GATE Engineering Mathematics questions with solutions

Q1. The smallest positive root of the equation x⁵ - 5x⁴ - 10x³ + 50x² + 9x - 45 = 0 lies in the range

1. 0 < x ≤ 2
2. 2 < x ≤ 4
3. 6 ≤ x ≤ 8
4. 10 ≤ x ≤ 100

Answer: 0 < x ≤ 2

The correct option is right because evaluating the polynomial at various points shows that it changes sign between 0 and 2, indicating a root exists in that interval. Specifically, the function is negative at 0 and positive at 2, confirming the presence of a root in the specified range.

Q2. The second-order differential equation in an unknown function u: u(x,y) is defined as ∂²u/∂x² = 2. Assuming g: g(x), f: f(y), and h: h(y), the general solution of the above differential equation is

1. u = x² + f(y) + g(x)
2. u = x² + x f(y) + h(y)
3. u = x² + x f(y) + g(x)
4. u = x² + f(y) + y g(x)

Answer: u = x² + x f(y) + h(y)

Integrating d2u/dx2 = 2 twice with respect to x gives u = x^2 + x*f(y) + h(y), where the two integration 'constants' are arbitrary functions of y. This is option idx 1; the stored idx 3 with y*g(x) is incorrect.

Q3. What are the eigenvalues of the matrix [2, 1, 1; 1, 4, 1; 1, 1, 2]?

1. 1, 2, 5
2. 1, 3, 4
3. −5, 1, 2
4. −5, −1, 2

Answer: 1, 2, 5

The eigenvalues of a matrix are the solutions to its characteristic polynomial, which in this case yields the values 1, 2, and 5. These values indicate the scaling factors for the eigenvectors associated with the matrix.

Q4. A vector field p and a scalar field r are given by p = (2x² - 3xy + z²) î + (2y² - 3yz + x²) ĵ + (2z² - 3xz + x²) k̂ r = 6x² + 4y² - z² - 9xyz - 2xy + 3xz - yz Consider the statements 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 p is zero. Which one of the following options is CORRECT?

1. Both P and Q are FALSE
2. P is TRUE and Q is FALSE
3. P is FALSE and Q is TRUE
4. Both P and Q are TRUE

Answer: Both P and Q are TRUE

The curl of the gradient of any scalar field is always zero, confirming statement P as true. Additionally, the divergence of the curl of any vector field is also zero, validating statement Q as true.

Q5. The sum of the following infinite series is: 1/1! + 1/2! + 1/3! + 1/4! + 1/5! +...

1. π
2. 1 + e
3. e − 1
4. e

Answer: e − 1

The series represents the Taylor expansion of the exponential function e^x evaluated at x=1, which sums to e. However, since the series starts from 1/1! instead of 1/0!, we subtract 1 from e, resulting in e - 1.

Q6. A circle with center at (x,y) = (0.5, 0) and radius = 0.5 intersects with another circle with center at (x,y) = (1,1) and radius = 1 at two points. One of the points of intersection (x,y) is:

1. (0,0)
2. (0.2,0.4)
3. (0.5,0.5)
4. (1,2)

Answer: (0.2,0.4)

The point (0.2, 0.4) lies on both circles, satisfying their equations, which confirms it as a point of intersection. The other options either do not satisfy the equations of both circles or lie outside their respective radii.

Q7. Suppose λ is an eigenvalue of matrix A and x is the corresponding eigenvector. Let x also be an eigenvector of the matrix B = A − 2I, where I is the identity matrix. Then, the eigenvalue of B corresponding to the eigenvector x is equal to

1. λ
2. λ + 2
3. 2λ
4. λ − 2

Answer: λ − 2

If x is an eigenvector of A corresponding to eigenvalue λ, then Ax = λx. For the matrix B = A - 2I, we have Bx = (A - 2I)x = Ax - 2Ix = λx - 2x = (λ - 2)x. This shows that x is also an eigenvector of B with eigenvalue λ - 2.

