JEE Advanced · Mathematics · 17. Properties of Triangles

Let $A B C$ be a triangle such that $\angle A C B=\frac{\pi}{6}$ and let $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively. The value(s) of $x$ for which $a=x^2+x+1, b=x^2-1$ and $c=2 x+1$ is (are)

A
$-(2+\sqrt{3})$
B
$1+\sqrt{3}$
C
$2+\sqrt{3}$
D
$4 \sqrt{3}$

Verified Solution

Answer & Solution

Correct Answer

(B)
$1+\sqrt{3}$

Step-by-step Solution

Detailed explanation

Using, $\cos C=\frac{a^2+b^2-c^2}{2 a b}$

\[
\begin{gathered}
\Rightarrow \frac{\sqrt{3}}{2}=\frac{\left(x^2+x+1\right)^2+\left(x^2-1\right)^2-(2 x+1)^2}{2\left(x^2+x+1\right)\left(x^2-1\right)} \\
\Rightarrow(x+2)(x+1)(x-1) x+\left(x^2-1\right)^2 \\
=\sqrt{3}\left(x^2+x+1\right)\left(x^2-1\right) \\
\Rightarrow x^2+2 x+\left(x^2-1\right)=\sqrt{3}\left(x^2+x+1\right) \\
\Rightarrow(2-\sqrt{3}) x^2+(2-\sqrt{3}) x-(\sqrt{3}+1)=0
\end{gathered}
\]

$\Rightarrow x=-(2+\sqrt{3})$ and $x=1+\sqrt{3}$
But, $x=-(2+\sqrt{3}) \Rightarrow c$ is negative. $\therefore \quad x=1+\sqrt{3}$ is the only solution.
Hence, (b) is the correct option.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a .$ Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 .$ Suppose the tangent to the hyperbola at $P$ passes through the point $(1, 0),$ and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $Δ$ denote the area of the triangle formed by the tangent at $P$ , the normal at $P$ and the $x$ -axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

JEE Advanced 2020 Medium

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty)

•

Column 1 contains information about zeros of

f (x)

f' (x)

and

f'' (x)

•

Column 2 contains information about the limiting behaviour of

f (x)

f' (x)

and

f'' (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f' (x)

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in $(0, 1)$
(II) $f^{'} (x) = 0$ for some $x i n (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in $(0, 1)$
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

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Let \(A B C\) be a triangle such that \(\angle A C B=\frac{\pi}{6}\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively. The value(s) of \(x\) for which \(a=x^2+x+1, b=x^2-1\) and \(c=2 x+1\) is (are)

Answer & Solution

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