Free Algebraic Expressions and Identities Subjective Test 01 Practice Test - 8th Grade

Question 1

Subtract (ap+bq-cr) from (a+b+c)(p+q+r) . [1 MARK]

SOLUTION

Solution : Concept: 1 Mark

(a + b + c)(p + q + r) - (ap + bq - cr)

= (ap + aq + ar + bp + bq + br + cp + cq + cr) - (ap + bq - cr)

Now,

= ap + aq + ar + bp + bq + br + cp + cq + cr - ap - bq + cr
= aq + ar + bp + br + cp + cq + 2cr

Question 2

Add the following:

$a (a + b), b (b + c), c (a + c) a n d a^{2} + b^{2} + c^{2}$ [2 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 1 Mark

$\Rightarrow a (a + b) + b (b + c) + c (a + c) + a^{2} + b^{2} + c^{2}$

$\Rightarrow a^{2} + a b + b^{2} + b c + c^{2} + c a + a^{2} + b^{2} + c^{2}$

$\Rightarrow 2 a^{2} + 2 b^{2} + 2 c^{2} + a b + b c + c a$

Question 3

Using the identities of algebraic expressions , find

$(i) 1002 \times 1003$

$(i i) 97 \times 103$ . [2 MARKS]

SOLUTION

Solution :
Application: 1 Mark each

$(i) 1002 \times 1003$

$= (1000 + 2) (1000 + 3) \Rightarrow 1000^{2} + (2 + 3) 1000 + (3) (2)$

$= 1000000 + 5000 + 6$

$= 1005006$

$(i i) 97 \times 103$

$= (100 - 3) (100 + 3)$

$= [{(100)}^{2} - {(3)}^{2}]$

$= [10000 - 9]$

$= 9991$

Question 4

Find the volume of rectangular boxes with the following length, breadth and height respectively

$(i) (a^{2}, a^{3}, a^{4})$

$(i i) (x^{2} y^{2}, x z, z^{4})$ [2 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 1 Mark

(i) Volume = $a^{2} \times a^{3} \times a^{4} = a^{9}$

(ii) Volume = $x^{2} y^{2} \times x z \times z^{4} = x^{3} y^{2} z^{5}$

Question 5

Sandeep wants to buy 5 chocolates of Rs 10 each. He has to pay Rs 20 for parking also. If the price of chocolate is increased by Rs x, he decides to buy y less chocolates and he has to pay Rs z less for the parking. Write the arithmatic expression of the money spent by Sandeep. [2 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 1 Mark

Price of chocolate = (10 + x)

No. of chocolates = (5 – y)

Parking fees = (20 – z)

Total money spent = (10 + x) (5 – y) + (20 – z)

Question 6

Simplify the following:

$(i) (1.5 x - 4 y) (1.5 x + 4 y + 3)$

$(i i) (x + y) (2 x + y) + (x + 2 y) (x - y)$

$(i i i) (a^{2} + 5) (b^{3} + 3) + 5$ [3 MARKS]

SOLUTION

Solution :
Application: 1 Mark each

$(i) (1.5 x - 4 y) (1.5 x + 4 y + 3)$

$2.25 x^{2} + 6 x y + 4.5 x - 6 x y - 16 y^{2} - 12 y$

$\Rightarrow 2.25 x^{2} - 16 y^{2} + 4.5 x - 12 y$

$(i i) (x + y) (2 x + y) + (x + 2 y) (x - y)$

$(2 x^{2} + x y + 2 x y + y^{2}) + (x^{2} + x y - 2 y^{2})$

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

$(i i i) (a^{2} + 5) (b^{3} + 3) + 5$

$a^{2} b^{3} + 3 a^{2} + 5 b^{3} + 15 + 5$

$\Rightarrow a^{2} b^{3} + 3 a^{2} + 5 b^{3} + 20$

Question 7

Using the identities of algebraic expressions, Find :

$(i) 102^{2}$

$(i i) 53^{2}$

$(i i i) 1001^{2}$ [3 MARKS]

