COURSE: STATISTICS, TRANSFORMS & NUMERICAL METHODS

COURSE CODE: 22MAT31B

MODULE – 1: CURVE FITTING & STATISTICS
Q.No Questions Marks CO’s BL
1. Fit the best possible curve of the form 𝑦 = 𝑎 𝑥 + 𝑏, using method of 6 1 1
least square for the data
x 1 3 4 6 8 9 11 14
y 1 2 4 4 5 7 8 9
2. Find the linear law 𝑃 = 𝑚 𝑊 + 𝑐 6 1 1
W 50 70 100 120
P 12 15 21 25

3. Fit a parabola 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 to the data 6 1 1

x 1 2 3 4 5
y 10 12 13 16 19

4. a) The revolution (r) and time (t) are related by quadratic 7 1 2

polynomial 𝑟 = 𝑎𝑡 2 + 𝑏𝑡 + 𝑐. Estimate the number of revolution for
time 3.5 units, given
t 1.2 1.6 1.9 2.1 2.4 2.6 3
r 5 10 15 20 25 30 35

5. Fit a curve of the form 𝑦 = 𝑎 𝑏 𝑥 for the data 7 2 2

x 2 3 4 6 10
y 50 47 46 45 44

6. Fit a curve of the form 𝑦 = 𝑎 𝑏 𝑥 for the data 7 2 2

x 10 20 30 40 50
y 8 10 15 21 30

7. Fit a curve of the form 𝑦 = 𝑎 𝑒 𝑏𝑥 for the data 7 3 3

x 5 6 7 8 9 10
y 133 55 23 7 2 2

8. Fit a curve of the form 𝑦 = 𝑎 𝑒 𝑏𝑥 for the data 7 3 3

x 0 1 2 3 4
y 1 1.8 1.3 2.5 2.3

9. Fit a least square geometric curve 𝑦 = 𝑎 𝑥 𝑏 for the following data 7 3 3

x 1 2 3 4 5 6
y 1 3 2 4 3 5

10. An experiment on lifetime 𝑡 of cutting tools at different speed 𝑣 (units) is 7 3 4

given below. Fit a least square geometric curve of the form 𝑣 = 𝑎 𝑡 𝑏
Speed(v) 350 400 500 600
Life (t) 61 26 7 2.6

1
11. 𝜎𝑥 2 + 𝜎𝑦 2 – 𝜎𝑥−𝑦 2 6 4 4
Establish the formula 𝑟 = 2 𝜎𝑥 𝜎𝑦

12. Calculate the correlation co-efficient for the following heights in inches of 6 4 3
fathers (x) and their sons (y).
x 65 66 67 67 68 69 70 72
y 67 68 65 68 72 72 69 71

13. Find the co-efficient of correlation between industrial production and 6 4 4

export using the following data and comment on the result.
Production (in crore tons) 55 56 58 59 60 60 62
Exports(in crore tons) 35 38 38 39 44 43 45

14. Obtain the regression lines of y on x and x on y for the following data: 7 4 3
𝑥 1 2 3 4 5
𝑦 2 5 3 8 7

15. Obtain the regression lines of y on x and x on y for the following data: 7 4 3
x 2 4 6 8 10
y 5 7 9 8 11

16. Given 𝑟 = 0.8, write down the equation of the lines of regression and 7 4 4
hence find the most probable value of y when x =70
𝑥 𝑦
Mean 18 100
S.D 14 20
17. The following results were obtained from records of age(x) and blood 7 4 4
pressure (y) of a group of 10 men, given Σ (𝑥 − 𝑥̅ )(𝑦 − 𝑦̅) = 1220. Find
the appropriate regression equation and use it to estimate the blood
pressure of a man whose age is 45
𝑥 𝑦
Mean 53 142
Variance 130 165

