Column-I (A) The value of the sum (3)/(1^2 ·2^2)+(5)/(2^2-3^2)+(7)/(3^2 ·4^2)+…. .+(29)/(14^2 ·1

step 1 So let's solve this question. So in this question we have, we have T a applying TR is equal to 2

Column-I (A) The value of the sum (3)/(1^2...

Column-I (A) The value of the sum $\frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2}-3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\ldots . .+\frac{29}{14^{2} \cdot 15^{2}}$ is (P) 1 (B) If the tangents to the graph of $f(x)=x^{3}+a x+b$ at $x=a$ and (Q) 0 $\mathrm{x}=\mathrm{b}(\mathrm{a} \neq \mathrm{b})$ are parallel then $\mathrm{f}(1)$ is equal to (C) If the sum of x coordinates of points of extrema and points of inflection (R) $\frac{224}{225}$ of $f(x)=3 x^{5}-250 x^{3}+735 x$ is $75 k$ then $k$ equals (D) If a right triangle is drawn in a semi circle of radius $\frac{1}{2}$ with one leg (S) $\frac{3 \sqrt{3}}{32}$ (not the hypotenuse) along the diameter, the maximum area of the triangle is

Column-I
(A) The value of the sum $\frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2}-3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\ldots . .+\frac{29}{14^{2} \cdot 15^{2}}$ is
(P) 1
(B) If the tangents to the graph of $f(x)=x^{3}+a x+b$ at $x=a$ and
(Q) 0
$\mathrm{x}=\mathrm{b}(\mathrm{a} \neq \mathrm{b})$ are parallel then $\mathrm{f}(1)$ is equal to
(C) If the sum of x coordinates of points of extrema and points of inflection
(R) $\frac{224}{225}$
of $f(x)=3 x^{5}-250 x^{3}+735 x$ is $75 k$ then $k$ equals
(D) If a right triangle is drawn in a semi circle of radius $\frac{1}{2}$ with one leg
(S) $\frac{3 \sqrt{3}}{32}$
(not the hypotenuse) along the diameter, the maximum area of the triangle is

Key Concepts

Optimization in Geometry

Optimization in geometry involves finding the maximum or minimum values of geometric quantities subject to given constraints. This is often achieved by expressing a quantity (such as the area of a shape) in terms of one or more variables and then applying calculus techniques, like taking derivatives and finding critical points, to determine the optimal solution.

Extrema and Inflection Points in Polynomials

Extrema and inflection points are critical features of a function’s graph. Extremum points (maximum or minimum) indicate where the function reaches a peak or a trough, while inflection points are where the curvature changes. These points are found using first and second derivatives, and their analysis is crucial in understanding the overall behavior of polynomial functions.

Parallel Tangents and Derivative Analysis

The concept of parallel tangents involves comparing the slopes of tangents to a function at different points. Since the slope of the tangent line is given by the derivative, setting the derivatives equal for two different points is essential to determine conditions under which the tangents will be parallel.

Partial Fractions

Partial fraction decomposition is an algebraic technique used to break down a complicated rational expression into a sum of simpler fractions. This is particularly useful in series summation and integration, as it can reveal hidden cancellations or simplify the analysis of the expression’s behavior.

Telescoping Series

A telescoping series is one in which many terms cancel out successively, allowing the sum to simplify to just a few terms. This method often uses algebraic manipulation or partial fraction decomposition to transform a complex series into a simpler form where intermediate terms cancel, leaving only the initial and final terms.

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