Exploring Algebraic Identities Class 9 Worksheet with Answers Maths Chapter 4

Teachers can assign these Ganita Manjari Class 9 Worksheet and NCERT Class 9 Maths Chapter 4 Exploring Algebraic Identities Worksheet with Answers Pdf for daily practice.

Exploring Algebraic Identities Worksheet Class 9

Class 9 Maths Exploring Algebraic Identities Worksheet

Worksheet On Exploring Algebraic Identities Class 9

Multiple Choice Questions

Question 1.
An algebraic identity is a statement that is true:
(a) For some values of variables
(b) For all values of variables
(c) Only for integers
(d) Only for positive numbers
Answer:
(b) For all values of variables

Question 2.
Which identity is used to factorise expressions of the form x² – y²?
(a) Square identity
(b) Cube identity
(c) Difference of squares
(d) Sum of cubes
Answer:
(c) Difference of squares

Question 3.
Which of the following is an identity?
(a) ² – 1 = 0
(b) x + 2 = 5
(c) (x + y)² = x² + 2xy + y²
(d) 2x + 3 = x
Answer:
(c) (x + y)² = x² + 2xy + y²

Question 4.
Which identity represents the square of the difference of two numbers?
(a) (a + b)² = a² – 2ab + b²
(b) (a – b)² = a² – 2ab + b²
(c) (a – b)² = a² + 2ab – b²
(d) (a – b)² = a² – b²
Answer:
(b) (a – b)² = a² – 2ab + b²

Question 5.
What is the expansion of (3x + 2)²?
(a) 9x² + 12x + 4
(b) 9x² + 4
(c) 6x² + 4x + 4
(d) 9x² 6x + 4
Answer:
(a) 9x² + 12x + 4

Question 6.
What is the value of (x + y)² – (x – y)²?
(a) 0
(b) 4xy
(c) 2xy
(d) x² + y²
Answer:
(b) 4xy

Question 7.
The factors of the expression x² + 5x + 6 is
(a) (x + 2)(x + 3)
(b) (x + l)(x + 6)
(c) (x + 5)(x + 6)
(d) None of these
Answer:
(a) (x + 2)(x + 3)

Question 8.
49² can be expressed as
(a) (50 – l)²
(b) (40 x 9)²
(c) (40 + 9) (40 – 9)
(d) All of these
Answer:
(a) (50 – l)²

Question 9.
In which of the following is one of a factor of (x + y)³ – (x³ + y³)?
(a) x² + y² + 2xy
(b) x² + y² – xy
(c) xy²
(d) 3xy
Answer:
(d) 3xy

Question 10.
The factorisation of x⁴ + 4 is ______.
(a) (x² + 2)²
(b) (x² + 2) (x² – 2)
(c) (x² + 2x + 2) (x² – 2x + 2)
(d) None of these
Answer:
(c) (x² + 2x + 2) (x² – 2x + 2)

Question 11.
If (a + b)² = a² + b², then which of the following must be true?
(a) a = b
(b) a = -b
(c) ac = 0
(d) a = 0
Answer:
(c) ac = 0

Question 12.
If x² + y² = 50 and xy = 24, then the value of (x – y)² is
(a) 2
(b) 4
(c) 6
(d) 8
Answer:
(a) 2

Question 13.
‘if x = y, then what is the value of (x – y)²?
(a) 0
(b) 1
(c) x²
(d) Cannot be determined
Answer:
(a) 0

Question 14.
The value of (50 + 2)² is
(a) 2604
(b) 2504
(c) 2704
(d) 2404
Answer:
(c) 2704

Question 15.
If a = 2 and b = -3, then the value of (a + b)² is
(a) 1
(b) 25
(c) 5
(d) 9
Answer:
(a) 1

Assertion-Reason

In Q. 1 to 5, two statements are given, one labelled as Assertion (A) and the other labelled as Reason (R). Select the correct answer from the options (a), (b), (c), and (d) given below.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.

1. Assertion (A): (a – b)² is always a non-negative value.
Reason (R): The square of any real number is always non-negative.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

2. Assertion (A): x² + 5x + 6 is a perfect square.
Reason (R): It can be factorised as (x + 2)(x + 3).
Answer:
(d) Assertion (A) is false, but Reason (R) is true.

3. Assertion (A): (x + y)² = (x – y)², if x = 0 or y = 0.
Reason (R): Subtracting the two expressions (x + y)² and (x – y)² gives 4xy.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

4. Assertion (A): (x + y)³ – (x – y)³ is divisible by y.
Reason (R): Every term in the expansion contains at least one factor of y.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

5. Assertion (A): (a – b)³ = a³ – b³ – 3ab(a – b).
Reason (R): The signs of all terms change when subtraction is involved.
Answer:
(c) Assertion (A) is true, but Reason (R) is false.

Exploring Algebraic Identities Class 9 Ganita Manjari Worksheet

Short Answer Type Questions – I

Question 1.
Using a suitable identity, evaluate 99².
Answer:
9801

Question 2.
Find the value of (a + b)³ – a³ – b³ in terms of a and b.
Answer:
3a²b + 3ab²

Question 3.
If x + \(\frac{1}{x}\) = 4, then find the value of x² + \(\frac{1}{x^2}\)
Answer:
14

Question 4.
If a + b = 6 and ab = 5, find a² + b².
Answer:
26

Question 5.
Determine whether x² + 12x + 36 is a perfect square. Justify.
Answer:
(x + 6)²

Question 6.
Factorise: 8a³ + 27.
Answer:
(2a + 3) (4a² – 6a + 9)

Question 7.
If x + y = 5 and x² + y² = 17,find xy.
Answer:
4

Question 8.
Using suitable identity, find the product of (3x + 4) and (3x – 5).
Answer:
9x² – 3x – 20

Question 9.
Expand using a suitable identity \(\left[\frac{1}{4} a-\frac{1}{2} b+1\right]^2\)
Answer:
[/latex]\frac{1}{16} a^2+\frac{1}{4} b^2+1-\frac{1}{4} a b-b+\frac{1}{2} a[/latex]

Question 10.
\(\frac{x}{y}+\frac{y}{x}\) = -1, find the value of x³ – y³.
Answer:
0

Short Answers Type Questions – ll

Question 1.
Factorise: 9a² – 9b² + 6a + 1.
Answer:
(3a + 3b + 1)(3a – 3b + 1)

Question 2.
Factorise: \(8 p^3+\frac{12}{5} p^2+\frac{6}{25} p+\frac{1}{125}\)
Answer:
\(\left(2 p+\frac{1}{5}\right)^3\)

Question 3.
Factorise each of the following:
(a) 27y³ + 125z³
(b) 64m³ – 343n³
Answer:
(a) (3y + 5z)(9y² — l5yz + 25z²)
(b) (4m – 7n)(16m² + 28mn + 49n²)

Question 4.
Give possible expressions for the length and breadth of each of the following rectangles whose areas are given as:
(a) Area: 25a² – 35a + 12
(b) Area: 35y² + 13y – 12
Answer:
(a) (5a – 3) and (5a – 4)
(b) (7y – 3) and (5y + 4)

Question 5.
Evaluate the following using suitable identities: \(\frac{4 x^2+4 x+1}{4 x^2-1}\)
Answer:
\(\frac{2 x+1}{2 x-1}\)

Question 6.
If the sum of a number and its reciprocal is equal to \(\frac{10}{3}\) find the number.
Answer:
\(\frac{1}{3}\) or 3

Question 7.
Using identities, evaluate (10l)³ – (99)³.
Answer:
60002

Long Answer Type Questions

Question 1.
Factorise: (x⁶ — y⁶).
Answer:
(x – y)(x + y) (x² + xy + y²)(x² – xy + y²)

Question 2.
A group of (a + fa) teachers, (a² + b²) girls and (a³ + b³) boys set out for an ‘Adult Education Mission’. If there are 10 teachers and 58 girls in the group, find the number of boys.
Answer:
370

Question 3.
Prove that (a + b + c)³ – a³ – b³ – c³ = 3(a + b)(b + c)(c + a).
Answer:

Question 4.
Show that: (a + b)³ + (a – b)³ = 2(a³ + 3ab²). Hence, find the value of (5 + 2)³ + (5 – 2)³.
Answer:
370

Question 5.
Ravita donated ₹ \(\left(x^3+\frac{1}{x^3}\right)\) to a blind school. Her friends wanted to know the amount donated by her. She did not disclose the amount but gave a hint that x + \(\frac{1}{x}\) = ₹ 7. Find the amount donated by Ravita to blind school.
Answer:
₹322

Case Based Questions

Question 1.
A square garden has side (a + b) metres. It is divided into a square of side a, a square of side b, and two identical rectangles.
(a) Write the total area of the garden in terms of (a + b).
Answer:
(a + b)² m²

(b) Write the area of the square garden as the sum of smaller parts. Hence, obtain an identity.
Answer:
(a² + b² + 2ab) m²

(c) Find the area of the garden, when a = 4, b = 2.
Answer:
36m²

OR
If one rectangle has the area 12 m² and a = 3 m, find the value of b.
Answer:
b = 4m

Question 2.
A municipal corporation is redesigning a square-shaped city park that currently measures 100 m x 100 m. To improve functionality, they decide to expand the park by adding 5 m to both the length and the width, making it a larger square. Later, to build a footpath around it, they take 2 m off the length and 2 m off the width of this expanded park, making it (105 – 2) × (105 – 2).
Using this information, answer the following questions:

(a) Which algebraic identity is most useful to calculate the area of the initially expanded park (105 m × 105 m)?
Answer:
(a + b)² = a² + b² + 2ab

(b) Find the exact area of the expanded park (105 m × 105 m) using suitable identities.
Answer:
11025m²

(c) If the park dimensions become (105 – 2) × (105 – 2), find its area using suitable identity.
Answer:
10609 m²

Competency Based Questions

Question 1.
Without expanding, determine which is greater:
50² or 49 × 51
Give reason for your answer.
Answer:
(50)²

Question 2.
A number is increased by 2 and then the result is squared. Another expression is obtained by squaring the number first and then adding 4 times the number and 4. Are the two expressions always equal? Justify your answer using an algebraic identity.
Answer:
Yes

Question 3.
A student observes that (3 + 4)³ = 343 and 3³ + 4³ = 91 Explain the reason for the difference.

Question 4.
Explain why the expression x² + 10x + 21 is not a perfect square, even though it can be factorised.

Activity

Question 1.
Visualise the algebraic identity (a – b)² geometrically using algebraic tiles. Justify, increasing both a and b always increases the value of (a – b)². Is this correct? Justify

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