HSC Science कक्षा ११ - Tamil Nadu Board of Secondary Education Question Bank Solutions

Using the Mathematical induction, show that for any natural number n ≥ 2,
`1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 +2 + 3 + 4) + .... + 1/(1 + 2 +  3 + ... + "n") = ("n" - 1)/("n" + 1)`

Using the Mathematical induction, show that for any natural number n,
`1/(1*2*3) + 1/(2*3*4) + 1/(3*4*5) + ... + 1/("n"("n" + 1)*("n" + 2)) = ("n"("n" + 3))/(4("n" + 1)("n" + 2))`

Using the Mathematical induction, show that for any natural number n,
`1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"n" - 1)(3"n" + 2)) = "n"/(6"n" + 4)`

Prove by Mathematical Induction that
1! + (2 × 2!) + (3 × 3!) + ... + (n × n!) = (n + 1)! − 1

Using the Mathematical induction, show that for any natural number n, x²ⁿ − y²ⁿ is divisible by x + y

By the principle of Mathematical induction, prove that, for n ≥ 1
`1^2 + 2^2 + 3^2 + ... + "n"^2 > "n"^2/3`

Use induction to prove that n³ − 7n + 3, is divisible by 3, for all natural numbers n

Use induction to prove that 5ⁿ⁺¹ + 4 × 6ⁿ when divided by 20 leaves a remainder 9, for all natural numbers n

Use induction to prove that 10ⁿ + 3 × 4ⁿ⁺² + 5, is divisible by 9, for all natural numbers n

Prove that using the Mathematical induction
`sin(alpha) + sin (alpha + pi/6) + sin(alpha + (2pi)/6) + ... + sin(alpha + (("n" - 1)pi)/6) = (sin(alpha + (("n" - 1)pi)/12) xx sin(("n"pi)/12))/(sin (pi/12)`

Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways

Choose the correct alternative:
If `""^("a"^2 - "a")"C"_2 = ""^("a"^2 - "a")"C"_4` then the value of a is

Choose the correct alternative:
Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is ______

Choose the correct alternative:
1 + 3 + 5 + 7 + · · · + 17 is equal to

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`1/(5 + x)`

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`2/(3 + 4x)^2`

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`(5 + x^2)^(2/3)`

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`(x + 2) - 2/3`

Find `root(3)(10001)` approximately (two decimal places

Prove that `root(3)(x^3 + 6) - root(3)(x^3 + 3)` is approximately equal to `1/x^2` when x is sufficiently large