Engineering Mathematics 1 Jan 2014

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First Semester B.E. Degree Examination, Dec.2013lJan.2Ul4Engineering Mathematics - I

Time: 3 hrs. Max. Marks:100Note: l. Answer any FIW full questions, choosing at leust twofrom each part.

2. Answer all objective type questions only in OMR sheet page 5 of the answer booklel3. Answer to objective tlpe questions on sheets other than OMR will not be valued.

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PART _ A1 u.,,.

,Choose the

3o;;ect.answels for the following :

:\ If. I t,

x+l(-l)n (n + 1)!

(x + l)n*1

ii) If y: (ax +b)- with m: n, then y, isA) n! an B)0

A) B) (-l)nnl(-.rt*i

C) (-1)'n!(x + l)n

C) n! bn

D) 1-1)n-rn!(". fli

D)n!iii) The geometriCal interpretation of Lagrange's mean value theorem is

f(b)-f(a) B) f,r.t_f(b)+f(a)C; l''lc)_ f(b)-f(a) D)noneof theseA)f'(c)=- b-a b-a g,(c) g(b)-g(a)

iv) The Maclaurin's series expansion of e-' is

,,:3! 2! 3!

C) *- x- * x3 D) * *4*d* (04 Marks)2t3!.213!

b. Ify:sinlog (x2 +2x+ l),provethat(x+ l)'yr,+ r+(2n+ 1)(x+ 1)yn+1n2+4)yn:0.

c. If x ispositive, showthatx> log (1 +x) r"-L*.d. Using Maclourin's series, expand log (1 + e*) upto the terms containing x

2 a. Choose the correct answers for the following :

. lim /, .--..\i) " -+l

|;J3!I lis equalto-( f;-x )

A) 2 B)-2 C) 1

ii) If $ be the angle between the tangent and radius vector at anyr = (e), then sin $ is equal to

(04 Marks)

(06 Marks)(06 Marks)

D)-1point on the curve

D) dsl

(04 Marks)

the curve meets the

(06 Marks)

(06 Marks)

A) drlds gt.do'ds C) ,do'dr

ct ? *,,'Y'11

t + z\ -rt2cosm0.

iii) The rate at which the curve is bending calledA) radius of curvature B) curvature C) circle of curvature D)

iv) The radius of curvature for polar curve r : (e) is given by

A) l'-tY s1 ''*,lY'' ,2*rl-rr2 'rl*x'-rr,b. Find the Pedal equation of the curve ,t : a'c. Find the radius of curvature for the curve v2=d9:*), where

xx - axis.

Evaluate lim (ax + t )x.*--*(.ax-l)d.

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i) If u: log (x2 + f + r'1,then I ,,0z

2x

x'+y'+z'2vts\ .,-+ )x'+v'+7,- *2 +y2 +12

1OMAT11

*2 +y2 -12

D) .none

of these(04 Marks)

are made

(04 Marks)

(06 Marks)

(06 Marks)

D) none of these

D)3

D) none of these(04 Marks)

(04 Marks)

- y)k is irrotational.

(06 Marks)

(06 Marks)

2z2zD)c)

ii) If u:(x, y) and y is a function x, then

. du au dudyA\^'d* Cd,n\du fu aury,l\ l\-r

-'dx ax q dx

^2. a2f a2fiii) If r={*.s=- &t=3dx' oxo\ dy':,,,

A) rr-s2?O g)rt-s2:o

-au du audvax dx dy dx

^.au du dufoI \\ut

-ax dx ar ax,' i','"''

, then the condition for the shddle point is

C) r1r* s'> O D)rt-s2+0/\

iv) Ifu:x*yr z,y=,!+z,z:z,then J[ ''u'l ]ir.qrulto\x'Y'zl

A)1 B) -1

d. If x: u(1 - v), y: uY, Ftove that JJ' :1.

a mirror is given by the formula I - I = Z . If .qual errors. 'e'u (t r)

in the determination of u and v. Show that the rosulting error in f is el - + - I .

^ +"^*1tu;^ ,ur-)'v)If u: f(2x-3y.3y -42.42-2x).prove that 1;.i n*i;=0.

c)0

b.

c.


i) Direotional derivative is maximum alongA) tangent to the surfaceC) any unit vector

,ii) Ifr: lx;+y1 + 2rl, then Vrn is

' iii) If f : 3x2 - lf + 4z2,thencurl (grad f) is

B) normal to the SuiftceD) coordinate axes

C) V.Vr'

A) 4x - 6y + 8z B) 4x1-6y; + 8z k C) b'iv) If the base vectors er and azzta orthogonal then I e1 x e2 | is

b.

c.

