Tag: maths

Questions Related to maths

If the volume of a prism is $1920$ $\sqrt{3} cm^3$ and the side of the equilateral base is $16$ $cm$, then the height (in cm) of the prism is?

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $19$

  2. $20$

  3. $30$

  4. $40$

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Correct Option: C
Explanation:
Volume of prism $=$ Area of equilateral triangle $\times $ height
Now, Area of triangle $= \dfrac{\sqrt{3}}{4}a^2$
= $\dfrac{\sqrt{3}}{4} 16^2$ = $64 \sqrt{3}$
$1920 \sqrt{3} = 64\sqrt{3}\times $ height

$\therefore $ height $=$ $\displaystyle \dfrac{1920}{64}$
$= 30 \ cm$

The corner of a cube_has  been cut by the plane passing through mid-point of  the three edges meeting at that corner. If the edge of  the cube is of 2 cm length,  then the volume of the  pyramid thus cut off is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $\dfrac{1}{24}cm^3$

  2. $\dfrac{1}{6}cm^3$

  3. $\dfrac{1}{48}cm^3$

  4. $6cm^3$

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Correct Option: B
Explanation:

The base of the pyramid thus out off will be a right angled triangle whose sides containing the right angle will be each equal to 1 cm. The height of the pyramid will also be equal to 1 cm. Hence, the volume will be equal to $\frac{1}{6}  cm^3$.

Each side of the base of a square pyramid is reduced by $20%$. By what percent must the height be increased so that the volume of the new pyramid is the same as the volume of the original pyramid?

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. 20

  2. 40

  3. 46.875

  4. 56.25

  5. 71.875

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Correct Option: D
Explanation:
Let $a$ be the side of the square.
Length of side of square when reduced by $20\% = a-\dfrac{20a}{100}=0.8a$
Let $a _1=0.8a$
Volume of pyramid $V=\dfrac { 1 }{ 3 } \times $ Area of base $\times height=\dfrac{1}{3}A\times h$
Area of base with side $a = { a }^{ 2 }$ 
${ a }^{ 2 }=0.8a$

${V} _{ 1 }=\dfrac { 1 }{ 3 } \times { \left( { a } _{1} \right)  }^{ 2 }\times { h } _{ 1 }$ 
${V} _{ 1 }=V$ ....... [Given]

$\Rightarrow \dfrac { 1 }{ 3 } { a }^{ 2 }\times h=\dfrac { 1 }{ 3 } { \left( 0.8 \right)  }^{ 2 }{ a }^{ 2 }\times { h } _{ 1 }$

$\therefore { h } _{ 1 }=1.5625h$ 

$\implies \dfrac { { h } _{ 1 }-h }{ h } =1.5625-1=56.25%$

$\therefore$  'h' need to be increase by $56.25$ 

A vertex of square is $(3,4)$ and diagonal's equation is given by $x+2y=1$,then the second diagonal which passes through given vertex will be 

maths drawing of different geometrical figures constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector

  1. $2x-y+2=0$

  2. $x+2y=11$

  3. $2x-y=2$

  4. None of these

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Correct Option: C
Explanation:
Diagonals of Square are perpendicular bisector of each other.

$x+2y=1 \Rightarrow y=\dfrac{-x}{2}+\dfrac{1}{2}$

Comparing with $y=mx+c$ we have $slope=m _1=\dfrac{-1}{2}$

As diagonals are perpendicular, so

 $m _1\times m _2=-1$

$\dfrac{-1}{2}\times m _2=-1$

$m _2=2$

equation of other diagonal is $y=2x+c$

as this passes through $(3,4)$ it satifies the equation

$\Rightarrow 4=2(3)+c \Rightarrow c=-2$

Thus equation of diagonal $\equiv 2x-y-2=0$

Hence, the answer is option $C$

With ruler and compasses,we can bisect any given line segment.

maths drawing of different geometrical figures constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. True

  2. False

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Correct Option: A

$A B C$  is a triangle. The bisectors of the internal angle  $\angle B$  and external angle $\angle C$  intersect at  $D.$  if  $\angle B D C = 60 ^ { \circ }$  then  $\angle A$  is

maths geometrical construction constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $120 ^ { \circ }$

  2. $180 ^ { \circ }$

  3. $60 ^ { \circ }$

  4. $150 ^ { \circ }$

Show answer
Correct Option: C
Explanation:

Consider $\triangle ABC$

Let $BC$ be extended to $E$
Since Angular bisectors Meet at $D$
$\angle ABD=\angle DBC\cdots(1)$
$\angle ACD=\angle DCE\cdots(2)$
Consider $ \triangle DBC$
By External sum property 
$\angle DCE=\angle BDC+\angle DBC$
$\implies 2\angle DCE=2(60^{\circ})+2\angle DBC$
$\implies \angle ACE=120^{\circ}+\angle ABC$
By external sum property of $\triangle ABC$
$\angle ACE=\angle BAC+\angle ABC$
$\implies \angle A=60^{\circ}$

The line segment connecting (x, 6) and (9, y) is bisected by the point (7, 3) Find the values of x and y

maths geometrical construction constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. 15, 6

  2. 33, 12

  3. 5, 0

  4. 14, 6

  5. none of these

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Correct Option: C
Explanation:

Since line segment connecting $(x,6)$ and $(9,y)$ is bisected by the point $(7,3)$


Therefore, $\dfrac {x+9}2=7\Rightarrow x=5$ and $\dfrac {6+y}2=3\Rightarrow y=0$

$\therefore x=5, y=0$

Option C is correct.

If $PQ$ is the perpendicular bisector of $AB$, then $PQ$ divides $AB$ in the ratio:

maths geometrical construction constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1:2$

  2. $1:3$

  3. $2:3$

  4. $1:1$

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Correct Option: D
Explanation:

Perpendicular bisector always divides a segment into two equal parts.
Therefore $PQ$ divides $AB$ into $1:1$.

For drawing the perpendicular bisector of $PQ$, which of the following radii can be taken to draw arcs from $P$ and $Q$?

maths geometrical construction constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $\dfrac{PQ}2$

  2. $\dfrac{PQ}3$

  3. $\dfrac{2PQ}3$

  4. $\dfrac{PQ}4$

Show answer
Correct Option: C
Explanation:

To draw a perpendicular bisector of a given side, take any length that is greater than half the length of the side. Draw the arcs from the edges of the base. The point where arcs meet is on the perpendicular bisector.


From the given options,

$\dfrac{2PQ}{3}$ can be considered to draw to draw arcs from edges $P, \ Q$

Remaining options has the value $\leq \dfrac{PQ}{2}$

With the help of a normal ruler and a compass only, which of the following line segment is possible to construct?

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. $2.1\ cm$

  2. $4.2\ cm$

  3. $5.43\ cm$

  4. $3.3\ cm$

Show answer
Correct Option: A,B,D
Explanation:

Least count of normal scale is $.1\ \ cm$ that is minimum length that can be measured using a normal scale is $.1\ \ cm$.

So the lengths that can be measured are
$2.1\ \ cm,4.2\ \ cm$ and $3.3\ \ cm$
To measure $5.43$ we need a scale whose least count is $.001$