2
d) 8s+16 / (s2+s−12)2

5. Find the L(t+sin(2t)).
a) 1/s+2/ (s2+4)
b) 1/s+3/(s2+4)
c) 1/s+2/(s2+2)
d) 2/s+2/(s2+4)

6. The L(te^-3t cos⁡(2t)cos⁡(3t)) is given by k[25−(s+3)2((s+3)2+25)2+(1−(s+3)2)((s+3)2+1)2]. Find the value of k.
a) 0
b) 1
c) 12
d) −1/2

7. Find the L(sinh(at)t).
a) 1/2log(s×a / s−a)
b) 1/2log(s−a / s+a)
c) 1/2log(s+a / s−a)
d) 1/3log(s+a / s−a)

8. Find the L(ddt(sintt)).
a) s×cot^-1 s-1
b) s×tan^-1 s-1
c) s×cot⁡(s)-1
d) s×tan⁡(s)-1

9. Find the L(∫t0sin(u)cos(2u)du).
a) 1/2s[3/s2+9−1/s2+1]
b) 1/2s[9/s2+9−1/s2+1]
c) 1/2s[3/s2+9+1/s2+1]
d) 1/s[3/s2+9−1/s2+1]

 

 

 

 

10. Which of the following is not a term present in the Laplace Transform of e^2t sin⁴ t.
a) 38s
b) 38(s−2)
c) s8((s−2)2+16)
d) s2((s−2)2+4)

11. If (erf(t√))=1ss√, then what is L(erf(2t√))?
a) 2 / √s
b) 1 / s√s
c) 2 / s√s
d) 4  / s√s

12. Find the value of L(3^2t).
a) 1/ s−2log(3)
b) 1 / s+2log(3)
c) 1 / s−3log(2)
d) 1  / s+3log(2)

 

 

Unit 3  fourier transform

 

1. In Fourier transform f(p)=∫∞−∞e(ipx)F(x)dx,e(ipx) is said to be Kernel function.
a) True
b) False
View Answer

Answer: a

2. Fourier Transform of e−|x|is 21+p2. Then what is the fourier transform of e−2|x|?
a) 4 / (4+p2)
b) 2 / (4+p2)
c) 2 / (2+p2)
d) 4 / (2+p2)
View Answer

Answer: a

3. What is the fourier sine transform of e^-ax?
a) 4 / (4+p2)
b) 4a / (4a2+p2)
c) p / (a2+p2)
d) 2p / (a2+p2)
View Answer

Answer: c

4. Find the fourier sine transform of x(a2+x2).
a) 2πe−ap
b) π/2e−ap
c) 2π/e−ap
d) πe−ap
View Answer

Answer: b

5. Find the fourier transform of F(x) = 1, |x|<a0, otherwise.
a) 2sin(ap)/p
b) 2asin(ap)/p
c) 4sin(ap)/p
d) 4asin(ap)/p
View Answer

Answer: a

 

6. In Finite Fourier Cosine Transform, if the upper limit l = π, then its inverse is given by ________
a) F(x)=2π∑∞p=1fc(p)cos(px)+1/πfc(0)
b) F(x)=2π∑∞p=1fc(p)cos(px)
c) F(x)=2π∑∞p=1fc(p)cos(pxπ)
d) F(x)=2π∑∞p=0fc(p)cos(px)+1/πfc(0)
View Answer

Answer: a

7. Find the Fourier Cosine Transform of F(x) = 2x for 0<x<4.
a) fc(p)=32 / (p2π2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=16
b) fc(p)=32 / (p2π2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=32
c) fc(p)=64 / (pπ2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=16
d) fc(p)=32 /(pπ2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=64
View Answer

Answer: a

8. If Fourier transform of e−|x|=21+p2, then find the fourier transform of t2e−|x|.
a) 4 /1+p2
b) −2 / 1+p2
c) 2 / 1+p2
d) −4 / 1+p2
View Answer

Answer: b

9. If Fc{e−ax}=pa2+p2, find the Fs{−ae−ax}.
a) 4p / a2+p2
b) −p^2 /ca2+p2
c) 4p2 / a2+p2
d) p / a2+p2
View Answer

Answer: b

10. Find the fourier transform of ∂2u∂x2 . (u’(p,t) denotes the fourier transform of u(x,t)).
a) (ip)² u’(p,t)
b) (-ip)² u’(p,t)
c) (-ip)² u(p,t)
d) (ip)² u(p,t)
View Answer

Answer: a

11. What is the fourier transform of e^-a|x| * e^-b|x|?
a) 4ab / (a2+p2)(b2+p2)
b) 2ab / (a2+p2)(b2+p2)
c) 4 / (a2+p2)(b2+p2)
d) a2b2 / (a2+p2)(b2+p2)
View Answer

Answer: a

12. What is the Fourier transform of e^ax? (a>0)
a) pa2+p2
b) 2aa2+p2
c) −2aa2+p2
d) cant’t be found

View Answer

Answer: d

13. F(x)=x(−12)is self reciprocal under Fourier cosine transform.
a) True
b) False
View Answer

Answer: a

14. Find the fourier cosine transform of e^-ax * e^-ax.
a) p2 / a2+p2
b) p2 / (a2+p2)2
c) 4p2 / (a2+p2)2
d) −p2 / (a2+p2)2
View Answer

Answer: b

15. Find the fourier sine transform of F(x) = -x when x<c and (π – x) when x>c and 0≤c≤π.
a) π/ccos(pc)
b) π/pcos(pc)
c) π/ccos(pπ)
d) pπ/ccos(pc)
View Answer

Answer: b

1. Find the Z-Transform of nCp.
a) (1-z^-1)ⁿ
b) (1+z^-1)ⁿ
c) (1-z^-1)^-n
d) (1+z^-1)^-n
View Answer

Answer: b

 

