DHARMSINH DESAI UNIVERSITY, NADIAD
FACULTY OF TECHNOLOGY
B.TECH. SEMESTER IV [EC, CH, CL, IT, IC]
SUBJECT: MATHEMATICS - IV
Examination : First Sessional
Seat No.
: ___________
Date
: 04/01/2021
Day
: Monday
Time
: 11.20 to 12.15
Max. Marks : 24
INSTRUCTIONS:
1.
Figures to the right indicate maximum marks for that question.
2.
The symbols used carry their usual meanings.
3.
Assume suitable data, if required & mention them clearly.
4.
Draw neat sketches wherever necessary.
Q.2 Attempt Any Three from the following questions.
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2
2
2
(a) If the directional derivative of  = ax y + by z + cz x at the point (1, 1, 1) has
maximum magnitude 15 in the direction of the line parallel to the line
𝑥−1
𝑦−3
= −2 = 𝑧 . Find the values of a, b, c.
2
(b) Solve: 𝑥𝑒 𝑥 − 2 = 0 using Regula Falsi method correct up to three decimal places.
(c) Find the angle between the surfaces 𝑥 2 + 𝑦 2 + 𝑧 2 = 9 and 𝑧 = 𝑥 2 + 𝑦 2 − 3 at the point
(2, -1, 2).
(d) Prove that: √1 + 𝛿 2 𝜇2 = 1 + 𝛿 2 = cosh(ℎ𝑑)
2
Q.3
(a) Prove that ∆𝑥 𝑚 − 1 ∆2 𝑥 𝑚 + 1∙3 ∆3 𝑥 𝑚 − 1∙3∙5 ∆4 𝑥 𝑚 +. . . . . . . . = (𝑥 + 1)𝑚 − (𝑥 − 1)𝑚
2
2∙4
2∙4∙6
2
2
[4]
(b) Find the approximate value of the real root of the equation 2𝑥 − log10 𝑥 − 7 = 0 using
[4]
Newton Raphson method correct up to three decimal places.
[4]
 ar 
 ar 
a
(c) For a constant vector 𝑎⃗ and position vector 𝑟⃗ prove that   
=
−
+
3
r
 5 
3 
3
 r

r
 r 
OR
Q.3
(𝑛−𝑥
1 ) ∆𝑦𝑛−1
2
n-𝑥 𝑛−𝑥
(a) Prove that: 𝑦𝑥 = 𝑦𝑛 −
+ (𝑛−𝑥
∆
𝑦𝑥
2 ) ∆ 𝑦𝑛−2 +. . . . . . + (−1)
(b) Obtain root of the equation 𝑥 3 − 3𝑥 + 1 = 0 using direct iteration method by applying 6
iterations.
1
(c) Find ∇2 ( ), where 𝑟⃗ is position vector.
𝑟
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