Algebraic Expressions And Identities(8th Grade > Mathematics ) Questions and answers for exam Preparation

8th Grade > Mathematics

ALGEBRAIC EXPRESSIONS AND IDENTITIES MCQs

Total Questions : 88 | Page 7 of 9 pages

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]

Discuss Question

Answer: Option A. ->
:

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 62.

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]

Discuss Question

Answer: Option A. ->
:

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 63.

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]

Discuss Question

Answer: Option A. ->
:

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 64.

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]

Discuss Question

Answer: Option A. ->
:

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 65.

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]

Discuss Question

Answer: Option A. ->
:

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 66.

Give three examples of the expressions:

(i) containing one variable

(ii) containing two variables

(iii) containing two terms. [4 MARKS]

Discuss Question

Answer: Option A. ->
:

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 67.

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]

Discuss Question

Answer: Option A. ->
:

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 68.

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]

Discuss Question

Answer: Option A. ->
:

Concept: 1 Mark
Application: 1 Mark each
$(i) 5 a^{2} b^{2} + 10 a b - 12 a^{2} + 15 b - 20 + 34 a b + 24 a^{2} + 22 b + 24$

     $= 5 a^{2} b^{2} + 44 a b + 12 a^{2} + 37 b + 4$
$(i i) a b c + 6 b c d + a b d - 9 a b + 4 b c - 9 a - 43 + 4 a c d + 10 a c - 54 b - 32 c$
     $= a b c + 6 b c d + a b d - 9 a b + 4 b c - 9 a - 43 + 4 a c d + 10 a c - 54 b - 32 c$
$(i i i) 5 b c + 9 a d - 3 c - 12 + 24 - 5 c - 5 b + 2 a b$
     $= 5 b c + 9 a d - 8 c + 12 - 5 b + 2 a b$

Question 69.

What must be subtracted from $(x^{3} - 3 x^{2} + 5 x - 1)$ to get $(2 x^{3} + x^{2} - 4 x + 2)$ ?

$- x^{3} + 4 x^{2} + 9 x - 3$
$- x^{3} - 4 x^{2} + 6 x - 3$
$- x^{3} - 4 x^{2} + 9 x - 9$
$- x^{3} - 4 x^{2} + 9 x - 3$

Discuss Question

Answer: Option D. ->

- x^{3} - 4 x^{2} + 9 x - 3

:
D

Let the expression to be subtracted be $y$ .
Hence,
$(x^{3} - 3 x^{2} + 5 x - 1) - y = 2 x^{3} + x^{2} - 4 x + 2$
On rearranging the terms,
$y = (x^{3} - 3 x^{2} + 5 x - 1) - (2 x^{3} + x^{2} - 4 x + 2)$
$y = x^{3} - 3 x^{2} + 5 x - 1 - 2 x^{3} - x^{2} + 4 x - 2$
$y = x^{3} - 2 x^{3} - 3 x^{2} - x^{2} + 5 x + 4 x - 1 - 2$
$\Rightarrow y = - x^{3} - 4 x^{2} + 9 x - 3$
So we have to subtarct $(- x^{3} - 4 x^{2} + 9 x - 3)$ to get the required value.