Mathematics · Reasoning

Algebraic Expressions

183 Questions

Algebraic expressions form the foundation of mathematics, involving variables, constants, and arithmetic operations. This topic includes evaluating coefficients, solving proportional relationships, and understanding boolean logic. It is widely tested across various competitive exams to assess analytical skills.

Variable evaluationExpression coefficientsBoolean logic variablesDirect proportionalityProgramming expressions

Algebraic Expressions Questions

Multiple choice statistics time series moving average and variation simple moving average

In simple linear regression, the numbers of unknown constants are: 

  1. One

  2. Two

  3. Three

  4. Four
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$  In simple linear regression, the number of unknown constants are $:Two$

$\Rightarrow$  The line of regression of $y$ on $x$ is given by $y=a+bx$ where $a$ and $b$ are unknown constants known as intercept and slope of the equation. This is used to predict the unknown value of variable $y$ when value of variable $x$ is known.

Multiple choice statistics time series moving average and variation simple moving average

For two variables $x$ and $y$ ,the following data are given as
$\Sigma x=125,\Sigma y=100, \Sigma x^2=1650,\Sigma y^2=1500,\Sigma xy=50,n=25$.Find the value of $x$ when $y=5$

  1. $5.231$
  2. $4.591$
  3. $6.564$
  4. $8.231$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the linear regression formula for x on y: x - mean_x = b_xy * (y - mean_y). Mean_x = 125/25 = 5. Mean_y = 100/25 = 4. Calculating b_xy and solving for x when y=5 yields approximately 4.591.

Multiple choice statistics time series moving average and variation simple moving average

Find the Value of $y$ from the following data when $x=70$ and coefficient of correlation $0.8$.

Series $x$ $y$
A.M $18$ $100$
Standard Deviation $14$ $20$
  1. $145.32$
  2. $44.23$
  3. $159.43$
  4. $561.12$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\bar x=18$, $\bar y=100$, $\sigma_x=14$, $\sigma_y=20$ and $r=0.8$
Regression line of y on x will be $y-\bar y=r\dfrac{\sigma_y}{\sigma_x}(x-\bar x)$
Subsitute all the abpve value in an equation of regression line, we get
$y-100=\dfrac{0.8\times 20}{14}(x-18)$
$y-100=1.143(x-18)$
$y=1.143x+79.43$
Now, Subsitute the value of x=70, we get
$y=1.143\times 70+79.43=159.43$

Multiple choice statistics time series moving average and variation simple moving average

If $4\bar {x}-5\bar y+33=0$ and $20\bar x-9\bar y=107$ are two lines of regression, then what are the values of $\bar { x } $ and $\bar { y } $ respectively.

  1. $12$ and $18$
  2. $18$ and $12$
  3. $13$ and $17$
  4. $17$ and $13$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
The given equations are $4\overline{x} - 5\overline{y} + 33 = 0$ .... $(i)$ and $20\overline{x} - 9\overline{y} -107 = 0$ ..... $(ii)$
Solving $(i)$ and $(ii)$, we get
$\overline{x} = 13$ 
Substituting $x$ in $(i)$, we get
$\overline{y} =17$
Hence, the answer is $13$ and $17$.
Multiple choice statistics time series moving average and variation simple moving average

Find the equation of $y$ on $x$ for the following data

$x$ $8$ $6$ $4$ $7$ $5$
$y$ $9$ $8$ $5$ $6$ $2$
  1. $y=2x -1.2$
  2. $y=1.2x +1.2$
  3. $y=1.2x -1.2$
  4. $y=1.2x -2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
 $x$  $y$  $x^2$  $x\times y$
 $8$  $9$  $64$  $72$
 $6$  $8$  $36$  $48$
 $4$  $5$  $16$  $20$
 $7$  $6$  $49$  $42$
 $5$  $2$  $25$  $10$

$\sum x=30$
$\sum y=30$
$\sum x^2=190$
$\sum x\times y=192$
So, $\bar x=\dfrac{\sum x}{n}=\dfrac{30}{5}=6$
$\bar y=\dfrac{\sum y}{n}=\dfrac{30}{5}=6$
$b_{yx}=\dfrac{\sum xy-\dfrac{\sum x \times \sum y}{n}}{\sum x^2-\dfrac{(\sum
x)^2}{n}}=\dfrac{192-180}{190-180}=\dfrac{12}{10}=1.2$
Regression line of y on x will be $y-\bar y=b_{yx}(x-\bar x)$
$y-6=1.2(x-6)$
$y=1.2x-7.2+6$
$y=1.2x-1.2$

Multiple choice statistics time series moving average and variation simple moving average

For the variables $x$ and $y$, the regression equations are given as $7x-3y-18=0$ and $4x-y-11=0$. Find the arithmetic means of $x$ and $y$ respectively.

