TED (15) -rcazEEVTSTON-201s) Signature

FIRST SEMESTER DIPLOMA EXAMINAIION IN ENGINEERING/TECHNOLOGY OCTOBERAIOVEMBER, 2016

ENGINEERING MATHEMATICS - IfTime :3 hours

(Ma.:rimum marks : 100)

PART - A(Maximum marks : 10)

Marks

Answer all questions. Each question carries 2 marks.

1. Prove that (sinA * cosA)2 : 1 + 2 sinA cosA.

2, If sinA : 3/- and A is acute find sin3A.5

3. Prove that in triangle ABC, abc :4RAdv4. Find ,i- if y : ClogdX

5, Find the velocity and acceleration at time t, of a particle moving according as

s : 5t3 - Ztz + 9t + 1. (5x2:10)

PART - B

Maxrnum marks : 30)

Answer any five questions from the following. Each question carries 6 marks.

1. Express {J *o * sirx in the form Rsin (x + a) where 'c[' is acute.

2. Prove that cos2Oo cos4Oo cos8Oo : *'3. Prove that R (u2 +bz + c2\: abc (cotA * cotB * cotC)

4. Differentiate 'cosx' by the method of first principles.

J.,

5. Frnd I lf xzy2 : *3 + y3 + 3xy.1Jox

6. Find the equafion to the tangent and normal to the curve y :

7. prove that sinA * ,i' (T * ) +

'in( + .o) : o.

at (4,3)

(5x6:30)

[45]

Reg. No.

II

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Marks

PART - C

(Maximum marks : 60)

(Answer one fiIl question from.each unit. 'Each full question carries 15 marks.)

UNrr - I

m (a) Prove that (cotA - 1)2 * (cotA + l)2 : 2cosec2A

(b) If sin0 : 1! and 0 is in the fourth quadran! calculate the value of25

30cos0 - 7tan0

3cos0 - sinO

(c) Prove that 2tan10o * tan4Oo : tan50"

On

rV (a)

(b)

4I t -sin0Prove that \lr 1*sin0

-12 24If cosA : +, cotB : + and A is in II quadrant and B is in quadrant I,

lJ t

find sin (A + B) and cos (A - B)

The horizontal between two towers is 50m and the angle of depression of the

first tower as seen from the second which is in 15Om height is 60o. Find the

height of the first tower.

UNrr - II

(a) prove that sinA + sin3-A + sin5A : tan3A.\-/ ---- cosA * cos3A * cos5A

(b) Prove that cot A - cot2A : cosec2A.

(c) Solve AngC given a: .4crrr, b : 5cm, c : Tcrn

On

vI (a) Prove ,rrut til3l * cos3A :4 cos2AsinA cosA

(b) Prove that cos3A * cos5A * cosgA * cosl7A : 4cos4Acos6AcosTA.

(c) Two angles of a triangular plot of.land are 53"17' and 67"09' and the side

between them is measured to be 100cm. How many metres of fencing is

required to fence the plot.

(c)

V

2

5

5

5

5

5

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UNIT, III MATKS

t+VII (a) Evaluate , *\o f#

dv Sin-tfi(b) Find * if : (i)dx \^/ ' *,(ii) x: a (0 - sing) ; y: a (t - cosg) (3+3:6;

(c) If y : Acospx * Bsinpx (A, B, p are constants),

strow tnat 921 is proportional to .y,. 5

dx-On

VIII (a) Evaluate , h^ I - c-os2x.

Jo-,3dy-.^

(b) Flrcl f if y : (*' * 1)ro secsx 3ox

(c) Using quotient rule, find the derivative of 'tanx'. 3

^2v dv(d) If y: acos (logx) *bsin(logx) show that *t" 1 + x* *y:0. 6dx2 dx

UNrr - IV

D( (a) Find the velocity and acceleration at time t - 4 secs. of a body whosedisplacernant s metre related to time 't' seconds is grven by the equatio:r

":;,t**ll 6

(b) A circular plate of radius 3 inches expands when heated at the rate of2 incVsec. Find the rate at which the area of the plate is increasing at the end

4of 3 secs.

(c) Find the maximum and minimum values of 2x3 - 3* - 36x + 10. 5

On

X (a) A balloon is spherical in shape. Gas is escaplng from it at the rate of 20 cclsec.How fast is the surface area shrinking when the radius is l5cm. 5

(b) For what values of 'x' is the tang€nt to the curve + parallel to the x-axis. 5x'* I

(c) An open box is to be made out of a sqrurc sheet of side i8cnr, by cutting offequal squares at each comer and turning up the sides. What size of thesqrures should be cut in order that the volume of the box may be maximum 2 5

I

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