Important Questions for CBSE Class 7 Maths Chapter 12 - Algebraic Expressions

1. Write an expression for a number \[\mathbf{7}\] is subtracted from sum of x and \[\mathbf{4}\].

Ans: According to the given statement the expression that is developed is given as $\left( x+4 \right)-7$

2. The difference of numbers p and q is subtracted from its product. Give equation.

Ans: According to the given statement the expression that is developed is given as\[pq-\left( p-q \right)\]

3. \[-12x\], $\frac{3}{4}x$ is an example for ___

Ans: As we can see that by dividing the first term by 16 we get the second term. So, we can say that the given two terms are like Term

4. Add 5pq and -12pq.

Ans:  Adding the coefficients of the given terms will give the sum of two terms that is,

\[\begin{align} & -12pq \\ & \underline{+\text{ }5pq} \\ & -\text{ }7pq \\ \end{align}\]

5. Subtract 12xy from -5xy.

Ans: Subtracting the coefficients of the given terms will give the difference of two terms that is,

\[\begin{align} & -\text{ }5xy \\ & \underline{-12xy} \\ & -17xy \\ \end{align}\]

6. Add \[x+y-5\], $y-x+5$, $x-y+5$

Ans: Adding the coefficients of the similar terms will give the sum of three terms that is,

\[\begin{align} & +x+y-5 \\ & -x+y+5 \\ & \underline{+x-y+5} \\ & x+y+5 \\ \end{align}\]

7. Add \[3{{a}^{2}}{{b}^{2}}-4ab+5,8{{a}^{2}}{{b}^{2}}+12ab-9,15-6ab-5{{a}^{2}}{{b}^{2}}\].

Ans: Adding the coefficients of the similar terms will give the sum of three terms that is,

\[\begin{align} & \text{  3}{{a}^{2}}{{b}^{2}}-4ab+5 \\ & +8{{a}^{2}}{{b}^{2}}+12ab-9 \\ & \underline{-5{{a}^{2}}{{b}^{2}}-6ab+15} \\ & \text{  }6{{a}^{2}}{{b}^{2}}+2ab+11 \\ \end{align}\]

8. Subtract \[\text{-}{{\text{x}}^{\text{2}}}\text{+6xy}\] from \[\text{8}{{\text{x}}^{\text{2}}}\text{-4xy+12}\].

Ans: Subtracting the coefficients of the similar terms will give the difference of two terms that is,

\[\begin{align} & +8{{x}^{2}}-4xy+12 \\ & \text{ }-{{x}^{2}}+6xy \\ & \underline{\text{  + - }} \\ & \text{  9}{{x}^{2}}+10xy+12 \\ \end{align}\]

9. Subtract \[{{a}^{2}}-4{{b}^{2}}+3ab-20\] from \[2{{a}^{2}}+6{{b}^{2}}+7ab+12\].

Ans: Subtracting the coefficients of the similar terms will give the difference of two terms that is,

\[\begin{align} & 2{{a}^{2}}+6{{b}^{2}}+7ab+12 \\ & \text{  }{{a}^{2}}-4{{b}^{2}}+3ab-20 \\ & \underline{-\text{    +       }-\text{      +      }} \\ & {{a}^{2}}+10{{b}^{2}}+4ab+32 \\ \end{align}\]

10. Find the value of the given equation $4{{x}^{2}}-3x+12$, if $x=-3$

Ans: We are given the quadratic equation of $\text{x}$ as,

$4{{x}^{2}}-3x+12$

Substituting the value $x=-3$,

\[\begin{align} & =4{{\left( -3 \right)}^{2}}-3\left( -3 \right)+12 \\ & =4\times 9+9+12 \\ & =36+9+12 \\ & =57 \\ \end{align}\]