18. In a partially destroyed laboratory record of correlation data, the 7 4 4

following result only are available, variance of x is 9, regression equation
y on x and x on y are 4𝑥 − 5𝑦 + 33 = 0, 20𝑥 − 9𝑦 − 107 = 0
respectively. Calculate the coefficient of correlation, 𝑥̅ , 𝑦̅ and 𝜎𝑦

19. If θ is the acute angle between the two regression lines relating the 6 5 5
𝜎𝑥 𝜎𝑦 1−𝑟 2
variables x and y, show that tan 𝜃 = 𝜎 2 +𝜎 2
( )
𝑥 𝑦 𝑟

20. Find the co-efficient of correlation between x and y given 2𝜎𝑥 = 𝜎𝑦 and 6 5 5
3
the angle between the lines of regression is tan−1 (5)

2
 COURSE: NUMERICAL METHODS FOR ODE AND PDE
COURSE CODE: 22MAT31B
MODULE – 2: NUMERICAL METHODS FOR ODE & PDE

Q.No Questions Marks CO’s BL

1. 𝑑𝑦 𝑥 6 1 1
Solve 𝑑𝑥 = 2𝑦 + 3𝑒 , y(0)=0. Using Taylor’s series method find y(0.1),
y(0.2)

2. 𝑑𝑦 6 1 1
Use Taylor’s series method to expand = 𝑥 2 + 𝑦, 𝑥 = 0, 𝑦 = 10 and use
𝑑𝑥
it to find y(0.1), y(0.2)

3. 𝑑𝑦 6 1 1
Given 𝑑𝑥 = 𝑥 − 𝑦 with y(0) = 1 find y(0.1), y(0.2) by applying Taylor’s
series method.

4. Employ Taylor’s series method to find the approximate solution to find y at 6 1 1

𝑑𝑦
x=0.1 given 𝑑𝑥 = 𝑥 − 𝑦 2 , y(0)=1 by considering upto 4th degree term.

5. RL circuit with ramp input differential equation governing by the current 6 1 2

𝑑𝑖 𝑅 𝑡 𝑅 1
flow it is given by + 𝑖 = , 𝑖(0) = 0, given = 0.1, = 0.1 with
𝑑𝑡 𝐿 𝐿 𝐿 𝐿
𝑖 = 0 at 𝑡 = 0. Solve by applying Taylor’s series method up to fourth
degree term to find 𝑖(0.5).

6. Apply Runge-kutta method to find an approximate value of y for x=0.2 in 7 2 2

𝑑𝑦
step of 0.2 for 𝑑𝑥 = 𝑥 + 𝑦, given that y=1 when x=0.

7. Apply Runge-kutta method of fourth order given 𝑦 ′ = 𝑥𝑦, y(1)=2, h=0.2, 7 2 2

find y(1.2).

8. Apply Runge-kutta method to find an approximate value of y for x=0.2 in 7 2 2

𝑑𝑦
step of 0.2 of 𝑑𝑥 = 𝑥 + 𝑦 2 , given that y=1 when x=0.

9. Apply Runge-kutta method to find an approximate value in the range 7 2 2

𝑑𝑦
0 ≤ 𝑥 ≤ 0.1 by taking h=0.1 for 𝑑𝑥 = 𝑥(1 + 𝑥𝑦), given that y=1 when x=0

10. Apply Runge-kutta method of 4th order, to compute y(0.3).Given that 7 2 2

𝑑𝑦
10 𝑑𝑥 = 𝑥 2 + 𝑦 2 y(0)=1,taking h=0.3.

11. Solve utt = 4uxx subject to u(0, t) = 0 = u(4, t) , u(x, 0) = x(4 − x) and 8 5 5
ut (x, 0) = 0 Take h = 1, k = 0.5 up to four steps in t.

12. x(4−x) 8 5 5
Solve utt = uxx subject to u(0, t) = 0, u(4, t) = 0, u(x, 0) = 2 and
ut (x, 0) = 0 .Take h = 1, and find solution upto 5 steps in t-direction.

