Tag: mathematics and statistics

Questions Related to mathematics and statistics

State the following statement is true(T) or false(F).
If two lines intersect at a point P, then P is called the point of concurrent of the two lines.

mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

For $2$ lines

      We call it as Intersection point
For $3$ lines
      We call it as point of concurrent

If $O$ is origin and $C$ is the mid point of $A(2,-1)$ and $B(-4,3)$. Then value of $\overrightarrow { OC } $ is

mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

  1. $\hat { i } +\hat { j } $

  2. $\hat { i } -\hat { j } $

  3. $-\hat { i } +\hat { j } $

  4. $-\hat { i } -\hat { j } $

Show answer
Correct Option: C
Explanation:

Since $C$ is the mid point of $A(2,-1)$ and $B(-4,3)$


$\therefore$ Coordinates of $C$ are
$\left( \cfrac { 2-4 }{ 2 } ,\cfrac { -1+3 }{ 2 }  \right) =\left( -1,1 \right) $
$\therefore \overrightarrow { OC } =-\hat { i } +\hat { j } $

The area of $\displaystyle \Delta $ whose vertices are (a,b+c),(b,c+a),(c,a+b) will be -

mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

  1. 0

  2. a+b+c

  3. ab+bc+ac

  4. None of these

Show answer
Correct Option: A
Explanation:

Area of $\displaystyle \Delta $
$\displaystyle =\frac { 1 }{ 2 } \left[ a\left( c+a \right) -b\left( b+c \right) +b\left( a+b \right) -c\left( c+a \right) +c\left( b+c \right) -a\left( a+b \right)  \right] $
$\displaystyle =\frac { 1 }{ 2 } \left[ ac+{ a }^{ 2 }-{ b }^{ 2 }-bc+ab+{ b }^{ 2 }-{ c }^{ 2 }-ac+bc+{ c }^{ 2 }-{ a }^{ 2 }-ab \right] $
$\displaystyle =\frac { 1 }{ 2 } \times 0$
$\displaystyle =0$

If $A,B,C$ are the vertices of a triangle whose position vectors are $\vec { a } ,\vec { b } ,\vec { c } $ and $G$ is the centroid of the $\triangle ABC$, then $\overrightarrow { GA } +\overrightarrow { GB } +\overrightarrow { GC } $ is

mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

  1. $\vec { 0 } $

  2. $\vec { A } +\vec { B } +\vec { C } $

  3. $\cfrac { a+b+c }{ 3 } $

  4. $\cfrac { a-b-c }{ 3 } $

Show answer
Correct Option: A
Explanation:

Given the position vectors of vertices $A,B$ and $C$ of the triangle $ABC$ are $\vec { a } ,\vec { b } $ and $\vec { c } $


ie $\overrightarrow { OA } =\vec { a } \quad \overrightarrow { OB } =\vec { b } \quad \overrightarrow { OC } =\vec { c } $

$\therefore$ Centroid of triangle $(G)=\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 } $

Now $\overrightarrow { GA } +\overrightarrow { GB } +\overrightarrow { GC } $

$=\left( \overrightarrow { OA } -\overrightarrow { OG }  \right) +\left( \overrightarrow { OB } -\overrightarrow { OG }  \right) +\left( \overrightarrow { OC } -\overrightarrow { OG }  \right) $

$=\left( \vec { a } -\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 }  \right) +\left( \vec { b } -\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 }  \right) +\left( \vec { c } -\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 }  \right) \quad $

$=\cfrac { 1 }{ 3 } \left( 3\vec { a } -\vec { a } -\vec { b } -\vec { c } +3\vec { b } -\vec { a } -\vec { b } -\vec { c } +3\vec { c } -\vec { a } -\vec { b } -\vec { c }  \right) $

$=\cfrac { 1 }{ 3 } \left[ \vec { 0 }  \right] =\vec { 0 } $

The compound interest on Rs. 50,000 at 4% per annum for two years compounded anually is :

mathematics and statistics interest applying compound interest compound interest formula with different successive rate of interest compound interest ( for different time period)

  1. $4000$

  2. $4080$

  3. $4280$

  4. $4050$

Show answer
Correct Option: B
Explanation:

