(www.entrance-exam.net)-karnataka public service commission mathematics sample paper 1

8/4/2019 (Www.entrance-exam.net)-Karnataka Public Service Commission Mathematics Sample Paper 1

1/15

2008MATHEMATICS

Paper 1

1'1/111' : 3 Hours 1

12/1

r Maximum Afar"'s : 30n

INSTRUCTIONS

Candidates should attempt all the questions in Parts A, B & c .However, they have to choose only three questions in Part D,Answers must be written In the mediu III opted (i.e. English orKiuinu da).

This paper has four parts:A 20 marksB ] 00 marksc 90 marksD gO marks

Mark allotted t(} each question are indicated in each part.


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1211 (2 )

PART AEach question carries 5 marks.

1. (a) Define concept of the dimension of a vector space and give anexample of an infinite dimensional vector space.

(1) Solve x dy - y dx = J x 2 + y2 dx, given that y = 1 whenx = . . / 3 .

(c) Let f be a function satisfyingf'{x + y) = f rx) f(y) - - J 4 - f'(y) and f(h) ~ 4 as h ..-')O.Discuss the continuity of f

(d) --,>If A = ----) _ ,i + 2 j 3k and _ , ----)B = ~ i .--~ - ,~j + 2k, then show_, ---) ----,! ,.. :that A + B is perpendicular to A - Band calculate the anglo

----) ----) ----) - - ,between 2A + B and A + 2B.


3/15

( 3) 12,/1PARTB

Each question carries 10 marks. 1 ( ) x 1 ( ) : = ; j O O2. Let W be the subspace of E ;4 spanned by the vectors

0'.1 = 0, 2, 2, 1), 0'.2 = (0, 2, 0, 1) and 0'.3 = (-2, 0, -4, ~n.Prove that 0'.1' u .2' (i3 form a basis for \V. Also, prove that

(1,0,2,0), (0,2,0, 1), (0,0,0,3)form a basis for \V.

a , State and prove the Cayley - Hamilton theorem.4. If y = (l + l/x!x, then evaluate y"(2),5. 2 2Determine the points on the curve 5x - 6xy -I- 5y = 4 that arc nearest

the origin.6. I,n) Find the asymptotes of the curve x2y2 - x2y - xy2 + x + y + 1 = {J .

ibJ Find the minimum value of x2 + y2 + z:2, given that x , y, z areall positive and xyz = 8.7. A variable' lint' is drawn through 0 to cut two fixed straight lines Ll and

L:.!;in Rand S. A point P is chosen 011 the variable line such thatm t n m n------ :::: ~ +OP on as

Show that tho locus of P IS a straight line passmg through the point ofintersection of L] and L-,.

8, elySo1\'(> x cos x + (x sin x + cos x: y = 1dx9, , 4 : '3 < ISolve (y + 2)') dx + (xy' ;-~ .Y - 4x I dy = 0

Turn ovc


4/15

, 2/1 (4 )10. (a)

---'I ---'; ---'; ---';If R :::: Xl + y j + zkcurl (r 100 R ) ---';= O.

---';and I R I = r. then show that

---'; ---'; ---';(b) Find the work done by the variable force F = 2y i + xy j on a

particle, when it is displaced from the origin to the point- - - ' ; , - > - - - ' ;R = ' 4 i + 2 j along the parabola y2 :::: x.

11. (a) A fluid motion is given by~ ---'; ---'; ---';v = (y + z) i + (z + x ) j + (x + y) k.

Is this motion irrot.ational ? Is the motion possible for anincompressible fluid?

(b) Evaluate f J (x dy dz + y dz dx + z dx dy) over the surface of ftsphere x2 + y2 + z2 ::; 4.


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( 5 ) 12/1PART C

Each question carries 15 marks. 6x15=:90

] 2. Let X be a subset of a vector space V over a field F. Prove that X is abasis for V if and only if any mapping of X into any vector space W overF can be uniquely extended to a linear transformation of V into W .

] 3. (a) Find the coordinates of the point on the curve x3 ::: y (x - a)2whose ordinate IS rmrnmum.

(b) If A > 0, B > and A + B ::: n/3, then find the maximumvalue of tan A tan B.

] 4. (a) Evaluateand y ::: x.

I f xy (x + y) dx dy over the area bounded by y ::: x2

(b) By using the transformation X + Y :::: u, y::: uv, evaluate1 I-xf f eY /x+y dx dyo 0

15. (a) Two circles have centres (a, 0) and (-a, 0) and radii band crespectively and a > b > c. Prove that the points of contact of thecommon tangents to the two circles lie on x2 + y2 ::: a2 be.