3
13. Solve uxx = utt subject to u(0, t) = 0, u(1, t) = 100sinπt, u(x, 0) = 0, 8 5 5
1
ut (x, 0) = 0, 0 ≤ t ≤ 1, by taking h = 4 .

14. Solve 25uxx = utt subject to u(0, t) = 0 = u(5, t), 8 5 5

2x , 0 ≤ x ≤ 2.5
u(x, 0) = { } , ut (x, 0) = 0, by taking h=1 upto
10 − 2x , 2.5 ≤ x ≤ 5
t=1.

1
15. Solve uxx = utt subject to u(0, t) = 0 = u(1, t), u(x, 0) = 2 x(1 − x), 8 5 5
ut (x, 0) = 0, 0 ≤ t ≤ 0.4 by taking h = k = 0.1 .

16. Using Bender- Schmidt method to solve uxx = 2ut subject to u(0, t) = 8 4 4
u(4, t) = 0 and the initial condition u(x, 0) = x(4 − x)taking h = k = 1
upto t=5.

17. Using Bender- Schmidt method to solve uxx = ut subject to u(0, t) = 8 4 4

u(5, t) = 0 and the initial condition u(x, 0) = x 2 (25 − x 2 ) taking
h = 1, k = 1/2 up to t=3.

18. Using Bender-Schmidt Method to solve 4uxx = ut subject to the 8 4 4

1
conditions u(0, t) = u(8, t) = 0, u(x, 0) = 4x − 2 x 2 , t ≥ 0 by taking
1
h = 1, k = 8 up to t=3/4

19. ∂u ∂2 u 8 4 4
Solve by using Bender- Schmidt method the equation ∂t = 5 ∂x2 subject
to the boundary condition u(0, t) = 0, u(5, t) = 60, u(x, 0) =
20x , 0 < 𝑥 ≤ 3
{ for 5 steps in t, by taking h = 1 & 𝑘 = 0.1
60, 3 < 𝑥 ≤ 5

20. Solve the equation ut = uxx subject to u(0, t) = u(1, t) = 0, u(x, 0) = 8 4 4

sinπx, t ≥ 0. Using Bender-Schmidt Method, carry out four levels in t,
1 1
taking h = 3 , k = 18

4
 COURSE: STATISTICS, NUMERICAL METHODS & TRANSFORMS
COURSE CODE: 22MAT31B
MODULE – 3: FOURIER SERIES
Q.No Questions Marks CO’s BL
1. Find the Fourier series for the function 𝑓(𝑥) = 𝑥, 0 ≤ 𝑥 ≤ 2𝜋, where 6 1 2
𝑓(𝑥) = 𝑓(𝑥 + 2𝜋)

2. 1, 0≤ 𝑥 ≤𝜋 6 1 2
Find the Fourier series for 𝑓(𝑥) = { }, where
−1 , 𝜋 ≤ 𝑥 ≤ 2𝜋
𝑓(𝑥) = 𝑓 (𝑥 + 2𝜋) .

3 Find the Fourier expansion of the function𝑓(𝑥) defined by the 6 2 2

𝑥 , 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 𝜋
𝑓(𝑥) = {
2𝜋 − 𝑥, 𝑓𝑜𝑟 𝜋 ≤ 𝑥 ≤ 2𝜋

4. Find the Fourier series expansion 𝑓(𝑥) = 𝑥 2 in the interval −𝜋 ≤ 𝑥 ≤ 𝜋. 6 1 2

5. Obtain the Fourier series for the function 𝑓(𝑥) = 𝑥 in the interval 6 1 2
−𝜋 ≤ 𝑥 ≤ 𝜋.

6. −𝑘 − 𝜋 ≤ x ≤ 0 6 2 2
Determine Fourier series of the function (𝑥) = { . Hence,
𝑘 0≤𝑥≤𝜋
𝜋 1 1 1
deduce that 4 = 1 − 3 + 5 − 7 + ⋯ … …

