Vector ⚫Introduction ⚪ Those quantities which have both magnitudes and direction are called Vector . Example : velocity ,accerelation , weight , force are some vector quantities. ⚫ Representation of vectors : ⚪ Geometrically a vector is represented by a line segment. For example , . Here A is called the initial point and B is terminal point. Magnitude or modulus of a is expressed as ⚫ Types of Vector ⚪ Zero vector or null vector : A vector whose magnitude is zero is called zero or null vector. ⚪ Unit Vector : A vector whose magnitude is unity (1). The unit vector in the direction of a vector a is denoted by . ⚪ Collinear or Parallel Vector : Vectors having the same or parallel supports are called collinear or parallel vectors ⚪ Co-initial Vector : Vectors having the same initial point. ⚪ Coplanar Vector : A system of vectors is said to be coplanar, if their support are p...

CSAT Practice Question Question no. 21 What is the least four digit number when divided by 3 , 4 , 5 and 6 leaves a remainder 2 in each case ? [CSAT 2020] looks_one 1012 looks_two 1022 looks_3 1122 looks_4 1222 Answer Option looks_two 1022 Solution Solution : Question no. 22 The recurring decimal representation 1.272727.... is equivalent to [CSAT 2020] looks_one 13/11 looks_two 14/11 looks_3 127/99 looks_4 137/99 Answer Option looks_two 14/11 Solution Solution : Question no. 23 What is the greatest length x such that and are integral multiplers of x ? [CSAT 2020] looks_one looks_two looks_3 looks_4 Answer Option looks_4 Solution Solution : Question no. 24 Consider the following data : Year birth rate death rate 1911 - 1921 48.1 35.5 1921 - 1931 46.4 36.3 1931 - 1941 45.2 31.2 1941 - 1951 39.9 27.4 1951 - 1961 ...

Linear Algebra Question no. 1 What are the eigen values of ? looks_one 1,4 looks_two 2,3 looks_3 0,5 looks_4 1,5 Answer Option looks_3 0,5 Solution Solution : ( 4 - λ)(1 - λ) - 4 = 0 λ 2 - 5λ + 4 - 4 = 0 λ 2 - 5λ = 0 λ(λ - 5) = 0 λ = 0 , 5 Question no. 2 What is the eigenvalues of the matrix ? looks_one 1, 4 ,4 looks_two 1 , 4 ,-4 looks_3 3 , 3 ,3 looks_4 1 , 2 ,6 Answer option looks_one 1, 4 ,4 Solution Solution : ( 3 - λ )[(3-λ) 2 -1] + 1(-3 + λ -1 ) - 1(1+3 -λ) = 0 ( 3 - λ )(9 + λ 2 - 6λ -1 ) + 2( λ - 4 ) = 0 ( 3 - λ )( λ 2 - 6λ + 8 ) + 2( λ - 4 ) = 0 ( 3 - λ )( λ - 4)( λ - 2) + 2( λ - 4 ) = 0 (λ - 4) [( 3 - λ )( λ - 2) + 2 ] = 0 (λ - 4) (λ 2 - 5λ + 4) = 0 (...

Eigen values and Eigen Vector ⚫Introduction ⚪ For a given square matrix A ► A - λI matrix is called Characteristic Matrix ► | A - λI | = 0 is Characteristic Equation ► The roots of | A - λI | = 0 are eigen values Example : Characteristic Equation : Eigen Values or Characteristics Equation Roots : λ 3 - 7λ 2 + 11λ - 5 = 0 (λ - 1)(λ-1)(λ - 5) =0 λ = 1 ,1 ,5 ⚫ Important Properties Of Eigenvalues ► Any Square matrix A and its transpose A' have the same eigenvalue ► The sum of eigenvalues is equal to the trace of the matrix (sum of principal diagonal element) ► Product of eigenvalues is equal to the determinant of the matrix ► If λ 1 , λ 2 , λ 3 ...... λ n are the eigenvalues of A , then the Eigenvalues of    ⚪ k A are k λ 1 , k λ 2 , k λ 3 ...... k λ n    ...

Indefinite integration Question no. 31 What is ∫ x/(1-xcotx) dx is equal to ? looks_one log(cosx- xsinx) + c looks_two log(xsinx-cosx) + c looks_3 log(sinx - xcosx) + c looks_4 none of these Answer Option looks_3 log(sinx - xcosx) + c Solution Solution : log(sinx - xcosx) + c Question no. 32 What is ∫ (x-2)/(x 2 -4x+3) dx equal to ? looks_one log√(x 2 -4x+3) + c looks_two xlog(x-3) - 2log(x-2) +c looks_3 log(x-3)(x-1) +c looks_4 none of these Answer option looks_one log√(x 2 -4x+3) + c Solution Solution : ∫ (x-2)/(x 2 -4x+3) dx Let x 2 -4x+3 = t Differentiate w.r.t x (2x-4)dx = dt Put the value of dx in following integration ∫ 1/2t dt (1/2)log(t) + c log(t) 1/2 + c log(x 2 -4x+3) 1/2 + c Question no. 33 What is ∫ 3x 2 /(x 6 +1) dx equal to ? looks_one log(1+x 6 ) + c looks_two tan -1 (x 3 ) + c looks_3 3tan -1 (x 3 ) + c l...