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u.s.a. Civil features IAS leading Written examination 2015: arithmetic Paper I

arithmetic is likely one of the optionals in the america Civil functions IAS main Written examination. The query paper of mathematics Paper I of united states IAS main Written exam 2015 has been provided right here.arithmetic not obligatory includes two papers – Paper I and Paper II. The Paper I of mathematics optional is of 250 marks. The length of the exam is 3 hours. Candidates can during the previous yr query paper to keep in mind the class of questions asked by means of usa in  u.s.a. Civil capabilities IAS leading Written exam greater:

mathematics Paper I (2015)

section – A

1. reply here questions:

(a) The vectors V₁ – (1, 1, 2, four), V2 – (2, – I , -5 , 2), V three = (1, -1 , -four , 0) and V₄ = (2, 1, 1 ,6 ) are linearly impartial. Is it actual? Justify your reply. (10)(b) reduce the following matrix to row echelon kind and hence locate its rank: (10)1b I(c) evaluate here limit: (10)1c I(d) evaluate the following imperative: (10)1d I(e) For what advantageous cost of a, the plane ax – 2y + z + 12 = 0 touches the sphere x² + y² + z² – 2x – 4y + 2z – three = 0 and hence discover the point of contact. (10)

2. answer the following questions:

(a) If matrix2a Ithen find a³⁰. (12)(b) A conical tent is of given potential. For the least amount of Canvas required, for it, locate the ratio of its peak to the radius of its base. (13)(c) find the eigen values and eigen vectors of the matrix:  (12)2c I(d) If 6x = 3y = 2z represents some of the three mutually perpendicular generators of the cone 5yt – 8zx – 3xy = 0 then reap the equations of the other two turbines. (13)

3. reply right here questions:

(a) Let V = R³ and T Є A(V), for all a₁ Є A(V), be described byT(a₁, a₂, a₃) = (2a₁ + 5a₂ + a₃, – 3a₁ + a₂ – a₃, – a₁ + 2a₂ + 3a₃)what's the matrix T relative to the basisV₁ = (1, 0, 1) V₂ = (-1, 2, 1) V₃ = (three, -1, 1)? (12)

(b) Which aspect of the sphere x² + y² + z² = 1 is at the maximum distance from the element (2, 1, 3)? (13)

(c) (i) reap the equation of the plane passing through the facets (2, three ,1 ) and (four, -5, 3) parallel to x-axis. (6)(c) (ii) assess if the strains:x – a + d/α – δ = y – a/α = z- a- d/α + δ and x – b + c/β – γ = y – b/β = z – b – c/β + γare coplanar. If yes, then locate the equation of the plane through which they lie.(7)

(d) evaluate the integralʃʃ(x – y)² cos² (x + y) dx dywhere R is the rhombus with successive vertices as (π, 0) (2 π , π) (π, 2 π) (0, π). (12)

four. reply right here questions:

(a) evaluate ʃʃR √ |y – x²| dx dyWhere R = = [-1, 1 ; 0, 2]. (13)

(b) locate the dimension of the subspace of R⁴, spanned by using the set(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1) (12)

(c) Two perpendicular tangent planes to the paraboloid x² + y² = 2z intersect in a straight line within the plane x = 0. reap the curve to which this straight line touches. (13)

(d) For the characteristic4d Iexamine the continuity and differentiability. (12)

part – B

5. answer right here questions:

(a) clear up the differential equation: (10)xcosx dy/dx+y(x sinx = cosx)= 1.(b) remedy the differential equation: (10)(2xy⁴eᵞ + 2xy³ + y)dx + (x²y²+eᵞ – x²y² – 3x)dy = 0,(c) A physique relocating beneath SHM has an amplitude ‘a’ and time length ‘T’. If the velocity is trebled, when the gap from imply place is ‘2/3a’ the length being unaltered, discover the new amplitude. (10)(d) A rod o f eight kg is movable in a vertical airplane a couple of hinge at one conclusion, yet another conclusion is fastened a weight equal to half of the rod, this conclusion is fixed with the aid of a string of size / to a point at a peak b above the hinge vertically. reap the tension within the string. (10)(e) locate the angle between the surfaces x² + y² + z² – 9 = 0 and z = x² + y² – three at (2, -1, 2). (10)

6. reply the following questions:

(a) discover the constant a so that (x + y)ᵃ us the Integrating element of (4x² + 2xy + 6y)dx+ (2x² + 9y + 3x)dy = 0 and hence clear up the differential equation. (12)(b) Two equal ladders of weight four kg each are positioned with a purpose to lean at A in opposition t every different with their ends resting on a tough ground, given the coefficient of friction is μ. The ladders at A make an perspective 60° with each and every different. discover what weight on the accurate would cause them to slip. (13)(c) locate the price of λ and μ in order that the surfaces λx² – μyz = (λ + 2)x and 4x²y + z³ = 14 may additionally intersect orthogonally at (1, -1, 2). (12)(d) A mass begins from leisure at a distance ‘a’ from the centre of drive which attracts inversely as the distance. discover the time of arriving on the centre. (13)

7. reply the following questions:

(a) (i) obtain Laplace Inverse transform of7ai I(a) (ii) the use of Laplace transform, resolve (6+6=12)y’’ + y = t, y(0) = 1, y’(0) = -2.

(b) A particle is projected from the base of a hill whose slope is that of a right round cone, whose axis is vertical. The projectile grazes the vertex and strikes the hill once again at some extent on the base. If the semi vertical perspective of the cone is 30°, h is peak, check the initial speed u of the projection and its angle of projection. (13)

(c) A vector container is given through7c Iverify that the box F is irrotational or no longer. find the scalar expertise. (12)

(d) clear up the differential equation (13)x = py – p² where p = dy/dx

8. answer the following questions:

(a) locate the size of an limitless chain so that it will cling over a round pulley of radius ‘a’ so as to keep up a correspondence with both-thirds of the circumference of the pulley. (12)(b) A particle strikes in a aircraft below a force, in opposition t a fixed centre, proportional to the space. If the route of the particle has two apsidal distances a, b (a > b), then find the equation of the direction. (13)(c) consider8c I(sin y dx +cosy dy), where C is the rectangle with vertices (0, 0), (π, 0), (π, π/2), (0, π/2). (12)(d) clear up: (13)x⁴ d⁴y/dx⁴ + 6x³ d³y/dx³ + 4x² d²y/dx² – 2x dy/dx – 4y = x² + 2cos (logₑx).

click on here to find PDF of arithmetic Paper  I question Paper- mathematics Paper I

value of mathematics query Paper

it's crucial for the candidates to have a glance on the outdated years question papers of arithmetic non-compulsory of usa IAS Civil functions leading Written exam.  it's going to assist the candidates in comprehending the level of capabilities in addition to discerning the most critical subject matters for mathematics non-compulsory of u.s. IAS Civil functions leading Written examination. Candidates should still always determine for news, updates and guidance for u.s. IAS Civil functions examination.

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