(b) Circles are drawn through the point (c, 0) touching the circlex2 + y2 ::: a2, If P is a point such that the tangents at theextremities of any chord passing through P intersect on the x-axis,Find the locus of P.

10. Solve the following :d2y dy

(a) -- + 2- + y ::: X cos xdx2 dx

(b) d2y2 + 16y ::: e-3x + cos 4xdx

[ Turn over


6/15

12/1 ( 6 )17. (a) Show that g.. dxl dxl IS an invariant, whereIJfundamental covariant function.

(b) If ~j is a skew symmetric tensor, then show that( o ~ o t + o ~ o f ) Aik = O .

(c) Prove that the magnitudes of associated vectors are equal.

IS the


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( 7 ) 1211PARTD

Answer any three of the following questions. Each question carries30 marks. 3x.30=90

18. (a) Find the point on the curve 4x2 + a2y2 = 4a2, where 4 < a2 < 8,that is farthest from the point (0, ~2).

(b) Show that the square roots of two successive natural numbersgreater than N2 differ by less than 1/2N.

(c) State Lagrange's mean value theorem and prove that_x~ < log (1 + x) < x for x > 0.1+ x

1lB. (a) Express f Xffi (1 - XO)P dx Interms of gamma function and

o1

hence evaluate f xS (1 - x3)10 dx.o

(b) Find the volume bounded by the cylinder x2 + y2 = 4 and theplanes y + z = 4 and z = O ,

(c) Find the centre of gravity of a plane lamina bounded byr = a (1 + cos 6).

20. (a) Suppose SY and S'Y' are the perpendiculars from the foci Sand S'upon the tangent at any point P of the ellipse

2 2x Y2+2=1.a bShow that Y and Y' lie on the circle with the major axis as diameter.

(b) Find the locus of the point of intersection of the tangents to theabove ellipse which are at right angles.

(c) Let P be a point on the above ellipse whose ordinate is y'. Provethat the angle between the tangent at P and the focal chordthrough P is tan-1 (b2/aey').

[ Turn eve.


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12/1 ( 8 )

21. (a) A uniform rod can move freely about one of its ends and is pulledaside from the vertical by a horizontal force acting at the other endof the rod equal to half its weight. Prove that the rod will rest atan inclination of 450 to the vertical.

(b) In a simple harmonic motion, the distances of a particle from themiddle point of its path at three consecutive seconds are observedto be x, y and z. Show that the time of complete oscillation is2n/cos-1 (x 2: z ) .

(c) A right circular cone of density p floats just immersed with itsvertex downwards in a vessel containing two liquids of densities a1and a2 respectively. Show that the plane of separation of two

liquids cuts off from the axis of the cone a fraction ( p - a2 J 1 I 3 ofa1 -a2its length.

---7 ---722. (a) Using divergence theorem evaluate f F. dS whereS

---7 3---7 3: 3---7F=xi +YJ +zkand S is the surface of the sphere x2 + y2 + z2 = a2.

(b) Establish the following:

(i) [ij, k 1 = g h k L h j )(ii) j ij, k ] + [jk, i] agi~a x J

(c) A uniform rod of length 1 rests in a vertical plane against asmooth horizontal bar at a height h, the lower end of the rod beingon the level ground. Show that if the rod is on the point ofslipping when its inclination to the horizon is e, then the coefficientf fricti b h d d h d' I sin 28 . sin 8o rictron etween t e ro an t e groun IS 4h -I sin 28 cos 8


9/15

2008MATHEMATICS

Paper 2

12/2

Time: 3 Hours] [Maximum Marks: 300

INSTRUCTIONS

Candidates should attempt all the questions in Parts A, B & C.However, they have to choose only three questions in Part D.Answers must be written in the medium opted (i.e. English orKannada).This paper has four parts:

A 20 marksB 100 marksc 90 marksD 90 marks

Marks allotted to each question are indicated in each part.


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12/2 ( 2 )PART A

Each question carries 5 marks. 1x5;;;;201. (a) Prove that the elements of a group G which commute with the

square of a given element b of G form a subgroup H of G .. 1 ( 1 1 1 )im r 1+ In + rn + ... + 1- .n- - - - * 0 0 v n v 2 " 3 v n

(c) Find the product of inertia of a rectangular plate of sides a, babout two adjacent sides.

(b) Find

(d) Apply Runge - Kutta method to find an approximate value of y forx ;;;;02 if dy = x + y2, y;;;; 1 when x;;;; O.dx


11/15

(3 ) 12/2PAUTB

Each question carries 10 marks. 10xl0~1()O2. Show that the set of non-zero residue classes modulo a prime integer p

forms an abelian group of order p - 1 with respect to multiplication ofresidue classes.

a. If Hand K are finite subgroups of a group G, then show thatO(HK) = O(H) O(K)O(H n K) ,

where O(H) stands for the order of H.4. (a) If {an} is monotonically increasing and bounded, then show that

{an} is convergent.