𝜋−𝑥
7. Find the Fourier series expansion of 𝑓(𝑥) = in 0 < 𝑥 < 2 . 6 3 3
2

8. Obtain the Fourier series for the function 𝑓(𝑥) = 2𝑥 − 𝑥 2 in the interval 6 3 3
0≤𝑥 ≤2.

9. Determine the Fourier series for the function 𝑓(𝑥), defined on [−2,2] , 7 3 3
−1 , −2 ≤ 𝑥 < 0
where 𝑓(𝑥) = { }.
2 , 0<𝑥≤2

10. Determine the Fourier series expansion of the function 8 3 3

𝑓(𝑥) = 1 − 𝑥 2 in −1 ≤ 𝑥 ≤ 1 . Hence show that
𝜋2 1 1 1 1
= 1 + 22 + 32 + 42 + 52 …. .
6

11. Obtain the Fourier series of 𝑓(𝑥) = 𝜋 − 𝑥 as half range cosine series in the 6 4 4
interval 0 ≤ 𝑥 ≤ 𝜋 .

12. Find the half range Fourier cosine series 𝑓(𝑥) = 𝑥 3 𝑖𝑛 0 ≤ 𝑥 ≤ 1 . 7 4 4

13. Find the half range cosine series for the function 𝑓(𝑥) = 𝑥 in the interval 7 4 4
0 < x < 2.

5
14. Determine the half range Fourier sine series 𝑓(𝑥) = 𝑥 3 𝑖𝑛 0 ≤ 𝑥 ≤ 1 . 7 4 4

15. Find the half range sine series for the function 𝑓(𝑥) = 𝑥 in the interval 7 4 4
0 < x < 2.

16. Analyse harmonically the data given below & express y as a Fourier series 8 5 5
up to 2nd harmonic
x 0 𝜋⁄ 2𝜋⁄ 𝜋 4𝜋⁄ 5𝜋⁄ 2𝜋
3 3 3 3
y 1.0 1.4 1.9 1.7 1.5 1.2 1.0

17. Determine the constant terms and the first & second cosine and sine terms 8 5 5
of the Fourier series expansion of y from fallowing table
x 0° 45° 90° 135° 180° 225° 270° 315°
y 2 3/2 1 1/2 0 1/2 1 3/2

18. The following table gives the variations of periodic current over a period 8 5 5
t(sec) 0 T/6 T/3 T/2 2T/3 5T/6 T
A(amp) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
Examine that there is a direct current part of 0.75 amp in the variable
current and also obtain the amplitude of the 1st harmonic.

19. Determine the Fourier coefficients up to the 2nd harmonics given the 8 5 5
fallowing values
x 0 1 2 3 4 5
y 9 18 24 28 26 20

20. Determine the Fourier coefficients up to the 2nd harmonics given the 8 5 5
fallowing values
x 0 1 2 3 4 5
y 8 16 22 28 26 18

6
 COURSE: SATISTICS, TRANSFORMS & NUMERICAL METHODS
COURSE CODE: 22MAT31B
MODULE – 4: FOURIER TRANSFORM
Q.No Questions Marks CO’s BL
1. −𝑎|𝑥| 6 1 1
Find the Fourier transform of 𝑒 . Where a>0

2 2
2. Find the Fourier transform of 𝑒 −𝑎 𝑥 , 𝑎 > 0 (−∞ < 𝑥 < ∞). Hence 6 1 1
2
prove that 𝑒 −𝑥 /2 is Self- reciprocal.