2. Find the function whose Z – Transform is 1z.
a) δ(n)
b) δ(n+1)
c) U(n)
d) U(n+1)
View Answer

Answer: b

3. Find the function whose Z transform is e1z.
a) log(n)
b) 1/n
c) 1/n!
d) 1/(n+1)!
View Answer

Answer: c

4. Find the inverse Z- Transform of (zz−a)3.
a) 1/2.(n+1)(n−2)an−2U(n)
b) 1/2.(n−1)(n−2)an−3U(n)
c) 1/2.(n−1)(n+2)an−1U(n)
d) 1/2.(n+1)(n+2)anU(n)
View Answer

Answer: d

5. Find the inverse Z – Transform of logzz+1.
a) (−1)^n / n
b) (−1)^n /n+1n
c) 1n
d) (−1)^n /n+1
View Answer

Answer: a

6. Find the Z – Transform of sinh ⁡nθ.
a) sinhθ / z^2−2zcoshθ+1
b) 1/2sinhθ / z^2−2zcoshθ+1
c) zsinhθ) / z^2−2zcoshθ+1
d) z(z−sinhθ) / z^2−2zcoshθ+1
View Answer

Answer: a

7. Find the value of u₃ if U(z)=3z2+2z+10(z−1)4.
a) 12
b) 13
c) 14
d) 15
View Answer

Answer: c

8. Find the Z – Transform of n^p.
a) −zd /dz(Z(np−1))
b) zd /dz(Z(np))
c) −zd /dz(Z(np+1))
d) zd /dz(Z(np+1))
View Answer

Answer: a

9. The Z – Transform of a function is given by U(z)=z3+6z2+9z+3(z−1)4. Find the Z-Transform of u_n+2.
a) 10z^3+3z^2+7z1−1/(z−1)4
b) 10z^4+3z^3+7z^2−z /(z−1)4
c) 10z^4+4z^3+7z^2−2z /(z−1)4
d) 10z^4+3z^3−4z /(z−1)4
View Answer

Answer: b

10. Find u₂ if U(z)=z3+6z2+9z+3(z−1)4.
a) 8
b) 9
c) 10
d) 11
View Answer

Answer: c

11. Find the order of the difference equation Δ³y_n – Δ²y_n – Δy_n = 3.
a) 3
b) 4
c) 2
d) 5
View Answer

Answer: a

12. Find the order of the difference equation y_n+3 -3 y_n+1 – y_n-2 = 4.
a) 3
b) 4
c) 5
d) 6
View Answer

Answer: c

 

 

13. Find the difference equation of y_n = A 3ⁿ + B 5ⁿ.
a) y_n+2 -16 y_n+1 + 15 y_n-1 = 0
b) y_n+3 -14 y_n+1 + 30 y_n = 0
c) 2 y_n+2 -14 y_n+1 + 15 y_n = 0
d) 2 y_n+2 -16 y_n+1 + 30 y_n = 0
View Answer

Answer: d

14. Find the difference equation of y = ax + b.
a) Δ²y = 0
b) Δ²y = 1
c) Δ²y + 3Δy = 2
d) Δ²y + 4Δy = 5
View Answer

Answer: a

15. Solve u_n+2 + 10 u_n+1 + 9 u_n = 2ⁿ.
a) un=2^n+1 / 33+(−9)^n+1 / 88+(−1)^n+1 / 24
b) un=2n^ / 33+(−9)^n / 88+(−1)^n−1 / 24
c) un=2^n+1 / 11+(−9)^n+1 / 88+(−1)^n / 24
d) un=2^n / 11+(−9)^n / 88+(−1)n−1^ / 24
View Answer

Answer: b

_________________________________________

 

Unit 4  partial differential equation and their appliction

 

1. Find ∂z∂x where z=ax2+2by2+2bxy.
a) 3by
b) 2ax
c) 3(ax+by)
d) 2(ax+by)
View Answer

Answer: d

2. Find ∂z∂x where z=sinx2×cosy2.
a) 2xsin⁡x²
b) x sin2x
c) 2xsin⁡x² cos⁡y²
d) 6xsin⁡x² cos⁡y²
View Answer

Answer: c

3. Find ∂u∂x where u=cos(x−−√+y√).
a) −12x√×tan(√x+√y)
b) −12x√×cos(√x+√y)
c) −12x√×sin(√x+√y)
d) −1x√×sin(√x+√y)
View Answer

Answer: c

4. If u=ex+yex−ey, what is ∂u∂x+u∂y?
a) 2((ex−ey)×ex+y)−(ex+y)(ex+ey) / (ex−ey)2
b) 2((ex−ey)×ex+y)−(ex+y)(ex+ey) / (ex+ey)2
c) 2((ex−ey)×ex+y)−(ex+y)(ex−ey) / (ex−ey)2
d) u
View Answer

Answer: a

5. If θ=tne−r22t, find the value of n that satisfies the equation, ∂θ∂t=1r2∂∂r(r2∂θ∂r).
a) 0
b) -1
c) 1
d) 3
View Answer

Answer: b

_____________________________________________

1. First order partial differential equations arise in the calculus of variations.
a) True
b) False
View Answer

Answer: a

2. The symbol used for partial derivatives, ∂, was first used in mathematics by Marquis de Condorcet.
a) True
b) False
View Answer

Answer: a

3. What is the order of the equation, xy3(∂y∂x)2+yx2+∂y∂x=0?
a) Third Order
b) Second Order
c) First Order
d) Zero Order

Answer: c

4. In the equation, y= x²+c,c is known as the parameter and x and y are known as the main variables.
a) True
b) False
View Answer

Answer: a

5. Which of the following is one of the criterions for linearity of an equation?
a) The dependent variable and its derivatives should be of second order
b) The dependent variable and its derivatives should not be of same order
c) Each coefficient does not depend on the independent variable
d) Each coefficient depends only on the independent variable
View Answer

Answer: d

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
View Answer

Answer: b

7. Which of the following is an example for first order linear partial differential equation?
a) Lagrange’s Partial Differential Equation
b) Clairaut’s Partial Differential Equation
c) One-dimensional Wave Equation
d) One-dimensional Heat Equation
View Answer

