Tag: maths

Questions Related to maths

Choose the correct answer from the alternatives given.
Water is flowing at the rate of $5$ km/hr through a pipe of diameter $14$ cm into a rectangular tank which is $50$ m long, $44$ m wide. The time taken (in hours) for the rise in the level of water in the tank to be $7$ cm is

maths line segment construction of line segment and circle of given radius construction related to lines constructing line segment circumscribing and inscribing a circle on a regular hexagon

  1. $2$

  2. $1\dfrac{1}{2}$

  3. $3$

  4. $2\dfrac{1}{2}$

Show answer
Correct Option: A
Explanation:

Water
flowed by the pipe in lh = $\pi r^2h$
= $\dfrac{22}{7} \times$ $\dfrac{7\times 7}{100\times100}$ $\times 5000 m^3 =77m^3$
Volume
of expected water in the tank = $\frac{50 \times 44 \times 7}{100} = 154
m^3$ 
Required
time= $154/77 = 2 hrs$.

If $z = \cos \dfrac{\pi }{6} + i\sin \dfrac{\pi }{6}$, then

maths introduction to three dimensional geometry midpoint of line segment mid point formula mid-point of a line segment

  1. $\left| z \right| = 1,\arg z = \dfrac{\pi }{4}$

  2. $\left| z \right| = 1,\arg z = \dfrac{\pi }{6}$

  3. $\left| z \right| = \dfrac{{\sqrt 3 }}{2},\arg z = \dfrac{{5\pi }}{{24}}$

  4. $\left| z \right| = \dfrac{{\sqrt 3 }}{2},\arg z = {\tan ^{ - 1}}\dfrac{1}{{\sqrt 2 }}$

Show answer
Correct Option: B
Explanation:

if $z=\cos \left(\dfrac{\pi}{6}\right)+i\sin \left(\dfrac{\pi}{6}\right)$ then.

As $z=|z|e^{i arq (z)}$
$\therefore z=\cos \left(\dfrac{\pi}{6}\right)+i\sin \left(\dfrac{\pi}{6}\right)=e^{i\left(\pi/6\right)}\quad [\because e^{i\theta}=\cos\theta+i\sin \theta]$
$\Rightarrow |z|=1,  arq=\dfrac{\pi}{6}$

STATEMENT - 1 : The coordinates of the point P(x, y) which divides the line segment joining the points A$(x _1,  y _1)$ and B$(x _2,  y _2)$ internally in the ration $m _1$  :  $m _2$ are $\left ( \dfrac{m _1 x _2 -m _2 x _1}{m _1 + m _2} ,  \dfrac{m _1 y _2 - m _2 y _1}{m _1 + m _2}\right )$


STATEMENT - 2 : The mid-point of the line segment joining the points P $(p _1 y _1)$ and Q$(x _2, y _2)$ is $\left ( \dfrac{x _1+x _2}{2} , \dfrac{y _1 + y _2}{2} \right )$

maths section and mid-point formula midpoint of line segment mid point formula mid-point of a line segment

  1. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1

  2. Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1

  3. Statement - 1 is True, Statement - 2 is False

  4. Statement - 1 is False, Statement - 2 is True

Show answer
Correct Option: D
Explanation:

Statement -1 is false,

As the formula is not for the internally it is when point divides externally.
Statement -2 is true.

The ratio in which the joining of (-3,2) and (5,6) is divided by the y-axis is

maths section and mid-point formula midpoint of line segment mid point formula mid-point of a line segment

  1. 3:5

  2. 2:5

  3. 1:3

  4. 2:3

Show answer
Correct Option: A
Explanation:

Let the ratio be k:1 Then
x-coordinate of P = 0

$\dfrac{k\times 5+1\times -3}{k+1}=0$
$k=\dfrac{3}{5}$

Consider points $A(-1,3), B(-1,2)$. Find point $P$ which divides $AB$ externally in $\dfrac{5}{4}$.

maths section and mid-point formula midpoint of line segment mid point formula mid-point of a line segment

  1. $(9,-22)$

  2. $(-1,2)$

  3. $(-1,-2)$

  4. $(9,22)$

Show answer
Correct Option: C
Explanation:
Let point P be (x,y)
$x=\cfrac { 5\times (-1)-4\times (-1) }{ 5-4 } \\ x=\cfrac { -5+4 }{ 1 } \\ x=-1\\ y=\cfrac { 5\times (2)-4\times (3) }{ 5-4 } \\ y=\cfrac { 10-12 }{ 1 } \\ y=-2\\ \therefore P=(-1,-2)$

In the sides $BC,CA,AB$ of a triangle $ABC$, three points $D,E,F$ are taken such that each of $BD,CE,AE$ is equal to one-third of the corresponding side, then
$\triangle DEF=\dfrac {1}{2}\triangle ABC$.

maths triangles relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

  1. True

  2. False

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Correct Option: B

In any triangle, medians meet at a point and divide each other as the ratio of $2:3$

maths triangles relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

  1. True

  2. False

Show answer
Correct Option: B

If $AD$ and $PM$ are medians of triangles $ABC$ and $PQR$, respectivetly where $\triangle ABC \sim \triangle PQR$, then  $\dfrac {AB}{PR}=\dfrac {AC}{PM}$.

maths triangles relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

  1. True

  2. False

Show answer
Correct Option: B

The standard unit of volume is in

maths perimeter and area of rectilinear figures region enclosed by a plane figure area of square and rectangle mensuration-i (area)

  1. $m$

  2. $m^2$

  3. $m^3$

  4. None of the above

Show answer
Correct Option: C
Explanation:

For eg., a cube is a $3D$ object.

$\therefore$ volume of a cuboid $=$ $l \times b \times h$
All the three dimensions are measures in "meter".
Hence, the unit of measure measurement of volume is in cubic meter or $m^3$.
So, option C is correct.

Area is measured in____

maths perimeter and area of rectilinear figures region enclosed by a plane figure area of square and rectangle mensuration-i (area)

  1. sq units

  2. cm units

  3. cu units

  4. none of these

Show answer
Correct Option: A
Explanation:

area is measured in sq units..(cm x cm or mt x mt)