HSC Commerce इयत्ता ११ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Business Mathematics and Statistics

By the principle of mathematical induction, prove the following:

5²ⁿ – 1 is divisible by 24, for all n ∈ N.

By the principle of mathematical induction, prove the following:

n(n + 1) (n + 2) is divisible by 6, for all n ∈ N.

By the principle of mathematical induction, prove the following:

2ⁿ > n, for all n ∈ N.

The term containing x³ in the expansion of (x – 2y)⁷ is:

If u = e^xy, then show that `(del^2"u")/(delx^2) + (del^2"u")/(del"y"^2)` = u(x² + y²).

Let u = x cos y + y cos x. Verify `(del^2"u")/(delxdely) = (del^"u")/(del"y"del"x")`

Verify Euler’s theorem for the function u = x³ + y³ + 3xy².

Let u = x²y³ cos`(x/y)`. By using Euler’s theorem show that `x*(del"u")/(delx) + y * (del"u")/(dely)`

Let u = `log (x^4 - y^4)/(x - y).` Using Euler’s theorem show that `x (del"u")/(del"x") + y(del"u")/(del"y")` = 3.

If u = 4x² + 4xy + y² + 4x + 32y + 16, then `(del^2"u")/(del"y" del"x")` is equal to:

If u = x³ + 3xy² + y³ then `(del^2"u")/(del "y" del x)`is:

If u = `e^(x^2)` then `(del"u")/(delx)` is equal to:

If q = 1000 + 8p₁ – p₂ then, `(del"q")/(del "p"_1)`is:

Evaluate the following using binomial theorem:

(101)⁴

Evaluate the following using binomial theorem:

(999)⁵

Expand the following by using binomial theorem.

(2a – 3b)⁴

Expand the following by using binomial theorem.

`(x + 1/y)^7`

Expand the following by using binomial theorem.

`(x + 1/x^2)^6`

Find the 5th term in the expansion of (x – 2y)¹³.

Find the middle terms in the expansion of

`(x + 1/x)^11`