Q8. Let A = [[1, 1], [1, 3], [−2, −3]] and b = [b1, b2, b3]. For Ax = b to be solvable, which one of the following options is the correct condition on b1, b2, and b3:

1. b1 + b2 + b3 = 1
2. 3b1 + b2 + 2b3 = 0
3. b1 + 3b2 + b3 = 2
4. b1 + b2 + b3 = 2

Answer: 3b1 + b2 + 2b3 = 0

The correct option is based on the requirement for the system of equations represented by Ax = b to have a solution. This condition arises from the need for the vector b to lie in the column space of matrix A, which is determined by the linear combinations of its columns. The equation 3b1 + b2 + 2b3 = 0 ensures that b can be expressed as a linear combination of the columns of A.

Q9. Which one of the following options is the correct Fourier series of the periodic function f(x) described below: f(x) = { 0 if −2 < x < −1 { 2k if −1 < x < 1; period = 4 { 0 if 1 < x < 2

1. f(x) = k/2 + (2k/π) (cos(πx/2) − (1/3) cos(3πx/2) + (1/5) cos(5πx/2) + …)
2. f(x) = k/2 + (2k/π) (sin(πx/2) − (1/3) sin(3πx/2) + (1/5) sin(5πx/2) + …)
3. f(x) = k + (4k/π) (cos(πx/2) − (1/3) cos(3πx/2) + (1/5) cos(5πx/2) + …)
4. f(x) = k + (4k/π) (sin(πx/2) − (1/3) sin(3πx/2) + (1/5) sin(5πx/2) + …)

Answer: f(x) = k + (4k/π) (cos(πx/2) − (1/3) cos(3πx/2) + (1/5) cos(5πx/2) + …)

The pulse has height 2k on -1<x<1 with period 4, so the DC term is (1/4)*2k*2 = k, and a_n = (4k/(n*pi))*sin(n*pi/2) giving 4k/pi for n=1. Since f is even the series uses cosines: f = k + (4k/pi)(cos(pi x/2) - (1/3)cos(3pi x/2) + ...). That is option C, not the stored choice.

Q10. For a flowing fluid, a dimensionless combination of velocity (V), length scale (l), and acceleration due to gravity (g) would be

1. V²/(gl)
2. Vg/l
3. gl²/V
4. l/(V² g)

Answer: V²/(gl)

The correct option, V²/(gl), is dimensionless because it combines the units of velocity squared (L²/T²) with the units of gravity (L/T²) and length (L), resulting in a ratio that cancels out all dimensions, leaving a pure number.

Q11. If the quadrantal bearing of a line is N 30° W, then the whole circle bearing of the line is

1. 120°
2. 210°
3. 300°
4. 330°

Answer: 330°

Quadrantal bearing N 30 W is measured 30 degrees toward west from north, so the whole circle bearing = 360 - 30 = 330 degrees. The stored 300 deg is incorrect.

Q12. Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:

1. ∂²u/∂t² = c²(∂²u/∂x² + ∂²u/∂y²) + xy
2. ∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² = 0
3. ∂u/∂t = c ∂u/∂x
4. (∂²u/∂t²)² = c² ∂²u/∂x²

Answer: ∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² = 0

The correct option is a second-order, linear, homogeneous partial differential equation because it consists of second derivatives of the function u, is linear in u and its derivatives, and equals zero, indicating that there are no additional terms that would make it non-homogeneous.

Q13. The value of lim_(x→∞) (x − √(x² + x)) is equal to

1. -1
2. -0.5
3. -2
4. 0

Answer: -0.5

As x approaches infinity, the expression can be simplified by factoring out x from the square root, leading to a limit that evaluates to -0.5.

Q14. Organic fraction of municipal solid waste (OFMSW) with bulk density of 315 kg/m³ and water content of 30% is mixed with municipal sludge of bulk density 700 kg/m³ and water content of 70%, such that the water content of the mixture is 40%. The amount (in kg) of sludge to be mixed per kg of OFMSW (rounded off to 2 decimal places) and the density of the mixture (in kg/m³) (rounded off to the nearest integer) are calculated. Which of the following options is/are true:

1. 0.33 kg of sludge added per kg of OFMSW
2. Density of the mixture is 365 kg/m³
3. 0.66 kg of sludge added per kg of OFMSW
4. Density of the mixture is 450 kg/m³

Answer: 0.33 kg of sludge added per kg of OFMSW

The correct option is accurate because the calculations for the mixture's water content and the proportions of OFMSW and sludge lead to the conclusion that 0.33 kg of sludge is needed for every kg of OFMSW to achieve the desired water content of 40%.

Q15. The solution of the equation x dy/dx + y = 0 passing through the point (1,1) is

1. x
2. x²
3. x⁻¹
4. x⁻²

Answer: x⁻¹

The equation is a separable differential equation that can be rearranged and integrated to yield the solution y = C/x. Given the point (1,1), we find that C = 1, leading to the solution y = 1/x, which is equivalent to x⁻¹.

Q16. A probability distribution with right skew is shown in the figure. The correct statement for the probability distribution is

1. Mean is equal to mode
2. Mean is greater than median but less than mode
3. Mean is greater than median and mode
4. Mode is greater than median

Answer: Mean is greater than median and mode

In a right-skewed distribution, the tail on the right side pulls the mean to the right, making it greater than both the median and the mode, which are closer to the peak of the distribution.

Q17. The matrix [2, -4; 4, -2] has

1. real eigenvalues and eigenvectors
2. real eigenvalues but complex eigenvectors
3. complex eigenvalues but real eigenvectors
4. complex eigenvalues and eigenvectors

Answer: complex eigenvalues and eigenvectors

The matrix has a negative determinant and a non-zero trace, which indicates that its eigenvalues are complex. Consequently, the eigenvectors associated with these complex eigenvalues are also complex.

Q18. The Laplace transform F(s) of the exponential function, f(t)=e^(at) when t≥0, where a is a constant and (s−a)>0, is

1. 1/(s+a)
2. 1/(s−a)
3. 1/(a−s)
4. ∞

Answer: 1/(s−a)

The Laplace transform of the function f(t)=e^(at) is derived by integrating e^(at) multiplied by e^(-st) over the interval from 0 to infinity, resulting in the expression 1/(s-a) when the condition (s-a)>0 is satisfied.

Q19. The rank of the following matrix is [1, 1, 0, -2; 2, 0, 2, 2; 4, 1, 3, 1]

1. 1
2. 2
3. 3
4. 4

Answer: 2

With R2->R2-2R1 = [0,-2,2,6] and R3->R3-4R1 = [0,-3,3,9], we find R3' = 1.5*R2', so the third row is linearly dependent. Only two independent rows remain, so the rank is 2.

Q20. Euclidean norm (length) of the vector [4 -2 -6]^T is

1. √12
2. √24
3. √48
4. √56

Answer: √56

The Euclidean norm is calculated as the square root of the sum of the squares of the vector's components. For the vector [4, -2, -6], this results in √(4² + (-2)² + (-6)²) = √(16 + 4 + 36) = √56.

Q21. The Laplace transform of sinh(at) is

1. a/(s² − a²)
2. a/(s² + a²)
3. s/(s² − a²)
4. s/(s² + a²)

Answer: a/(s² − a²)

The Laplace transform of sinh(at) is a/(s^2 - a^2), valid for s > |a|. The option s/(s^2 - a^2) is the transform of cosh(at), not sinh(at).

Q22. The following inequality is true for all x close to 0. 2 − x²/3 < x sin x/(1 − cos x) < 2 What is the value of lim(x→0) x sin x/(1 − cos x) ?

1. 0
2. 1/2
3. 1
4. 2

Answer: 2

As x approaches 0, the expression x sin x/(1 - cos x) simplifies to 2, which can be confirmed using L'Hôpital's rule or Taylor series expansion, showing that the limit converges to 2.

Q23. What is curl of the vector field 2x²y i + 5z² j − 4yzk ?

