HSC Arts इयत्ता १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions

If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k"` and `"c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, prove that c is the geometric mean of a and b

Let `vec"a",  vec"b",  vec"c"` be three non-zero vectors such that `vec"c"` is a unit vector perpendicular to both `vec"a"` and `vec"b"`. If the angle between `vec"a"` and `vec"b"` is `pi/6`, show that `[(vec"a", vec"b", vec"c")]^2 = 1/4|vec"a"|^2|vec"b"|^2`

Write the Maclaurin series expansion of the following functions:

e^x

Write the Maclaurin series expansion of the following functions:

sin x

Write the Maclaurin series expansion of the following functions:

cos x

Write the Maclaurin series expansion of the following functions:

log(1 – x); – 1 ≤ x ≤ 1

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)

Evaluate the following:

`int_0^1 x^3"e"^(-2x)  "d"x`