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# IMS Maths Optional Test Series

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## IMS Maths Optional Test Series

Contents

#### Cover Test Series Topics

1. IMS Maths Optional Test Series 1 2020
2. IMS Maths UPSC Optional Test Series 2 2020
3. IMS Maths IAS Optional Test Series 3 2020
4. IMS Maths Optional Test Series 4 2020
5. IMS Maths Optional Test Series 5  2020
6. IMS Maths Optional Test Series 6  2020
7. IMS Maths Optional Test Series 7  2020
8. IMS Maths Optional Test Series 8 2020
9. IMS Maths Optional Test Series 9 2020
10. IMS Maths Optional Test Series 20 2020

### Click This Link Pdf Notes: IMS Maths Notes Complete Study materials

UPSC Maths Optional Syllabus For Paper I
PAPER – I

(1) Linear Algebra: Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; linear transformations, rank and nullity, matrix of a linear transformation. Algebra of Matrices; row and column reduction, echelon form, congruence and similarity; rank of a matrix; the inverse of a matrix; solution of system of linear equations; eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.

(2) Calculus: Real numbers, functions of a real variable, limits, continuity, differentiability, mean value theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes; curve tracing; functions of two or three variables: limits, continuity, partial derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian. Riemann’s definition of definite integrals; indefinite integrals; infinite and improper integrals; double and triple integrals (evaluation techniques only); areas, surface and volumes.

IMS Maths Optional UPSC Test Series

(3) Analytic Geometry: Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to canonical forms, straight lines, shortest distance between two skew lines; plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

(4) Ordinary Differential Equations: Formulation of differential equations; equations of first order and first degree, integrating factor; orthogonal trajectory; equations of first order but not of first degree, Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, complementary function, particular integral and general solution. Second order linear equations with variable coefficients, Euler-Cauchy equation; determination of complete solution when one solution is known using method of variation of parameters. Laplace and inverse Laplace transforms and their properties; Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.

(5) Dynamics & Statics: Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion; work and energy, conservation of energy; Kepler’s laws, orbits under central forces. Equilibrium of a system of particles; work and potential energy, friction; common catenary; principle of virtual work; stability of equilibrium, equilibrium of forces in three dimensions.

(6) Vector Analysis: Scalar and vector fields, differentiation of vector field of a scalar variable; gradient, divergence and curl in cartesian and cylindrical coordinates; higher order derivatives; vector identities and vector equations. Application to geometry: curves in space, curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.

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