Engineering Mathematics (XE-A) - GATE XE (Engineering Sciences) Subject-Wise PYQ Practice

Question 1

Let 𝑋 and 𝑌 be two random variables with mean 0, variance 1, and correlation coefficient 1 3 . Then the value of Var(𝑋+ 3𝑌) is equal to

Accepted Answer

12

Answer

9

Answer

10

Answer

11

Question 2

Consider the second order Partial Differential Equation (PDE) 4𝑥2 𝜕2𝑢 𝜕𝑥2 + 4(𝑥+ 𝑦) 𝜕2𝑢 𝜕𝑥𝜕𝑦+ (𝑥2 + 𝑦2) 𝜕2𝑢 𝜕𝑦2 − 𝑢= 0. Then which one of the following statements is correct ?

Accepted Answer

The PDE is hyperbolic in the region {(𝑥, 𝑦) ∈ℝ2: −1 < 𝑥< 0, 𝑦< 0}

Answer

The PDE is hyperbolic in the region {(𝑥, 𝑦) ∈ℝ2: 1 < 𝑥< ∞, 𝑦< 0}

Answer

The PDE is elliptic in the region {(𝑥, 𝑦) ∈ℝ2: 0 < 𝑥< 1, 𝑦> 0}

Answer

The PDE is parabolic in the region {(𝑥, 𝑦) ∈ℝ2: 1 < 𝑥< ∞, 𝑦< 0}

Question 3

Consider the infinite series (𝑃): ∑ 1 (𝑛log 𝑛)1/𝑛 ∞ 𝑛=2 (𝑄): ∑𝑛𝑛 (2𝑛)! ∞ 𝑛=1 Then which one of the following statements is correct ?

Accepted Answer

Series (𝑃) diverges and series (𝑄) converges

Answer

Series (𝑃) and (𝑄) both converge

Answer

Series (𝑃) converges and series (𝑄) diverges

Answer

Series (𝑃) and (𝑄) both diverge

Question 4

Suppose the polynomial 𝑎+ 𝑏𝑥+ 𝑐𝑥2 + 𝑑𝑥3 interpolates the data, (−1, 1), (0, 3), (1, 2) and (2, 4). Then which one of the following statements is correct ?

Accepted Answer

𝑎= −2𝑐, 𝑑= −2𝑏

Answer

𝑎= 2𝑐, 𝑑= 2𝑏

Answer

𝑏= 3𝑐, 𝑎= 2𝑑

Answer

𝑏= 2𝑐, 𝑎= 3𝑑

Question 5

Let 𝐶 be the positively oriented boundary of the domain bounded by the curves 𝑦= 2𝑥2 and 𝑦2 = 4𝑥. Then the value of the line integral ∮(2𝑦2 + 2𝑥𝑦+ 4𝑦)𝑑𝑥+ (𝑥2 + 4𝑥𝑦+ 8𝑥)𝑑𝑦 𝐶 is equal to

Accepted Answer

8 3

Answer

2 3

Answer

4 3

Answer

1 3

Question 6

Let 𝑦(𝑥) be the solution of the initial value problem, 𝑥2𝑦′′ + 𝑥𝑦′ −𝑦= 0, 𝑥> 0, 𝑦(1) = 0, 𝑦′(1) = 2. Then the value of 𝑦′ ( 1 2) is equal to (Answer in integer) _______

Accepted Answer

5

Question 7

Suppose that 2 is an eigenvalue of the matrix [ 0 3 −𝛼 0 1 0 1 −1 3 ] Then the value of 𝛼 is equal to (Answer in integer) __________

Accepted Answer

2

Question 8

Let 𝑓(𝑧) be an analytic function such that 𝑅𝑒(𝑓′(𝑧)) = 3𝑥2 −4𝑦−3𝑦2, 𝑓(𝑖) = 0, 𝑓′(0) = 0, where 𝑖= √−1. Then the value of 𝑓(1) is equal to

Accepted Answer

1 + 5𝑖

Answer

4 + 2𝑖

Answer

1 −𝑖

Answer

4 −2𝑖

Question 9

Consider the function 𝑓(𝑥, 𝑦) = 𝑥2𝑦+ 2𝑥𝑦2 −2𝑥2𝑦2. Then which one of the following statements is correct ?