Answer: a

8. What is the nature of Lagrange’s linear partial differential equation?
a) First-order, Third-degree
b) Second-order, First-degree
c) First-order, Second-degree
d) First-order, First-degree
View Answer

Answer: d

9. Find the general solution of the linear partial differential equation, yzp+zxq=xy.
a) φ(x²-y² – z² )=0
b) φ(x²-y², y²-z² )=0
c) φ(x²-y², y²-x² )=0
d) φ(x²-z², z²-x² )=0
View Answer

Answer: b

10. The equation 2dydx–xy=y−2, is an example for Bernoulli’s equation.
a) False
b) True
View Answer

Answer: b

11. A particular solution for an equation is derived by eliminating arbitrary constants.
a) True
b) False
View Answer

Answer: b

12. 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
View Answer

Answer: b

 

 

1. Which of the following is an example of non-linear differential equation?
a) y=mx+c
b) x+x’=0
c) x+x²=0
d) x”+2x=0
View Answer

Answer: c

2. Which of the following is not a standard method for finding the solutions for differential equations?
a) Variable Separable
b) Homogenous Equation
c) Orthogonal Method
d) Bernoulli’s Equation
View Answer

Answer: c

·  

3. Solution of a differential equation is any function which satisfies the equation.
a) True
b) False
View Answer

Answer: a

 

4. A solution which does not contain any arbitrary constants is called a general solution.
a) True
b) False
View Answer

Answer: a

5. 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
View Answer

Answer: b

6. A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
a) True
b) False
View Answer

Answer: a

7. Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.
a) True
b) False
View Answer

Answer: a

8. Which of the following equations represents Clairaut’s partial differential equation?
a) z=px+f(p,q)
b) z=f(p,q)
c) z=p+q+f(p,q)
d) z=px+qy+f(p,q)
View Answer

Answer: d

9. Which of the following represents Lagrange’s linear equation?
a) P+Q=R
b) Pp+Qq=R
c) p+q=R
d) Pp+Qq=P+Q
View Answer

Answer: b

 

 

 

10. A partial differential equation is one in which a dependent variable (say ‘x’) depends on an independent variable (say ’y’).
a) False
b) True
View Answer

Answer: a

11. What is the complete solution of the equation, q=e−pα?
a) z=ae−aαy
b) z=x+e−aαy
c) z=ax+e−aαy+c
d) z=e−aαy
View Answer

Answer: c

12. A particular solution for an equation is derived by eliminating arbitrary constants.
a) True
b) False
View Answer

Answer: b

 

 

1. Which of the following is an example of non-linear differential equation?
a) y=mx+c
b) x+x’=0
c) x+x²=0
d) x”+2x=0
View Answer

Answer: c

2. Which of the following is not a standard method for finding the solutions for differential equations?
a) Variable Separable
b) Homogenous Equation
c) Orthogonal Method
d) Bernoulli’s Equation
View Answer

Answer: c

3. Solution of a differential equation is any function which satisfies the equation.
a) True
b) False
View Answer

Answer: a

4. A solution which does not contain any arbitrary constants is called a general solution.
a) True
b) False
View Answer

Answer: a

5. 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
View Answer

Answer: b

6. A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
a) True
b) False
View Answer

Answer: a

7. Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.
a) True
b) False
View Answer

Answer: a

8. 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
View Answer

Answer: c

9. Which of the following represents Lagrange’s linear equation?
a) P+Q=R
b) Pp+Qq=R
c) p+q=R
d) Pp+Qq=P+Q
View Answer

Answer: b

 

 

 

10. A partial differential equation is one in which a dependent variable (say ‘x’) depends on an independent variable (say ’y’).
a) False
b) True
View Answer

Answer: a

11. What is the complete solution of the equation, q=e−pα?
a) z=ae−aαy
b) z=x+e−aαy
c) z=ax+e−aαy+c
d) z=e−aαy
View Answer

Answer: c

12. A particular solution for an equation is derived by eliminating arbitrary constants.
a) True
b) False
View Answer

Answer: b

 

1. Non-homogeneous which may contain terms which only depend on the independent variable.
a) True
b) False
View Answer

Answer: a

2. Which of the following is a non-homogeneous equation?
a) ∂2u∂t2−c2∂2u∂x2=0
b) ∂2u∂x2+∂2u∂y2=0
c) ∂2u∂x2+(∂2u∂x∂y)2+∂2u∂y2=x2+y2
d) ∂u∂t−T∂2u∂x2=0
View Answer

Answer: c

3. What is the general form of the general solution of a non-homogeneous DE (u_h(t)= general solution of the homogeneous equation, u_p(t)= any particular solution of the non-homogeneous equation)?
a) u(t)=u_h (t)/u_p (t)
b) u(t)=u_h (t)*u_p (t)
c) u(t)=u_h (t)+u_p (t)
d) u(t)=u_h (t)-u_p (t)
View Answer

Answer: c

4. While an ODE of order m has m linearly independent solutions, a PDE has infinitely many.
a) False
b) True
View Answer

Answer: b

5. Which of the following methods is not used in solving non-homogeneous equations?
a) Exponential Response Formula
b) Method of Undetermined Coefficients
c) Orthogonal Method
d) Variation of Constants
View Answer

Answer: c

6. What is the order of the non-homogeneous partial differential equation,
∂2u∂x2+(∂2u∂x∂y)2+∂2u∂y2=x2+y2?
a) Order-3
b) Order-2
c) Order-0
d) Order-1
View Answer

Answer: b

7. What is the degree of the non-homogeneous partial differential equation,
(∂2u∂x∂y)5+∂2u∂y2+∂u∂x=x2−y3?
a) Degree-2
b) Degree-1
c) Degree-0
d) Degree-5
View Answer

Answer: d

8. The Integrating factor of a differential equation is also called the primitive.
a) True
b) False
View Answer

Answer: b

9. A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
a) True
b) False
View Answer

Answer: a

 

10. What is the complete solution of the equation, q=e−pα?
a) z=ae−aαy
b) z=x+e−aαy
c) z=ax+e−aαy+c
d) z=e−aαy
View Answer