(b) Show that f (x) = 1x is not uniformly continuous on (0, 1).5. State and prove Cauchy's integral formula for analytic functions.

6. (a) Form a partial differential equation by eliminating the arbitraryfunction f [rom the relation

2 1z = y + 2 [( - + log y)x(b) Solve q2y2 = z (z - px), Further, show that there IS no singular

solution. .

7. Find the moment of inertia of a circular plate about a tangent.

H. Describe the procedure to solve an equation by Regula Falsi method. Findthe root of the equation 2x - loglO x = 7 which lies between 3[) and 4correct to 5 places of decimals using the method of false position.

i TUITI ever


12/15

12/2 ( 4)

9. State and prove Newton's forward interpolation formula.Using Newton's forward interpolation formula and given the table ofvalues,

x 11 13 15 17 19y 021 069 125 189 261

obtain the values of y, when x :::::14.10. Explain Trapezoidal rule for numerical integration.

Evaluate Idxf 1+ xosub-intervals of the interval [0, 1 1 . Hence find an approximate of log 2.

by Trapezoidal rule by considering eight

11. (a) Employing Euler's method, find the approximate solution of theinitial value problem

dy y-x= y e o ) : : : : :1dx y+xat x = 01, by taking h = 002.

(b) By using the Milne's predictor-corrector method, find anapproximate solution of the equation

dy = 2Ydx x' X - : : t . 0at the point x::: 2, given that y(1)::: 2, y(1'25) = 313, yf l-fi) = 45and y(175) = 613.


13/15

( 5 ) 12/2PARTe

Each question carries 15 marks. 6x15:=9012. State and prove Cauchy's theorem for a triangle or a rectangle.

13. Let f be a bounded function defined on E = [a, b] x [c, d]. Let f : E - - - - 7 Rbe continuous. Then, show that

14. Show that every integral domain can be imbedded in a field.15. (a) Find the interpolating polynomial that approximates the function

given byx: o

316

211

318

427(x) :

(b) A function y = f'(x) is specified by the following tablex : 1 12 14 16 18 200y : 000 o - 128 0544 1296 2432 400

Find the approximate values of ['(11) and f"(1 1).Hi. If (u, v, w) are orthogonal curvilinear coordinates, then show that

( ar ar ar 1 are reciprocal to the vectors (Vu, Vv, Vw), with usualau' Jv' aw)notations.

17. Find the parabola of the form y = a + bx + cx2 which fits most closelywith the following observation:

X' -3 -2 -1 o 1 2y: 463 211 067 009 063 215 458

I Tum over


14/15

12/2 (6 )

PARTDAnswer any three of the following questions. Each question carries30 marks. 3x30=.90

18. (a) State and prove Cauchy's residue theorem.2rt dt(b) Evaluate f J5 + cos taoo dx(c) Evaluate f 1+ x40

19. (a) Show that closed subset of a complete metric space is complete.(b) Show that a metric space is complete if and only if every nestedsequence of closed sets (Fnl with diameter, d(Fn) ~ 0 has a

non-empty intersection containing precisely one point.(c) Let (X , d) be a complete metric space. If T is a contraction on X ,

then show that there is only one x in X , such that 'I'(x) z: x.20. (a) If H IS a subgroup of G and N is a normal subgroup of G, then

show that H n N is a normal subgroup of H.(b) If a cyclic subgroup N of G is normal in G, then show that every

subgroup of N is a normal subgroup of G.(c) Let G be a group and a E G. Then Nra) = lx E G I ax = xa] IS

a subgroup of G.(d) Let G be a group and Z = (z E G I zx = XZ, \;j X E G). Then show

that Z is a normal subgroup of G.21. (a) Describe Euler's method of solving the initial value problem

dy- ::::f(x, y), y (xo) = Yodx2Use the Euler's method to solve the differential equation s ' = -2xy

subject to the condition yeO) = 1for values of x from 025 to 1 at stepsof length 025.


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( 7 ) 12/2

(b) Employing modified Euler's method solve the initial value problemy' ::: ~y; yCO)::: 1 for x::: 02 and 04.

(c) By Runge ~ Kutta method of order 4, solve the equationdy :::3x + Y with y(O)::: 1 for yCO2), taking step length h::: 01.dx 2

22. (a) Derive Poisson Distribution from Binomial Distribution.(b) Find the mean and standard deviation of the Poisson Distribution.(c) Show that in a Poisson Distribution with Unit Mean, the mean

deviation about the mean is [~] times the standard deviation.

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