Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU) B-Tech 3rd Semester (Third Semester) 2018-2019 Mathematics Iii Ras 301 Question Paper
Paper Id: 199311 Roll No:
, B TECH
(SEM III) THEORY EXAMINATION 2018-19
MATHEMATICS-III
Time: 3 Hours T otal Marks: 70
Notes: Assume any Missing Data.
SECTION - A
1. Attempt ALL parts of the following: 7 X 2 = 14
a) The function f (x)? ? ex 2(cos y + 15m y) is holomorphic or not.
b) Find the residue of W at p01ez= 2.
c) Formula of Measure of Kurtosis [92 =
d) The ?rst three central moments of a distribution are 0, 15,-31.Find the moment coef?cient of
skewness.
e) Obtain the function whose ?rst difference is 9x2 + 11x + 5.
f) F ind the normal equation of a curve y = ax + bx2
g) Let f(2) = u(r, 9) + iv(r, 6) be an analytic function. Ifu = -r3 sin 36, then ?nd 12.
SECTION? B .
2. Attempt any T HREE parts of the following: 3 X 7 = 21
a) F tom the following table of values of x and gobtam ?-? :for x? ? 1.2, 2.2, 1.6 '1 I
x: 1.0 1.2 1.4 ?18 x.20 2.2
33.?, . x
y: 2.7183 3.3201 40552 3413?2530 6.0496 73891 90250
b) Using Runga- Kutta method effburth order 11nd y(O. 8) correct to 4 decimal p1aces it:?: - ?y ?
x2,y(0. 6)?- ?? 1. 7379 takingl'l" 0. 1
e
C) Using complexintegrat1?ln method evaluatef?212 6?52
5+4 6056
d) The equations of tng regression lines, obtained 1n a eorreiatlon analysis of 60 observations
are:
5x ? 6y = 24, 768x ? lOOy = 3608. What 1s the cofreiation coef?cient? Show that the ratio
of coef?cient of variability of x to that of y is ?S?. What?is the ratio of variances of x and y?
e) The pressure of the gas corresponding to various volumes V is measured, given by the
following data: .
V(cm3) 50 60 70 ?:?29? ? 100
i?"
P(kg cm?z) 64.7 51.3 141105 "25.9 78
SECTION ? C
3. Attempt any T W0 parts 0fthe?f5110wing: 2 X 3.5 = 07
a) F ind the unique polynomial P(x) of degree 2 such that: P(l) = 1, PB) = 27, PM) = 64,
use Lagrange method of interpolation.
b) Using Simplson?s 3/8thru1e on integration, evaluate fo?l ?dx
c) Expand 22-3 +2 in the region 1 < |z| < 2.
. Attempt any T W0 parts of the following: 2 X 3.5 = 07
a) If the probability of hitting a target is 10% and 10 shots are ?red independently. What is the
probability that the target will be hit at least once?
b) A die is thrown 276 times and the results of these throws are given below:
No. appeared on the die 1 2 3 4 5 6
Frequency 40 32 29 59 57 59
Test whether the die is biased or not.[Tabulated value of )(2 at 5% level of signi?cance for 5
degree of freedom is 11.09]
c) By Residue method, ?nd the inverse Z-transform of
z2+7z+10
. Attempt any T W0 parts of the following: 2 X 3.5 = 07
a) The following data regarding the heights (y) and weights (x) of 100 college students are given:
2x = 15000,Z x2 = 227500.23; = 6800,2312 = 1163025,ny = 1022250
b) Solve x3 ? 5x+3=0 by using Regula-Falsi method correct up to four decimal places.
c) From the table, estimate the number of students who obtained marks between 40 and 45. .\
Marks: 30-40 40-50 50-60 60-70? .70'80
No.0f Students: 31 42 51
.,_) 3.
1 Q? 331
. Attempt any T W0 parts of the following: 2 X851? 07
W 1*:
a) Find the residue of f (z) W at its pole and hence evaluate f ffCZDdZ where C
is the circle Izl? ? 5/2 @1111! ?1 :1
b) Determine the largest Eign value and corresponding eigem ;y'ector of the matrix
2 -1 0 ?1 i
?41?ng ?*1
A = ?1 2 ?l tl?y?u?ee approximation. ?5?1
0 -1 2 ??1??? "
c) Verify Cauchy theorem by integrating e Zalong the boundary of the triangle with the vertices.?
at the points 1 +i, ?1 + 1' and ? 1 ?-i.
- Attempt any TWO parts of the following
-1 "?7
2 X 3. 5= 07
a) Use Picard 5 method to obtain y tor x? ? 0.?G1ven -d?y = x ? y with initial condition y? -? 1
dx
when x- ? 0 correct up to four decimal p?ie?s
b) In a normal distribution, 31% of mews are under 45 and 8% are over 64. Find the mean and
111
standard deviation of the dlstrlblmci?n It IS given that 1f f (t) 7x2 dx then f (0.5) ?
0.19,f(1. 4): 0.42 ?1"
c) Prove that hD- ? ?log(1 ?51% ? sin h? (#6)
nfoe
This post was last modified on 29 January 2020