Accepted Answer

( 3 2 , 3 4) is a saddle point of 𝑓.

Answer

( 3 2 , 0) is a point of local maxima of 𝑓.

Answer

(0, 3 4) is a point of local minima of 𝑓.

Answer

( 3 2 , 3 4) is a point of local maxima of 𝑓.

Question 10

For 𝑎, 𝑏∈ℝ, consider the system of linear equations 𝑥+ 𝑦+ 𝑎𝑧= 2 2𝑦+ 2𝑧= 1 𝑎𝑥+ 2𝑧= 𝑏 If the system has infinitely many solutions, then which of the following statements is/are correct ?

Accepted Answer

𝑎= 2, 𝑏= 3

Accepted Answer

𝑎= −1, 𝑏= − 3 2

Answer

𝑎= 2, 𝑏= 5

Answer

𝑎= 3, 𝑏= 5

Question 11

Let 𝑢(𝑥, 𝑡) be the solution of the initial boundary value problem 𝜕𝑢 𝜕𝑡−𝜕2𝑢 𝜕𝑥2 −𝑢= 0, 0 < 𝑥< 𝜋, 𝑡> 0, 𝑢(𝑥, 0) = 2 sin (3𝑥 2 ) cos (𝑥 2) , 0 < 𝑥< 𝜋, 𝑢(0, 𝑡) = 𝑢(𝜋, 𝑡) = 0, 𝑡> 0. Then the value of lim 𝑡→∞𝑢( 3𝜋 4 , 𝑡) is equal to (rounded off to two decimal places)___________

Accepted Answer

0.68 to 0.73

Question 12

Let 𝑓(𝑥) = {𝜋+ 𝑥,⁡⁡⁡⁡⁡⁡⁡ −𝜋≤𝑥< 0, ⁡⁡⁡0,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡0 ≤𝑥< 𝜋, with 𝑓(𝑥+ 2𝜋) = 𝑓(𝑥). If 𝐹(𝑥) represents the Fourier series of 𝑓(𝑥), then the value of 𝐹(− 𝜋 2) + 𝐹(0) is

Accepted Answer

𝜋

Answer

0

Answer

𝜋 2

Answer

3𝜋 2

Question 13

Let 𝑦 be a non-zero quadratic polynomial satisfying the differential equation (2 + 𝑥2) 𝑑2𝑦 𝑑𝑥2 + 𝑥𝑑𝑦 𝑑𝑥−𝑘𝑦= 0, where 𝑘 is a real constant. If 𝑦(1) = 1, then the value of the integral ∫2𝑦⁡𝑑𝑥 1 0 ⁡ is

Accepted Answer

4 3

Answer

1 3

Answer

2 3

Answer

1

Question 14

There are four cities, namely, 𝐶1, 𝐶2, 𝐶3 and 𝐶4. The cities are directly connected by four roads as shown in the picture given below, that is, 𝐶1 is connected with 𝐶2, 𝐶2 is connected with 𝐶3, 𝐶3 is connected with 𝐶4, and 𝐶4 is connected with 𝐶1. The probability of any road getting independently blocked is 1 3 . Let 𝐸1 be the event of travelling from 𝐶1 to 𝐶3 via 𝐶2 and 𝐸2 be the event of travelling from 𝐶1 to 𝐶3 via 𝐶4. Then, which of the following statements is correct ?

