JEE Advanced · Mathematics · 4. P&C

If $r, s, t$ are prime numbers and $p, q$ are the positive integers such that LCM of $p, q$ is $r^2 s^4 t^2$, then the number of ordered pairs $(p, q)$ is

A 252
B 254
C 225
D 224

Verified Solution

Answer & Solution

Correct Answer

(C) 225

Step-by-step Solution

Detailed explanation

Since, $r, s, t$ are prime numbers.
$\therefore$ Selection of $p$ and $q$ are as under

$\therefore$ Total number of ways to select $s=9$.
Similarly, the number of ways to select $t=5$.
$\therefore$ Total number of ways $=5 \times 9 \times 5=225$.

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

f (x) = \sin (π \cos x)

and

g (x) = \cos (2 π \sin x)

be two functions defined for

x > 0 .

Define the following sets whose elements are written in the increasing order:

X = \{x : f (x) = 0\}, Y = \{x : f^{'} (x) = 0\}

Z = \{x : g (x) = 0\}, W = \{x : g^{'} (x) = 0\}

List-

I

contains the sets

X, Y, Z

and

W .

List-

I I

contains some information regarding these sets.

List- $I$	List- $I I$
$(I)$	$X$	$(P)$	$\supseteq \{\frac{π}{2}, \frac{3 π}{2}, 4 π, 7 π\}$
$(I I)$	$Y$	$(Q)$	an arithmetic progression
$(I I I)$	$Z$	$(R)$	NOT an arithmetic progression
$(I V)$	$W$	$(S)$	$\supseteq \{\frac{π}{6}, \frac{7 π}{6}, \frac{13 π}{6}\}$
		$(T)$	$\supseteq \{\frac{π}{3}, \frac{2 π}{3}, π\}$
		$(U)$	$\supseteq \{\frac{π}{6}, \frac{3 π}{4}\}$

Which of the following is the only correct combination?

JEE Advanced 2019 Easy

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Paragraph:
A small block of mass $M$ moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$

Question:
The speed of the block at point $C$, immediately before it leaves the second incline is

JEE Advanced 2008 Hard

Match the chemical substance in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.

Column I	Column II
A. $C _6 H _5 CHO$	p. Gives precipitate with 2, 4-dinitrophenylhydrazine
B. $CH _3 \equiv CH$	q. Gives precipitate with $AgNO _3$
C. $CN ^{-}$	r. Is a nucleophile
D. $I ^{-}$	s. Is involved in cyanohydrin formation

JEE Advanced 2007 Medium

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If \(r, s, t\) are prime numbers and \(p, q\) are the positive integers such that LCM of \(p, q\) is \(r^2 s^4 t^2\), then the number of ordered pairs \((p, q)\) is

Answer & Solution

Step-by-step Solution

See the Complete Solution

Column I	Column II
A. \(C _6 H _5 CHO\)	p. Gives precipitate with 2, 4-dinitrophenylhydrazine
B. \(CH _3 \equiv CH\)	q. Gives precipitate with \(AgNO _3\)
C. \(CN ^{-}\)	r. Is a nucleophile
D. \(I ^{-}\)	s. Is involved in cyanohydrin formation