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Calculus & Differential Equations Exam

This document appears to be an exam paper for a second semester mathematics course covering topics like calculus, differential equations, and geometry. It contains 8 questions across 4 sections (A-D). The questions involve calculating integrals, finding asymptotes of curves, solving differential equations, and other mathematical problems. Students are required to attempt 5 questions total, selecting at least one from each section, with their fifth question coming from any section.

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Aashish Kohli
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126 views3 pages

Calculus & Differential Equations Exam

This document appears to be an exam paper for a second semester mathematics course covering topics like calculus, differential equations, and geometry. It contains 8 questions across 4 sections (A-D). The questions involve calculating integrals, finding asymptotes of curves, solving differential equations, and other mathematical problems. Students are required to attempt 5 questions total, selecting at least one from each section, with their fifth question coming from any section.

Uploaded by

Aashish Kohli
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Exam. Code 103202


Subject Code: 1025
B.A.lB.Sc. 2nd Semester
MATHEMATICS
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Paper-I
(Calculus and Differential Equations)
Time Allowed-Three Hours] [Maximum Marks-50
Note :- Paper consists of/our Sections A, B, C and D.
Each section contains two questions. Students are
required to attemptfive questions, selecting at least
one question from each section. The fifth question
may be attempted from any section.
SECTION-A
1. (a) Find the intervals in which the curve
y = (cosx + sin x)eX is concave upwards or concave
downwards in (0, 21t). Also find the points of
inflexsion.
(b) Find the centre of curvature at any point (x, y) of
x2 y2
the ellipse 2 + 2 = 1. Also find the evolute of
a b

5+5=10

2. (a) Find all the asymptotes of the curve


(x - y + 1) (x - Y - 2) (x + y) = 8x - 1.
(b) Find the position and nature of the double points
on the curve y2 = (x - 1) (x - 2)2. 5+5=10

2539(2519)/EBH-599 1 (Contd.)

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SECTION-B

3. (a) Integrate f sinh x sinh 2x sinh 3x dx .


(b) Find the area of the region bounded by the curves
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y2 = 4a(x + a), y2 = 4b(b - x) where a > 0,


b > O. 5+5=10

4. (a) If Im,n = f sin" x cos" x dx then prove that

sinm+lx cosn+1 X m + n + 2
Im,n m+ l + m+I Im+2,n· Hence

dx
evaluate f sin 4 x cos2 X .

(b) Find the length of a loop of the curve


9ay2 = x(x - 3a)2 , a > 0 5+5=10

SECTION-C
5. (a) Find the necessary and sufficient condition that
the equation Mdx + Ndy = 0 may be exact where
M, N are functions of x and y with the condition

t hat M, N,-, BMBN- .


are continuous functi
unction 0f
By ax
x and y.
(b) Find the orthogonal trajectories of the system of
circles touching a given straight line at a given
point. 5+5=10

2S39(2S19)/EBH-S99 2 (Contd.)

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6. (a) Solve the differential equation

dy
(8p3 - 27)x - 12p2y = 0 where p = dx

and investigate whether a singular solution exists.


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(b) Solve 2: -P=f(~ - :2) where p = :~.

5+5=10
SECTION-D
7. (a) Solve the differential equation

d
(D4 + 2D2 + l)y = x2 cos x where D = -.
dx
(b) Solve the differential equation
(D2+ 3D + 2)y = sinre-) by the method of variation
of parameters. 5+5=10
8. (a) Solve in series the differential equation

d2y dy 2 2 .
X2_2 +x-+(x -n )y=O where 2n IS a non
dx dx .
integer.
(b) Solve:

d2 d
J;. -{ + 2x__X_+3y= x, x > O. 5+5=10
dx dx

2539(2519)IEBH-599 3 9500

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