11. Simplify $3\left( 2x+1 \right)+4x+15$ when $x=-1$.

Ans: We are given the quadratic equation of $\text{x}$ as,

$3\left( 2x+1 \right)+4x+15$

Substituting $x=-1$,

\[\begin{align} & =3\left[ 2\left( -1 \right)+1 \right]+4\left( -1 \right)+15 \\ & =3\left( -2+1 \right)-4+15 \\ & =-3-4+15 \\ & =-7+15 \\ & =8 \\ \end{align}\]

12. Find the value of ${{a}^{2}}-{{b}^{2}}$ for $a=-2$ and $b=3$.

Ans: We are given the quadratic equation of $\text{a,b}$ as,

${{a}^{2}}-{{b}^{2}}$

Substituting $a=-2$ and $b=3$,

$\begin{align} & ={{\left( -2 \right)}^{2}}-{{\left( 3 \right)}^{2}} \\ & =4-9 \\ & =-5 \\ \end{align}\]

13. Identify monomials and binomials in the following:

$\text{4xy,-a+8,}{{\text{p}}^{\text{2}}}\text{,xy+4x}$.

Ans: 

Monomials: the expressions that have only one variable. From the given set of expressions the monomials are $\text{-a+8,}{{\text{p}}^{\text{2}}}$

Binomials: the expressions that have two variables. From the given set of expressions the binomials are  $\text{4xy,xy+4x}$.

14. Define

(a) Like Terms

(b) Unlike Terms

Ans: 

(a) Terms having the same algebraic factors are called like terms.

Example: \[3pq\] and \[7pq\]

(b) Terms having different algebraic factors are called unlike terms.

Example: $2xy$ and $-3x$

15. Find the value of equation $3{{x}^{2}}-4x+8$, when $x=8$.

Ans: We are given the quadratic equation of $\text{x}$ as,

$3{{x}^{2}}-4x+8$

Substituting $x=8$,

\[\begin{align} & =3{{\left( 8 \right)}^{2}}-4\left( 8 \right)+8 \\ & =3\left( 64 \right)-32+8 \\ & =192-32+8 \\ & =168 \\ \end{align}\]

16. What should be taken away from $3{{x}^{2}}+2{{y}^{2}}-5xy-25$ to get $-{{x}^{2}}-{{y}^{2}}+2xy+10$.

Ans: Let the term required be $p$.

\[\begin{align} & \left( 3{{x}^{2}}+2{{y}^{2}}-5xy-25 \right)-p=-{{x}^{2}}-{{y}^{2}}+2xy+10 \\ & \Rightarrow p=\left( 3{{x}^{2}}+2{{y}^{2}}-5xy-25 \right)-\left( -{{x}^{2}}-{{y}^{2}}+2xy+10 \right) \\ & \Rightarrow p=3{{x}^{2}}+2{{y}^{2}}-5xy-25+{{x}^{2}}+{{y}^{2}}-2xy-10 \\ & \Rightarrow p=4{{x}^{2}}+3{{y}^{2}}-7xy-35 \\ \end{align}\]

Hence, the required number is \[4{{x}^{2}}+3{{y}^{2}}-7xy-35\].

17. From the sum of $7p+3q+11$ and $4p-2q-5$, subtract $3p-q+11$.

Ans: By adding coefficients of similar terms of the first two expressions we get,

\[\begin{align} & 7p+3q+11 \\ & \underline{4p-2q-\text{ }5} \\ & 11p+q+6 \\ \end{align}\]

By subtracting coefficients of similar terms of the above expression and third expression we get,

\[\begin{align} & 11p+q+6 \\ & 3p-q+11 \\ & \underline{-\text{   +    }-\text{   }} \\ & 8p+2q-5 \\ \end{align}\]

18. From the sum of $8a-5b+3$ and $6a+3b+5$, subtract the difference of $2a-3b+8$ and $a+2b+6$.

Ans: By adding coefficients of similar terms of the first two expressions we get

\[\begin{align} & 8a-5b+3 \\ & \underline{6a+3b+5} \\ & 14a-2b+8 \\ \end{align}\]