  1. $3$ and $1$
  2. $1$ and $3$
  3. $2$ and $4$
  4. $4$ and $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since the two lines of regression interest at the point $(\bar{X}, \bar{Y} )$


Replace $x$ and $y$ by $\bar{X}$ and $\bar{Y}$ respectively in the given regression equations.

We get,
$7\bar{X}- 3\bar{Y} - 18 = 0$ and $4\bar{X}-\bar{Y} - 11 = 0$ 


Solving these equations, we get $\bar{X} = 3$ and $\bar{Y} = 1$ 

Thus the arithmetic mean of $x$ and $y$ is given by $3$ and $1$ respectively.

Multiple choice maths lines graphs of linear equations equations of lines parallel to the x-axis and y-axis graph of linear equations in two variables

If $2^{2x-y}=32$ and $2^{x+y}=16$ then $x^2+y^2$ is equal to

  1. 9

  2. 10

  3. 11

  4. 13
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given,

${ 2 }^{ 2x-y }=32$
${ 2 }^{ x+y }=16$
these equations can be writtn as
${ 2 }^{ \left( 2x-y \right)  }={ 2 }^{ 5 }$
$\Rightarrow 2x-y=5$.......................eq1
${ 2 }^{ x+y }={ 2 }^{ 4 }$
$\Rightarrow x+y=4$...........................eq2
on adding both equations
$ x+y=4$
$2x-y=5$
_______________________
$3x=9$
$x=3$
putting the value of x we get the value of y
$3+y=4$
$y=1$
putting the values of x and y we get
${ x }^{ 2 }+{ y }^{ 2 }$
${ 3 }^{ 2 }+{ 1 }^{ 2 }=9+1=10$

Multiple choice maths lines graphs of linear equations equations of lines parallel to the x-axis and y-axis graph of linear equations in two variables

In the equation $2x - 3y = 12$. The value of $x$ and $y$ that satisfy the equation are-

  1. $(5, 3)$
  2. $(2, 3)$
  3. $(6, 0)$
  4. $(3,12)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Put $\displaystyle x=6$ and $\displaystyle y=0$ in given equation.
$\displaystyle 2x-3y=2\times 6-3\times 0$
=12-0
=12
Hence, the value of x and y that satisfy the equation are (6,0).

Multiple choice combining transformations transformations vectors and transformations maths

By translating the axes the equation $xy-x+2y=6$ has changed to $XY=C$, then $C=$

  1. 4

  2. 5

  3. 6

  4. 7
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We have

$xy-x+2y=6$


Now,

$ xy-x+2y=6 $

$ \Rightarrow xy-x+2y-6=0 $

$ \Rightarrow xy-x+2y-2=4 $

$ \Rightarrow x\left( y-1 \right)+2\left( y-1 \right)=4 $

$ \Rightarrow \left( x+2 \right)\left( y-1 \right)=4 $


This equation is the form of $XY=C$

Then,

$C=4$


Hence, this is the answer.

Multiple choice statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

Calculate Pearson's coefficient of correlation between the values of $X$ and $Y$.

X 1 2 3 4 5
Y 7 6 5 4 3
  1. $-0.3$
  2. $0.3$
  3. $1$
  4. $-1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$X\\ 1\\ 2\\ 3\\ 4\\ 5\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \overline { x } \cfrac { 15 }{ 5 } =3$                $Y\\ 7\\ 6\\ 5\\ 4\\ 3\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \overline { y } =\cfrac { 25 }{ 5 } =5$                  $Y=y-\overline { y } \\ \quad 02\\ \quad 01\\ \quad 00\\ -1\\ -1\\ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 0$                    $XY\\ -4\\ -1\\ \quad 0\\ -1\\ -4\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { XY=-10 } $                 ${ X }^{ 2 }\\ 4\\ 1\\ 0\\ 1\\ 4\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { { X }^{ 2 }=10 } $        

${ Y }^{ 2 }\\ 4\\ 1\\ 0\\ 1\\ 4\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { { Y }^{ 2 }=10 } $

Therefore, $r=\cfrac { \sum { XY }  }{ \sqrt { \sum { { X }^{ 2 }\sum { { Y }^{ 2 } }  }  }  } \\ =\cfrac { -10 }{ \sqrt { 10*10 }  } \\ =\cfrac { -10 }{ 10 } \\ =-1.$
Multiple choice statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

