List of top Mathematics Questions on Absolute maxima and Absolute minima

List-I	List-II
(A) Absolute maximum value	(I) 3
(B) Absolute minimum value	(II) 0
(C) Point of maxima	(III) -5
(D) Point of minima	(IV) 4

Two vertices of the parallelogram \( ABCD \) are given as \( A(-1, 2, 1) \) and \( B(1, -2, 5) \). If the equation of the line passing through \( C \) and \( D \) is: \[ \frac{x - 4}{1} = \frac{y + 7}{-2} = \frac{z - 8}{2}, \] find the distance between sides \( AB \) and \( CD \). Hence, find the area of the parallelogram \( ABCD \).

The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that the volume of the cylinder so formed is maximum.

Let \( \mathbb{R}_+ \) denote the set of all non-negative real numbers. Then the function \( f : \mathbb{R}_+ \to \mathbb{R}_+ \) defined as \( f(x) = x^2 + 1 \) is:

A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated.

The differential equation representing the growth of bacteria is given as: \[ \frac{dP}{dt} = kP, \] where \( P \) is the population of bacteria at any time \( t \). bf{Based on the above information, answer the following questions:}
[(i)] Obtain the general solution of the given differential equation and express it as an exponential function of \( t \).
[(ii)] If the population of bacteria is 1000 at \( t = 0 \), and 2000 at \( t = 1 \), find the value of \( k \).

A relation \( R \) on set \( A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\} \) is defined as: \[ R = \{(x, y) : x + y \text{ is an integer divisible by 2}\}. \]

Show that \( R \) is an equivalence relation. Also, write the equivalence class \([2]\).

If \( x \cos(p + y) + \cos p \sin(p + y) = 0 \), prove that \( \cos p \frac{dy}{dx} = -\cos^2(p + y) \), where \( p \) is a constant.

It is given that the function \( f(x) = x^4 - 62x^2 + ax + 9 \) attains a local maximum value at \( x = 1 \). Find the value of \( a \), and hence obtain all other points where the given function \( f(x) \) attains local maximum or local minimum values.

Let \( A = \mathbb{R} - \{3\} \) and \( B = \mathbb{R} - \{a\} \). Find the value of \( a \) such that the function \( f: A \to B \) defined by \( f(x) = \frac{x-2}{x-3} \) is onto. Also, check whether the given function is one-one or not.

A function \( f(x) = |1 - x + |x| | \) is:

Rohit, Jaspreet, and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is \( \frac{1}{5} \), Jaspreet's selection is \( \frac{1}{3} \), and Alia's selection is \( \frac{1}{4} \). The events of selection are independent of each other.

Based on the above information, answer the following questions:

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}| = 1 \), \( |\vec{b}| = 2 \), and \( \vec{a} \cdot \vec{b} = \sqrt{3} \), then the angle between \( 2\vec{a} \) and \( -\vec{b} \) is:

Using integration, find the area bounded by the ellipse \( 9x^2 + 25y^2 = 225 \), the lines \( x = -2, x = 2 \), and the X-axis.

Sketch the graph of \( y = x|x| \) and hence find the area bounded by this curve, the X-axis, and the ordinates \( x = -2 \) and \( x = 2 \), using integration.

Find the distance between the line \( \frac{x}{2} = \frac{2y - 6}{4} = \frac{1 - z}{-1} \) and another line parallel to it passing through the point \( (4, 0, -5) \).

Let \( A = \mathbb{R} - \{5\} \) and \( B = \mathbb{R} - \{1\} \). Consider the function \( f : A \to B \), defined by \( f(x) = \frac{x - 3}{x - 5} \). Show that \( f \) is one-one and onto.

A card from a well-shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.

The Cartesian equation of a line passing through the point with position vector \( \vec{a} = \hat{i} - \hat{j} \) and parallel to the line
\( \vec{r} = \hat{i} + \hat{k} + \mu(2\hat{i} - \hat{j}) \), is:

The lines \( \frac{1 - x}{2} = \frac{y - 1}{3} = \frac{z}{1} \) and \( \frac{2x - 3}{2p} = \frac{y}{-1} = \frac{z - 4}{7} \) are perpendicular to each other for \( p \) equal to:

Let \[ A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \] and \[ B = \frac{1}{3} \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & \lambda \end{bmatrix}. \] If \( AB = I \), then the value of \( \lambda \) is:

If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \) and \( A^2 + 7I = kA \), then the value of \( k \) is:

The vectors \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - 3\hat{j} - 5\hat{k} \), and \( \vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k} \) represent the sides of:

An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at \( O(0,0,0) \) and the three stars have their locations at points \( D, A, \) and \( V \), having position vectors: \[ 2\hat{i} + 3\hat{j} + 4\hat{k}, \quad 7\hat{i} + 5\hat{j} + 8\hat{k}, \quad -3\hat{i} + 7\hat{j} + 11\hat{k} \] respectively. Based on the above information, answer the following questions: