Equation of the line of the shortest distance between the lines x ⁡ 1 = y ⁡ - 1 = z ⁡ 1 and x ⁡ - 1 0 = y ⁡ + 1 - 2 = z ⁡ 1 is

x ⁡ - 1 1 = y ⁡ + 1 - 1 = z ⁡ - 2

x ⁡ 1 = y ⁡ - 1 = z ⁡ - 2

x ⁡ - 1 1 = y ⁡ + 1 - 1 = z ⁡ 1

Find the shortest distance between the lines r → = ( 4 i ^ - j ^ ) + λ ( i ^ + 2 j ^ - 3 k ^ ) , and r → = ( i ^ - j ^ + 2 k ^ ) + μ ( 2 i ^ + 4 j ^ - 5 k ^ )

We know that the shortest distance between the lines r → = a → 1 + λ b → 1 and r → = a → 2 + λ b → 2 is given by d = a → 2 - a → 1 · b → 1 × b → 2 b → 1 × b → 2 Here, we have, a → 1 = 4 i ^ - j ^ , a → 2 = i ^ - j ^ + 2 k ^ , b → 1 = i + 2 j - 3 k ^ and b → 2 = 2 i ^ + 4 j ^ - 5 k ^ Now, a → 2 - a → 1 = i ^ - j ^ + 2 k ^ - 4 i ^ + j ^ = - 3 i ^ + 0 j ^ + 2 k ^ And b → 1 × b → 2 = i ^ j ^ k ^ 1 2 - 3 2 4 - 5 = - 10 + 12 i ^ - - 5 + 6 j ^ + 4 - 4 k ^ = 2 i ^ - j ^ + 0 k ^ Therefore, a → 2 - a → 1 · b → 1 × b → 2 = ( - 3 i ^ + 0 j ^ + 2 k ^ ) · ( 2 i ^ - j ^ + 0 k ^ ) = - 6 + 0 + 0 = - 6 And b → 1 × b → 2 = 4 + 1 + 0 = 5 Therefore, d = a → 2 - a → 1 · b → 2 × b → 1 b → 2 × b → 1 = - 6 5 = 6 5 5 units. Hence, the shortest distance between the lines is 6 5 5 units.

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Important Questions on Application of Derivatives

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Find two positive numbers whose sum is

15

and the sum of whose squares is minimum.

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2 m$ and volume is $8 m^{3}$ . If building of tank costs $₹ 70$ per square metre for the base and $₹ 45$ per square metre for the sides, what is the cost of least expensive tank $?$

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height

h

and radius

r

is one-third of the height of the cone, and the greatest volume of the cylinder is

\frac{4}{9}

times the volume of the cone.

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

What does an ecological pyramid represent? State any two limitations that these pyramids have.

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The point on the curve

x^{2} = 2 y

which is nearest to the point

(0, 5)

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Describe an inverted pyramid of biomass with the help of an example.

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ?

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The shortest distance between the lines

\frac{x}{2} = \frac{y}{2} = \frac{z}{1}

and

\frac{x + 2}{- 1} = \frac{y - 4}{8} = \frac{z - 5}{4},

lies in the interval:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

S_{1}

and

S_{2}

are respectively the sets of local minimum and local maximum points of the function,

f (x) = 9 x^{4} + 12 x^{3} - 36 x^{2} + 25, x \in R,

then

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

f (x)

is a non-zero polynomial of degree four, having local extreme points at

x = - 1, 0, 1;

then the set

S = {x \in R : f (x) = f (0)}

contains exactly

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Give a diagrammatic representation of trophic levels in an ecosystem.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

From the top of a

64

metres high tower, a stone is thrown upwards vertically with the velocity of

48 m / s .

The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration

g = 32 m / s^{2}

, is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Acid used in Lead Batteries is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Which of the following ecological pyramids is generally inverted?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Equation of the line of the shortest distance between the lines

\frac{x}{1} = \frac{y}{- 1} = \frac{z}{1}

and

\frac{x - 1}{0} = \frac{y + 1}{- 2} = \frac{z}{1}

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Find the shortest distance between the lines

\vec{r} = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 3 \hat{k})

, and

\vec{r} = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (2 \hat{i} + 4 \hat{j} - 5 \hat{k})

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Suppose

f (x)

is a polynomial of degree four having critical points at

- 1, 0, 1

. If

T = \{x \in R | f (x) = f (0)\}

, then the sum of squares of all the elements of

T

is :

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Let

M

and

m

be respectively the absolute maximum and the absolute minimum values of the function,

f (x) = 2 x^{3} - 9 x^{2} + 12 x + 5

in the interval

[0, 3]

. Then

M - m

is equal to

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If the function

f

given by

f (x) = x^{3} - 3 (a - 2) x^{2} + 3 a x + 7

, for some

a \in R

is increasing in

(0, 1]

and decreasing in

[1, 5)

, then a root of the equation,

\frac{f (x) - 14}{{(x - 1)}^{2}} = 0, (x \neq 1)

is :

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

How solar energy is used in a spacecraft?

Which of the following is the primary advantage of air transport?

Important Questions on Application of Derivatives

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