Accepted Answer

𝑃(𝐸1 ∪𝐸2) = 56 81⁡

Answer

𝑃(𝐸1 ∪𝐸2) = 8 9

Answer

𝑃(𝐸1|𝐸2) ≠𝑃(𝐸1) = 4 9

Answer

𝑃(𝐸1 ∩𝐸2) = 0

Question 15

Assume that 𝑓: [0, 1] →ℝ is continuous on [0, 1] and differentiable on (0, 1) such that 𝑓(𝑥+ ℎ) = 𝑓(𝑥) + ℎ𝑓′(𝑥+ 𝜃ℎ) for some 0 < 𝜃< 1. If 𝑓(𝑥) = 𝑥2(1 + 𝑥), and 𝜃 is expressed in terms of 𝑥 and ℎ, then the value of lim ℎ→0 𝜃(𝑥, ℎ) is

Accepted Answer

1 2

Answer

1 3

Answer

1 4

Answer

4 5

Question 16

Let 𝐴 be a 3 × 3 matrix whose eigenvalues are 2, 3, 4 and let 𝐼 be the identity matrix of order 3. If 𝐴−1 = 1 2𝑘(𝐴2 −9𝐴) + 13 𝑘𝐼 for some integer 𝑘≠0, then the value of 𝑘 is ______

Accepted Answer

12

Question 17

For some integer 𝑘, the differential equation 𝑥2 𝑑2𝑦 𝑑𝑥2 −3𝑥𝑑𝑦 𝑑𝑥+ (𝑘+ 2)𝑦= 0 is transformed into (𝐷−2)2𝑦= 0, where 𝐷= 𝑑 𝑑𝑡 and 𝑡= log𝑒𝑥. Then, the value of 𝑘 is _____

Accepted Answer

2

Question 18

The approximate value (rounded off to two decimal places) of the integral ∫𝑒−𝑥2⁡𝑑𝑥 1/2⁡ 0 , using the Trapezoidal rule with step-size ℎ= 1 8 , is ______

Accepted Answer

0.42 to 0.55

Question 19

Consider 𝑓(𝑧) = 𝑒𝑧, where 𝑧= 𝑥+ 𝑖𝑦⁡and 𝑖= √−1. Which of the following statements is correct ?

Accepted Answer

𝑓 is periodic

Answer

𝑓 is not periodic

Answer

|𝑓| = 1

Answer

arg(𝑓) = 𝑦± 𝑛𝜋⁡⁡for⁡all⁡𝑛= 0, 1, 2, …

Question 20

Let 𝑃 and 𝑄 be two square matrices of the same order. Then, which of the following matrices is/are necessarily equal to (𝑃+ 2𝑄)2 ?

Accepted Answer

𝑃(𝑃+ 2𝑄) + 𝑄(2𝑃+ 4𝑄)

Accepted Answer

(𝑃+ 2𝑄)(2𝑄+ 𝑃)

Accepted Answer

𝑃2 + 2𝑃𝑄+ 2𝑄𝑃+ 4𝑄2

Answer

𝑃2 + 4𝑃𝑄+ 4𝑄2

Question 21

If ∫ ∫𝑒𝑦3 1 √𝑥/𝛼⁡ 𝛼 0 𝑑𝑦𝑑𝑥= 𝑒−1, 𝛼> 0, then the value (in integer) of 𝛼 is _____

Accepted Answer

3

Question 22

Consider the vector field 𝐹⁡⃗⃗⃗ = (2𝑥+ 𝑦2)⁡𝑖̂ + (2𝑥𝑦+ 3𝑦)⁡𝑗̂ and 𝛼𝑚= ∫ 𝐹⁡⃗⃗⃗ 𝐶𝑚 ⋅𝑑𝑟 , 𝑚= 1, 2, where 𝐶1 is an arc of the unit circle connecting the points (1, 0) and (0, 1) and 𝐶2 is the straight line connecting the points (1, 0) and (0, 1). Then, the value (in integer) of 2(𝛼1 2 + 3𝛼2 2) is ______

Accepted Answer

2

Question 23

Let 𝐴 be a 3 × 3 real matrix having eigenvalues 1, 2, and 3. If 𝐵= 𝐴2 + 2𝐴+ 𝐼, where 𝐼 is the 3 × 3 identity matrix, then the eigenvalues of 𝐵 are

Accepted Answer

4, 9, 16

Answer

1, 2, 3

Answer

1, 4, 9

Answer

4, 16, 25

Question 24

Let 𝑓: ℝ2 →ℝ be a function defined by 𝑓(𝑥, 𝑦) = { 𝑥𝑦 |𝑥| + 𝑦 , 𝑦≠−|𝑥| 0 , otherwise. Then which one of the following statement is TRUE?

