Remedial Mathematics — Summer 2025

Define modulus function and identity of function.

Differentiate f(x) = x²sin(x) using the product rule.

Solve the following system of linear equations using Cramer's Rule: • x + y + z = • 2x - y + z = • x + 2y + 3z =

Find the equation of the line passing through (-3,5) and perpendicular to the line through the points (2,5) and (-3,6)

1 2 0 1 Prove A.B ≠ B. Afor A = ( ) & B = ( )

3 1 If A = [ ] ,sℎow tℎat A2 −5A+7I. −1 Hence find A−1

dy

Find , if x = acos𝜃, y = asin𝜃 dx (ii) Derive the trigonometry identity: sin2x = 2sinxcosx

sinx+sin3x Prove tℎat: = tan2x cosx+cos3x

ax +xcosx Find tℎe limit of lim x→0 bsinx

i) Find the distance between the parallel lines 3x - 4y + 7 = 0 and 3x - 4y + 5 = 0 ii) The slop of a line is double of the slop of another line. If tangent of the1 angle between them is , find the slop of the lines3 OR1

Define the matrices (i) Zero matrix (ii) Scalar matrix (iii) Transpose of a matrix

State the Quotient (division) rule of derivative and compute the derivative of x+1 x −1

i)Evaluate: lim(cosecx−cotx ) x→0 x+cosx ii) Find tℎe derivate of ∶ tanx

Determine the order of the differential equations:

dy +sinx = 10 ii) y" −secx3 = y′ dx

Find the general solution of: dy = (1+ x2 ) (1 + y2 ) dx

Find a particular solution of the differential equation dy +ycotx = 4x cosec x (x ≠ 0) dx given that y = 0 when x = π/2

Find the second derivative of the function (x+1). sinx

Solve the differential equation: extany dx +(1−ex)sec2y dy = 0

Show that the differential equation (x- y) dy = (x+2y) dx is homogeneous and solve it.

Evaluate: ∫(2x2+ex)dx

Evaluate the definite integral ∫ 2 (4x3-5x2+6x +9) dx1

1 2x+3 Evaluate the definite integral: ∫ dx 0 5x2+1

Find ∫ 1−cosx dx 1+cosx

Integrate the function: xsin3x (By part method)

3x −1 ∫ dx (x−1)(x−2)(x−3)