JEE Advanced · Mathematics · 30. Vector Algebra

Paragraph:

Let $O$ be the origin, and $\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}$ be three unit vectors in the directions of the sides $\overrightarrow{Q R}, \overrightarrow{R P}$, $\overrightarrow{P Q}$, respectively, of a triangle $P Q R$.

Question:

$|\overrightarrow{O X} \times \overrightarrow{O Y}|=$

A $\sin (P + Q)$
B $\sin 2 R$
C $\sin (P + R)$
D $\sin (Q + R)$

Verified Solution

Answer & Solution

Correct Answer

(A) $\sin (P + Q)$

Step-by-step Solution

Detailed explanation

\vec{O X}, \vec{O Y}, \vec{O Z}

are unit vectors in the directions of sides

\vec{O R}, \vec{R P}

and

\vec{P Q}

respectively,

∴ \vec{O X} = \frac{\vec{Q R}}{|\vec{Q R}|}, \vec{O Y} = \frac{\vec{R P}}{|\vec{R P}|}, \vec{O Z} = \frac{\vec{P Q}}{|\vec{P Q}|}

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

E_{1} =

{

x \in ℝ : x \neq 1

and

\frac{x}{x - 1} > 0

} and

E_{2} =

{

x \in E_{1} : \sin^{- 1} (\log_{e} (\frac{x}{x - 1}))

is a real number} (Here the inverse trigonometric function

\sin^{- 1} (x)

assumes values in

[- \frac{π}{2} . \frac{π}{2}]

.)
Let

f : E_{1} \to ℝ

be the function defined by

f (x) = \log_{e} (\frac{x}{x - 1})

And

g : E_{2} \to R

be the function defined by

g (x) = \sin^{- 1} (\log_{e} (\frac{x}{x - 1}))

LIST-I	LIST-II
A. The range of $f$ is	P. $(- \infty, \frac{1}{1 - e}] \cup [\frac{e}{e - 1}, \infty)$
B. The range of $g$ contains	Q. (0, 1)
C. The domain of $f$ contains	R. $[- \frac{1}{2}, \frac{1}{2}]$
D. The domain of $g$ is	S. $(- \infty, 0) \cup (0, \infty)$
	T. $(- \infty, \frac{e}{e - 1}]$
	U. $(- \infty, 0) \cup (\frac{1}{2}, \frac{e}{e - 1}]$

The correct option is:

JEE Advanced 2018 Hard

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 $(e, e^{2})$

Which of the following options is the only Correct combination?

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Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of $n$ for which the function $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$,is	(1)	8
(Q)	The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}$,is an increasing function on $\mathbb{R}$,is	(2)	9
(R)	The smallest natural number $n$ which is greater than 5,such that $x=3$ is a point of local minima of $h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)$,is	(3)	5
(S)	Number of $x_0 \in \mathbb{R}$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}$,is NOT differentiable at $x_0$,is	(4)	6
		(5)	10

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The green colour produced in the borax bead test of a chromium(III) salt is due to -

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	LIST - I		LIST - II
(P)	The minimum value of \(n\) for which the function \(f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]\) is continuous on the interval \([1,2]\),is	(1)	8
(Q)	The minimum value of \(n\) for which \(g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}\),is an increasing function on \(\mathbb{R}\),is	(2)	9
(R)	The smallest natural number \(n\) which is greater than 5,such that \(x=3\) is a point of local minima of \(h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)\),is	(3)	5
(S)	Number of \(x_0 \in \mathbb{R}\) such that \(l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}\),is NOT differentiable at \(x_0\),is	(4)	6
		(5)	10