Accepted Answer

𝑓 is NOT continuous at (0,0).

Answer

𝜕𝑓 𝜕𝑥(0,0) = 0, and 𝜕𝑓 𝜕𝑦(0,0) = 1.

Answer

𝜕𝑓 𝜕𝑥(0,0) = 1, and 𝜕𝑓 𝜕𝑦(0,0) = 0.

Answer

𝜕𝑓 𝜕𝑥(0,0) = 1, and 𝜕𝑓 𝜕𝑦(0,0) = 1.

Question 25

If the quadrature formula ∫𝑓(𝑥)𝑑𝑥 1 −1 ≈1 9 (𝑐1𝑓(−1) + 𝑐2𝑓(1 2) + 𝑐3𝑓(1)) is exact for all polynomials of degree less than or equal to 2, then

Accepted Answer

𝑐1 + 𝑐2 4 + 𝑐3 = 6

Answer

𝑐1 + 𝑐2 3 + 𝑐3 = 4

Answer

𝑐1 + 𝑐2 2 + 𝑐3 = 2

Answer

𝑐1 + 𝑐2 + 𝑐3 = 5

Question 26

The second smallest eigenvalue of the eigenvalue problem 𝑑2𝑦 𝑑𝑥2 + (𝜆−3)𝑦= 0, 𝑦(0) = 𝑦(𝜋) = 0, is

Accepted Answer

7

Answer

4

Answer

3

Answer

9

Question 27

Which one of the following functions is differentiable at 𝑧= 0 but NOT differentiable at any other point in the complex plane ℂ?

Accepted Answer

𝑓(𝑧) = 𝑧|𝑧|, 𝑧∈ℂ

Answer

𝑓(𝑧) = sin(𝑧), 𝑧∈ℂ

Answer

𝑓(𝑧) = {𝑒 1 𝑧, 𝑧≠0 0, 𝑧= 0 for 𝑧∈ℂ

Answer

𝑓(𝑧) = 𝑒−𝑧2, 𝑧∈ℂ

Question 28

If the polynomial 𝑃(𝑥) = 𝑎0 + 𝑎1𝑥+ 𝑎2𝑥(𝑥−1) + 𝑎3𝑥(𝑥−1)(𝑥−2) interpolates the points (0, 2), (1, 3), (2, 2), and (3, 5), then the value of 𝑃( 5 2) is _________ (round off to 2 decimal places).

Accepted Answer

2.60 to 2.65

Question 29

The value of 𝑚 for which the vector field 𝐹⃗(𝑥, 𝑦) = (4𝑥𝑚𝑦2 −2 𝑥 𝑦𝑚)𝑖̂ + (2𝑥4𝑦−3𝑥2𝑦2)𝑗̂ is a conservative vector field, is _________________ (in integer).

Accepted Answer

3

Question 30

Let 𝑃= [ 4 −2 2 6 −3 4 3 −2 3 ], and 𝑄= [ 3 −2 2 4 −4 6 2 −3 5 ] . The eigenvalues of both 𝑃 and 𝑄 are 1, 1, and 2. Which one of the following statements is TRUE?

Accepted Answer

𝑃 is diagonalizable but 𝑄 is NOT diagonalizable

Answer

Both 𝑃 and 𝑄 are diagonalizable

Answer

𝑃 is NOT diagonalizable but 𝑄 is diagonalizable

Answer

Both 𝑃 and 𝑄 are NOT diagonalizable

Question 31

The surface area of the portion of the paraboloid 𝑧= 𝑥2 + 𝑦2 that lies between the planes 𝑧= 0 and 𝑧= 1 4 is

Accepted Answer

𝜋 6 (2√2 −1)

Answer

𝜋 2 (2√2 −1)

Answer

𝜋(2√2 −1)

Answer

𝜋 3 (2√2 −1)

Question 32

The probability of a person telling the truth is 4 6 . An unbiased die is thrown by the same person twice and the person reports that the numbers appeared in both the throws are same. Then the probability that actually the numbers appeared in both the throws are same is __________ (round off to 2 decimal places).