A)0 B)-1 c)+1

-+If F =(x+y+l)i+ j-(x+y)k, showthat F.crulF =0.

Find constants,a, and,b, such that i = (axy + r31i+ (3x2 - z)j+ (brr2

Also find a scalar function $such that i - V0.Prove that a spherical coordinate system is orthogonal.

1^{Ad.






PART _ B


1l

il Jsin

7 x dx is equal to0

A) zero Bt 32n'35

?)C):35

C) x - axis

)c){'2)/7 .: a-," rS

Cy,za

1OMAT11

(06 Marks)

(04 Marks)

(04 Marks)

(06 Marks)(06 Marks)

ii) The asymptote of (2 - x)l : *3 isA) x:2 B) y - axis

iii) The ar€a of the cordioid r : a(1 - cos0) is.2JfiAA)_

2

1(trD) =:"'"a^

-)z

D) none of these

n

D) JA'2

D) a. (04 Marks)

b' Evaluate Iar,t + acosx)dx uy:airrrr.*iating0'

B)I2

iv) The entire length f the asteroid x2l3 + f '3

A) 6a B) 3a

ir) The equationy - 2x: c represents the orthogonal trajectories of the familyA) y: ae-" B) x2 + 2f : u c) xy:.a D) x + 2y: a

under the integral si

2a

c. Evaluate I"'.lZu- *2dr. using reduction formula.

d. Find the volume of generated by the revolution of the curveline.

6 a. Choose the corrbct'u.rr*.r, for the following : .

D The general solution of the differential equation dyldx : (y/x,) + tan (y/x) is

.A) sin (y/x) : c B) sin (y/x) : cx C) cos(y/x) : ci , ,:

,,D) cos (y/x) : c

ii) ''.'the family of straight lines passing through the origin is represented by the differentialequation:A) ydx+xdy:0 B)xdy-ydx:0 C) xdx+ydy:0 D)ydy-xdx:0

iii) The homogeneous differential equation Mdx + Ndy : 0 can be reduced to a

differential equation, in which the variables are separated by the substitutionA) v: vx B)x*y:v c) xy: v D)x-y:v

b.

c.

d.

Solve 1, + fl9- y = e3*1x + l)2.dx

Solve (1 + xy) ydx+ (1 -xy) xdy:0.Find the orthogonal trajectory of the cordioids r: a(l - cos 0).

3 of 4






1OMATl1


i) If every minor of order 'r' of a matrix A i, ,..o, then rank of A isA) greater than r B) equal r C) less than or

"q*l,o . D) lessthan r.

-:

-r----- -" / rvo'

ii) The trivial solution for the given system of equations x + ly * 3z:0, 3x + 4y+ 42* 0,7x+ 10 y + l2z:0 is J

A) (1, 1, t) B) (1,0,0) c) (0, 1,0) D) (0,0,0)iii) Matrix has a value. This statement / \"> -' "''

. . , 11) is always true B) depends upon the matrices C) is false D; none of theseiv){.ai1singularandp(A):p(A:B)thenthesystemhas

A) unique solution B) infinitely many solution c) trivial sorution D) no solution.

b.

d.

8a.

c.

Using elementary transformations, find the rank of the matrix '; ' ,.',' :,'

ttl:41 21.I ^ _l

(04Marks)l-2 3 2 s_l

Showthat the system* i y'* z:4;2x+ y - z: l; x- y *22:2 is consistent, solve the(06 Marks)Apply Gauss - Jordan method to sorve the system of equation :

2x+ 5y t 7z : 52 ; 2x-r y - z: 0; i + S, + ;': ;.- -

(06 Marks)

Choose the correct answers lor the following :

i) A square matrix A is called orthosonai ifA)A:AL B)A':A-I I ,,,gy AA-r:I

ii) The eigen vatues ofrhe ,nu,ri* | | fl ^r,142 2)A)lrJ6' B)ltJ5 c)Js D)l

ii0 T11e^iTex;d

signature of the quadratic form xf +2xf-3xj are respecrivelyA) 2,.t " B) 1,2 C) 1, 1 -: . njrron. of theseiv) Two square matrices A and B are similar. if ::. .:.' A) A: B B) B : p-rAI, C) Ar : Br D) A-r : B-r.

Reduce the quadratic form gx2 + 7y2 * 322 - 12yz + 4zx_gxy to tt . .urronfuii Orljo Marks)

Determine the characteristic roots and eigen vectors of

I s -6 21

A=l -6 7 -41 .

l, -4 3j

b.

c.

D) none of these

(04 Marks)

(06 Marks)

d' Reduce the quadratic form xl + 2xl - 7"1 - 4xp2+ 8x2x3 into sum of squares. (06 Marks)

***rF*






Engineering Mathematics 1 Jan 2014

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