Calculus | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) PDF Download

Q1: The expression for computing the effective interest rate (i_eff) using continuous compounding for a nominal interest rate of 5% is

The effective interest rate (in percentage) is ______(rounded off to 2 decimal places).  [2024 , Set-ll]
Ans: 5.11 to 5.15



Q2: Three vectors  are given as

Which of the following is/are CORRECT? [2024 , Set-ll]
(a)
(b)
(c)
(d)
Ans: (a, c, d)
(A)
(This is always true for any three given vectors)
(B) We know that is always true but because
This can be true only when



(Hence proved)

Q3: The function f(x) = x³ − 27x + 4 , 1 ≤ x ≤ 6 has [2024 , Set-ll]
(a) Inflection point 
(b) Saddle point 
(c) Minima point 
(d) Maxima point
Ans:(c)

3x² - 27 = 0


So at x = 3 function has point at local minima.

Q4: The second derivative of a function F is computed using the fourth-order Central Divided Difference method with a step length h.
The CORRECT expression for the second derivative is [2024 , Set-ll]
(a)
(b)
(c)

(d)
Ans: (d)
The second derivative of a function of using fourth order central divided difference method is given by


Q5: A vector fieldand a scalar field r are given by

Consider the statement P and Q : 
P : Curl of the gradient of the scalar field r is a null vector. 
Q : Divergence of curl of the vector field  is zero. 
Which one of the following options is CORRECT? [2024 , Set-l]
(a) P is TRUE and Q FALSE
(b) P is FALSE and Q is TRUE 
(c) Both P and Q are TRUE
(d) Both P and Q are FALSE
Ans: (c)

Hence both are true. Hence option (D).

Q6: The smallest positive root of the equation x⁵ − 5x⁴ −10x³ + 50x² + 9x − 45 = 0 lies in the range [2024 , Set-l]
(a) 10 ≤ x ≤ 100 
(b) 6 ≤ x ≤ 8 
(c) 2 < x ≤ 4 
(d) 0 < x ≤ 2
Ans: (d)
Taking option (A) 0 ≤ x ≤ 2 
f(0) = 0 − 45 < 0  
f(2) = 2⁵ − 5(2)⁴ − 10(2)³ + 50 (2)² + 9 × 2 − 45 
= + 45 > 0 

hence there will be one root in this interval which will be smallest root as per the given option.

Q7: Two vectors [2 1 0 3]^τand [1 0 1 2]^τbelong to the null space of a 4 × 4 matrix of rank 2. Which one of the following vectors also belongs to the null space? [2023, Set-ll]
(a) [11−11]^τ
(b) [2 0 1 2]^τ
(c) [0 − 2 1 − 1]^τ
(d) [3 1 1 2]^τ
Ans:(a)
Given matrix is 4 × 4 4×4 and rank of matrix is 2. 
Therefore, rank of matrix  = No. of variables Thus, there two linearly dependent vectors & two linearly independent vectors are present.
X₁ = [2 1 0 3]^τ
X₂ = [1 0 1 2]^τ

$[11-11]$^τ 

Q8: Let ϕ be a scalar field, and u be a vector field. Which of the following identities is true for div (ϕu) ? [2023, Set-II]
(a) div(ϕu) = ϕ div(u) + u ⋅ grad(ϕ) 
(b) div(ϕu) = ϕ div(u) + u × grad(ϕ) 
(c) div (ϕu) = ϕ grad(u) + u⋅grad(ϕ) 
(d) div(ϕu) = ϕ grad(u) + u × grad(ϕ)
Ans: (a)
div(ϕu) = ϕdiv(μ) + ugrad(ϕ)

Q9: For the function f (x) = e^x∣sin x∣ ; , which of the following statements is/are TRUE? (a) The function is continuous at all x [2023, Set-I]
(b) The function is differentiable at all x 
(c) The function is periodic 
(d) The function is bounded
Ans: (a)

From above graph its clear that for every

lim_h+0 f(x − h) = lim_h+0 f(x + h) = f(x) 
So, function is always continuous but in the graph there are corner points so function is not differentiable.