Accepted Answer

0.26 to 0.30

Question 33

Let 𝑢(𝑥, 𝑡) be the solution of the initial boundary value problem 𝜕𝑢 𝜕𝑡− 𝜕2𝑢 𝜕𝑥2 = 0, 𝑥∈(0, 2), 𝑡> 0 𝑢(𝑥, 0) = sin(𝜋𝑥), 𝑥∈(0, 2) 𝑢(0, 𝑡) = 𝑢(2, 𝑡) = 0. Then the value of 𝑒𝜋2 (𝑢( 1 2 , 1) −𝑢( 3 2 , 1)) is __________ (in integer).

Accepted Answer

2

Question 34

The value of lim ௫→଴ 1 𝑥 න ቀ𝑡൅ ඥ𝑡ଶ൅5 ቁ 𝑑𝑡 ଶା௫ ଶ is

Accepted Answer

5

Answer

0

Answer

4

Answer

6

Question 35

Let Χ ൌ ൛𝑧ൌ𝑥൅𝑖𝑦∶𝑥and 𝑦are real numbers, 𝑖ൌ√െ1 ൟ be the set of complex numbers. Let the function 𝑓ሺ𝑧ሻൌ𝑢ሺ𝑥, 𝑦ሻ൅𝑖 𝑣ሺ𝑥, 𝑦ሻ for 𝑧ൌ𝑥൅𝑖𝑦 Î Χ be analytic in Χ, where 𝑢ሺ𝑥, 𝑦ሻൌ𝑥 𝑦ଷെ𝑦 𝑥ଷ and 𝑣ሺ𝑥, 𝑦ሻൌ 𝑥ସ 4 ൅ 𝑦ସ 4 െ 3 2 𝑥ଶ𝑦ଶ. If 𝑓ᇱሺ𝑧ሻ denotes the derivative of 𝑓ሺ𝑧ሻ, then

Accepted Answer

|𝑓ᇱሺെ1 ൅𝑖ሻ|ଶൌ8

Answer

|𝑓ᇱሺെ1 ൅𝑖ሻ|ଶൌ1

Answer

|𝑓ᇱሺെ1 ൅𝑖ሻ|ଶൌ7

Answer

|𝑓ᇱሺെ1 ൅𝑖ሻ|ଶൌ10

Question 36

If the partial differential equation ሺ𝑥൅2ሻ𝜕ଶ𝑢 𝜕𝑥ଶ൅2ሺ𝑥൅𝑦ሻ𝜕ଶ𝑢 𝜕𝑥 𝜕𝑦൅2ሺ𝑦െ1ሻ𝜕ଶ𝑢 𝜕𝑦ଶെ3𝑦ଶ 𝜕𝑢 𝜕𝑦ൌ0 is parabolic on the circle ሺ𝑥െ𝑎ሻଶ൅ሺ𝑦െ𝑏ሻଶൌ𝑟ଶ, then the values of 𝑎, 𝑏 and 𝑟 are given by

Accepted Answer

𝑎ൌെ1, 𝑏ൌ2, 𝑟ൌ1

Answer

𝑎ൌ1, 𝑏ൌ2, 𝑟ൌ1

Answer

𝑎ൌ1, 𝑏ൌെ2, 𝑟ൌ1

Answer

𝑎ൌെ1, 𝑏ൌെ2, 𝑟ൌ1

Question 37

Let G be the positively oriented circle 𝑥ଶ൅𝑦ଶൌ9 in the 𝑥𝑦-plane. If ර൫3𝑦൅𝑒௫ୱ୧୬௫൯ 𝑑𝑥൅ ൫7𝑥൅ √𝑒௬൅2 ൯ 𝑑𝑦 ൌ𝛼 𝜋 G , where 𝛼 is a real constant, then 𝛼 is equal to ________.