C.I. = Amount - Principle

=> $P((1+\dfrac{r}{100})^{T} _{} - 1)$
CI = $50,000(1+\dfrac{4}{100})^{2} _{} - 1)$
C.I.= $4080.$

At what rate percent per annum will a sum of Rs. $2000$ amount to Rs. $2205$ in $2$ years at compound interest?

mathematics and statistics interest applying compound interest compound interest formula with different successive rate of interest compound interest ( for different time period)

  1. $3$%

  2. $2$%

  3. $5$%

  4. $4$%

Show answer
Correct Option: C
Explanation:

$\Rightarrow$  $P=$Rs.$2000,\,A=$Rs.$2205$ and $T=2\,$years.

$\Rightarrow$  $A=P(1+\dfrac{R}{100})^T$

$\Rightarrow$  $2205=2000\times (1+\dfrac{R}{100})^2$

$\Rightarrow$  $\dfrac{441}{400}=(1+\dfrac{R}{100})^2$

$\Rightarrow$  $\dfrac{21}{20}=(1+\dfrac{R}{100})$

$\Rightarrow$  $R=\dfrac{1}{20}\times 100=5\%$

The cost of a car purchased 2 years ago depreciates at the rate of 20% per annum. If its present value is Rs 3, 15,600, find the value of the car after 2 years

mathematics and statistics interest applying compound interest compound interest formula with different successive rate of interest compound interest ( for different time period)

  1. Rs 2,01,994

  2. Rs 50,496

  3. Rs 2,01,984

  4. Rs 10,09,920

Show answer
Correct Option: C
Explanation:

$Depreciation = A=P\left (1-\frac {r}{100}\right )^n=3,15,600\left (1-\frac {20}{100}\right )^2$
$=3,15,600\left (\frac {4}{5}\right )^2=3,15,600\times \frac {16}{25}=Rs 2,01,984$

What sum will amount to Rs. $32,967$ in $2$ years C.I., if the rates are $10$ per cent and $11$ per cent for the successive years?

mathematics and statistics interest applying compound interest compound interest formula with different successive rate of interest compound interest ( for different time period)

  1. Rs. $25,000$

  2. Rs. $26,000$

  3. Rs. $27,000$

  4. Rs. $28,000$

Show answer
Correct Option: C
Explanation:

$\Rightarrow$  $P=$Rs.$32,967,\,R _1=10\%,\,R _2=11\%$


$\Rightarrow$  $A=P\times (1+\dfrac{R _1}{100})\times (1+\dfrac{R _2}{100})$

$\Rightarrow$  $32967=P\times (1+\dfrac{10}{100})\times (1+\dfrac{11}{100})$

$\Rightarrow$  $32967=P\times \dfrac{11}{10}\times \dfrac{111}{100}$

$\Rightarrow$  $P=\dfrac{32967\times 1000}{1221}=27\times 1000=$Rs. $27,000$

What sum will amount to Rs. $65,934$ in $2$ years C.I., if the rates are $10$ per cent and $11$ per cent for the successive years?

mathematics and statistics interest applying compound interest compound interest formula with different successive rate of interest compound interest ( for different time period)

  1. $53,000$

  2. $54,000$

  3. $55,000$

  4. $60,000$

Show answer
Correct Option: B
Explanation:
$A _1=P(1+\cfrac{10}{100})^1=P\times 1.1$
$A _2=A _1(1+\cfrac{11}{100})^1=P\times 1.221$
$\implies 65934=1.221P\\ \implies p=54,000$
Hence sum$=Rs.54,000$

Read the following statement carefully and select the correct option 
statement - i : An article marked at rs 800 is sold at successive discount of 15% and 25%. If the buyer desires to sell. it off at a profit of 20% after allowing a 10% discount, then his marked price will be rs 680.
Statement- ii; if the amount of Rs p at 10% per annum for 3 year compounded annually becomes rs q, then q ; p is 125 ; 216.

mathematics and statistics interest applying compound interest compound interest formula with different successive rate of interest compound interest ( for different time period)

  1. Both statement- i and statement- ii are false

  2. Both statement- i and statement- ii are true

  3. Both statement- 1 is true but statement- 11 are false

  4. statement - 1 is false but statement -11 is true

Show answer
Correct Option: A