Question

1. The third term of a G.P is 10,
Then
the product of first five
terms is
O 1000
O
10000
O 100
o
100000
2. If A is an empty set and B is a
set of vowels, then AnB is *
ОА
A
B
O A or B
O None of these​

Answer 1

Question 1) :- The third term of a G.P is 10,

Then the product of first five terms is ?

A) 1000

B) 10000

C) 100

D) 100000

Solution :-

Let us assume that, the common ratio be r and the first term of given GP is a .

Than,

→ nth term of GP = a * r^(n - 1).

So,

→ First five terms are = a, ar, ar²,ar³ ,ar⁴ .

Now, we have given that, the third term of GP is 10.

So,

→ T(n) = a*r^(n - 1)

→ T(3) = a * r^(3 - 1)

→ ar² = 10 ------------ Eqn.(1)

Therefore,

→ Product of first 5 terms = a * ar * ar² * ar³ * ar⁴ = a⁵ * r¹⁰

→ a * ar * ar² * ar³ * ar⁴ = (a * r²)⁵

Putting value of Eqn.(1) , we get,

→ a * ar * ar² * ar³ * ar⁴ = (10)⁵

→ a * ar * ar² * ar³ * ar⁴ = 100000 (Option D) (Ans.)

_______________________

Question 2) :- If A is an empty set and B is a

set of vowels, then A n B is :-

A) А

B) AB

C) A or B

D) None of these.

Solution :-

Given that, A is an empty set.

So,

→ A = { } or Ф .

Also, given, B is a set of vowels.

So ,

→ B = { a, e , i , o , u .}

we have to find A n B . ( Elements which are common in both sets .)

Since, A is an empty set.

we can conclude that, their is no element common between set A and B.

Therefore, A n B is also an Empty set.

Hence,

→ A n B = { } or Ф .

→ A n B = A . (Option A). (Ans.)

____________________

Answer 2

$Let \: 'a'\: and \: 'r' \: are \: first \:term \:and \\common \:ratio \: of \: a \: G.P$

$Third \: term (a_{3}) = 10 \: (given)$

$\boxed{ \pink{ n^{th} \:term (a_{n}) = ar^{n-1} }}$

$\implies ar^{2} = 10 \: --(1)$

$\red{ Product \: of\: first \: 5 \: terms }$

$= a \times ar \times ar^{2} \times ar^{3} \times ar^{4}$

$= a^{5} \times r^{10}$

$= (ar^{2})^{5}$

$= 10^{5} \: [ From \: (1) ]$

$= 100000$

Therefore.,

$\red{ Product \: of\: first \: 5 \: terms }\green { = 100000}$

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