Accepted Answer

36

Question 38

Let 𝑦ଵሺ𝑥ሻ and 𝑦ଶሺ𝑥ሻ be two linearly independent solutions of 𝑥ଶ 𝑑ଶ𝑦 𝑑𝑥ଶെ 2𝑥 𝑑𝑦 𝑑𝑥൅2𝑦ൌ0, 𝑥 ൐0. Let 𝑊ሺ𝑦ଵ, 𝑦ଶሻሺ𝑥ሻ denote the Wronskian of 𝑦ଵሺ𝑥ሻ and 𝑦ଶሺ𝑥ሻ at 𝑥. If 𝑊ሺ𝑦ଵ, 𝑦ଶሻሺ1ሻൌ1 then 𝑊ሺ𝑦ଵ, 𝑦ଶሻሺ2ሻ is equal to _______.

Accepted Answer

4

Question 39

Let 𝐴ൌ ቎ 2 0 1 2 1 1 5 െ5 0 0 0 0 3 0 1 3 ቏. Then the sum of the geometric multiplicities of the distinct eigenvalues of 𝐴 is equal to ________.

Accepted Answer

2

Question 40

In a cosmopolitan city, the population comprises of 30% female and 70% male. Suppose that 5% of female and 30% of male in the population are foreigners. A person is selected at random from this population. Given that the selected person is a foreigner, the probability that the person is a female is ________ (round off to three decimal places).

Accepted Answer

0.066 to 0.068

Question 41

Let 𝑓: ሺ0, ∞ሻ→ℝ be the continuous function such that 𝑓ሺ𝑥ሻൌ2 ൅ ௚ሺ௫ሻ ௫ for all 𝑥൐0, where 𝑔ሺ𝑥ሻൌ ׬ 𝑓ሺ𝑡ሻ ௫ ଵ 𝑑𝑡 for all 𝑥൐0. Then 𝑓ሺ2ሻ is equal to

Accepted Answer

2 ൅ log௘4

Answer

2 ൅ log௘2

Answer

2 െ log௘2

Answer

2 െ log௘4

Question 42

Let 𝐴 and 𝐵 be 𝑛ൈ𝑛 matrices with real entries. Consider the following statements: P: If 𝐴 is symmetric then 𝑟𝑎𝑛𝑘ሺ𝐴ሻൌ Number of nonzero eigenvalues (counting multiplicity) of 𝐴. Q: If 𝐴𝐵ൌ𝟎 then 𝑟𝑎𝑛𝑘ሺ𝐴ሻ൅𝑟𝑎𝑛𝑘ሺ𝐵ሻ൑𝑛. Then

Accepted Answer

both P and Q are TRUE

Answer

P is TRUE and Q is FALSE

Answer

P is FALSE and Q is TRUE

Answer

both P and Q are FALSE

Question 43

Let 𝑓: ℝଶ→ℝ be given by 𝑓ሺ𝑥, 𝑦ሻൌ4𝑥𝑦െ2𝑥ଶെ𝑦ସ൅1. The number of critical points where 𝑓 has local maximum is equal to ________.

Accepted Answer

2

Question 44

If the quadrature rule න𝑓ሺ𝑥ሻ𝑑𝑥 ଵି ଵ ൎ𝑓ሺ𝛼ሻ൅ 𝛾 𝑓ሺ𝛽ሻ, where 𝛼, 𝛽 and 𝛾 are real constants, is exact for all polynomials of degree ൑3, then 𝛾൅3 ሺ𝛼ଶ൅ 𝛽ଶሻ൅ሺ𝛼ଷ൅𝛽ଷሻ is equal to _______.