Answer: c

11. In recurrence relation, each further term of a sequence or array is defined as a function of its succeeding terms.
a) True
b) False
View Answer

Answer: b

12. What is the degree of the differential equation, x³-6x³ y³+2xy=0?
a) 3
b) 5
c) 6
d) 8
View Answer

Answer: c

 

 

1. What is the general form of second order non-linear partial differential equations (x and y being independent variables and z being a dependent variable)?
a) F(x,y,z,∂z∂x,∂z∂y,∂2z∂x2,∂2z∂y2,∂2z∂x∂y)=0
b) F(x,z,∂z∂x,∂z∂y,∂2z∂x2,∂2z∂y2)=0
c) F(y,z,∂z∂x,∂z∂y)=0
d) F(x,y)=0
View Answer

Answer: a

2. The solution of the general form of second order non-linear partial differential equation is obtained by Monge’s method.
a) False
b) True
View Answer

Answer: b

 

 

 

 

3. What is the reason behind the non-existence of any real function which satisfies the differential equation, (y’)² + 1 = 0?
a) Because for any real function, the left-hand side of the equation will be less than, or equal to one and thus cannot be zero
b) Because for any real function, the left-hand side of the equation becomes zero
c) Because for any real function, the left-hand side of the equation will be greater than, or equal to one and thus cannot be zero
d) Because for any real function, the left-hand side of the equation becomes infinity
View Answer

Answer: c

4. What is the order of the partial differential equation, ∂2z∂x2−(∂z∂y)5+∂2z∂x∂y=0?
a) Order-5
b) Order-1
c) Order-4
d) Order-2
View Answer

Answer: d

5. 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

6. Which of the following represents the canonical form of a second order parabolic PDE?
a) ∂2z∂η2+⋯=0
b) ∂2z∂ζ∂η+⋯=0
c) ∂2z∂α2+∂2z∂β2…=0
d) ∂2z∂ζ2+⋯=0
View Answer

Answer: a

7. The condition which a second order partial differential equation must satisfy to be elliptical is
b²-ac=0.
a) True
b) False
View Answer

Answer: b

 

8. 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

9. Which of the following represents the family of the characteristic curves for parabolic equations?
a) aζ_x+bζ_y=0
b) aζ_x+b=0
c) a+ζ_y=0
d) a(ζ_x+ζ_y)=0
View Answer

Answer: a

10. The condition that a second order partial differential equation should satisfy to be parabolic is b²-ac=0.
a) True
b) False
View Answer

Answer: a

11. Elliptic equations have no characteristic curves.
a) True
b) False
View Answer

Answer: a

12. Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.
a) True
b) False
View Answer

Answer: a

13. In the formation of differential equation by elimination of arbitrary constants, after differentiating the equation with respect to independent variable, the arbitrary constant gets eliminated.
a) False
b) True
View Answer

Answer: a

 

 

 

APPLICATION

1. By using the method of separation of variables, the determination of solution to P.D.E. reduces to determination of solution to O.D.E.
a) True
b) False
View Answer

Answer: a

2. Separation of variables, in mathematics, is also known as Fourier method.
a) False
b) True
View Answer

Answer: b

3. Which of the following equations cannot be solved by using the method of separation of variables?
a) Laplace Equation
b) Helmholtz Equation
c) Alpha Equation
d) Biharmonic Equation
View Answer

Answer: c

4. The matrix form of the separation of variables is the Kronecker sum.
a) True
b) False
View Answer

Answer: a

5. For a partial differential equation, in a function φ (x, y) and two variables x, y, what is the form obtained after separation of variables is applied?
a) Φ (x, y) = X(x)+Y(y)
b) Φ (x, y) = X(x)-Y(y)
c) Φ (x, y) = X(x)Y(y)
d) Φ (x, y) = X(x)/Y(y)
View Answer

Answer: c

6. What is the solution of, ∂2u∂x2=2xet, after applying method of separation of variables (u(0,t)=t,∂u∂x(0,t)=et)?
a) u=x33et+xet
b) u=x33et+xet+t
c) u=x33et+et+t
d) u=x22et+xet+t
View AnswerAnswer: b

7. Which of the following is true with respect to formation of differential equation by elimination of arbitrary constants?
a) The given equation should be differentiated with respect to independent variable
b) Elimination of the arbitrary constant by replacing it using derivative
c) If ‘n’ arbitrary constant is present, the given equation should be differentiated ‘n’ number of times
d) To eliminate the arbitrary constants, the given equation must be integrated with respect to the dependent variable
View Answer

Answer: d

8. In the formation of differential equation by elimination of arbitrary constants, after differentiating the equation with respect to independent variable, the arbitrary constant gets eliminated.
a) False
b) True
View Answer

Answer: a

9. u (x, t) = e − 2π*2t*sin πx is the solution of the two-dimensional Laplace equation.
a) True
b) False
View Answer

Answer: b

10. The symbol used for partial derivatives, ∂, was first used in mathematics by Marquis de Condorcet.
a) True
b) False
View Answer

Answer: a

11. Separation of variables was first used by L’Hospital in 1750.
a) False
b) True
View Answer

Answer: b

12. A particular solution for an equation is derived by eliminating arbitrary constants.
a) True
b) False
View Answer

Answer: b

 

 

1. The partial differential equation of 1-Dimensional heat equation is ___________
a) u_t = c²u_xx
b) u_t = pu_xx
c) u_tt = c²u_xx
d) u_t = – c²u_xx
View Answer

Answer: a

2. When using the variable separable method to solve a partial differential equation, then the function can be written as the product of functions depending only on one variable. For example, U(x,t) = X(x)T(t).
a) True
b) False
View Answer

Answer: a

3. The one dimensional heat equation can be solved using a variable separable method. The constant which appears in the solution should be __________
a) Positive
b) Negative
c) Zero
d) Can be anything
View Answer