Q10: The following function is defined over the interval [−L, L]:
f(x) = px⁴ + qx⁵
If it is expressed as a Fourier series,

which options amongst the following are true?
(a) a_n, n = 1,2,...,∞ depends on p
(b) a_n, n = 1,2,...,∞ depends on q
(c) b_n, n = 1,2,...,∞ depends on p
(d) b_n, n = 1,2,...,∞ depends on q
Ans: (b, c)
f(x) = px⁴ + qx

Q11: What is curl of the vector field 2x²yi + 5z²j - 4yzk? [2019 : 1 Mark, Set-ll]
(a)-14zi-2x²k
(b)6zi + 4x²j - 2x²k
(c)- 14zi + 6yj + 2x²k
(d)6zi - 8xyj + 2x²yk
Answer: (a)

Solution:

Question 12: Euclidean norm (length) of the vector [4 -2 -6]^r is    [2019 : 1 Mark, Set-ll]

Answer: (b)

Solution:

Euclidean norm length

Question 13: The following inequality is true for all x close to 0.

What is tha value of [2019 : 1 Mark, Set-ll]

(a) 1

(b) 0

(c) 1/2

(d) 2

Answer: (d)

Solution:

Question 14: Consider the functions: x = y In φ and y = φ In y. which one of the following the correct expression for [2019 : 2 Mark, Set-l]

Answer: (a)

Solution:

......(i)

Question 15: Which one of the following is NOT a correct statement?    [2019 : 2 Marks, Set-I]

(a) The functionhas the global minima at x = e

(b) The function has the global maxima at x = a

(c) The function x³ has neither global minima nor global maxima

(d) The function |x| has the global minima at x = 0

Answer: (a)

Solution:Let y = x^1/x

log y = logx/x

y maximum (or) minimum when,

is maximum (or) minimum

Question 16: For a small value of h, the Taylor series expansion for f(x +h) is [2019 : 1 Mark, Set-I]

Answer:(c)

Solution:Taylor series of f (x + h) at x.

f(x + h) = f(x) + (x + h - x)

Question 17: Which of the following is correct?    [2019 : 1 Mark, Set-I]

Answer: (d)

Solution:

Question 18: The value (up to two decimal places) of a line along C which is a straight line joining (0, 0) to (1, 1) is _____.    [2018 : 2 Marks, Set-II]

Solution:

(0, 0) to (1, 1) line is y = x

Question 19: The value of the integral [2018 : 2 Marks, Set-I]

(d) ∏²

Answer: (b)

Solution: 

Question 20: At the point x = 0, the function f(x) = x³ has    [2018 : 1 Marks, Set-I]

(a) local maximum

(b) local minimum

(c) both local maximum and minimum

(d) neither local maximum nor local minimum

Answer:(d)

Solution:

Question 21: Consider the following definite integral:

The value of the integral is    [2017 : 2 Marks, Set-II]

Answer:(a)

Solution:

Question 22: The tangent to the curve represented by y = x In x is required to have 45° inclination with the x-axis. The coordinates of the tangent point would be    [2017 : 2 Marks, Set-II]

(a) (1,0)

(b) (0,1)

(c) (1,1)

(d)

Answer: (a)

Solution:

tan 45° = In x + 1

1 = lnx + 1

⇒ Inx = 0

∴ x = 1

Putting x = 1 in the eq. of curve, we get y = 0.

Question 23:  The divergence of the vector field V = x²i + 2y³j + z⁴k at x = 1, y = 2, z = 3 is _________.    [2017 : 1 Mark, Set-II]
Solution:


Question 24: Let w= f(x, y), where x and yare functions of t. Then, according to the chain rule dw/dt    [2017 : 1 Mark, Set-II]




Answer: (c)
Solution:W = f(x, y)
By Chain rule,


Question 25: Let x be a continuous variable defined over the interval (-∞, ∞), and The integral is equal to    [2017 : 1 Mark, Set-I]




Answer:(b)
Solution:

Let e^-x = t

Question 26:[2017 : 1 Mark, Set-I]
Solution: (Applying L'Hospital rule)

 = 

Question 27: The quadratic approximation of
f(x) = x³ - 3x² - 5 a the point x = 0 is    [2016 : 2 Marks, Set-II]
(a) 3x² - 6x - 5
(b) -3x² - 5
(c) -3x² + 6x - 5
(d) 3x² - 5
Answer:(b)
Solution:The quadratic approximation of f{x) at the point x = 0 is,