By subtracting coefficients of similar terms of the above expression and third expression we get,

\[\begin{align} & 2a-3b+8 \\ & \text{  }a+2b+6 \\ & \underline{-\text{   }-\text{    }-\text{  }} \\ & \text{  }a-5b+2 \\ \end{align}\]

By subtracting,

\[\begin{align} & 14a-2b+8 \\ & \text{    }a-5b+2 \\ & \underline{-\text{     +     }-\text{    }} \\ & \text{ }13a+3b+6 \\ \end{align}\]

19. Find the value of

(a) $3{{p}^{2}}+4{{q}^{2}}-5$, when $p=3$ and $q=-2$

(b) ${{x}^{3}}-3{{x}^{2}}y+2x{{y}^{2}}+8xy+9$, when $x=-3$ and $y=1$

Ans: 

(a) $3{{p}^{2}}+4{{q}^{2}}-5$ 

Substituting $p=3,q=-2$,

\[\begin{align} & =3{{\left( 3 \right)}^{2}}+4{{\left( -2 \right)}^{2}}-5 \\ & =27+16-5 \\ & =38 \\ \end{align}\]

(b) ${{x}^{3}}-3{{x}^{2}}y+2x{{y}^{2}}+8xy+9$

Substituting $x=-3,y=1$,

\[\begin{align} & ={{\left( -3 \right)}^{3}}-3{{\left( -3 \right)}^{2}}\left( 1 \right)+2\left( -3 \right){{\left( 1 \right)}^{2}}+8\left( -3 \right)\left( 1 \right)+9 \\ & =-27-27-6-34+9 \\ & =-54-30+9 \\ & =-75 \\ \end{align}\]

20. What should be the value of ‘p’, $3{{m}^{2}}+m+p=12$ when $m=0$.

Ans: We are given the quadratic equation of $\text{m}$ as,

$3{{m}^{2}}+m+p=12$

Substituting $m=0$,

\[\begin{align} & 3{{\left( 0 \right)}^{2}}+\left( 0 \right)+p=12 \\ & p=12 \\ \end{align}\]

Important Questions for CBSE Class 7 Maths Chapter 12 - Free PDF Download

A Quick Recap of All the Basic Terminologies

1. Variable- An unknown entity is called a variable when it changes with a change in the situation. 

2. Constant- Constant is the value that never changes, it stays fixed.

3. Terms- The quantities that are added or subtracted are called terms.

4. Coefficient- The number with which a variable is multiplied. 

5. Like Terms- Terms having the same variables.

For example 3y, 7y, 45y

6. Unlike Terms- Terms that have different variables.

For example 3xy, 6x, 7y

Addition and Subtraction in Algebra

Like Terms

The coefficients of all the terms are added or subtracted.

3x + 8x - 4x - 2x =?

3x + 8x - 4x - 2x = 5x

Unlike Terms

All the terms having similar variables when taken together the coefficients can be added or subtracted.

7xy - 2x + 8x + 6y - 4xy=?

(7xy - 4xy) + (-2x + 8x) + 6y = 3xy + 6x + 6y

Number Patterns

• The successor of a natural number, n = (n+1). 

• Example: If n = 12, then the successor = (n+1)= 13

• 2n is an even number (if n is a natural number) 

• (2n+1) is an odd number

Some Important Definitions

Algebraic Expression: An algebraic expression is formed with the help of operators i.e. addition, subtraction, multiplications, and divisions.

Equation: When an equality sign “=” is used between two expressions, then it is called an equation.

Example: 3+4x=15.

Problem: 

Form an algebraic expression: x is first multiplied by 6 and then 7 subtracted from the product.

Solution: 

(6*x)-7= 6x-7

Problem: 

Let an algebraic expression be 3m²-4m+2 with variable m. Find the value of the expression if m=2.

Solution: 

3(2)²-4(2)+2 = 3*4 – 8+2 = 12-8+2 = 6

Practical Use of Expression

2 boys go to a shop and both of them buy a notebook and a pen. The cost of the notebook and pen is $5 and $1 respectively. What is the total cost?

