Tag: maths

Questions Related to maths

In $\triangle ABC, D$ is a point on AB and E is a point on BC such that DE || AC and $ar (DBE) = \dfrac {1}{2} ar (ABC)$. Find $\dfrac{AD}{AB}$

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

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

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

  3. $\dfrac{\sqrt 2 - 1}{2}$

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

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Correct Option: B
Explanation:

In $\triangle ABC$

$DE \parallel BC$
$ \cfrac{AD}{BD} = \cfrac {AE}{EC} $ Basic proportionality theorm
$ AD = \cfrac{BD}{2}$
$ \cfrac {AD}{BD} = \cfrac{1}{3}$
$ \cfrac {area(DBE)}{area(ABC)} = \left (\cfrac {AD}{AB} \right)^{2} $

$ \sqrt {\cfrac {area(DBE)}{area(ABC)}} = \left (\cfrac {AD}{AB} \right) $
$ 1 - \cfrac{AD}{AB}$
$ = 1- \cfrac{1}{\sqrt {2}}$
$ = \cfrac {\sqrt{2} - 1}{\sqrt{2}}$

In any triangle ABC state whether following statements are true or false:
(1) the bisectors of the angles A, B, and C meet in a point,
(2) the medians, i.e. the lines joining each vertex to the middle point of the opposite side, meet in a point, and
(3) the straight lines through the middle points of the sides perpendicular to the sides meet in a point.

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. True

  2. False

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Correct Option: A

D,E,F are midpoints of sides BC, CA and AB of $\Delta ABC$. If perimeter of $\Delta ABC$ is 12.8 cm, then perimeter of $\Delta DEF$ is :

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. $17 cm$

  2. $38.4 cm$

  3. $25.6 cm$

  4. $6.4 cm$

Show answer
Correct Option: D
Explanation:

Given in $\triangle ABC$, $D,E,F$ are the mid points of sides $AB,BC$ and $CA$ respectively

Now using mid point theorem line segment joining the mid points of two sides is parallel to third side and also half of it.
$\therefore DF=\dfrac{1}{2}BC$
$\Rightarrow \dfrac{DF}{BC}=\dfrac{1}{2}.......(i)$
Similarly $\dfrac{DE}{AC}=\dfrac{1}{2}.........(ii)$
$\dfrac{EF}{AB}=\dfrac{1}{2}...........(iii)$
Using $(i),(ii)$ and $(iii)$
$\dfrac { DF }{ BC } =\dfrac { DE }{ AC } \dfrac { EF }{ AB } =\dfrac { 1 }{ 2 } $
$\dfrac { DF }{ BC } =\dfrac { DE }{ AC } \dfrac { EF }{ AB } =\dfrac { 1 }{ 2 } \ \therefore \triangle ABC\sim \triangle DEF$
Now if triangles are similar then ratio of their perimeter is equal to ratio of  of their corresponding sides.
$\ \dfrac { Perimeter(\triangle DEF) }{ Perimeeter(\triangle ABC) } =\dfrac { 1 }{ 2 } $
$\Rightarrow $ Perimeter of $\triangle ABC=\dfrac{1}{2}\times 12.8=6.4$ cm

If A, B and C are the midpoint of the sides PQ, QR and PR of $\triangle $PQR respectively, then the area of $\triangle $ABC equals if area of $\triangle PQR$ is $4$ units

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. $1$

  2. $2$

  3. $3$

  4. $4$

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Correct Option: A

The sides $AB, BC$ and $CA$ of a triangle $ABC$ have $3, 4$ and $5$ interior points respectively on them.The number of triangles that can be constructed using these interior points as vertices is 

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. 60

  2. 205

  3. 115

  4. 405

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Correct Option: B
Explanation:

No. of ways $={^{ 3 }{ { C } _{ 1 } }}\times {^{ 4 }{ { C } _{ 1 }} }\times {^{ 5 }{ { C } _{ 1 } }}+{^{ 3 }{ { C } _{ 2 } }}\left( ^{ 4 }{ { C } _{ 1 } }+{^{ 5 }{ { C } _{ 1 } }}\right)+{^{ 4 }{ { C } _{ 2 }} } \left(^{ 3 }{ { C } _{ 1 } }+{^{ 5 }{ { C } _{ 1 }} }\right) +{^{ 5 }{ { C } _{ 2 }} }\left( ^{ 3 }{ { C } _{ 1 } }+{^{ 4 }{ { C } _{ 1 } }}\right)$

$\Rightarrow$ No of ways $=60+3\left( 4+5\right) +6\left( 3+5\right) +10 \left( 3+4\right)$
$\Rightarrow$ No of ways $=60+27+48+70=205.$
Hence, the answer is $205.$

In $\triangle ABC, D$ and $E$ are the mid point of $\bar {BC}$ and $\bar {AC}$ respectively. $\bar {AD}$ and $\bar {BE}$ intersect each other in $G.A$ line $m$ passing through $D$ and parallel to $\overleftrightarrow { BE } $ intersects $\bar {AC}$ in $K$.
then $AC=4CK$

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. True

  2. False

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Correct Option: A

An example of a production overhead would be:

maths profit-loss overhead expenses introduction to total cost price of the article profit and loss with overhead expenses

  1. materials

  2. labour cost

  3. supervisory costs

  4. rent

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Correct Option: C
Explanation:

An example of a production overhead would be supervisory costs.
Materials and labour costs would be directly attributable to the product and would be classed as direct costs. Rent is a non production overhead but salaries of supervisors are related to production and are an overhead as they do not vary directly with output.

Repairs, taxes, insurance, rent are all examples of

maths profit-loss overhead expenses introduction to total cost price of the article profit and loss with overhead expenses

  1. overhead expenses

  2. interest

  3. company profit

  4. cost price

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Correct Option: A
Explanation:
Overhead expenses are all costs on the income statement except for direct labour, direct materials, and direct expenses. 
Overhead expenses include accounting fees, advertising, insurance, interest, legal fees, labour burden, rent, repairs, supplies, taxes, telephone bills, travel expenditures, and utilities.

An ______ is the additional cost which is added to the cost price of an item.

maths profit-loss overhead expenses introduction to total cost price of the article profit and loss with overhead expenses

  1. interest

  2. cost

  3. overhead

  4. profit

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Correct Option: C
Explanation:

By definition, Overhead is the additional cost which is added to the cost price of an item. 

Overhead expenses are all costs on the income statement except for direct labour, direct materials, and direct expenses. 
Overhead expenses include accounting fees, advertising, insurance, interest, legal fees, labour burden, rent, repairs, supplies, taxes, telephone bills, travel expenditures, and utilities.

Absorption costing is closely related to which of the following cost elements?

maths profit-loss overhead expenses introduction to total cost price of the article profit and loss with overhead expenses

  1. Direct labour

  2. Overheads

  3. Total cost

  4. Machine

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Correct Option: B
Explanation:

Absorption costing is closely related to overhead expenses.