Question
Statement-I : If sin x + cos x = 
then 
Statement-II : AM ≥ GM
- Statement-I is True, Statement-II is True; Statement-II is a correct explanation for Statement-I.
- Statement-I is True, Statement-II is True; Statement-II is NOT a correct explanation for Statement-I
- Statement-I is True, Statement-II is False
- Statement-I is False, Statement-II is True
Hint:
In this question, given two statements. It is like assertion and reason. Statement1 is assertion and statement 2 is reason, Find the statement 1 is correct or not and the statement 2 correct or not if correct then is its correct explanation.
The correct answer is: Statement-I is True, Statement-II is True; Statement-II is a correct explanation for Statement-I.
Here, we have to find the which statement is correct and if its correct explanation or not.
Firstly,
Statement-I: If sinx + cosx = 
, x ∈ [ 0, π ] then x = 
, y = 1
Here, we have,
sinx + cosx =
we know
cosx + sinx ≤ 2 and 
,
assuming y>0
and in x∈[0,π] equality only exit
if x= and y=1 as
sin +cos =
and 
=2 … (1)
when substituted in given equation
sinx + cosx =
⇒ √2 = √2 for these values it satisfies the equation
thus, solution is, x=, y=1
Therefore, statement-I is correct.
Now,
Statement-II: AM ≥ GM
For all,
We know that the inequality relation between AM and GM is
Let a1, a2…, can be n positive real numbers. The Arithmetic Mean and Geometric Mean defined as:
AM =
(a1+a2+⋯+an)/n
GM =
The AM–GM inequality states that
AM ≥ GM
Therefore, Statement-II is correct, but it is not explanation of statement-I.
The correct answer is, Statement-I is true, Statement-II is true; Statement-II is NOT a correct explanation for Statement-I
In this question, we have to find the statements are the correct or not and statement 2 is correct explanation or not, is same as assertion and reason. Here, Start solving first Statement and try to prove it. Then solve the Statement-II. Always, the AM–GM inequality states that AM ≥ GM.
Related Questions to study
Statement-I : The number of real solutions of the equation sin x = 2x + 2–x is zero
Statement-II : Since |sin x| ≤ 1
In this question, we have to find the statements are the correct or not and statement 2 is correct explanation or not, is same as assertion and reason. Here, -1 ≤ sinx ≤ 1 for all value, remember that.
Statement-I : The number of real solutions of the equation sin x = 2x + 2–x is zero
Statement-II : Since |sin x| ≤ 1
In this question, we have to find the statements are the correct or not and statement 2 is correct explanation or not, is same as assertion and reason. Here, -1 ≤ sinx ≤ 1 for all value, remember that.
if
if
if
if
if
if
if
if
if
In this question, we have to find the general solution of x. Here more than one option will correct. Remember the rules for finding the general solution.
if
In this question, we have to find the general solution of x. Here more than one option will correct. Remember the rules for finding the general solution.
Number of ordered pairs (a, x) satisfying the equation 
is
Number of ordered pairs (a, x) satisfying the equation is
The general solution of the equation, 
is :
The general solution of the equation, is :
The noble gas used in atomic reactor ,is
The noble gas used in atomic reactor ,is
Which compound does not give 
on heating?
Which compound does not give on heating?
In the interval 
the equation 
has
In this question, we have to find type of solution, here, we know if 
then we can take antilog, b = ca, and cos2 θ = 1 – 2 sin2 θ . Remember these terms and find the solution easily.
In the interval the equation has
In this question, we have to find type of solution, here, we know if then we can take antilog, b = ca, and cos2 θ = 1 – 2 sin2 θ . Remember these terms and find the solution easily.
If 
,then
In this question, we have to find the where is θ lies. Here, always true for sinx ≥ 0 otherwise it is true for x=0,
,
,2π and since we also need x≠, for tanx and x≠0,2π for cotx all the solutions.
If ,then
In this question, we have to find the where is θ lies. Here, always true for sinx ≥ 0 otherwise it is true for x=0,,,2π and since we also need x≠, for tanx and x≠0,2π for cotx all the solutions.
Number of solutions of the equation 
in the interval [0, 2
] is :
Number of solutions of the equation in the interval [0, 2] is :
If 
,then
In this question, we have given equation, where we have to find where the θ belongs. For all value of sin θ cos θ = 
so θ always lies between ( 0 , ) .
If ,then
In this question, we have given equation, where we have to find where the θ belongs. For all value of sin θ cos θ = so θ always lies between ( 0 , ) .
