Mathematics · Reasoning

Algebraic Expressions

157 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 maths use of brackets order operations and algebra using brackets in algebraic expressions order of operations

$ x + y -(z - x -[y + z - (x + y - \left { z + x - (y + z + x) \right})])$ is equal to

  1. 3x

  2. 2y

  3. x

  4. 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

On simplifying, we get
$x + y -(z - x -[y + z - (x + y - \left { z + x - (y + z + x) \right})])\$ 
$=  x + y -(z - x -[y + z - (x + y - \left { z + x - y - z - x \right})])\$ 
$=  x + y -(z - x -[y + z - (x + y - \left { - y \right})])\$ 
$=  x + y -(z - x -[y + z - (x + y + y)])\$ 
$= x + y -(z - x -[y + z - x - y - y)])\$ 
$= x + y -(z - x -[ z - x - y])\$ 
$= x + y -(z - x - z + x + y)\$ 
$= x + y -(y)\$ 
$= x.$

Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

If x and y are positive numbers such that x + y = 1, which of the following could be the value of 100x + 200y?
I. 80
II. 140
III. 199

  1. II only

  2. III only

  3. I and II

  4. I and III

  5. II and III
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Given x+y=1  So 100x+100y=100 and 200x+200y =200

So       100 < 100x + 200y <200
II and III satisfies this situation So correct answer will be Option E

Multiple choice business maths linear programming problems structure of linear programming model linear programming problem operations research

If  x + y = 3 and xy = 2, then the value of $\displaystyle x^{3}-y^{3}$ is equal to 

  1. 6

  2. 7

  3. 8

  4. 0
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Formula used:
$x^3-y^3=(x-y)(x^2+xy+y^2)$
               $=(\sqrt{(x+y)^2-4xy})[(x+y)^2-xy]$
               $=(\sqrt{(3)^2-4(2)})[(3)^2-2]$
               $=(\sqrt 1)(7)=7$
Option B is correct.

Multiple choice graphs to solve linear and non linear equations functions and their graphs curved graphs sets, relations and functions maths

If $x^{2}+y^{2}=a^{2}$ touches the line $y=3x+10$, then $a=$ 

  1. $\sqrt{10}$
  2. $10$
  3. $\sqrt{5}$
  4. $\dfrac{10}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle x^2 + y^2 = a^2 to touch a line y = mx + c, the perpendicular distance from the center (0,0) to the line must equal the radius a. The distance is |c| / sqrt(m^2 + 1) = |10| / sqrt(3^2 + 1) = 10 / sqrt(10) = sqrt(10).

Multiple choice graphs to solve linear and non linear equations functions and their graphs curved graphs sets, relations and functions maths

$(x,y)$ satisfies the given set of the equations , find the value of ${x}^{2}$.
${x}^{2}+{y}^{2}=153$ and $y=-4x$

  1. $-51$
  2. $3$
  3. $9$
  4. $144$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let $x^{2}+y^{2}=153$.......(1)
and $y=-4x$......(2)
Put the value $y=-4x$ in equation (1), we get
$x^{2}+(-4x)^{2}=153$
$\Rightarrow x^{2}+16x^{2}=153$
$\Rightarrow 17x^{2}=153$
$\Rightarrow x^{2}=9$
Multiple choice graphs to solve linear and non linear equations functions and their graphs curved graphs sets, relations and functions maths

If $8x+8y=18$ and $x^2-y^2=-\displaystyle\frac{3}{8}$, calculate the value of $2x-2y$.

  1. $-\displaystyle\frac{1}{3}$
  2. $-\displaystyle\frac{1}{6}$
  3. $\displaystyle\frac{1}{3}$
  4. $\displaystyle\frac{1}{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $8x+8y=18$

$\Rightarrow 8(x+y)=18$
$\Rightarrow (x+y)=\dfrac{18}{8}$
$\Rightarrow (x+y)=\dfrac{9}{4}$.......(1)
Also given, $(x^{2}-y^{2})=-\dfrac{3}{8}$
$\Rightarrow (x+y)(x-y)=-\dfrac{3}{8}$.......(2)
Put the value as per equation (1), $(x+y)=\dfrac{9}{4}$, we get
$\dfrac{9}{4}\left ( x-y \right )=-\dfrac{3}{8}$
$\Rightarrow (x-y)=-\dfrac{3}{8}\times \dfrac{4}{9}$
$\Rightarrow (x-y)=-\dfrac{12}{72}$
$\Rightarrow 2(x-y)=-2\times \dfrac{12}{72}$
$\Rightarrow 2x-2y=-\dfrac{1}{3}$

Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

It is known that $x+y=10,$ then $x+y+z=10+z$. The Euclid's axiom that illustrates this statement is

  1. first axiom

  2. second axiom

  3. third axiom

  4. fourth axiom
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Here z has been added to two quantities  which are equal to each other.
So, by Euclid's second axiom, which states that
if equals are added to equals the wholes are equal,
x+y+z=10+z is true.
Ans- Option B.
Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

Define the Euclid's axiom which contains following equation
If $x=9$ and $y=1$, then $x-y=8$.

  1. Axiom 5

  2. Axiom 3

  3. Axiom 2

  4. Axiom 1
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
If $x= 9$ and $y= 1$ then $x-y = 8$
The Euclid's axiom used in this statement is Axiom 3 which states that "If equal be subtracted from equal the remainders are equal".
Multiple choice maths solving equations numerically finding roots by iteration fundamental theorem of algebra complex numbers and linear inequations

If $x=2+\sqrt{3}$, $xy=1$, then $\cfrac { x }{ \sqrt { 2 } +\sqrt { x }  } +\cfrac { y }{ \sqrt { 2 } +\sqrt { y }  } =$.......

  1. $\sqrt{2}$
  2. $\sqrt{3}$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given $x=2+\sqrt{3}$ and $xy=1$
$\Rightarrow\,y=\dfrac{1}{x}=\dfrac{1}{2+\sqrt{3}}=\dfrac{2-\sqrt{3}}{4-3}=2-\sqrt{3}$
Let $\sqrt{x}=\sqrt{a}+\sqrt{b}$
then $x=a+b+2\sqrt{ab}$ by squaring both sides
We have $x=2+\sqrt{3}=a+b+2\sqrt{ab}$
$\Rightarrow\,a+b=2,\,\sqrt{ab}=\dfrac{\sqrt{3}}{2}$ or $ab=\dfrac{3}{4}$
$\Rightarrow\,{\left(a-b\right)}^{2}={\left(a+b\right)}^{2}-4ab=4-4\times \dfrac{3}{4}=4-3=1$
$\Rightarrow\,a-b=1$
$\Rightarrow\,a+b=2,\,a-b=1$
$\Rightarrow\,2a=3$
$\Rightarrow\,a=\dfrac{3}{2}$
Put $a=\dfrac{3}{2}$ in $a+b=2$
$b=2-a=2-\dfrac{3}{2}=\dfrac{4-3}{2}=\dfrac{1}{2}$
$\therefore\,a=\dfrac{3}{2},b=\dfrac{1}{2}$
So,$\sqrt{x}=\dfrac{\sqrt{3}+1}{\sqrt{2}}$
$\sqrt{y}=\dfrac{1}{\sqrt{x}}=\dfrac{\sqrt{2}}{\sqrt{3}+1}\\$
$\sqrt{y}=\dfrac{\sqrt{2}}{\sqrt{3}+1}\times\dfrac{\sqrt{3}-1}{\sqrt{3}-1}\\$
$=\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{3-1}=\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{2}\times\dfrac{\sqrt{2}}{\sqrt{2}}\\$
$=\dfrac{2\left(\sqrt{3}-1\right)}{2\sqrt{2}}=\dfrac{\sqrt{3}-1}{\sqrt{2}}\\$
Now substitute in the given expression:
$\dfrac{x}{\sqrt{2}+\sqrt{x}}=\dfrac{\left(2+\sqrt{3}\right)\times\sqrt{2}}{\sqrt{2}\times \sqrt{2}+\sqrt{3}+1}\\$
$=\dfrac{\left(2+\sqrt{3}\right)\times\sqrt{2}}{2+\sqrt{3}+1}=\dfrac{\sqrt{2}\left(2+\sqrt{3}\right)}{3+\sqrt{3}}\\$
$=\dfrac{\sqrt{2}\left(2+\sqrt{3}\right)}{3+\sqrt{3}}\times\dfrac{3-\sqrt{3}}{3-\sqrt{3}}=\dfrac{\sqrt{2}\left(3+\sqrt{3}\right)}{6}\\$
$\dfrac{y}{\sqrt{2}-\sqrt{y}}=\dfrac{\left(2+\sqrt{3}\right)\times\sqrt{2}}{\sqrt{2}\times \sqrt{2}-\sqrt{3}+1}\\$
$=\dfrac{\sqrt{2}\left(2-\sqrt{3}\right)}{2-\sqrt{3}+1}=\dfrac{\sqrt{2}\left(2-\sqrt{3}\right)}{3-\sqrt{3}}\\$
$=\dfrac{\sqrt{2}\left(2-\sqrt{3}\right)}{3-\sqrt{3}}\times\dfrac{3+\sqrt{3}}{3+\sqrt{3}}=\dfrac{\sqrt{2}\left(3-\sqrt{3}\right)}{6}\\$
Now,$\dfrac{x}{\sqrt{2}+\sqrt{x}}+\dfrac{y}{\sqrt{2}-\sqrt{y}}=\dfrac{\sqrt{2}\left(3+\sqrt{3}\right)}{6}+\dfrac{\sqrt{2}\left(3-\sqrt{3}\right)}{6}$
$=\dfrac{\sqrt{2}\left(3+\sqrt{3}+3-\sqrt{3}\right)}{6}$
$=\dfrac{6\sqrt{2}}{6}=\sqrt{2}$
Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

If $x=\sqrt {12}-\sqrt {9},y=\sqrt {13}-\sqrt {10}$ and $z=\sqrt {11}-\sqrt {8}$, then which of the following is true?

  1. $z > x > y$
  2. $z > y > x$
  3. $y > x > z$
  4. $y > z> x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x=\sqrt{12}-\sqrt{9}$

$x=2\sqrt{3}-3$
$x=2(1.73)-3$
$x=3.46-3$
$\therefore$   $x=0.46$                    ----- ( 1 )
Now,
$y=\sqrt{13}-\sqrt{10}$
$y=3.60-3.16$
$\therefore$  $y=0.44$                ---- ( 2 )
Now,
$z=\sqrt{11}-\sqrt{8}$
$z=3.32-2.83$
$\therefore$  $z=0.49$             ---- ( 3 )
From ( 1 ), ( 2 ) and ( 3 )
$\Rightarrow$  $0.49>0.46>0.44$
$\therefore$  $z>x>y$

Multiple choice maths 5-digit numbers expanded form introduction to numbers and number systems numbers in general form

$202\times315$ can be expanded as.............

  1. $200\times300+300\times15+2\times200+2\times15$
  2. $200\times300+200\times15+2\times300+2\times15$
  3. $200\times3+300\times2+200\times300+2\times15$
  4. $200\times2+300\times15+200\times300+2\times15$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$202\times 315=63,630$

It can be expanded as,
            $63,630=60,000+3000+600+30$
            $300\times 200+200\times 15+300\times 2+15\times 2$
Therefore, option (B) is correct.