Mathematics is one of the optionals in the UPSC Civil Services IAS Main Written Exam. The catechism cardboard of Mathematics Cardboard I of UPSC IAS Main Written Assay 2015 has been provided here.Mathematics alternative consists of two affidavit – Cardboard I and Cardboard II. The Cardboard I of Mathematics alternative is of 250 marks. The continuance of the assay is 3 hours. Candidates can through the antecedent year catechism cardboard to accept the blazon of questions asked by UPSC in UPSC Civil Services IAS Main Written Assay 2015.Read More:

Mathematics Cardboard I (2015)

SECTION – A

1. Acknowledgment the afterward questions:

(a) The vectors V₁ – (1, 1, 2, 4), V2 – (2, – I , -5 , 2), V 3 = (1, -1 , -4 , 0) and V₄ = (2, 1, 1 ,6 ) are linearly independent. Is it true? Justify your answer. (10)(b) Reduce the afterward cast to row degree anatomy and appropriately acquisition its rank: (10)(c) Evaluate the afterward limit: (10)(d) Evaluate the afterward integral: (10)(e) For what absolute bulk of a, the even ax – 2y z 12 = 0 touches the apple x² y² z² – 2x – 4y 2z – 3 = 0 and appropriately acquisition the point of contact. (10)

2. Acknowledgment the afterward questions:

(a) If matrixthen acquisition A³⁰. (12)(b) A conical covering is of accustomed capacity. For the atomic bulk of Canvas required, for it, acquisition the arrangement of its acme to the ambit of its base. (13)(c) Acquisition the eigen ethics and eigen vectors of the matrix: (12)(d) If 6x = 3y = 2z represents one of the three mutually erect generators of the cone 5yt – 8zx – 3xy = 0 afresh access the equations of the added two generators. (13)

3. Acknowledgment the afterward questions:

(a) Let V = R³ and T Є A(V), for all a₁ Є A(V), be authentic byT(a₁, a₂, a₃) = (2a₁ 5a₂ a₃, – 3a₁ a₂ – a₃, – a₁ 2a₂ 3a₃)What is the cast T about to the basisV₁ = (1, 0, 1) V₂ = (-1, 2, 1) V₃ = (3, -1, 1)? (12)

(b) Which point of the apple x² y² z² = 1 is at the best ambit from the point (2, 1, 3)? (13)

(c) (i) Access the blueprint of the even casual through the credibility (2, 3 ,1 ) and (4, -5, 3) alongside to x-axis. (6)(c) (ii) Verify if the lines: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, afresh acquisition the blueprint of the even in which they lie.(7)

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

4. Acknowledgment the afterward questions:

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

(b) Acquisition the ambit of the subspace of R⁴, spanned by the set{(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)} (12)

(c) Two erect departure planes to the paraoid x² y² = 2z bisect in a beeline band in the even x = 0. Access the ambit to which this beeline band touches. (13)

(d) For the functionExamine the alternation and differentiability. (12)

SECTION – B

5. Acknowledgment the afterward questions:

(a) Break the cogwheel equation: (10)xcosx dy/dx y(x sinx = cosx)= 1.(b) Break the cogwheel equation: (10)(2xy⁴eᵞ 2xy³ y)dx (x²y² eᵞ – x²y² – 3x)dy = 0,(c) A anatomy affective beneath SHM has an amplitude ‘a’ and time aeon ‘T’. If the acceleration is trebled, back the ambit from beggarly position is ‘2/3a’ the aeon actuality unaltered, acquisition the new amplitude. (10)(d) A rod o f 8 kg is adaptable in a vertical even about a articulation at one end, addition end is attached a weight according to bisected of the rod, this end is attached by a cord of breadth / to a point at a acme b aloft the articulation vertically. Access the astriction in the string. (10)(e) Acquisition the bend amid the surfaces x² y² z² – 9 = 0 and z = x² y² – 3 at (2, -1, 2). (10)

6. Acknowledgment the afterward questions:

(a) Acquisition the connected a so that (x y)ᵃ us the Integrating agency of (4x² 2xy 6y)dx (2x² 9y 3x)dy = 0 and appropriately break the cogwheel equation. (12)(b) Two according ladders of weight 4 kg anniversary are placed so as to angular at A adjoin anniversary added with their ends comatose on a asperous floor, accustomed the accessory of abrasion is μ. The ladders at A accomplish an bend 60° with anniversary other. Acquisition what weight on the top would account them to slip. (13)(c) Acquisition the bulk of λ and μ so that the surfaces λx² – μyz = (λ 2)x and 4x²y z³ = 14 may bisect orthogonally at (1, -1, 2). (12)(d) A accumulation starts from blow at a ambit ‘a’ from the centre of force which attracts inversely as the distance. Acquisition the time of accession at the centre. (13)

7. Acknowledgment the afterward questions:

(a) (i) Access Laplace Inverse transform of(a) (ii) Using Laplace transform, break (6 6=12)y’’ y = t, y(0) = 1, y’(0) = -2.

(b) A atom is projected from the abject of a acropolis whose abruptness is that of a appropriate annular cone, whose arbor is vertical. The projectile grazes the acme and strikes the acropolis afresh at a point on the base. If the semi vertical bend of the cone is 30°, h is height, actuate the antecedent acceleration u of the bump and its bend of projection. (13)

(c) A agent acreage is accustomed byVerify that the acreage F is irrotational or not. Acquisition the scalar potential. (12)

(d) Break the cogwheel blueprint (13)x = py – p² area p = dy/dx

8. Acknowledgment the afterward questions:

(a) Acquisition the breadth of an amaranthine alternation which will adhere over a annular caster of ambit ‘a’ so as to be in acquaintance with the two-thirds of the ambit of the pulley. (12)(b) A atom moves in a even beneath a force, appear a anchored centre, proportional to the distance. If the aisle of the atom has two apsidal distances a, b (a > b), afresh acquisition the blueprint of the path. (13)(c) Evaluate(sin y dx cosy dy), area C is the rectangle with vertices (0, 0), (π, 0), (π, π/2), (0, π/2). (12)(d) Solve: (13)x⁴ d⁴y/dx⁴ 6x³ d³y/dx³ 4x² d²y/dx² – 2x dy/dx – 4y = x² 2cos (logₑx).

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Importance of Mathematics Catechism Paper

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