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...

Indefinite integration Question no. 16 What is ∫ sin -1 x + cos -1 x dx is equal to ? looks_one 0.5πx + c looks_two x(sin -1 x + cos -1 x) + c looks_3 x(sin -1 x - cos -1 x) + c looks_3 π/2 + x + c Answer Option looks_one 0.5πx + c Solution Solution : ∫ sin -1 x + cos -1 x dx ∫ π/2 dx 0.5πx + c Question no. 17 What is ∫ (x-1)/(x+1) 2 dx equal to ? looks_one log(x+1) + 2/(x+1) +c looks_two log(x+1) - 2/(x+1) +c looks_3 -log(x+1) + 2/(x+1) +c looks_4 none of these Answer Option looks_one log(x+1) + 2/(x+1) +c Solution Solution : ∫ (x-1)/(x+1) 2 dx x + 1 = t ⇒ x = t -1 ⇒ dx = dt ∫ (t-2)/t 2 dt ∫ 1/t - 2/t 2 dt log t + 2/t + c log(x+1) + 2/(x+1) +c Question no. 18 What is ∫ (1+ 2x +3x 2 +4x 3 ...........) dx equal to ? looks_one (1+x) -1 + c looks_two (1-x) -1 + c looks_3 (1-x) -1 ...

Circle Question no. 31 If the chord of contact of tangents drawn from a point on the circle x 2 + y 2 = a 2 to the circle x 2 + y 2 = b 2 touches the circle x 2 + y 2 = c 2 , then a ,b ,c are in : looks_one G.P. looks_two H.P. looks_3 A.P. looks_4 none of these Answer looks_one G.P. Solution Solution : The point (x 1 , y 1 ) from which the tangent are drawn is on circle x 2 + y 2 = a 2 , it means x 1 2 + y 1 2 = a 2 Equation of chord : xx 1 + y y 1 = b 2 The chord will be the tangent to the circle x 2 + y 2 = c 2 distance from the origin to that tangent is equal to radius 'c' Distance of line (xx 1 + y y 1 = b 2 ) from origin : Question no. 32 what is the equation to circle which touches both the axes and has centre on the line x+y-4=0 ? looks_one x 2 + y 2 +4x -4y +4 = 0 looks_two x 2 + y 2 +4x +4y +4 = 0 looks_3 x 2 + y 2 -4x -4y +4 = 0 looks_4 x 2 + y 2 -4x +4y ...