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Calculus I Fall 2023

This document outlines an examination for a Bachelor-level Calculus I course, scheduled for Fall 2023, with a total of 100 marks. It includes various mathematical problems related to continuity, differentiability, theorems, differential equations, and integration. Candidates are instructed to provide answers in their own words and attempt all questions within a 3-hour time frame.

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Brijesh Yadav
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80 views2 pages

Calculus I Fall 2023

This document outlines an examination for a Bachelor-level Calculus I course, scheduled for Fall 2023, with a total of 100 marks. It includes various mathematical problems related to continuity, differentiability, theorems, differential equations, and integration. Candidates are instructed to provide answers in their own words and attempt all questions within a 3-hour time frame.

Uploaded by

Brijesh Yadav
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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π

2
d)∫ sin 3 x cos 4 xdx
Level: Bachelor Semester –FALL Year : 2023 0

Programme: BE Marks: 100


Course: Calculus I Pass Marks: 45 4.a) Find the volume of the solid in region bounded
Time : 3hrs. by the curve y= x2 +1 and the line y= -x+3 revolved
Candidates are required to give their answers in their own words as far as
about the X-axis. [8]
practicable.
The figures in the margin indicate full marks. b) State and prove Euler’s theorem on homogeneous function of
Attempt all the questions. Three independent variables of degree n.
1.a) Define continuity and differentiability of a function. Show that ∂ u ∂u
2
x y
2 x + y =3 tan u
Differentiability of a function f(x) at x=a implies continuity but sin u =, x+ y show that ∂x ∂ y [7]
Converse may not be always true. [7] If
b) State and prove Role's theorem. Interpret Geometrically. [8] 5.a) Find the extreme values of the function f(x,y,z)= x 2+ y 2+ z 2
OR subject to the constraints ax+by+cz=k. [7]
tan−1 x
a) If y=e , show that [7]
2
b) Show that the substitution y=y1+u where y1 is a solution of Riccati’s
i) (1+x )y2 +(2x-1) y1 = 0
Equation reduces the Riccati’s equation to a Bernoulli’s equation. [8]
ii) (1+x2)yn+2+(2nx+2x-1)yn+1 n( n+1)yn =0
b) Trace the curve: y 2 ( a−x ) =x2 (a+ x) [8] 6.a) Find the general solution of the differential equation
2.a) Find the asymptotes of the curve: y"-y'-2y= 3e 2 x , y(0)=0 , y'(0)= -2 [7]

x3+2x2y- xy2-2y3+4y2+2xy-5y+6=0 [8] b) Find general solution of differential equation by using method of
b) Find the perimeter of the asteroid: x 2/ 3 + y2 /3 =a2 /3 [7] Parameters: y"+2y +y= e− xcosx [8]
3. Integrate any THREE of the following: 5*3=15 OR
3 a) Solve Second order differential equation of the series RLC circuit
x
a) ∫ dx 2
( x−2 ) (x−3) d Vc dVc 1 Vin
L 2
+R + Vc= , [7]
dt dt C C
1
b)∫ dx
3 sinx +4 sin x where R=10  ,L=1 H, C=1610−4 F Vin=0,VC(0)=6V, V’c (0)=6A
¿4
b) Solve the following initial value problem:
c) Prove that: ∫ log (1+tanx ) dx= ❑ log 2
0 8 2 3
x y"- 2xy'+2y= 0 , y(1) = , y'(1) = 1 [8]
2
Attempt all the questions: 4*2.5=10
a) Find yn if x n ,where n is positive integ
b)Find the radius of curvature:y2= 4ax
c) Show that the function f(x,y) = x 3 + y 3−3 xy has a saddle point at (0,0).
dy 1−cos 2 y
d) Solve: + =0
dx 1−cos 2 x

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