Answer: 

x= cost of a notebook

Y= cost of a pen

Total cost = 2(x + y)= 2(5 + 1) = 2*6 = 12

Hence, the total cost of a pen and notebook for 2 girls is $12.

Types of Algebraic Expression

Based on the number of terms;

• Monomial Expression - Expression that has only one term. Example: 9xy, 3x², 22y

• Binomial Expression - Expression with only two terms. Example: xyz + 8y², x²y² + 9xy

• Trinomial Expression - Expression with more than two terms. Example: 65xyz + 10x²y + 31y²z + 42z²

General Formulae for Solving Sums on Algebraic Expression

1. (a + b)² = a²+ 2ab + b²

2. (a – b) ² = a² – 2ab + b²

3. (a + b) (a – b) = a²– b²

4. (a + b)³ = a³ + b³+ 3ab (a + b)

5. (a – b) ³ = a³ – b³– 3ab (a – b)

6. x³ + y³ = (x + y) (x² – xy + y²)

7. x³ – y³ = (x – y) (x² + xy + y²)

Algebraic expressions can be used to find out the perimeter of various figures.

If l is considered to be the length of each side, then the perimeter of

1. Square= 4l

2. Equilateral Triangle= 3l

3. Regular Pentagon= 5l

4. Regular Hexagon= 6l, and so on

Algebraic expressions can also be used to find out the area of various figures.

Square: area= l² (l= length of each side)

Rectangle: area= l*b (l is the length and b is the breadth)

Triangle: area= (b*h)/2 (b and h are the base and height of the triangle respectively)

Diagonal: Number of diagonals drawn  by choosing one vertex of a polygon = n-3 (n = number of sides of the polygon)

Some Important Questions from Algebraic Expression

1. Subtract 12xy from -5xy. ( 1 mark )

Solution:

-5xy

-12xy

-17xy   

2. Subtract - x² + 6xy from 8x² - 4xy + 12 (2 marks)

Solution:

8x² - 4xy + 12

- x² + 6xy

+       - 

9x² - 10xy + 12

3. Simplify 3(2x+1) + 4x + 15 When x=1 (2 marks)

Solution:

3(2x+1) + 4x + 15

= 3[2(-1) +1] + 4(-1) + 15

= -3-4+15

= -7+15

= 8

4. Find the Value of 

a) 3p² + 4q² - 5 when p=3 and q= -2

b) x³ - 3x²y + 2xy² + 8xy + 9 when x= -3 and y= 1. (3 marks)

Solution:

a) 3p² + 4q² - 5

= 3(3)² + 4(-2)² - 5

= 3*9 + 16 – 5

= 38

b) x³ - 3x²y + 2xy² + 8xy + 9

= (-3)³ – 3(-3)²(1)+ 2(-3)(1)²+ 8(-3)(1)+ 9

= -27 – 27 – 6 – 24 + 9

= - 75

5. What should be the Value of ‘p’, 3m² + m + p= 12 When m= 0. (3 marks)

Solution: 

3m² + m + p= 12

=˃ 3(0) + 0 + p = 12 [m=0]

=˃ p= 12  

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The notes are written after very careful study so that students can understand the topic well enough to see successful outcomes in their examinations. Vedantu allows the student to see studying as a source of fun and not a burden.

Hence, this article has provided an outline of Chapter 12 (Algebraic Expression) and covered all the basic terminologies in Algebra with a few examples of the method of solving algebraic expressions. Some important questions with solutions are also provided which one can easily download.

Conclusion

Algebraic Expressions is an integral part of Class 7 Maths and plays a crucial role from an examination perspective. The important questions for Class 7 Maths, as discussed by NCERT, cover a wide range of topics within the subject. They also provide a concise guide to critical points and details related to the topic.

A solid understanding of each section of Class 7 Maths is fundamental as it forms the basis for higher-level studies. However, this section primarily focuses on important questions within the context of Class 7 Maths.