For $ n=25,\sum x=125,\sum x^2=650,\sum y=100,\sum y^2=460,\sum xy=508$, correlation coefficient is 

  1. $0.99$
  2. $0.207$
  3. $0.66$
  4. $0.89$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$n=25$

$\sum x=125,\ \sum{{x}^{2}}=650,\ \sum{xy}=334$  $\sum y=100,\ \sum{{y}^{2}}=508$
$r=\cfrac { n\sum { xy } -(\sum { x } \times \sum { y } ) }{ \sqrt { (n\sum { { x }^{ 2 } } -\sum { { x }^{ 2 } } )(n\sum { { y }^{ 2 } } -\sum { { y }^{ 2 } } ) }  } =\cfrac { 25\times 508 -100\times 125 }{ \sqrt { (25\times 650 -{ 125 }^{ 2 }  )(25\times 460 -{100}^{2}) }  }=0.207 $

Multiple choice statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

$n=25,\sum x=125,\sum x^2=650, \sum y=100,\sum y^2=460,\sum xy=508$. It was observed that two pair of values of $(x,y)$ were copied as $ (6,14)$ and $(8,6) $ instead of $(8,12),(6,8).$ The correct correlation coefficient is 

  1. $0.667$
  2. $0.87$
  3. $-0.25$
  4. $0.356$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Corrected $\sum x=125-6-8+8+6=125$

Corrected $\sum y=100-14-6+12+8=100$
Corrected $\sum x^2=650-(6)^2-(8)^2+(6)^2+(8)^2=650$
Corrected $\sum y^2=460-(14)^2-(6)^2+(12)^2+(8)^2=436$
Corrected $\sum xy=508-6\times 18-8\times 6+8\times 12+6\times 8=520$
The formula for Pearson product moment correlation is
$r=\dfrac{n\sum xy-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}$

    $=\dfrac{25\times 520-125\times 100}{\sqrt{[25\times 650-(125)^2][25\times 436-(100)^2]}}$

   $=\dfrac{13000-12500}{\sqrt{[16250-15625][10900-10000]}}$

   $=\dfrac{500}{\sqrt{[625][900]}}$

   $=\dfrac{500}{(25)(30)}$

   $=\dfrac{2}{3}$

   $=0.667$

Multiple choice business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

For the variables $x$ and $y$, the two regression lines are $6x+y=30$ and $3x+2y=25$. What are the value of $\bar{x},\bar{y}$ and $r$ respectively ?

  1. $\dfrac{20}{3},\dfrac{35}{9},-0.5$
  2. $\dfrac{20}{3},\dfrac{35}{9},0.5$
  3. $\dfrac{35}{9},\dfrac{20}{3},-0.5$
  4. $\dfrac{35}{9},\dfrac{20}{3},0.5$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$6x+y=30$                    ----- ( 1 )

$3x+2y=25$                  ----- ( 2 )
Multiplying  equation ( 1 ) by $2$ we get,
$\Rightarrow$  $12x+2y=60$         ---- ( 3 )
Subtracting equation ( 2 ) from equation ( 3 ) we get,
$\Rightarrow$  $9x=35$
$\therefore$  $x=\dfrac{35}{9}$
Substituting value of $x$ in equation ( 1 ) we get
$\Rightarrow$  $6\times\dfrac{35}{9}+y=30$
$\Rightarrow$  $\dfrac{70}{3}+y=30$
$\Rightarrow$  $y=30-\dfrac{70}{3}$
$\Rightarrow$  $y=\dfrac{90-70}{3}$
$\therefore$  $y=\dfrac{20}{3}$

$\therefore$  $\overline{x}=\dfrac{35}{9}$ and $\overline{y}=\dfrac{20}{3}$
Now, we are going to find $r:$
$6x+y=30$                            [ Given ]
$6x=30-y$
$x=\dfrac{30}{6}-\dfrac{y}{6}$
$\Rightarrow$  $b _{xy}=\dfrac{-1}{6}$
$3x+2y=25$              [ Given ]
$2y=25-3x$
$y=\dfrac{25}{2}-\dfrac{3}{2}x$
$\Rightarrow$  $b _{yx}=\dfrac{-3}{2}$
$\Rightarrow$  Coefficient of correlation $(r)=\sqrt{b _{xy}\times b _{yx}}$
                                                        $=\sqrt{\dfrac{-1}{6}\times \dfrac{-3}{2}}$

                                                        $=\sqrt{\dfrac{1}{4}}$

                                                        $=\dfrac{1}{2}$

                                                        $=0.5$
$\therefore$  $r=0.5$