Mathematics

Mathematics

Problem

Match the following:
List-IList-II
(P) Total number of values of xx, of the form 1n\frac{1}{n}, n∈Nn \in \mathbb{N} in the interval [115,110]\left[ \frac{1}{15}, \frac{1}{10} \right] which satisfies {x}+{2x}+⋯+{12x}=78x\{x\} + \{2x\} + \dots + \{12x\} = 78x where {}\{\} denotes fractional part of xx(1) 1
(Q) Let (2x2+3x+4)10=∑r=020arxr(2x^2 + 3x + 4)^{10} = \sum_{r=0}^{20} a_r x^r then the value of 2a9a11\frac{2a_9}{a_{11}} is equal to(2) 2
(R) Let GG be centroid of △ABC\triangle ABC whose side length are a,b,ca, b, c. If PP is a point in the plane of △ABC\triangle ABC such that PA=1,PB=2,PC=1PA = 1, PB = 2, PC = 1 and PG=1PG = 1 then the value of a2+b2+c29\frac{a^2 + b^2 + c^2}{9} is equal to(3) 3
(S) If a derivable function f:R+→Rf: \mathbb{R}^+ \rightarrow \mathbb{R} satisfies f(xy)=f(x)+f(y)∀x,y∈R+f(xy) = f(x) + f(y) \quad \forall x, y \in \mathbb{R}^+. If f(16)=3f(16) = 3 then 1f(2)+1f(4)\frac{1}{f(2)} + \frac{1}{f(4)} is equals to(4) 4; (5) 10
&emsp;(A) P→(3),Q→(4),R→(1),S→(2)P \rightarrow (3), Q \rightarrow (4), R \rightarrow (1), S \rightarrow (2)<br>&emsp;(B) P→(2),Q→(5),R→(1),S→(3)P \rightarrow (2), Q \rightarrow (5), R \rightarrow (1), S \rightarrow (3)<br>&emsp;(C) P→(4),Q→(3),R→(1),S→(2)P \rightarrow (4), Q \rightarrow (3), R \rightarrow (1), S \rightarrow (2)<br>&emsp;(D) P→(2),Q→(4),R→(1),S→(5)P \rightarrow (2), Q \rightarrow (4), R \rightarrow (1), S \rightarrow (5) (A) P→(3),Q→(4),R→(1),S→(2)P \rightarrow (3), Q \rightarrow (4), R \rightarrow (1), S \rightarrow (2) (B) P→(2),Q→(5),R→(1),S→(3)P \rightarrow (2), Q \rightarrow (5), R \rightarrow (1), S \rightarrow (3) (C) P→(4),Q→(3),R→(1),S→(2)P \rightarrow (4), Q \rightarrow (3), R \rightarrow (1), S \rightarrow (2) (D) P→(2),Q→(4),R→(1),S→(5)P \rightarrow (2), Q \rightarrow (4), R \rightarrow (1), S \rightarrow (5)
Intuitive explanation

This is a matching problem where you need to evaluate four distinct mathematical expressions and pair each with its numerical answer. The key is to recognize the type of technique each problem requires: (P) uses properties of fractional parts and algebraic constraints; (Q) uses polynomial coefficient extraction and ratio simplification; (R) uses the centroid property and the distance constraint to find a relationship between side lengths; (S) recognizes a functional equation as logarithmic and applies it to find a reciprocal sum. Work through each independently, then match your four answers to the given options.

Intuitive explanation

This is a matching problem where you need to evaluate four distinct mathematical expressions and pair each with its numerical answer. The key is to recognize the type of technique each problem requires: (P) uses properties of fractional parts and algebraic constraints; (Q) uses polynomial coefficient extraction and ratio simplification; (R) uses the centroid property and the distance constraint to find a relationship between side lengths; (S) recognizes a functional equation as logarithmic and applies it to find a reciprocal sum. Work through each independently, then match your four answers to the given options.

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