HSC Science (General) 11th Standard - Maharashtra State Board Question Bank Solutions

Answer the following:

show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2

Answer the following:

If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x² + y² = 1

Answer the following:

Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number

Answer the following:

Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real

Answer the following:

Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`

Answer the following:

Simplify: `("i"^65 + 1/"i"^145)`

Answer the following:

Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`

Answer the following:

Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`

Check whether the following sequence is G.P. If so, write t_n.

2, 6, 18, 54, …

Check whether the following sequence is G.P. If so, write t_n.

1, –5, 25, –125 …

Check whether the following sequence is G.P. If so, write t_n.

`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...

Check whether the following sequence is G.P. If so, write t_n.

3, 4, 5, 6, …

Check whether the following sequence is G.P. If so, write t_n.

7, 14, 21, 28, …

For the G.P. if r = `1/3`, a = 9 find t₇

For the G.P. if a = `7/243`, r = 3 find t₆.

For the G.P. if r = − 3 and t₆ = 1701, find a.

For the G.P. if a = `2/3`, t₆ = 162, find r.

Which term of the G.P. 5, 25, 125, 625, … is 5¹⁰?

For what values of x, the terms `4/3`, x, `4/27` are in G.P.?

If for a sequence, t_n = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio