C 14-met-mng-aei-102-engg maths-1

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A/AA/CH/CHST/AEI/FW/MNG/
MET/IT/TT/PKG–102
4002
BOARD DIPLOMA EXAMINATION, (C–14)
OCT/NOV—2016
FIRST YEAR ( COMMON) EXAMINATION
ENGINEERING MATHEMATICS—I
Time : 3 hours ] [ Total Marks : 80
PART—A 4×10=40
Instructions : (1) Answer all questions.
(2) Each question carries four marks.
(3) Answers should be brief and straight to the point.
1. (a) Define proper fraction and give an example.
(b) Resolve
x
x( )+12
into partial fraction.
2. (a) Find sin 75°.
(b) If tan A =
1
2
and tan B =
1
3
, find tan( )A B+ .
3. (a) Find tan 22
1
2
°.
(b) Find 8 20 6 203
cos cos° °
- .
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4. (a) Define symmetric and skew-symmetric matrices.
(b) If A =
é
ë
ê
ù
û
ú
1 2
3 4
, find A2
.
5. (a) Write the formula to find A-1
.
(b) Define singular and non-singular matrices.
6. (a) Find the intercepts made by the straight line 2 3 5 0x y+ - = on
the coordinate axes.
(b) Write the formula to find the distance between parallel lines.
7. (a) Find the radius of the circle
x y x y2 2
6 4 12 0+ - + - =
(b) Write the equation of tangent to the circle x y r2 2 2
+ = at
( , )x y1 1 .
8. (a) Find the amplitude of 1 3+ i .
(b) State de Moivre’s theorem.
9. (a) Evaluate :
lt
x
xx ®¥
+
+
1
2
(b) Evaluate :
lt
x
xx ®0
7sin
10. (a) Differentiate :
x xx3
3 3+ +
(b) Differentiate :
x
x
+
+
1
3
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PART—B 10×4=40
Instructions : (1) Answer any four questions.
(2) Each question carries ten marks.
(3) Answers should be comprehensive and the criterion
for the valuation is the content but not the length of
the answer.
11. (a) Prove that
1
1
1
2
2
2
a a
b b
c c
a b b c c a= - - -( )( )( )
(b) Solve
x y z+ + = 6, x y z- + = 2, 2 1x y z+ - =
using matrix-inversion method.
12. (a) In any triangle ABC, prove that
cos cos cos cos cos cos2 2 2 1 4A B C A B C+ + = - -
(b) Prove that
tan tan- -æ
è
ç
ö
ø
÷ +
æ
è
ç
ö
ø
÷ =1 11
4
3
5 4
p
13. (a) Solve cos sinq q+ =3 1.
(b) In a triangle ABC, prove that ( )cosa b C a b c+ = + +å .
14. (a) Find the equation of the parabola whose focus is (3, 4) and
directrix is 2 3 4 0x y- + = .
(b) Find the eccentricity, centre, vertices, foci, length of latus
rectum and equation of directrices of the ellipse
3 4 12 8 4 02 2
x y x y+ - + + =
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15. (a) Find the derivative of tan x using first principle.
(b) Find the derivative of e x4 2
w.r.t. 1 4 2
+ x .
16. (a) Find
dy
dx
, if y x x x= + + + ¥3 3 3
L .
(b) If y x= sin(log ), prove that
x y xy y2
2 1 0+ + =
17. (a) Find the lengths of the tangent, normal, sub-tangent and
sub-normal to the curve y x x= - +3
2 5 at (1, 4).
(b) The side of an equilateral triangle is increasing at the rate of
2 cm/sec. Find the rate of increase of its area and perimeter
given, its side is 25 cm.
18. (a) Prove that the maximum rectangle that can be inscribed in a
circle is a square.
(b) Find approximately the value of 1233 .
H H H
/4002 4 AA6(T)—PDF