3. 1, |𝑥| ≤ 𝑎 7 1 2
Find the Fourier transform 𝑓(𝑥) = { a>0 evaluate
0, |𝑥| > 𝑎
∞ 𝑠𝑖𝑛𝑎𝑥
∫0 𝑑𝑥
𝑥

4. 1 − |𝑥| 𝑓𝑜𝑟 |𝑥| ≤ 1 7 1 2

Find a Fourier transform of 𝑓(𝑥) = { and
0 𝑓𝑜𝑟 |𝑥| > 1
∞ 𝑠𝑖𝑛2 𝑥
evaluate∫0 𝑑𝑥
𝑥2

5. 1 − 𝑥 2 𝑓𝑜𝑟 |𝑥| ≤ 1 7 1 2
Find a Fourier transform of 𝑓(𝑥) = { and evaluate
0 𝑓𝑜𝑟 |𝑥| > 1
∞ 𝑥𝑐𝑜𝑠𝑥−𝑠𝑖𝑛𝑥
∫0 ( 𝑥3
) 𝑑𝑥

6. Find the Fourier transform of 𝑥𝑒 −𝑎|𝑥| . Where a>0 7 1 1

7. Find the inverse Fourier transform of 𝑒 −𝑎|𝑢| where a>0 6 3 3

2
8. Find the inverse Fourier transform of 𝑒 −𝑢 6 3 3

9. 𝑥 0<𝑥<2 5 3 3
Find Fourier Cosine transformation of 𝑓(𝑥) = {
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

10. ∞ 1 − 𝑢, 0<𝑢<1 7 3 3
Solve Integral equation ∫0 𝑓(𝑥)𝑐𝑜𝑠𝑢𝑥 𝑑𝑥 = { Hence
0, 𝑢≥1
∞ 𝑠𝑖𝑛2 𝑥 𝜋
deduce that ∫0 𝑥 2 𝑑𝑥 = 2

∞ 𝑐𝑜𝑠⋋𝑥
11. Find the Fourier Cosine transform of 𝑒 −𝑎𝑥 , 𝑎 ≥ 0, hence find ∫0 𝑑𝑥 7 3 3
𝑎2 + 𝑥 2

12. Find the finite Fourier Sine transform of 𝑓(𝑥) = 2𝑥 𝑖𝑛 0 ≤ 𝑥 ≤ 4. 6 3 3

13. 1 𝑓𝑜𝑟 0 ≤ 𝑢 < 1 6 3 3

∞
Solve the integral equation ∫0 𝑓(𝑥)𝑠𝑖𝑛𝑢𝑥 𝑑𝑥 = { 2 𝑓𝑜𝑟 1 ≤ 𝑢 < 2
0 𝑓𝑜𝑟 𝑢≥2

14. Find the Fourier Sine transform of 𝑒 −𝑥 . Hence prove that 7 3 3

∞ 𝑥𝑠𝑖𝑛𝑚𝑥 𝜋
∫0 1+𝑥 2 𝑑𝑥 = 2 𝑒 −𝑚 , 𝑚 > 0

7
15. Find the inverse Fourier Cosine transform of 𝑒 −2𝑢 6 4 4

1
16. Find the inverse Fourier Sine transform of 𝑢 𝑒 −𝑎𝑢 where a>0 6 4 4

2
17. Employ Convolution theorem to find 𝐹(𝑓 ∗ 𝑔) 𝑔𝑖𝑣𝑒𝑛 𝑓(𝑥) = 𝑔(𝑥) = 𝑒 −𝑥 6 5 5

18. Employ Convolution theorem to find 𝐹(𝑓 ∗ 𝑔) given 6 5 5

1, |𝑥| ≤ 1
𝑓(𝑥) = 𝑔(𝑥) = {
0, |𝑥| > 1

∞ 𝑑𝑡 𝜋
19. Using Parseval’s identities prove that ∫0 = 2𝑎𝑏(𝑎+𝑏) 6 5 5
(𝑎2 +𝑡 2 )(𝑏 2 +𝑡 2 )

