Tag: measures and relations

Questions Related to measures and relations

If $\dfrac{2+3}{x}=\dfrac{2+x}{3}$
What one value for $x$ can be correctly entered into the answer grid?

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. -5

  2. 3

  3. -3

  4. 2

Show answer
Correct Option: B
Explanation:

Given, $\dfrac{2+3}{x}=\dfrac{2+x}{3}$

$\Rightarrow 2x+x^2=15$
$\Rightarrow x^2+2x-15=0$
$\Rightarrow x^2+5x-3x-15=0$
$\Rightarrow x(x+5)-3(x+5)=0$
$\Rightarrow (x+5)(x-3)=0$
$\Rightarrow x=-5,3$
Value of $x$ is not negative, so $x=3$.

Some situations are given below. State true or false:
The temperature of a day is variable.

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. True

  2. False

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Correct Option: A
Explanation:

Since the temperature of a day depends on the weather. For example, in summers, the temperature will be higher in Celsius whereas in winters, the temperature is low in Celsius. Therefore, the temperature always varies.


Hence, the given statement "The temperature of a day is variable." is true.

Some situations are given below. State true or false:
Length of your classroom is constant.

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Since the length, breadth and height of any area (Room, park, hall, etc) is measured and decided before its construction and then it gets constructed. Therefore, the length of any area will always remain the same.


Hence, the given statement "Length of your classroom is constant." is true.

Some situations are given below. State true or false:
Height of growing plant is constant.

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. True

  2. False

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Correct Option: B
Explanation:

Since plants have the unique ability to grow indefinitely throughout their life due to the presence of ‘meristems’ in their body. Meristems in the roots and shoots of plants are responsible for ‘primary growth of the plant’. These increase the height of the plant. Therefore, the height of any growing plant varies.


Hence, the given statement "Height of growing plant is constant." is false.

Some situations are given below.State true or false.
The number of days in the month of January are varying.

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Since there are $31$ days in January in every year, therefore the number of days in the month of January doesn't vary.


Hence, the given statement "The number of days in the month of January are varying." is False.

Solve: $(3x-5)^2 +(3x+5)^2$ = $(18x+10)(x-2)$

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. $\dfrac{-35}{13}$

  2. $\dfrac{-25}{13}$

  3. $\dfrac{-15}{13}$

  4. $\dfrac{-45}{13}$

Show answer
Correct Option: A
Explanation:

We will solve the given expression $(3x-5)^2+(3x+5)^2=(18x+10)(x-2)$ as shown below:


$(3x-5)^{ 2 }+(3x+5)^{ 2 }=(18x+10)(x-2)\ \Rightarrow [(3x)^{ 2 }+(5)^{ 2 }-(2\times 3x\times 5)]+[(3x)^{ 2 }+(5)^{ 2 }+(2\times 3x\times 5)]=18x(x-2)+10(x-2)\ (\because \quad (a+b)^{ 2 }=a^{ 2 }+b^{ 2 }+2ab,\quad (a-b)^{ 2 }=a^{ 2 }+b^{ 2 }-2ab)\ \Rightarrow 9x^{ 2 }+25-30x+9x^{ 2 }+25+30x=18x^{ 2 }-36x+10x-20\ \Rightarrow 18x^{ 2 }+50=18x^{ 2 }-26x-20\ \Rightarrow 18x^{ 2 }+50-18x^{ 2 }+26x+20=0$
$\Rightarrow 26x+70=0\ \Rightarrow 26x=-70\ \Rightarrow x=-\dfrac { 70 }{ 26 } \ \Rightarrow x=-\dfrac { 35 }{ 13 }$ 

Hence, $x=-\dfrac { 35 }{ 13 }$.

For $|x| < 1$ the constant terms in the expressions of $\dfrac {1}{x-1(^{2})(x-2)}$ is

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. $1$

  2. $2$

  3. $0$

  4. $-1/2$

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Correct Option: B

If ${x}^{3}+m{x}^{2}+nx+6$ has $(x-2)$ as factor and leaves a remainder $3$ when divided by $(x-3)$ find the values of $m,\,n$

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. $m=2,n=2$

  2. $m=2,n=-2$

  3. $m=-2,n=1$

  4. $m=-3,n=-1$

Show answer
Correct Option: D
Explanation:

$x-2$ is factor 


$\Rightarrow x=2$
$f\left( 2 \right) =14+4m+2n$
Remainder is zero

$\Rightarrow 7+2m+n=0\quad \longrightarrow \left( i \right) $
Now, $x-3=0$
gives remainder $3$

$\Rightarrow f\left( 3 \right) =3$
$\Rightarrow 33+9m+3n=3$
$\Rightarrow 10+3m+n=0\quad \longrightarrow \left( ii \right) $

From $(i)$ & $(ii)$
$m=-3$
$n=-1$

The constant term in expression $5xy-4x+8$ is

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. $3$

  2. $-5$

  3. $8$

  4. $1$

Show answer
Correct Option: C
Explanation:
In $5xy-4x+8$

$\therefore$ The constant term is $=+8$.

If the point (2, -3) lies on $\displaystyle kx^{2}-3y^{2}+2x+y-2=0$ then k is equal to 

maths unchanging relations algebra aid introduction to unknowns measures and relations

  1. $\displaystyle \frac{1}{7}$

  2. 16

  3. 7

  4. 12

Show answer
Correct Option: C
Explanation:

As the point lies on the given line, it should satisfy the equation of the line, if we substitute $ x = 2 $ and $ y = -3 $ in it.

So, $k({ 2) }^{ 2 }3{ (-3) }^{ 2 }+2(2)-32=0$
$ => 4k -27 + 4 - 3 -2 = 0 $
$ 4k = 28 $
$ k = 7 $