JEE Advanced · Mathematics · 21. ITF

The value of $c o t (Σ_{n = 1}^{23} {cot}^{- 1} (1 + Σ_{k = 1}^{n} 2 k))$ is

A $\frac{23}{25}$
B $\frac{25}{23}$
C $\frac{23}{24}$
D $\frac{24}{23}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{25}{23}$

Step-by-step Solution

Detailed explanation

c o t (Σ_{n = 1}^{23} {cot}^{- 1} (1 + Σ_{k = 1}^{n} 2 k)) = cot Σ_{n = 1}^{23} {cot}^{- 1} (1 + 2 + 4 + 6 + \dots + 2 n)

=cot Σ {cot}^{- 1} (1 + n (n + 1))

=cot Σ {tan}^{- 1} \frac{(n + 1) - n}{1 + n (n + 1)}

(c o t^{- 1} x = \tan^{- 1} \frac{1}{x})

=cot Σ_{n = 1}^{23} ({tan}^{- 1} (n + 1) - {tan}^{- 1} n) [{tan}^{-1} (\frac{x-y}{1 + x y}) {=tan}^{-1} {x-tan}^{-1} y]

Now

{tan}^{- 1} (n + 1) - {tan}^{- 1} {n=tan}^{- 1} (2) - {tan}^{- 1} {1+tan}^{- 1} (3) - {tan}^{- 1} {2+....+tan}^{- 1} (24) - {tan}^{- 1} 23

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

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

If a tangent to a suitable conic (Column 1) is found to be

y = x + 8

and its point of contact is

(8, 16)

, then which of the following options is the only Correct combination?

JEE Advanced 2017 Hard

Let

p, q, r

be non-zero real numbers that are, respectively, the

10^{th}, 100^{th}

and

1000^{th}

terms of a harmonic progression. Consider the system of linear equations

x + y + z = 1

10 x + 100 y + 1000 z = 0

q r x + p r y + p q z = 0

	List- $I$		List- $II$
$(I)$	If $\frac{q}{r} = 10$ , then the system of linear equations has	$(P)$	$x = 0, y = \frac{10}{9}, z = - \frac{1}{9}$ as a solution
$(II)$	If $\frac{p}{r} \neq 100$ , then the system of linear equations has	$(Q)$	$x = \frac{10}{9}, y = - \frac{1}{9}, z = 0$ as a solution
$(III)$	If $\frac{p}{q} \neq 10$ , then the system of linear equations has	$(R)$	infinitely many solutions
$(IV)$	If $\frac{p}{q} = 10$ , then the system of linear equations has	$(S)$	no solution
		$(T)$	at least one solution

The correct option is:

JEE Advanced 2022 Medium

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Step-by-step Solution

The value of $c o t (Σ_{n = 1}^{23} {cot}^{- 1} (1 + Σ_{k = 1}^{n} 2 k))$ is