C14 ee-102-engg maths-1

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C14–EE–102
4041
BOARD DIPLOMA EXAMINATION, (C–14)
OCT/NOV—2016
DEEE—FIRST YEAR 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
and shall not exceed five simple sentences.
1. (a) Write the types of fractions.
(b) If
x
x x
A
x( ) ( )+
=
+
+
+2
1
2 22 2
then find A.
2. (a) If
A =
é
ë
ê
ù
û
ú
1 2
3 4
then find A2
.
(b) Write any four types of matrices.
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3. (a) Find the value of
1 2 3
4 5 6
7 8 9
(b) Find the value of
i i
i i
-
4. (a) Find cos75°.
(b) If sin A = 3
5
, then find cos A.
5. (a) Write the formula for tan 3A.
(b) Simplify :
1 2
2
+ cos
sin
A
A
6. (a) If z i= -2 3 , then find z z+ .
(b) Find the amplitude of 4 3+ i.
7. (a) If the slope of the line kx y+ + =2 3 0 is 3, then find k.
(b) Find the distance between two parallel lines 2 3 5 0x y+ + = and
2 3 9 0x y+ + = .
8. (a) Find the equation of circle having ( , )x y1 1 and ( , )x y2 2 as the
ends of diameter.
(b) Find the centre of the circle x y x y2 2
4 8 0+ + - = .
9. (a) Find
Lt
x
x
x®
-
-2
3
8
2
(b) Find
Lt
x
x
x®
-
0
5 1
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10. (a) Differentiate e xx
tan w.r.t. x.
(b) Differentiate 2 3sin logx x+ w.r.t. x.
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 valuation is the content but not the length of
the answer.
11. (a) Find the inverse of the matrix
- - -é
ë
ê
ê
ê
ù
û
ú
ú
ú
4 3 3
1 0 1
4 4 3
(b) Solve the equations
x y z
x y z
x y z
+ + =
- + =
+ - =
6
2
2 1
by using Cramer’s rule.
12. (a) In any triangle ABC, prove that
sin sin sin sin sin sin2 2 2 4A B C A B C+ + =
(b) Show that
2
1
3
1
7 4
1 1
tan tan- -æ
è
ç
ö
ø
÷ +
æ
è
ç
ö
ø
÷ =
p
13. (a) Solve sin cosq q+ = 2.
(b) Solve the triangle ABC with a =1, b = 2 and c = 3.
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14. (a) Find the equation of the parabola whose focus is the point
( , )3 4- and the directrix is the line x y- + =5 0.
(b) Find the equation of the ellipse whose centre is origin, whose
axes are the axes of coordinates and which passes through the
points (1, –3) and (–2, 2).
15. (a) Find the derivative of
sin
cos
x
x1 +
w.r.t. x.
(b) Find
dy
dx
if y x x x= + + +cos cos cos L
16. (a) Find
dy
dx
if x y xy3 3
9+ = .
(b) If y a x b x= +cos(log ) sin (log ), then prove that
x y xy y2
2 1 0+ + =
17. (a) Find the lengths of tangent, normal, subtangent, subnormal
to the curve y x= 3
at ( , )1 1.
(b) A circular path of oil spreads out on water, the area growing
at the rate of 6 square centimetre per second. How fast is the
radius increases when the radius is 2 centimetres?
18. (a) Show that semivertical angle of the cone of maximum volume
and of given slant height is tan ( )-1
2 .
(b) If the length of a simple pendulum is decreased by 2%, then
find the percentage error in its period T, where T l
g
= 2p .
H H H
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