HSC Arts Class 11 - Tamil Nadu Board of Secondary Education Question Bank Solutions

In a ∆ABC, prove that (a² – b² + c²) tan B = (a² + b² – c²) tan C

An Engineer has to develop a triangular shaped park with a perimeter 120 m in a village. The park to be developed must be of maximum area. Find out the dimensions of the park

A rope of length 42 m is given. Find the largest area of the triangle formed by this rope and find the dimensions of the triangle so formed

Derive Projection formula from Law of sines

Derive Projection formula from Law of cosines

Choose the correct alternative:
In a ∆ABC, if
(i) `sin  "A"/2 sin  "B"/2 sin  "C"/2 > 0`
(ii) sin A sin B sin C > 0 then

By the principle of mathematical induction, prove that, for n ≥ 1
1³ + 2³ + 3³ + ... + n³ = `(("n"("n" + 1))/2)^2`

By the principle of mathematical induction, prove that, for n ≥ 1
1² + 3² + 5² + ... + (2n − 1)² = `("n"(2"n" - 1)(2"n" + 1))/3`

Prove that the sum of the first n non-zero even numbers is n² + n

By the principle of Mathematical induction, prove that, for n ≥ 1
1.2 + 2.3 + 3.4 + ... + n.(n + 1) = `("n"("n" + 1)("n" + 2))/3`

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

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