SOLUTION

Solution :
Application: 1 Mark each

Using the identity :
$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

$(i) (102)^{2}$

$\Rightarrow (100 + 2)^{2}$

$\Rightarrow (100)^{2} + (2)^{2} + 2 (100) (2)$

$\Rightarrow 10000 + 4 + 400$

$\Rightarrow 10404$

$(i i) (53)^{2}$

$\Rightarrow (50 + 3)^{2}$

$\Rightarrow (50)^{2} + (3)^{2} + 2 (50) (3)$

$\Rightarrow 2500 + 9 + 300$

$\Rightarrow 2809$

$(i i i) (1001)^{2}$

$\Rightarrow (1000 + 1)^{2}$

$\Rightarrow (1000)^{2} + (1)^{2} + 2 (1000) (1)$

$\Rightarrow 1000000 + 1 + 2000$

$\Rightarrow 1002001$

Question 8

Find the distance travelled with the following pairs of monomials as their speed and time respectively:

$(i) (25 a^{2} b^{2}, 50 b)$

$(i i) (12 a b c, 23 c d e)$

$(i i i) (5 c^{2} d^{2}, 4 a^{2} b^{2})$ [3 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 2 Marks

Distance Travelled

$= S p e e d \times T i m e$

(i) Distance Travelled $= 25 a^{2} b^{2} \times 50 b = 1250 a^{2} b^{3}$

(ii) Distance Travelled $= 12 a b c \times 23 c d e = 276 a b c^{2} d e$

(iii) Distance Travelled $= 5 c^{2} d^{2} \times 4 a^{2} b^{2} = 20 a^{2} b^{2} c^{2} d^{2}$

Question 9

Write the algebraic expression for:

(i) The volume of cube if its side(a) is reduced by 5.

(ii) Simple interest if time is reduced by 2 years and rate is increased by $5 %$ .

(iii) Distance travelled if speed is increased by 5 units and time by 10 units. [3 MARKS]

SOLUTION

Solution :
Concept: 1 Mark each

(i) Volume

$= (a - 5)^{3}$

Here a is the initial length of the side of the cube.

(ii) Simple interest

$= \frac{[P \times (R + 5) \times (T - 2)]}{100}$

P=Principal Amount

R=rate

T=Time

(iii) Distance = (s+5) (t+10)

s=speed

t=time

Question 10

Construct:

(i) 2 binomials with 3 variables.

(ii) 3 polynomials with 3 terms and 2 variables.

(iii) 2 trinomials with x as the only variable.

(iv) 3 binomials with x and y as variable. [4 MARKS]

SOLUTION

Solution :
Application: 1 Mark each

$(i) x y + x y z, 4 x^{2} y + 5 x y z$

$(i i) 4 a b^{2} + 5 a b + 10 a, 5 x y + 4 y - 24 a n d 10 x^{2} y^{2} + 12 x^{2} y - 5 x y^{2}$

$(i i i) 4 x^{2} + 5 x - 16, 5 x^{3} - 2 x^{2} + 24$

$(i v) 10 x^{2} y^{2} + 12 x^{2} y, 8 x y - 5 x a n d 5 x y + 8 x$

Question 11

Using the identites of algebriac expression , find the following :

$(i) 403 \times 397$

$(i i) {8.2}^{2}$

$(i i i) 9.7 \times 9.8$

$(i v) {22.9}^{2} - {7.1}^{2}$ [4 MARKS]

SOLUTION

Solution :
Application: 1 Mark each

$(i) 403 \times 397$

$= (400 + 3) (400 - 3)$

$= [(400)^{2} - (3)^{2}]$

$= 160000 - 9$

$= 159991$

$(i i) {8.2}^{2}$

$= (8 + 0.2)^{2}$

$= (8)^{2} + (0.2)^{2} + 2 (8) (0.2)$

$= 64 + 0.04 + 3.2$

$= 67.24$

$(i i i) 9.7 \times 9.8$

$= (9 + 0.7) (9 + 0.8)$

$= (9)^{2} + (0.7 + 0.8) 9 + (0.7) (0.8)$

$= 81 + 13.5 + 0.56$

$= 95.06$

$(i v) {22.9}^{2} - {7.1}^{2}$

$= (22.9 + 7.1) (22. 9 - 7.1)$

$= (30) (15.8)$

$= 474$

Question 12

A rectangular box having length, breadth and height as $(a + b + c), (a^{2} + b^{2}) a n d (2 a + 2 b + 2 c)$ respectively. Find the volume of the box. [4 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 3 Marks