1. 6z i + 4x j − 2x² k
2. 6z i − 8xy j + 2x² y k
3. −14z i + 6y j + 2x² k
4. −14z i − 2x² k

Answer: −14z i − 2x² k

The curl of a vector field measures the rotation at a point in the field. In this case, calculating the curl of the given vector field results in the components that match the correct option, indicating that the field has a specific rotational behavior characterized by the terms involving z and x.

Q24. A closed thin-walled tube has thickness, t, mean enclosed area within the boundary of the centreline of tube’s thickness, Aₘ, and shear stress, τ. Torsional moment of resistance, T, of the section would be

1. 0.5 τ Aₘ t
2. τ Aₘ t
3. 2 τ Aₘ t
4. 4 τ Aₘ t

Answer: 2 τ Aₘ t

The torsional moment of resistance for a closed thin-walled tube is derived from the shear stress acting over the mean enclosed area and the thickness of the tube. The factor of 2 accounts for the contribution of shear stress across both walls of the tube, leading to the formula T = 2 τ Aₘ t.

Q25. The probability density function of a continuous random variable distributed uniformly between x and y (for y > x) is

1. 1/(x - y)
2. 1/(y - x)
3. x - y
4. y - x

Answer: 1/(y - x)

The probability density function for a uniform distribution is defined as the reciprocal of the range of the distribution, which is (y - x). This ensures that the total area under the density function equals 1, satisfying the properties of a probability distribution.

Q26. Consider the hemi-spherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m³) when the depth of water at the centre of the tank is 6 m?

1. 78π
2. 156π
3. 396π
4. 468π

Answer: 396π

For a hemispherical bowl R=13 with water depth h=6, cap volume = pi*h^2*(3R-h)/3 = pi*36*(39-6)/3 = pi*36*11 = 396*pi. Stored 156*pi is wrong; correct is 396*pi.

Q27. An ordinary differential equation is given below. (dy/dx)(x ln x) = y The solution for the above equation is (Note: K denotes a constant in the options)

1. y = Kx ln x
2. y = Kxe^x
3. y = Kxe^(-x)
4. y = K ln x

Answer: y = K ln x

From (x ln x) dy/dx = y, separate to dy/y = dx/(x ln x). The right side integrates to ln|ln x|, so ln y = ln(ln x) + C, giving y = K ln x. The stored answer y = Kx ln x is wrong.

Q28. Q.10 In an equilateral triangle PQR, side QR is divided into six equal parts. The length of each subdivided part is 4. The minimum area of the triangle is

1. 18
2. 24
3. 48√3
4. 144√3

Answer: 144√3

The area of an equilateral triangle can be calculated using the formula A = (sqrt(3)/4) * a², where 'a' is the length of a side. Since each subdivided part of side QR is 4, the total length of side QR is 6 * 4 = 24, leading to an area of (sqrt(3)/4) * 24² = 144√3.

Q29. If k is a constant, the general solution of dy/dx - y/x = 1 will be in the form of

1. y = kx
2. y = kx + x ln x
3. y = kx + 1
4. y = kx + x

Answer: y = kx + x ln x

The correct option is derived from solving the first-order linear differential equation using an integrating factor, which leads to a solution that includes a term involving the natural logarithm of x, specifically x ln x, in addition to the homogeneous solution.

Q30. Consider the following equations of straight lines: Line L1: 2x - 3y = 5 Line L2: 3x + 2y = 8 Line L3: 4x - 6y = 5 Line L4: 6x - 9y = 6 Which one among the following is the correct statement?

1. L1 is parallel to L2 and L1 is perpendicular to L3
2. L2 is parallel to L4 and L2 is perpendicular to L1
3. L3 is perpendicular to L4 and L3 is parallel to L2
4. L4 is perpendicular to L2 and L4 is parallel to L3

Answer: L4 is perpendicular to L2 and L4 is parallel to L3

Line L4 is perpendicular to L2 because the product of their slopes equals -1, indicating they intersect at a right angle. Additionally, L4 is parallel to L3 since both lines have the same slope, confirming they never intersect.