Question 28: The area between the parabola x² = 8y and the straight line y = 8 is______.    [2016 : 2 Marks, Set-II]
Solution:Parabola is x² = 8y
and straight is y = 0
At the point of intersection, we have,

⇒


Question 29: The angle of intersection of the curves x² = 4y and y² = 4x at point (0, 0) is     [2016 : 2 Marks, Set-II]
(a) 0°
(b) 30°
(c) 45°
(d) 90°
Answer:(d)
Solution:Given curve,
x² = 4y .......(i)
and y² = 4x ........(ii)






Question 30: The area of the region bounded by the parabola y = x² + 1 and the straight line x + y = 3 is
(a) 59/6
(b) 9/2
(c) 10/3
(d) 7/6
Answer:(b)
Solution:At the point of intersection of the curves,
y = x² + 1 and x + y = 3 i.e., y = 3 - x , we have,
x² + 1 - 3 - x ⇒ x² + x - 2 = 0
⇒ x = -2, 1 and 3 - x > x² + 1



Question 31: The value of [2016 : 2 Marks, Set-I]
(a) π/2
(b) π
(c) 3π/2
(d) 1
Answer:(b)
Solution:



(Using "division by x")


(Using definition of Laplace transform)
Put s - 0, we get


Question 32: What is the value of [2016 : 1 Mark, Set-II]
(a) 1
(b) -1
(c) 0
(d) Limit does not exit
Answer: (d)
Solution:

(i.e., put x = 0 and then y = 0)

which depends on m.

Question 33: The optimum value of the function f(x) = x² - 4x + 2 is    [2016 : 1 Mark, Set-II]
(a) 2 (maximum)
(b) 2 (minimum)
(c) -2 (maximum)
(d) -2 (minimum)
Answer:(d)
Solution:
f'(x) = 0
⇒ 2x — 4 = 0
⇒x = 2 (stationary point)
f"(x) = 2 > 0

⇒ f(x) is minimum at x = ?
i.e., (2)² - 4(2) + 2 = -2
∴ The optimum value of f(x) is -2 (minimum)

Question 34: While minimizing the function f(x), necessary and sufficient conditions for a point x₀ to be a minima are    [2015 : 1 Mark, Set-II]
(a) f' (x₀) > 0 and f" (x₀) = 0
(b) f'(x₀)< 0 an d f"(x₀) = 0
(c) f' (x₀) = 0 and f" (x₀) < 0
(d) f' (x₀) = 0 and f" (x₀) > 0
Answer: (d)
Solution: f(x) has a local minimum at x = x₀
if f'(x₀) = 0 and f''(x₀) > 0

Question 35:is equal to    [2015 : 1 Mark, Set-II]
(a) e^-2
(b) e
(c) 1
(d) e²
Answer:(d)
Solution:

⇒
Which is in the form of
To convert this into 0/0  form, we rewrite as,
⇒
Now it is in 0/0 form.
Using L’Hospital’s rule,


∴ y = e²

Question 36: The directional derivative of the field u(x, y, z) = x² - 3yz in the direction of the vector at point (2, - 1, 4) is _________.    [2015 : 2 Marks, Set-I]
Solution:


Directional derivative


Question 37: The expression is equal to    [2014 : 2 Marks, Set-II]
(a) ln x
(b) 0
(c) x ln x
(d) ∞
Answer:(a)
Solution:



Question 38:[2014 : 1 Marks, Set-I]
(a) -∞
(b) 0
(c) 1
(d) ∞
Answer: (c)
Solution: Put



Question 39: The value of [2013 : 2 Marks]
(a) 0
(b) 1/15
(c) 1
(d) 8/3
Answer:(b)
Solution:

⇒


Alternative Method:




Question 40: For the parallelogram OPQR shown in the sketch,  The area of the parallelogram is    [2011 : 2 Marks]

(a) ad - bc
(b) ac + bd
(c) ad + bc
(d) ab - cd
Answer:(a)
Solution:

The area of parallelogram OPQR in figure shown above, is the magnitude of the vector product