Total volume =

$(a + b + c) (a^{2} + b^{2}) (2 a + 2 b + 2 c)$

$= (a^{3} + a b^{2} + a^{2} b + b^{3} + c a^{2} + c b^{2}) (2 a + 2 b + 2 c)$

$= (2 a^{4} + 2 a^{3} b + 2 a^{3} c + 2 a^{2} b^{2} + 2 a b^{3} + 2 a b^{2} c + 2 a^{3} b + 2 a^{2} b^{2} + 2 a^{2} b c + 2 a b^{3} + 2 b^{4} + 2 b^{3} c + 2 a^{3} c + 2 a^{2} b c + 2 a^{2} c^{2} + 2 a b^{2} c + 2 b^{3} c + 2 b^{2} c^{2})$

$= (2 a^{4} + 4 a^{3} b + 4 a^{3} c + 4 a^{2} b^{2} + 4 a b^{3} + 4 a b^{2} c + 4 a^{2} b c + 2 b^{4} + 4 b^{3} c + 2 a^{2} c^{2} + 2 b^{2} c^{2})$

Question 13

Classify the following polynomials as monomials, binomials, trinomials.

$(i) 4 x^{2}$

$(i i) 5 x y^{2} + 6 x y z$

$(i i i) 5 x y z$

$(i v) a + b + c$

$(v) 5 x^{2} + 6 x + 25$

$(v i) x + y$ [4 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 0.5 Mark each

Monomials -: $4 x^{2}, 5 x y z$

Binomials -: $5 x y^{2} + 6 x y z, x + y$

Trinomials -: $a + b + c, 5 x^{2} + 6 x + 25$

Question 14

Give three examples of the expressions:

(i) containing one variable

(ii) containing two variables

(iii) containing two terms. [4 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 1 Mark each

$(i) 4 x; 5 x + 2; 4 x^{2} + 4 x + 2$

$(i i) x + y; 5 x - 3 y; 4 x^{2} y^{2} + 4 x^{2} y + 5 y + 2$

$(i i i) 5 x - 4 y; 4 a^{2} + 5 b^{2}; 10 y + 2$

Question 15

Add the following:

$(i) 5 a^{2} b^{2} + 10 a b - 12 a^{2} + 15 b - 20$ and $34 a b + 24 a^{2} + 22 b + 24$

$(i i) a b c + 6 b c d + a b d - 9 a b + 4 b c - 9 a - 43$ and $4 a c d + 10 a c - 54 b - 32 c$

$(i i i) 5 b c + 9 a d - 3 c - 12$ and $24 - 5 c - 5 b + 2 a b$ [4 MARKS]

SOLUTION

Solution :
Concept: 1 Mark
Application: 1 Mark each

$(i) 5 a^{2} b^{2} + 10 a b - 12 a^{2} + 15 b - 20 + 34 a b + 24 a^{2} + 22 b + 24$

     $= 5 a^{2} b^{2} + 44 a b + 12 a^{2} + 37 b + 4$

$(i i) a b c + 6 b c d + a b d - 9 a b + 4 b c - 9 a - 43 + 4 a c d + 10 a c - 54 b - 32 c$

     $= a b c + 6 b c d + a b d - 9 a b + 4 b c - 9 a - 43 + 4 a c d + 10 a c - 54 b - 32 c$

$(i i i) 5 b c + 9 a d - 3 c - 12 + 24 - 5 c - 5 b + 2 a b$

     $= 5 b c + 9 a d - 8 c + 12 - 5 b + 2 a b$

Free Algebraic Expressions and Identities Subjective Test 01 Practice Test - 8th Grade

Question 1

SOLUTION

Question 2

SOLUTION

Question 3

SOLUTION

Question 4

SOLUTION

Question 5

SOLUTION

Question 6

SOLUTION

Question 7

SOLUTION

Question 8

SOLUTION

Question 9

SOLUTION

Question 10

SOLUTION

Question 11

SOLUTION

Question 12

SOLUTION

Question 13

SOLUTION

Question 14

SOLUTION

Question 15

SOLUTION

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