Answer: b

4. When solving the 1-Dimensional heat equation for the conduction of heat along the rod without radiation with conditions:
i) u(x,t) is finite for t tends to infinite
ii) u_x(0,t) = 0 and u_x(l,t) = 0
iii) u(x,t) = x(l-x) for t=0 between x=0 and x=l, which condition is the best to use in the first place?
a) u_x(0,t) = u_x(l,t) = 0
b) u(x,t) = x(l-x) for t=0 between x=0 and x=l.
c) u(x,t) = x(l-x) for x=0 between t=0 and t=l.
d) u(0,t) = u(l,t) = 0
View Answer

Answer: a

5. Solve the 1-Dimensional heat equation for the conduction of heat along the rod without radiation with conditions:
i) u(x,t) is finite for t tends to infinite
ii) u_x(0,t) = 0 and u_x(l,t) = 0
iii) u(x,t) = x(l-x) for t=0 between x=0 and x=l.
a) U(x,t) =l23/2+∑cos(nπxl)e−c2n2π2tl2−4l2(2m)2+π2
b) U(x,t) =l23+∑cos(nπxl)e−c2n2π2tl2−4l2(2m)2+π2
c) U(x,t) =l23+∑cos(nπxl)e−c2n2π2tl24l2(2m)2+π2
d) U(x,t) =l23/2+∑cos(nπxl)e−c2n2π2tl24l2(2m)2+π2
View AnswerAnswer: a

6. A rod of 30cm length has its ends P and Q kept 20°C and 80°C respectively until steady state condition prevail. The temperature at each point end is suddenly reduced to 0°C and kept so. Find the conditions for solving the equation.
a) u(0,t) = 0 = u(30,t) and u(x,0) = 20 + 60/10 x
b) u_x(0,t) = 0 = u_x(30,t) and u(x,0) = 20 + 60/30 x
c) u_t(0,t) = 0 = u_t(30,t) and u(x,0) = 20 + 60/10 x
d) u(0,t) = 0 = u(30,t) and u(x,0) = 20 + 60/30 x
View Answer

Answer: d

7. Is it possible to have a solution for 1-Dimensional heat equation which does not converge as time approaches infinity?
a) Yes
b) No
View Answer

Answer: b

8. Solve the equation u_t = u_xx with the boundary conditions u(x,0) = 3 sin (nπx) and u(0,t)=0=u(1,t) where 0<x<1 and t>0.
a) 3∑∞n=1 e^-n2 π² t cos⁡(nπx)
b) ∑∞n=1 e^-n2 π² t sin⁡(nπx)
c) 3∑∞n=1 e^-n2 π² t sin⁡(nπx)
d) ∑∞n=1 e^-n2 π² t cos(nπx)
View Answer

Answer: c

9. If two ends of a bar of length l is insulated then what are the conditions to solve the heat flow equation?
a) u_x(0,t) = 0 = u_x(l,t)
b) u_t(0,t) = 0 = u_t(l,t)
c) u(0,t) = 0 = u(l,t)
d) u_xx(0,t) = 0 = u_xx(l,t)
View Answer

Answer: a

10. The ends A and B of a rod of 20cm length are kept at 30°C and 80°C until steady state prevails. What is the condition u(x,0)?
a) 20 + ⁵⁄₂ x
b) 30 + ⁵⁄₂ x
c) 30 + 2x
d) 20 + 2x
View Answer

Answer: b

 

 

1. Solve ∂u∂x=6∂u∂t+u using the method of separation of variables if u(x,0) = 10 e^-x.
a) 10 e^-x e^-t/3
b) 10 e^x e^-t/3
c) 10 e^x/3 e^-t
d) 10 e^-x/3 e^-t
View Answer

Answer: a

2. Find the solution of ∂u∂x=36∂u∂t+10u if ∂u∂x(t=0)=3e−2x using the method of separation of variables.
a) −32e−2xe−t/3
b) 3exe−t/3
c) 32e2xe−t/3
d) 3e−xe−t/3
View Answer

Answer: a

3. Solve the partial differential equation x3∂u∂x+y2∂u∂y=0 using method of separation of variables if u(0,y)=10e5y.
a) 10e52x2e5y
b) 10e−52y2e5x
c) 10e−52y2e−5x
d) 10e−52x2e5y
View Answer

Answer: d

4. Solve the differential equation 5∂u∂x+3∂u∂y=2u using the method of separation of variables if u(0,y)=9e−5y.
a) 9e175xe−5y
b) 9e135xe−5y
c) 9e−175xe−5y
d) 9e−135xe−5y
View Answer

Answer: a

 

 

 

 

 

 

5. Solve the differential equation x2∂u∂x+y2∂u∂y=u using the method of separation of variables if u(0,y)=e2y.
a) e−3ye2x
b) e3ye2x
c) e−3xe2y
d) e3xe2y
View Answer

Answer: c

6. While solving a partial differential equation using a variable separable method, we assume that the function can be written as the product of two functions which depend on one variable only.
a) True
b) False
View Answer

Answer: a

7. While solving a partial differential equation using a variable separable method, we equate the ratio to a constant which?
a) can be positive or negative integer or zero
b) can be positive or negative rational number or zero
c) must be a positive integer
d) must be a negative integer
View Answer

Answer: b

8. When solving a 1-Dimensional wave equation using variable separable method, we get the solution if _____________
a) k is positive
b) k is negative
c) k is 0
d) k can be anything
View Answer

Answer: b

9. When solving a 1-Dimensional heat equation using a variable separable method, we get the solution if ______________
a) k is positive
b) k is negative
c) k is 0
d) k can be anything
View Answer

Answer: b

 

10. While solving any partial differentiation equation using a variable separable method which is of order 1 or 2, we use the formula of fourier series to find the coefficients at last.
a) True
b) False
View Answer

Answer: a

 

1. Who was the first person to develop the heat equation?
a) Joseph Fourier
b) Galileo Galilei
c) Daniel Gabriel Fahrenheit
d) Robert Boyle
View Answer

Answer: a

2. Which of the following is not a field in which heat equation is used?
a) Probability theory
b) Histology
c) Financial Mathematics
d) Quantum Mechanics
View Answer

Answer: b

3. Under ideal assumptions, what is the two-dimensional heat equation?
a) u_t = c∇² u = c(u_xx + u_yy)
b) u_t = c² u_xx
c) u_t = c² ∇² u = c² (u_xx + u_yy)
d) u_t = ∇² u = (u_xx + u_yy)
View Answer

Answer: c

4. In mathematics, an initial condition (also called a seed value), is a value of an evolving variable at some point in time designated as the initial time (t=0).
a) False
b) True
View Answer

Answer: b

5. What is another name for heat equation?
a) Induction equation
b) Condenser equation
c) Diffusion equation
d) Solar equation
View Answer

Answer: c

6. Heat Equation is an example of elliptical partial differential equation.
a) True
b) False
View Answer

Answer: b

7. What is 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

8. Which of the following represents the canonical form of a second order parabolic PDE?
a) ∂2z∂η2+⋯=0
b) ∂2z∂ζ∂η+⋯=0
c) ∂2z∂α2+∂2z∂β2…=0
d) ∂2z∂ζ2+⋯=0
View Answer

Answer: a

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

10. What is the order of the partial differential equation, ∂2z∂x2−(∂z∂y)5+∂2z∂x∂y=0?
a) Order-5
b) Order-1
c) Order-4
d) Order-2
View Answer

Answer: d

 

 

 

 

 

 

 

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

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

Answer: b

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

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

Answer: a

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

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

Answer: b

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

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

Answer: a

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

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

 

 

 

 

 

UNIT 6:- Function of complex variable (integral Calculus)

 

1. Find ∫π20sin6(x)dx.
a) 0
b) ^π⁄₈
c) ^π⁄₄
d) 15π/96
View Answer

Answer: d

2. Find ∫π20sin10(x)cos(x)dx.
a) 1
b) 0
c) 13π/1098
d) 21π/2048
View Answer

Answer: d

3. Find ∫π40tan3(x)dx.
a) 0
b) 1
c)-1
d) None of the mentioned
View Answer

Answer: b

4. Find the value of ∫π20cos11(x).sin9(x)dx.
a) ¹⁄_10!
b) ^5!6!⁄_11!
c) ^10!⁄_5!6!
d) 0
View Answer

Answer: b

5. Find ∫2√1(x85−1x25)52dx.
a) -1
b) 1
c) 0
d) ¹⁄₅ – ¹⁄₃ + ¹⁄₁ – ^π⁄₄
View Answer

Answer: d

 

 

6. Find ∫π40x4.sin(x)dx.
a) -1
b) 1
c) 0
d) 4((^π⁄₂)³ – 3π + 1)
View Answer

Answer: b

7. Find ∫0−∞x5.exdx.
a) 1
b) 199
c) -5!
d) 5!
View Answer

Answer: c

8. Find ∫π20cos3(x).cos(2x)dx.
a) 0
b) 5
c) 87
d) ^-16⁄₁₀₅
View Answer

Answer: d

 

 

1. What is meant by quadrature process in mathematics?
a) Finding area of plane curves
b) Finding volume of plane curves
c) Finding length of plane curves
d) Finding slope of plane curves
View Answer

Answer: a

2. What is the formula used to find the area surrounded by the curves in the following diagram?

a) ∫baydx
b) ∫ba−ydx
c) ∫baxdy
d) ∫ba−xdy
View Answer

Answer: a

3. What is the formula used to find the area surrounded by the curves in the following diagram?

a) ∫baydx
b) ∫ba−ydx
c) ∫baxdy
d) ∫ba−xdy
View Answer

Answer: b

4. What is the formula used to find the area surrounded by the curves in the following diagram?

a) ∫dcydx
b) ∫dc−ydx
c) ∫dcxdy
d) ∫dc−xdy
View Answer

Answer: c

5. What is the formula used to find the area surrounded by the curves in the following diagram?

a) ∫dcydx
b) ∫dc−ydx
c) ∫dcxdy
d) ∫dc−xdy
View Answer

Answer: d

6. Find the area bounded in the following diagram.

a) 6
b) 12
c) 8
d) 10
View Answer

Answer: b

 

 

 

 

 

 

 

 

7. What is the area bounded by the curve y = x² – 5x + 4, x = 2, x = 3, x-axis in the following diagram?

a) 13
b) 6
c) 136
d) 613
View Answer

Answer: c

 

 

1. Rectification is determining ____________
a) Length of a line
b) Length of a curve
c) Area of an object
d) Perimeter of an object
View Answer

Answer: b

2. Which one of the following is an infinite curve?
a) Hyperbola
b) Koch curve
c) Gaussian curve
d) Parabola
View Answer

Answer: b

3. The expression for arc length in rectangular form is_________________
a) ds=∫baxy√1+(dy/dx)^2
b) ds=∫ba√1-(dy/dx)^2dx
c) ds=∫ba√(dy/dx)^2dx
d) ds=∫ba(√1- (dy/dx)^2dx
View Answer

Answer: d

4. The expression for arc length in parametric form is_________________
a) ds=∫baxy√(dx /dt)2+(dy/dt)2ds
b) ds=∫ba√(dx/dt)2+(dy/dt)2dt
c) ds=∫ba√(dx/dt)2+(dy/dt)2dx
d) ds=∫ba√(dx /dt)2+(dy/dt)2
View Answer

Answer: b

5. The expression for arc length in polar form is_________________
a) ds=∫bar2+(drdθ)2−−−−−−−−√
b) ds=∫bar2+(drdθ)2−−−−−−−−√dθ
c) ds=∫ba(drdθ)2−−−−√dθ
d) ds=∫bar2+(drdθ)2−−−−−−−−√dr
View Answer

Answer: b

6. Closed form solutions are absent for ellipses.
a) True
b) False
View Answer

Answer: a

7. What is the length of a circular curve when θ is in degrees and ‘r’ is the radius?
a) s = rθ
b) s = r
c) s=πrθ180
d) s = πr
View Answer

Answer: c

8. The length of curve r = e^θ, θ value ranges between 0 to π is________
a) 2–√
b) 2–√(e2π–1)
c) 2–√(e2π+1)
d) 2–√(e2π)
View Answer

