Week-2 Mathematics For DS

Notations:

$\mathbb{R}$ = Set of real numbers $\mathbb{Q}$ = Set of rational numbers
$\mathbb{Z}$ = Set of integers
$\mathbb{N}$ = Set of natural numbers

The set of natural numbers includes 0.

1) A bird is flying along the straight line $2y-6x=40$. In the same plane, an aeroplane starts to fly in a straight line and passes through the point $(4,12)$. Consider the point where aeroplane starts to fly as origin. If the bird and plane collides then enter the answer as 1 and if not then 0. Note: Bird and aeroplane can be considered to be of negligible size.

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 0

1 point

2) A rock is thrown in a pond, and creates circular ripples whose radius increases at a rate of 0.2 meter per second. What will be the value of $\frac{A}{\pi }$, where $A$ is the area (in square meter) of the circle after 5 seconds?
$\underset{‾}{Hint:}$ The area of a circle = $\pi r^2$, where $r$ is the radius of the circle.

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 1.0

1 point

1 point

3) State Government wants to connect the state road to the national highway from a town. There are 3 possible locations in the town A,B and C to connect to the National Highway whose locations are given by coordinates $(3,8)$,$(5,7)$,$(6,9)$ . The National Highway connects the 2 points $(2,1)$,$(10,7)$ and You, being the contractor, have the freedom to select any one of the 3 possible locations in the town.

Hint   Always select the shortest path to construct the road.

Note: 1 unit = 100 meter


What point will you select to build the road?

A

B

C

None

Yes, the answer is correct.
Score: 1

Accepted Answers:

B

4) The total expenses of mess consists of fixed cost and the variable cost, Variable cost is proportional to the number of inmates of the mess, The total expenses are 16000 rupees when 12 members in the mess,and 20000 rupees when 20 members in the mess, find the Fixed cost of the mess.

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 10000

1 point

1 point

5) If $p$ is the length of perpendicular from the origin to the line whose intercept of the axes are $a$ and $b$, then which of the following options always holds?

$\frac{1}{p^2}=\frac{1}{a^2}-\frac{1}{b^2}$

$\frac{1}{p^2}=\frac{1}{(a+b{)}^2}+\frac{1}{(a-b{)}^2}$

$\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$

$\frac{1}{p^2}=(\frac{1}{a}+\frac{1}{b}{)}^2$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$

1 point

6) Let $ABCD$ be a parallelogram with vertices $A(x_1,y_1)$, $B(x_2,y_2)$, and $C(x_3,y_3)$. Which of the following always denotes the coordinate of the fourth vertex $D$?

$(x_1+x_2+x_3,y_1+y_2+y_3)$

$(x_1-x_2+x_3,y_1-y_2+y_3)$

$(x_1+x_2-x_3,y_1+y_2-y_3)$

$(x_1-x_2-x_3,y_1-y_2-y_3)$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$(x_1-x_2+x_3,y_1-y_2+y_3)$

1 point

7) A line perpendicular to the line segment joining the points $A(1,0)$ and $B(2,3)$, divides it at $C$ in the ratio of $1:3$. Then the equation of the line is

$2x+6y-9=0$

$2x+6y-7=0$

$2x-6y-9=0$

$2x-6y+7=0$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$2x+6y-7=0$

8) Consider a triangle $△$$ABC$, whose co-ordinates are $A(-3,3),B(1,7)$and $C(2,-2)$. Let the point $M$ divides the line $AB$ in 1:3, the point $N$ divides the line $AC$ in 2:3 and the point O is the mid-point of $BC$. Find out the area of triangle $△$$MNO$ (in sq. unit).

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 4.5

1 point

1 point

9) Suppose that $P_1$ and $P_2$ are points in a Cartesian coordinate system, with $P_1$ located at $(3,-2)$ and $P_2$ located at $(-1,5)$. Also, let $L_1$ and $L_2$ be lines passing through $P_1$ and $P_2$ respectively.

If the $x-$intercept of the line $L_1$ is $1$ and $y-$ intercept of the line $L_2$ is -1 and If $\theta$ is the angle between $L_1$ and $L_2$, then $\tan \theta$ is equal to

$\frac{-5}{7}$

$\frac{5}{7}$

$\frac{5}{3}$

$\frac{4}{7}$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$\frac{5}{7}$

1 point

10) Consider the following three straight lines:
$l_1:2x+3y=2$
$l_2:x+3y=5$
$l_3:3x+9y=7$

Which of the following option(s) is(are) true?

$l_1$ and $l_2$ are parallel.

$l_3$ and $l_2$ are parallel.

$l_1$ and $l_3$ are parallel.

$l_1$ is equidistant from the line $l_2$ and $l_3$.

Yes, the answer is correct.
Score: 1

Accepted Answers:

$l_3$ and $l_2$ are parallel.