ICSE Class 8 Maths Algebra Unit Test Paper

Unit Test Paper

Algebra

Question 1: Write the Degrees of Algebraic Expressions

Write the degrees of the following algebraic expressions and name them.

(i) \(5 \times x \times x \times y + y \times y \times z - z \times z\)

(ii) \(\frac{x^4 y^5 z^2}{3}\)

(iii) \(\frac{x^4 y^5 z^2}{5} - \frac{x^3 y z}{4}\)

(iv) \(\frac{3x^6 + 4y^2 - 2z^2 + 6}{7}\)

Teacher's Note

Understanding polynomial degrees helps in analyzing real-world situations like calculating areas and volumes in construction projects.

Question 2: Add the Following

(i) \((5x^2 - 6) + (4y^2 + 4)\)

(ii) \((3x^2y^2 + 2y^2x^2 + 8) + (5y^2x^2 + 3z^2y^2 - 12)\)

(iii) \((8x^5) + 9y^4x + 6) + (x^2) - 11y^4x + 5xy - 3)\)

(iv) \((5x^5y^2 - 6x^2z^2 - 2y^2z^2 + 29) + (8x^2z^2 + 6y^2z^2 - 7x^2y^2 - 38)\)

Question 3: Subtract the Following

(i) \((6y^2 + 9) - (y^2 + 9)\)

(ii) \((3x^2y^2 + y^2z^2 + 2) - (3y^2z^2 + x^2y^2 - 3)\)

(iii) \((8x^3) + 9y^4x + 4) - (7y^4x - z^3y - 6)\)

(iv) \((12x^2yz - 11y^4xz + 4z^2yx - 17) - (5x^2yz + y^4xz - z^2xy - 11)\)

Question 4: Multiply the Following

(i) \(\frac{3x^2 y}{z} \left(\frac{z^2}{xy} + \frac{2x^2}{3x^2} + \frac{yz}{9x^2}\right)\)

(ii) \(\frac{x^3 y}{7z} \left(\frac{21z^2}{x^2 y} - \frac{2xy}{x^3} + \frac{14z}{x^3 y}\right)\)

(iii) \((6x^2 - 4)(2x^2 + 3)\)

(iv) \((5x - 2y)(3x + 4y)\)

(v) \((5x^2 + 2y^2 + z)(5x^2 - 2y^2 + z)\)

(vi) \((3x + 2y + 1)(9x^2 + 4y^2 - 6xy - 3x - 2y + 1)\)

Question 5: Divide the Following

(i) \(\left(\frac{xz}{6y} + \frac{3x^2z^2}{6y^2} + \frac{5y}{x} + \frac{x^2}{4y^2}\right) \div \frac{x^2}{2y^2}\)

(ii) \(\left(6x^2 y^2 - \frac{3x^2 y}{4z^2} + \frac{2y^2 x}{3z^2} - \frac{9xy}{8z}\right) \div \frac{3xy}{4z}\)

(iii) \((18x^2 + 18x - 20) \div (6x - 4)\)

(iv) \((8x^2 + 18xy - 35y^2) \div (2x + 7y)\)

Question 6: Simplify the Following

(v) \((3x^2 - x^2y - 23y^2x + 21y^3) + (3x - 7y)\)

(vi) \((8x^2 - 64y^2) \div (2x - 4y)\)

(i) \(\frac{1}{3x} \left[5x^2 + 2x - 3 - (x + 3)\left(3x + y - \frac{1}{3}\right) - (6x - y + 9 - 3x - 4y + 6)\right]\)

(ii) \([(3x - 5)(9x^2 + 15x + 25) + 125] + 9x^2] - 3x\)

Teacher's Note

Simplifying algebraic expressions is essential in engineering and physics to reduce complex formulas into manageable forms.

Question 7: Find Values Using Concept of Special Products

(i) \(289 \times 311\)

(ii) \(693 \times 707\)

(iii) \(567^2 - 565^2\)

(iv) \(1399^2 - 1398^2\)

Question 8: Find the Continued Product

(i) \(\left(\frac{2}{x} - 1\right)\left(\frac{2}{x} + 1\right)\left(\frac{4}{x^2} + 1\right)\)

(ii) \((2x - y)(2x + y)(4x^2 + y^2)(16x^4 + y^4)\)

Question 9: Perfect Square Problem

What should be subtracted from \(36x^2 - 42xy + 16y^2\) to make it a perfect square?

Question 10: Perfect Square Problem

What should be added to \(\frac{x^2}{4} + xy + 9y^2\) to make it a perfect square?

Question 11: Algebraic Identity

Given that \(y + \frac{1}{y} = 9\), find the value of \(y^2 + \frac{1}{y^2}\).

Question 12: Algebraic Identity

Given that \(y + \frac{1}{y} = 1\), find the value of \(y^4 + \frac{1}{y^4}\).

Question 13: Algebraic Identity

Given that \(y^2 + \frac{1}{y} = 98\), find the value of \(y + \frac{1}{y}\).

Question 14: Factorise the Following Expressions

(i) \(15x^3y^2 - 5x^5y^3 + 10x^3y^2\)

(ii) \(15xy - 12y + 15x - 12\)

(iii) \(6x^5y^2 + 2y^2x^2 - 36x^3y^2 - 12y^4x\)

(iv) \(8x^3y + 16x^2y^2 + 24x^2y - 12x^2y - 24y^2x - 36xy\)

(v) \(3x^2 - 5xy - 12y^2\)

(vi) \(10x^2 + 6xy - 28y^2\)

Teacher's Note

Factorization helps us solve real-world problems by breaking down complex expressions into simpler components.

Question 15: Given Formula Application

Given \(K = \frac{1}{2} mv^2\), find v given that \(K = 312.5\) and \(m = 0.01\).

Question 16: Solve Equations

(i) \(3(x + 7) - 4(x - 3) = 2x\)

(ii) \(5(x - 3) + 2(x + 4) = 4(x - 7)\)

Question 17: Word Problem - Number

Ramya thought of a number, halved it and then added 2. The sum divided by 3 gave the quotient 5. What was the number Ramya thought of?

Question 18: Word Problem - Consecutive Numbers

Find 3 consecutive odd numbers such that the sum of twice the least number and thrice the greatest number is 107.

Question 19: Word Problem - Ages

Mr Yadav is 3 years elder to Mrs Yadav. If the sum of \(\frac{1}{5}\) of his age and \(\frac{1}{6}\) of her age is 27 years, find their ages.

Question 20: Word Problem - Division of Money

Divide Rs 8030 between A and B such that \(\frac{2}{5}\) of A's share equals \(\frac{1}{6}\) of B's share.

Question 21: Word Problem - Fraction

The denominator of a common fraction exceeds the numerator by 3. If 1 is added to the numerator as well as denominator, the fraction \(\frac{4}{5}\) is obtained. Find the original common fraction.

Question 22: Word Problem - Two-digit Number

The sum of the digits of a 2-digit number is 11. When the digits are reversed, the number decreases by 9. Find the original 2-digit number.

Question 23: Word Problem - Coins

A piggy bank has Rs 45.60 made of 50 p, 25 p, and 20 p coins. If six-sevenths of the number of 50 p coins are 20 p coins and \(\frac{2}{3}\) of the number of 20 p coins are 25 p coins, find the number of coins of each denomination.

Question 24: Inequation Solution

Given the replacement set as \(x \in Z\), solve \(-57 \le 3(4x - 3) \le 15\) and represent the solution on the number line.

Question 25: Word Problem - Cost

3 calculators and 12 batteries cost Rs 585 while a calculator and 5 batteries cost Rs 202.50. How much does a calculator and a battery cost?

Question 26: Simultaneous Equations

(i) \(5x = 19 + 3y; 3x = 38 - 2y\)

(ii) \(8x - y + 19 = 0; 2x + 5y - 11 = 0\)

(iii) \(\frac{y}{3} - \frac{x}{5} = 5; \frac{y}{6} + \frac{x}{3} = 8\)

(iv) \(\frac{3}{x} + \frac{1}{y} = 7; \frac{12}{x} - \frac{2}{y} = 10\)

Teacher's Note

Solving simultaneous equations is used in business to find break-even points and optimize production costs.

Question 27: Word Problem - Bus Passengers

There are 12 more passengers on the lower deck of a double decker bus than the upper deck. If 24 passengers from the lower deck climb upstairs, the upper deck will have 3 times as many passengers as the lower deck. How many passengers are there on each deck of the bus?

Question 28: Word Problem - Speed and Distance

A tug-boat first travels 40 km upstream and 36 km downstream in 8 hours. Then it travels 54 km downstream and 20 km upstream in 7 hours. Find the speed of the tug-boat in still water and the speed of the water current.

Question 29: Quadratic Equations

How many values of x will the equation \(4x^2 + 12x + 9 = 0\) have? Why?

Question 30: Solve Quadratic Equations

(i) \(9x^2 - 36 = 0\)

(ii) \(x^2 - 13x + 40 = 0\)

(iii) \(7x^2 - 32x = 15\)

(iv) \(30x^2 + 13x = 10\)

Question 31: Graph - Speed and Distance

Given that speed = 3 m/s, draw the graph showing the mapping between time (t) and distance (d).

Question 32: Mapping - Profit Percentage

The C.P. and S.P. of 7 different articles bought and sold by a trader are as under:

Is the relation between the profit per cent of each article and its S.P. a mapping? Represent the relation by an arrow diagram and explain why.

Question 33: Draw Graphs

(i) \(5x + y = 15\)

(ii) \(3x - y = 14\)

(iii) \(3x + 5y = 20\)

(iv) \(x - \frac{y}{6} - 2 = 0\)

Question 34: Simultaneous Equation Graphs

Can a solution be found that satisfies the equations \(2x = 16\) and \(\frac{x}{4} - 1 = 0\) simultaneously? Draw graphs for both equations and explain why.

Question 35: Magic Squares

By now you must have figured out that there is less of 'magic' and more of Algebra in a 'Magic Square'. Work out the numbers in the following magic squares such that the sums of all the rows, columns, and diagonals in each are the same.

Teacher's Note

Magic squares demonstrate how algebra creates patterns and balances in mathematical structures, similar to how equations balance in chemistry.

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