HSC Science Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

Write the Maclaurin series expansion of the following functions:

tan^–1 (x); – 1 ≤ x ≤ 1

Write the Maclaurin series expansion of the following functions:

cos²x

Write down the Taylor series expansion, of the function log x about x = 1 upto three non-zero terms for x > 0

Expand sin x in ascending powers `x - pi/4` upto three non-zero terms

Expand the polynomial f(x) = x² – 3x + 2 in power of x – 1

Evaluate `lim_((x,  y) -> (1,  2))  "g"(x, y)`, if the limit exists, where `"g"(x, y) = (3x2 - xy)/(x^2 + y^2 + 3)`

Evaluate `lim_((x,  y) -> (0,  0)) cos((x^3 + y^2)/(x + y + 2))` If the limits exists

Let f(x, y) = `(y^2 - xy)/(sqrt(x) - sqrt(y))` for (x, y) ≠ (0, 0). Show that `lim_((x,  y) -> (0,  0)) "f"(x,  y)` = 0

Evaluate `lim_((x,  y) -> (0,  0)) cos(("e"^x sin y)/y)`, if the limit exists

Let g(x, y) = `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) = 0. Show that `lim_((x,  y) -> (0,  0)) "g"(x,  y)` = 0 along every line y = mx, m ∈ R

Let g(x, y) = `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) = 0. Show that `lim_((x,  y) -> (0,  0)) "g"(x,  y) = "k"/(1 + "k"^2)` along every parabola y = kx², k ∈ R\{0}

Show that f(x, y) = `(x^2 - y^2)/(y - 1)` s continuous at every (x, y) ∈ R² 

Let g(x, y) =  `("e"^y  sin x)/x` for x ≠ 0 and g(0, 0) = 1 shoe that g is continuous at (0, 0)

Find the differential equation of the family of all non-vertical lines in a plane

Find the differential equation of the family of all non-horizontal lines in a plane 

Form the differential equation of all straight lines touching the circle x² + y² = r²

Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis

Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis

Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin

Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be ^–8x, where A and B are arbitrary constants