Column - I Column - II (A) If x^2+y^2=1, then minimum...

Column - I $\quad$ Column - II (A) If $x^{2}+y^{2}=1$, then minimum value $x+y$ is (P) $-3$ (B) If the maximum value of $y=a \cos x-\frac{1}{3} \cos 3 x$ occurs (Q) $-\sqrt{2}$ when $x=\frac{\pi}{6}$, then the value of $^{\prime} a$ ' is (C) If $f(x)=x-2 \sin x, 0 \leq x \leq 2 \pi$ is increasing in the interval (R) 3 $[a \pi, b \pi]$, then $a+b$ is (D) If equation of the tangent to the curve $\mathrm{y}=-\mathrm{e}^{-\mathrm{u}^{2}}$ where it (S) 2 crosses the y-axis is $\frac{x}{p}+\frac{y}{q}=1$, then $p-q$ is

Column - I $\quad$ Column - II
(A) If $x^{2}+y^{2}=1$, then minimum value $x+y$ is
(P) $-3$
(B) If the maximum value of $y=a \cos x-\frac{1}{3} \cos 3 x$ occurs
(Q) $-\sqrt{2}$
when $x=\frac{\pi}{6}$, then the value of $^{\prime} a$ ' is
(C) If $f(x)=x-2 \sin x, 0 \leq x \leq 2 \pi$ is increasing in the interval
(R) 3 $[a \pi, b \pi]$, then $a+b$ is
(D) If equation of the tangent to the curve $\mathrm{y}=-\mathrm{e}^{-\mathrm{u}^{2}}$ where it
(S) 2 crosses the y-axis is $\frac{x}{p}+\frac{y}{q}=1$, then $p-q$ is

Key Concepts

Tangent Line to Curves

Determining the tangent line to a curve involves finding the derivative of the function to obtain the slope at the point of tangency, and then writing the equation of the line in a suitable form. Understanding how to convert between different representations of the line (for example, point-slope form to intercept form) is crucial for problems that require analyzing relationships between the line’s parameters and its intercepts.

Derivative and Monotonicity Analysis

This concept centers around using the first derivative of a function to assess where the function is increasing or decreasing. By computing the derivative and analyzing its sign over an interval, one can identify regions of monotonic behavior. This method is fundamental in calculus when determining intervals of increase or decrease for functions that include trigonometric as well as polynomial components.

Trigonometric Transformation and Maximum Values

This involves rewriting complex trigonometric expressions into simpler forms using identities, which facilitates the determination of their maximum or minimum values. For example, using the cosine addition formula and understanding the behavior of multiple angle functions are key steps in analyzing problems where a trigonometric function’s parameter is adjusted to achieve a desired peak value at a specified point.

Optimization on a Constrained Domain

This concept involves determining the extreme values of a function when the variables are subject to a constraint. A typical example is maximizing or minimizing a linear expression like x+y under a circular constraint such as x²+y²=1. Techniques such as using parametrization (e.g., x=cos t, y=sin t) or the method of Lagrange multipliers are commonly employed to solve such problems.

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