If only (a^2+b^2+c^2) is written for ( (a+b+c)^2 ), which terms are missing?
Correct answer: B
The square of three terms also includes twice the pair products. Exam tip: always write (2ab+2bc+2ca).
View question detailsMathematics
Algebraic identities
बीजीय सर्वसमिकाएँ
In this Class 9 Mathematics topic from “Exploring Algebraic Identities,” students learn how standard algebraic relationships remain true for all permitted values of the variables. They study and apply identities such as (a + b)², (a − b)², and a² − b² to expand expressions, simplify calculations, and factorise algebraic forms. The topic also develops skill in recognising suitable patterns, substituting values to verify results, and using identities to solve expressions efficiently and accurately.
TOPIC PRACTICE
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Hard · Level 64 · common error,three term square,missing terms
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- (ab+bc+ca)
- (2ab+2bc+2ca)
- (3abc)
- (a+b+c)
Hard · Level 64 · algebraic identities,perfect square trinomial,binomial square,class 9 mathematics,quadratic expressions
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- \(x^2+12x+36\)
- \(x^2+12x+18\)
- \(x^2+6x+36\)
- \(x^2-12x-36\)
Hard · Level 64 · algebraic identities,sum of squares,mental mathematics,class 9 mathematics
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- 5000
- 5002
- 4902
- 5100
Hard · Level 64 · algebraic identities,difference of squares,mathematics class 9,mental calculation,numerical expressions
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- 100
- 200
- 400
- 2
Hard · Level 64 · algebraic identities, square of trinomial, expression evaluation, class 9 mathematics, mixed terms, error analysis
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- The claim is correct; the value is \(74\).
- The claim is correct; the value is \(109\).
- The claim is incorrect; the value is \(4\).
- The claim is incorrect; the value is \(214\).
Hard · Level 64 · algebraic identities,difference of squares,polynomial simplification,class 9 mathematics,conjugate binomials
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- \(x^4-1\)
- \(x^4+1\)
- \(x^2-1\)
- \(x^4-2x^2+1\)
Hard · Level 64 · algebraic identities,difference of squares,polynomial simplification,class 9 mathematics
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- \(8x+8\)
- \(4x+4\)
- \(8x+4\)
- \(4\)
Hard · Level 64 · algebraic identities,difference of squares,polynomial simplification,class 9 mathematics
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- 36x+24
- 18x+24
- 36x+26
- 24x+36
Hard · Level 64 · algebraic identities,perfect square trinomial,square of difference,class 9 mathematics,binomial square
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- \(9p^2-24pq+12q^2\)
- \(9p^2-24pq+16q^2\)
- \(9p^2+24pq+16q^2\)
- \(6p^2-24pq+16q^2\)
Hard · Level 64 · algebraic identities, square of a sum, substitution, exponents, numerical evaluation
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- 20
- 50
- 100
- 10
Hard · Level 64 · algebraic identities, perfect square trinomial, binomial square, grade 9 mathematics, polynomial identification
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- \(4a^2+12ab+9b^2\)
- \(4a^2+12ab-9b^2\)
- \(4a^2+6ab+9b^2\)
- \(4a^2-12ab-9b^2\)
Hard · Level 64 · algebraic identities, square of sum, binomial square, class 9 mathematics, perfect square trinomial
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- \((a+b)^2=a^2+2ab+b^2\)
- \((a-b)^2=a^2-2ab+b^2\)
- \(a^2-b^2=(a+b)(a-b)\)
- \(a^3+b^3=(a+b)(a^2-ab+b^2)\)
Hard · Level 64 · factorisation,perfect square,difference of squares
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- ( (x+3-y)^2 )
- ( (x+3)(x-y) )
- ( (x+3+y)(x+3-y) )
- ( (x+y+3)(x-y-3) )
Medium · Level 62 · algebraic identities,perfect square trinomial,binomial square,class 9 mathematics,polynomials
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- \(x^2+14x+49\)
- \(x^2+14x+42\)
- \(x^2+7x+49\)
- \(x^2+49\)
Medium · Level 62 · difference of squares,product identity,two variables
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- \(9a^2+16b^2\)
- \(3a^2-4b^2\)
- \(9a^2-24ab+16b^2\)
- \(9a^2-16b^2\)
Medium · Level 62 · algebraic identities,perfect square trinomial,binomial square,grade 9 mathematics
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- \((x+6)^2\)
- \((x-6)^2\)
- \((x+12)^2\)
- \((x+36)^2\)
Medium · Level 62 · algebraic identities,difference of squares,mental multiplication,class 9 mathematics,numerical algebra
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- 9984
- 10016
- 9985
- 9604
Medium · Level 62 · algebraic identities,perfect square,binomial expansion,class 9 mathematics,polynomials
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- \(p^2+2pq+q^2\)
- \(p^2-2pq+q^2\)
- \(p^2-q^2\)
- \(p^2-2pq-q^2\)
Easy · Level 67 · algebraic identities, square of sum, polynomial expansion, class 9 mathematics
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- \(x^2+2xy+y^2\)
- \(x^2-2xy+y^2\)
- \(x^2-y^2\)
- \(x^2+xy+y^2\)
Easy · Level 67 · algebraic identities,difference of squares,factorisation,class 9 mathematics,polynomials
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- \((x+6)(x-6)\)
- \((6+x)(6-x)\)
- \((36+x)(36-x)\)
- \((6-x)^2\)
Which of the following trinomials can be written as the perfect square of a binomial?