20. ∞ 𝑡2 𝜋 6 5 5
Using Parseval’s identities prove that ∫0 𝑑𝑡 =
(𝑡 2 +1)2 4

8
 COURSE: STATISTICS, TRANSFORMS & NUMERICAL METHODS
COURSE CODE: 22MAT31B
MODULE – 5: Z – TRANSFORMS

Q.No Questions Marks CO’s BL

1. Find the Z- transform of 𝑛 and hence find 𝑍𝑇 (𝑘 𝑛 𝑛3 )
3
5 1 1

2. 𝑍(𝑍−cos 𝜃) 5 1 2
Prove that 𝑍𝑇 (cos 𝑛𝜃) = 𝑍 2−2𝑍 cos 𝜃+1 and hence deduce Z-Transform of
(𝑘 𝑛 𝑐𝑜𝑠𝑛𝜃)

3. Find the Z-Transform of sinh 𝑛𝜃 and hence find 𝑍𝑇 (𝑎𝑛 sinh 𝑛𝜃 ). 6 1 1

4. nπ π 6 1 2
a) Find the Z-Transforms of (𝑖)(𝑛 − 1)2 (𝑖𝑖)cos ( 2 + 4 )

1 1 𝜋𝑛
5. Find the Z-Transforms of (i) (2)𝑛 + (3)𝑛 (ii) 3𝑛 cos( ). 6 1 2
4

𝑛𝜋
6. Find the Z-Transforms of (i) (2𝑛 − 1)2 (ii)3𝑛 − 4𝑠𝑖𝑛 5 1 2
4

7. 2𝑧 2 +5𝑧+14 6 4 4
If 𝑢̅(z)= evaluate 𝑢2 and 𝑢3 .
(𝑧−1)4

8. 2𝑧 2 +3𝑧+4 8 4 4
Given that 𝑍(𝑢𝑛 ) = , |𝑧| > 3, show that 𝑢1 = 2, 𝑢2 = 21 and
(𝑧−3)3
𝑢3 = 139.

9. 5𝑧 2 +3𝑧+12 8 4 4
If 𝑢̅(z)= Show that 𝑢2 = 5 and 𝑢3 = 23.
(𝑧−1)4

10. 2𝑧 2 +3𝑧+12 8 4 4
If 𝑢̅(z)= evaluate 𝑢2 and 𝑢3 .
(𝑧−1)4

11. 𝑧(𝑧+3) 6 3 4
Find inverse Z-transform of (𝑧+1)(𝑧−2)

12. 𝑧2 6 3 4
Obtain the Inverse Z- transform of (𝑧−1)(𝑧+3)
.

13. 2𝑧 2 +3𝑧 6 3 4
Find the Inverse Z- Transform of (𝑧+2)(𝑧−4) .

14. 𝑧 3 −20𝑧 7 3 4
Find the Inverse Z- Transform of (𝑧−2)3 (𝑧−4)

𝑧
15. Find the Inverse Z- Transform of (𝑧+1)2 (𝑧−1) 7 3 4

16. 8𝑧−𝑧 3 7 3 4
Find the Inverse Z-transform of (4−𝑧)3

9
17. Find the response of the system 𝑦𝑛+2 − 5𝑦𝑛+1 + 6𝑦𝑛 = 𝑢𝑛 , with 7 5 5
𝑦0 = 0, 𝑦1 = 1 and 𝑢𝑛 = 1 for 𝑛 = 0,1,2,3, … .. by Z–transform method.

18. Solve the difference equation 𝑦𝑛+2 + 6𝑦𝑛+1 + 9𝑦𝑛 = 2𝑛 ; 7 5 5

𝑦0 = 0, 𝑦1 = 0 using z-transforms.

19. Solve the difference equation using Z-transform 7 5 5

𝑈𝑛+2 − 2𝑈𝑛+1 + 𝑈𝑛 = 2𝑛 ; 𝑈0 = 2, 𝑈1 = 1

20. Solve the difference equation using Z-transform 7 5 5

𝑦𝑛+2 − 3𝑦𝑛+1 + 2𝑦𝑛 = 0 given that 𝑦0 = 0, 𝑦1 = 1

10