Accepted Answer

3

Question 45

The largest interval in which the initial value problem ( ) ( ) () () 2 2 1 ln , (1) = 0 and 1 1, 5 x d y dy dy e x y x y dx x dx dx + + = = − has a unique solution is

Accepted Answer

(0, 5)

Answer

( , ) −∞∞

Answer

( 5, 5) −

Answer

(0, ) ∞

Question 46

The sum of the roots of the indicial equation at 0 x = of the differential equation ( ) 2 3 2 sin (tan ) 0, 0, d y dy x x x x y x dx dx + − = > is

Accepted Answer

0

Answer

1

Answer

2

Answer

2 −

Question 47

Let f be a three times continuously differentiable real valued function on (0, 5) such that its third derivative 1 ( ) 100 f x ′′′ = for all (0,5). x∈ If ( ) P x is a polynomial of degree 2 ≤ such that (1) (1), P f = (2) (2) P f = and (3) (3) P f = then | (4) (4) | f P − equals _______________

Accepted Answer

0.01

Question 48

For real numbers 1 α and 2, α if the formula () () 1 1 1 1 2 2 2 1 ( ) f x dx f f α α − − = + ∫ is exact for all polynomials of degree 1 ≤ then 1 2 2 3 α α + equals _______________

Accepted Answer

5

Question 49

Raju has four fair coins and one fair dice. At first Raju tosses a coin. If the coin shows head then he rolls the dice and the number that dice shows is taken as his score. If the coin shows tail then he tosses three more coins and the total number of tails shown (including the first one) is taken as his score. If Raju tells that his score is 2 then the probability that he rolled the dice is (up to two decimal places) _______________

Accepted Answer

0.29 to 0.32

Question 50

Let f be a continuously differentiable real valued function defined by 2 1, ( ) 5 1. bx a if x f x x if x + < = ≥ Then the value of 2 a b is _______________

Accepted Answer

250

Question 51

A rectangular box without top cover having a square base is to be made from a sheet of 108 square meters. Then the largest possible volume of the box in cubic meters is ______________

Accepted Answer

108

Question 52

Let 5 3 . 6 4 A − = − Then the trace of 1000 A equals

Accepted Answer

1000 2 1 +

Answer

1000 2 1 −

Answer

1

Answer

1000 2

Question 53

Let denote the set of complex numbers and 2 1. i =− Let γ be the simple positively oriented circle | | 1 z = and { } 0 | | 2 S z z = ∈ < < . If : f S → is analytic in S and is given by 2 0 1 7 ( ) , 8 2 n n n f z a z z S z z ∞ = = + + ∈ ∑ then the value of the contour integral 1 ( ) cos ze f z dz i z γ π + ∫ is

Accepted Answer

7

Answer

0

Answer

1 8

Answer

7 2

Question 54

Let 3 denote the three dimensional Euclidean space and ˆ ˆ ˆ ( , , ) x y z yi x j z k =− + + F for all 3 ( , , ) . x y z ∈ If C is the curve described by the parametric equation 2 ˆ ˆ ˆ ( ) cos sin 2 , 0 1, t t i t j t k t = + + ≤≤ r then the value of the line integral d C ⋅ ∫F r is_______________

Accepted Answer

3

Question 55

Let ( , ) u x t satisfy the initial and boundary value problem 2 2 2 , 0 , 0, u u x t t x π ∂ ∂ = < < > ∂ ∂ (0, ) 0 ( , ), 0, u t u t t π = = > ( ,0) sin 2sin 4 , 0 . u x x x x π = + < < Then the value of , ln(5) 2 u π is_______________

Accepted Answer

0.04

Question 56

A company records heights of all employees. Let X and Y denote the errors in the average height of male and female employees respectively. Assume that 𝑋𝑋 ~ 𝑁𝑁(0, 4) and 𝑌𝑌 ~ 𝑁𝑁(0, 9) and they are independent. Then the distribution of 𝑍𝑍= (𝑋𝑋+ 𝑌𝑌)/2 is

Accepted Answer

𝑁𝑁(0, 3.25)

Answer

𝑁𝑁(0, 6.5)

Answer

𝑁𝑁(0, 2)

Answer

𝑁𝑁(0, 1)

