Remedial Mathematics — Winter 2024

Find the equation of line and slope the points (-1,1) and (2,-4).

x3−x2+5x+6 Find the limits: lim x→1 x2−5x+6

dy

Find if x = 2cost −cos2t and y = 2sint −sin2t dx 2 3 3 −1 (ii) Find A.Bif A = [ ] and B = [ ] 1 4 1 −6

Derive the trigonometry identity: sin3x = 3sinx−4sin3x

4 −2 1 Find the Inverse of Matrix A = [ 5 0 3] −1 2

Solve the system of linear equations 2x −3y+5z = 11,3x +2y−4z = −5,x+y−2z = −3

(i) Define (a) Exponential function (b) Modulus function (c) Domain of function.04 (ii) Find the distance of the point (-3, 5) from the line 4x – 3y – 26 = 0.

State the multiplication rule of derivative and compute the derivative of f(x)= (x2+2x+3)(tanx)

Find the angle between the lines x +2y−5 = 0 and 3y−x +6 = 0

1 −2 0

Find the determinant of A = [2 −4 5] (ii) Verify (AB)T = BTAT for A = [ ],B = [ ]

Define (a) Symmetric Matrix (b) Skew symmetric matrix (c) Null Matrix

d2y dy If y = log (sinx) then prove that +( ) +1 = 0 dx2 dx

(i) If y = log (sinx2) then find dy dx x (ii) If f(x)= logx prove that (i)f(xy) = f(x)+f(y) (ii) f( )= f(x)−f(y) y

Determine the order and degree of the following differential equations d4y d2y dy d2y 4 dy

+( )−3 +y = 9 (ii) ( ) + = dx4 dx2 dx dx2 dx (iii) d3y +3x dy = ey dx3 dx

Solve the differential equations (x−xy2)dx = (y−x2y)dy

Solve the following equations: dy

Solve +ytanx = sin2x dx (ii) Solve (2x−3y)dx+(1−3x)dy = 0

Define (i) Differential equation (ii) Order (iii) Degree with example.

dy 1 Find the solution of = dx e2y+ x

Solve the following equations:

(1−3x)2 Find the integrals ∫ dx x3

𝜋 Evaluate the integrals: (i) ∫x logx dx (ii) ∫4cos2xdx0

x2 ∫ dx (x2−1)(x2−3)

14x3−5x+3 Find the integrals ∫ dx 0 √x

𝜋 sinx Find ∫3 dx 0 3+4cosx

Find ∫ 8 x dx 3 (x−1)(x−2)