Question

Write the quantified symbol for
1) All square are rectangle
2) for every integer x, x² is a non negative​

Answer 1

Answer:

²x²-3abx+2b²=0

or, {(ax)²-2.ax.3b/2+(3b/2)²}+2b²-9b²/4=0

or, (ax-3b/2)²+(8b²-9b²)/4=0

or, (ax-3b/2)²-(b/2)²=0

or, (ax-3b/2+b/2)(ax-3b/2-b/2)=0

or, (ax-b)(ax-2b)=0

Either, ax-b=0

or, ax=b

Answer:

SOLUTION :–

• Let the function –

\begin{gathered} \\ \implies\bf y = { \tan}^{ - 1} (x) \\ \end{gathered}

⟹y=tan

−1

(x)

• Let the function –

\begin{gathered} \\ \implies\bf x= \tan( \theta) \\ \end{gathered}

⟹x=tan(θ)

• Differentiate with respect to 'θ' –

\begin{gathered} \\\implies\bf \dfrac{dx}{d \theta}= \sec^{2} ( \theta) \\ \end{gathered}

⟹

dθ

dx

=sec

2

(θ)

\begin{gathered} \\\implies\bf \dfrac{dx}{d \theta}= 1 + \tan^{2} ( \theta) \\ \end{gathered}

⟹

dθ

dx

=1+tan

2

(θ)

\begin{gathered} \\\implies\bf \dfrac{dx}{d \theta}= 1 +x^{2} \: \: \: \: \: - - - eq.(1)\\ \end{gathered}

⟹

dθ

dx

=1+x

2

−−−eq.(1)

• Now –

\begin{gathered} \\ \implies\bf y = { \tan}^{ - 1} ( \tan( \theta) ) \\ \end{gathered}

⟹y=tan

−1

(tan(θ))

\begin{gathered} \\ \implies\bf y = \theta\\ \end{gathered}

⟹y=θ

• Differentiate with respect to 'θ' –

\begin{gathered} \\ \implies\bf \dfrac{dy}{d \theta} = \dfrac{d\theta}{d\theta}\\ \end{gathered}

⟹

dθ

dy

=

dθ

dθ

\begin{gathered} \\ \implies\bf \dfrac{dy}{d \theta} =1\\ \end{gathered}

⟹

dθ

dy

=1

• We should write this as –

\begin{gathered} \\ \implies\bf \dfrac{dy}{dx} \times \dfrac{dx}{d \theta} =1\\ \end{gathered}

⟹

dx

dy

×

dθ

dx

=1

• Using eq.(1) –

\begin{gathered} \\ \implies\bf \dfrac{dy}{dx} (1 + {x}^{2})=1\\ \end{gathered}

⟹

dx

dy

(1+x

2

)=1

\begin{gathered} \\ \large \implies{ \boxed{\bf \dfrac{dy}{dx} = \dfrac{1}{(1+{x}^{2})}}}\\ \end{gathered}

⟹

dx

dy

=

(1+x

2

)

1

or, x=b/a

Or, ax-2b=0

or, ax=2b

or, x=2b/a

∴, the roots of the given equations are: b/a, 2b/a. Ans.

\huge\mathcal{\fcolorbox{cyan}{black}{\orange{Høpe thìs helps u࿐}}}

Høpe th

ı

ˋ

s helps u࿐