Question 57

The volume of the solid obtained by revolving the curve 𝑦𝑦2 = 𝑥𝑥 , 0 ≤𝑥𝑥≤1 around 𝑦𝑦-axis is

Accepted Answer

𝜋𝜋 5

Answer

𝜋𝜋

Answer

2

Answer

𝜋𝜋 2

Question 58

Let 𝑦𝑦(𝑥𝑥) be the solution of the initial value problem 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑+ 2𝑥𝑥𝑥𝑥= 𝑥𝑥; 𝑦𝑦(0) = 0. Find the value of lim𝑥𝑥 → ∞𝑦𝑦(𝑥𝑥).

Accepted Answer

0.5

Question 59

Which of the following is a quasi-linear partial differential equation?

Accepted Answer

𝜕𝜕2𝑢𝑢 𝜕𝜕 𝑡𝑡2 + 𝑢𝑢2 = 0

Answer

ቀ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕ቁ 2 + 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕= 0

Answer

ቀ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕ቁ 2 − ቀ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕ቁ 2 = 0

Answer

ቀ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕ቁ 4 − ቀ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕ቁ 3 = 0

Question 60

Let 𝑃𝑃(𝑥𝑥) and 𝑄𝑄(𝑥𝑥) be the polynomials of degree 5, generated by Lagrange and Newton interpolation methods respectively, both passing through given six distinct points on the 𝑥𝑥𝑥𝑥-plane. Which of the following is correct?

Accepted Answer

𝑃𝑃(𝑥𝑥) ≡𝑄𝑄(𝑥𝑥)

Answer

𝑃𝑃(𝑥𝑥) − 𝑄𝑄(𝑥𝑥) is a polynomial of degree 1

Answer

𝑃𝑃(𝑥𝑥) − 𝑄𝑄(𝑥𝑥) is a polynomial of degree 2

Answer

𝑃𝑃(𝑥𝑥) − 𝑄𝑄(𝑥𝑥) is a polynomial of degree 3

Question 61

The Laurent series of 𝑓𝑓(𝑧𝑧) = 1/(𝑧𝑧3 − 𝑧𝑧4) with center at 𝑧𝑧= 0 in the region |𝑧𝑧| > 1 is

Accepted Answer

−∑ 1 𝑧𝑧𝑛𝑛+4 ∞ 𝑛𝑛=0

Answer

∑ 𝑧𝑧𝑛𝑛−3 ∞ 𝑛𝑛=0

Answer

∑ 𝑧𝑧𝑛𝑛 ∞ 𝑛𝑛=0

Answer

∑ 1 𝑧𝑧𝑛𝑛 ∞ 𝑛𝑛=0

Question 62

The value of the surface integral F dS Γ ⋅ ∫∫ n υρ over the sphere Γ given by 𝑥𝑥2 + 𝑦𝑦2 + 𝑧𝑧2 = 1, where ˆ ˆ = 4 F x i - z k ρ , and ndenotes the outward unit normal, is

Accepted Answer

4 𝜋𝜋

Answer

𝜋𝜋

Answer

2 𝜋𝜋

Answer

3 𝜋𝜋

Question 63

A diagnostic test for a certain disease is 90% accurate. That is, the probability of a person having (respectively, not having) the disease tested positive (respectively, negative) is 0.9. Fifty percent of the population has the disease. What is the probability that a randomly chosen person has the disease given that the person tested negative?

Accepted Answer

0.099 to 0.101

Question 64

Let 𝑀𝑀= ቀ1 1 0 1ቁ. Which of the following is correct?

Accepted Answer

1 is the only eigenvalue and 𝑀𝑀 is not diagonalizable

Answer

Rank of 𝑀𝑀 is 1 and 𝑀𝑀 is not diagonalizable

Answer

Rank of 𝑀𝑀 is 2 and 𝑀𝑀 is diagonalizable

Answer

1 is the only eigenvalue and 𝑀𝑀 is diagonalizable

Engineering Mathematics (XE-A) - GATE XE (Engineering Sciences) Subject-Wise PYQ Practice

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