Answer: b

9. The arc length of y = cosh⁡x where x varies from 0 to 1 is ____________
a) e2–12e
b) e2–12e+12e
c) e2–12e
d) 2e
View AnswerAnswer: a

1. Find the length of the curve given by the equation.
x23+y23=a23
a) 3a/2
b) −7a2
c) −3a/4
d) −3a/2
View Answer

Answer: d

2. Find the length of one arc of the given cycloid.

x=a(θ-sinθ)

y=a(1+cosθ)

a) a
b) 4a
c) 8a
d) 2a
View Answer

Answer: c

 

 

1. The volume of solid of revolution when rotated along x-axis is given as _____________
a) ∫baπy^2dx
b) ∫baπy^2dy
c) ∫baπx^2dx
d) ∫baπx^2dy
View Answer

Answer: a

2. The volume of solid of revolution when rotated along y-axis is given as ________
a) ∫baπy^2dx
b) ∫baπy^2dy
c) ∫baπx^2dx
d) ∫baπx^2dy
View Answer

Answer: d

 

 

 

 

3. What is the volume generated when the ellipse x2a2+y2b2=1 is revolved about its minor axis?
a) 4 ab cubic units
b) 4/3a2b cubic units
c) 4/3ab cubic units
d) 4 cubic units
View Answer

Answer: b

4. What is the volume generated when the region surrounded by y = x−−√, y = 2 and y = 0 is revolved about y – axis?
a) 32π cubic units
b) 32 /5 cubic units
c) 32π /5 cubic units
d) 5π /32 cubic units
View Answer

Answer: c

5. What is the volume of the sphere of radius ‘a’?
a) 4/3πa
b) 4πa
c) 4/3πa2
d) 4/3πa3
View Answer

Answer: d

6. Gabriel’s horn is formed when the curve ____________ is revolved around x-axis for x≥1.
a) y = x
b) y = 1
c) y = 0
d) y = 1/x
View Answer

Answer: d

 

 

 

1. Integration of function is same as the ___________
a) Joining many small entities to create a large entity
b) Indefinitely small difference of a function
c) Multiplication of two function with very small change in value
d) Point where function neither have maximum value nor minimum value
View Answer

Answer: a

2. Integration of (Sin(x) + Cos(x))e^x is______________
a) e^x Cos(x)
b) e^x Sin(x)
c) e^x Tan(x)
d) e^x (Sin(x)+Cos(x))
View Answer

Answer: b

3. Integration of (Sin(x) – Cos(x))e^x is ___________
a) -e^x Cos(x)
b) e^x Cos(x)
c) -e^x Sin(x)
d) e^x Sin(x)
View Answer

Answer: a

4. Value of ∫ Cos² (x) Sin² (x)dx.
a) 1/8[x−Cos(2x) /2]
b) 1/4[x−Cos(2x) /2]
c) 1/8[x−Sin(2x) /2]
d) 1/4[x−Sin(2x) /2]
View Answer

Answer: c

5. If differentiation of any function is zero at any point and constant at other points then it means?
a) Function is parallel to x-axis at that point
b) Function is parallel to y-axis at that point
c) Function is constant
d) Function is discontinuous at that point
View Answer

Answer: a

6. If differentiation of any function is infinite at any point and constant at other points then it means ___________
a) Function is parallel to x-axis at that point
b) Function is parallel to y-axis at that point
c) Function is constant
d) Function is discontinuous at that point
View Answer

Answer: a

7. Integration of function y = f(x) from limit x1 < x < x₂ , y₁ < y < y₂, gives ___________
a) Area of f(x) within x1 < x < x₂
b) Volume of f(x) within x1 < x < x₂
c) Slope of f(x) within x1 < x < x₂
d) Maximum value of f(x) within x1 < x < x₂
View Answer

Answer: a

8. Find the value of ∫ ^ln⁡(x)⁄_{x dx}.
a) 3a²
b) a²
c) a
d) 1
View Answer

Answer: a

9. Find the value of ∫^t⁄_(t+3)(t+2) dt, is?
a) 2 ln(t+3)-3 ln(t+2)
b) 2 ln(t+3)+3 ln(t+2)
c) 3 ln(t+3)-2 ln(t+2)
d) 3 ln(t+3)+2ln(t+2)
View Answer

Answer: c

10. Find the value of ∫ cot³(x) cosec⁴ (x).
a) –[cot4(x) /4+cosec6(x) /6]
b) –[cosec4(x) /4+cosec6(x) /6]
c) –[cot4(x) /4+cot6(x) /6]
d) –[cosec4(x) /4+cot6(x) /6]
View Answer

Answer: c

11. Find the value of ∫sec4(x)tan(x)√dx.
a) 2/5√tan(x) [5+sec^2(x)]
b) 2/5√sec(x) [5+tan^2(x)]
c) 2/5√tan(x) [6+tan^2(x)]
d) 2/5√tan(x) [5+tan^2(x)]
View Answer

Answer: d

 

 

 

12. Find the value of ∫14x2+4x+5dx.
a) ¹⁄₈ sin^(-1)⁡(x + ¹⁄₂)
b)¹⁄₄ tan^(-1)⁡(x + ¹⁄₂)
c) ¹⁄₈ sec^(-1)⁡(x + ¹⁄₂)
d) ¹⁄₄ cos^(-1)⁡(x + ¹⁄₂)
View Answer

Answer: b

13. Find the value of ∫√4x2+4x+5dx.
a) 2[1/2(x+1/2) √(x+1/2)^2+1)]+ln[(x+1/2)+ √(x+1/2)2+1]

b) 2[1/2)√(x+1/2)^2+1]+1/2ln[(x+1/2)+√(x+12)^2+1]

c) 2[1/2(x+1/2)√(x+1/2)^2+1)]+1/2ln[(x+1/2)+√(x+1/2)^2+1]

d) 2[(x+12)(x+12)2+1)−−−−−−−−−−−√]+12ln[(x+12)+(x+12)2+1−−−−−−−−−−√]

View Answer

Answer: c

 

1. Find the value of ∫tan^-1⁡(x)dx.
a) sec^-1 (x) – ¹⁄₂ ln⁡(1 + x²)
b) xtan^-1 (x) – ¹⁄₂ ln⁡(1 + x²)
c) xsec^-1 (x) – ¹⁄₂ ln⁡(1 + x²)
d) tan^-1 (x) – ¹⁄₂ ln⁡(1 + x²)
View Answer

Answer: b 

2. Integration of (Sin(x) + Cos(x))e^x is?
a) e^x Cos(x)
b) e^x Sin(x)
c) e^x Tan(x)
d) e^x (Sin(x) + Cos(x))
View Answer

Answer: b

3. Find the value of ∫x³ Sin(x)dx.
a) x³ Cos(x) + 3x² Sin(x) + 6xCos(x) – 6Sin(x)
b) – x³ Cos(x) + 3x² Sin(x) – 6Sin(x)
c) – x³ Cos(x) – 3x² Sin(x) + 6xCos(x) – 6Sin(x)
d) – x³ Cos(x) + 3x² Sin(x) + 6xCos(x) – 6Sin(x)
View Answer

Answer: d

 

4. Value of ∫uv dx,where u and v are function of x.
a) ∑ni=1(−1)iuivi+1
b) ∑ni=0uiv^i+1
c) ∑ni=0(−1)iuiv^i+1
d) ∑ni=0(−1)iuiv^n−i
View Answer

Answer: c

5. Find the value of ∫x⁷ Cos(x) dx.
a) x⁷ Sin(x) + 7x⁶ Cos(x) + 42x⁵ Sin(x) + 210x⁴ Cos(x) + 840x³ Sin(x) + 2520x² Cos(x) + 5040xSin(x) + 5040Cos(x)
b) x⁷ Sin(x) – 7x⁶ Cos(x) + 42x⁵ Sin(x) – 210x⁴ Cos(x) + 840x³ Sin(x) – 2520x² Cos(x) + 5040xSin(x) – 5040Cos(x)
c) x⁷ Sin(x) + 7x⁶ Cos(x) + 42x⁵ Sin(x) + 210x⁴ Cos(x) + 840x³ Sin(x) + 2520x² Cos(x) + 5040xSin(x) + 5040Cos(x)
d) x⁷ Sin(x) + 7x⁶ Cos(x) + 42x⁵ Sin(x) + 210x⁴ Cos(x) + 840x³ Sin(x) + 2520x² Cos(x) + 5040xSin(x) + 10080Cos(x)

Answer: a

6. Find the value of ∫x³ e^x e^2x e^3x….e^nx dx.
a) 2 /n(n+1)e^n(n+1)2x[x3+3x2[2 /n(n+1)]1+6x[2 /n(n+1)]2+6[2 /n(n+1)]^3]
b) 2 /n(n+1)e^n(n+1)2x[x3+3x2[2 /n(n+1)]1+6x[2 /n(n+1)]2+6[2 /n(n+1)]^3]
c)2 /n(n+1)e^n(n+1)2x[x3+3x2[2 /n(n+1)]1+6x[2 /n(n+1)]2+6[2 /n(n+1)]^3]
d)2 /n(n+1)e^n(n+1)2x[x3+3x2[2 /n(n+1)]1+6x[2 /n(n+1)]2+6[2 /n(n+1)]^3]
View Answer

Answer: a

7. Find the area of a function f(x) = x² + xCos(x) from x = 0 to a, where, a>0.
a) ^a2⁄₂ + aSin(a) + Cos(a) – 1
b) ^a3⁄₃ + aSin(a) + Cos(a)
c) ^a3⁄₃ + aSin(a) + Cos(a) – 1
d) ^a3⁄₃ + Cos(a) + Sin(a) – 1
View Answer

Answer: c

8. Find the area ^ln(x)⁄_x from x = x = ae^b to a.
a) ^b2⁄₂
b) ^b⁄₂
c) b
d) 1
View Answer

Answer: a

 

 

 

9. Find the area inside a function f(t) = t(t+3)(t+2)dt from t = -1 to 0.
a) 4 ln⁡(3) – 5ln⁡(2)
b) 3 ln⁡(3)
c)3 ln⁡(3) – 4ln⁡(2)
d) 3 ln⁡(3) – 5 ln⁡(2)
View Answer

Answer: d

10. Find the area inside integral f(x)=sec4(x)tan(x)√ from x = 0 to π.
a) π
b) 0
c) 1
d) 2
View Answer

Answer: b

11. Find the area inside function (2x3+5x2−4)x2 from x = 1 to a.
a) ^a2⁄₂ + 5a – 4ln(a)
b) ^a2⁄₂ + 5a – 4ln(a) – ¹¹⁄₂
c) ^a2⁄₂ + 4ln(a) – ¹¹⁄₂
d) ^a2⁄₂ + 5a – ¹¹⁄₂
View Answer

Answer: b

12. Find the value of ∫(x⁴ – 5x² – 6x)⁴ 4x³ – 10x – 6 dx.
a) (x4−5x2−6x)^4 /4
b) (x4−5x2−6x)^5 /5
c) (4x3−10x−6)^5 /5
d) (4x3−10x−6)^4 /4
View Answer

Answer: b

13. Temperature of a rod is increased by moving x distance from origin and is given by equation T(x) = x² + 2x, where x is the distance and T(x) is change of temperature w.r.t distance. If, at x = 0, temperature is 40 C, find temperature at x=10.
a) 473 C
b) 472 C
c) 474 C
d) 475 C
View Answer

Answer: a

14. Find the value of ∫1 /16x2+16x+10dx.
a) ¹⁄₈ sin^-1(x + ¹⁄₂)
b) ¹⁄₈ tan^-1(x + ¹⁄₂)
c) ¹⁄₈ sec^-1(x + ¹⁄₂)
d) ¹⁄₄ cos^-1(x + ¹⁄₂)
View AnswerAnswer: b