Correct answer: A
\(x^2+12x+36=(x+6)^2\) because the middle term is \(2\times x\times6=12x\). In option B, the constant term should be \(36\), not \(18\). Exam tip: check whether \(b^2=4ac\).
View question detailsWhat value is obtained by simplifying ( 51^2+49^2 ) using identity?
Correct answer: B
Here, \(51=50+1\) and \(49=50-1\). Using the identity \((a+b)^2+(a-b)^2=2a^2+2b^2\), with \(a=50\) and \(b=1\), we get \(51^2+49^2=2(50)^2+2(1)^2=5000+2=5002\). The value 5000 includes only \(2\times 50^2\); \(2\times 1^2\) must also be added. Exam tip: apply this identity directly when two numbers are equally spaced from a middle number.
View question detailsWhat is the value of ( 51^2-49^2 )?
Correct answer: B
Use the identity \(a^2-b^2=(a+b)(a-b)\). Here, \(51^2-49^2=(51+49)(51-49)=100\times2=200\). Option 100 is only the sum of the two numbers, while 2 is their difference; their product gives the required value. Exam tip: for the difference of squares of nearby numbers, apply this identity directly.
View question detailsA student claims that if \(a+b+c=12\) and \(ab+bc+ca=35\), then \(a^2+b^2+c^2=74\). Which option about the student’s claim is correct?
Correct answer: A
Using \((a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\), we get \(a^2+b^2+c^2=12^2-2\times35=144-70=74\). Thus, the claim is correct. \(109\) results from forgetting the factor \(2\). Exam tip: always check the coefficient of mixed terms.
View question detailsWhat is the simplified form of \( (x-1)(x+1)(x^2+1) \)?
Correct answer: A
First, use the identity \((x-1)(x+1)=x^2-1\). The expression then becomes \((x^2-1)(x^2+1)\), which again has the form \((a-b)(a+b)=a^2-b^2\). Therefore, it simplifies to \(x^4-1\). \(x^4+1\) is not the result of this difference-of-squares identity. Exam tip: identify conjugate binomial pairs and apply identities step by step.
View question detailsWhat is the simplified form of \( (2x+3)^2-(2x+1)^2 \)?
Correct answer: A
This is a difference of squares: \(a^2-b^2=(a+b)(a-b)\). Here, \(a=2x+3\) and \(b=2x+1\). Thus, \(a+b=4x+4\) and \(a-b=2\), so the expression becomes \((4x+4)\times2=8x+8\). The expression \(4x+4\) is only \(a+b\), not the final simplified form. Exam tip: first identify the two bases of the squares, then apply the difference-of-squares identity.
View question detailsWhat is the simplified form of ( (3x+5)^2-(3x-1)^2 )?
Correct answer: A
Use the identity \(a^2-b^2=(a+b)(a-b)\). Here, \(a=3x+5\) and \(b=3x-1\). Thus, \(a+b=6x+4\) and \(a-b=6\). Therefore, \((6x+4)\times6=36x+24\). Option B results from an incorrect multiplication of the \(6x\) term by 6. Exam tip: identify the two squared expressions first and apply the identity directly instead of expanding both squares.
View question detailsWhich of the following expressions can be written as the square of a difference of two terms?
Correct answer: B
Here \(9p^2=(3p)^2\) and \(16q^2=(4q)^2\). The middle term is \(-24pq=-2(3p)(4q)\), so the expression is \((3p-4q)^2\). Exam tip: check the sign of the middle term carefully.
View question detailsIf (a+b+c=10), what is the complete value of ( (a+b+c)^2 )?
Correct answer: C
Given \(a+b+c=10\), we get \((a+b+c)^2=10^2=100\). Option 10 is only the value of the sum, not its square. Exam tip: when a power is outside brackets, substitute the bracket's value first and then apply the power.
View question detailsWhich of the following expressions is a perfect-square trinomial for every a and b?
Correct answer: A
Here \(4a^2=(2a)^2\) and \(9b^2=(3b)^2\). The required middle term is \(2\times2a\times3b=12ab\), so the expression is \((2a+3b)^2\). Exam tip: always check both the sign and coefficient of the middle term.
View question detailsWhich of the following algebraic identities represents the square of the sum of two terms?
Correct answer: A
Squaring a sum gives the square of the first term, twice their product, and the square of the second term. Hence \((a+b)^2=a^2+2ab+b^2\). Exam tip: the middle term is \(+2ab\) for a sum, not \(-2ab\).
View question detailsWhich is the correct factor form of ( x^2+6x+9-y^2 )?
Correct answer: C
First identify (x^2+6x+9=(x+3)^2). Then apply the difference of squares identity to ( (x+3)^2-y^2 ).
View question detailsWhich of the following trinomials can be written as the square of a binomial?
Correct answer: A
In \(x^2+14x+49\), the last term is \(49=7^2\) and the middle term is \(14x=2\cdot x\cdot7\). Hence it is \((x+7)^2\). In exams, always check whether the middle term equals \(2ab\).
View question detailsWhat is the simplified form of ( (3a+4b)(3a-4b) )?
Correct answer: D
This is the form \( (u+v)(u-v)=u^2-v^2 \). With \(u=3a\) and \(v=4b\), it gives \(9a^2-16b^2\).
View question details( x^2+12x+36 ) is the perfect square of what?
Correct answer: A
Use the identity \((a+b)^2=a^2+2ab+b^2\). Here, \(x^2+12x+36=x^2+2\cdot x\cdot6+6^2\), so it equals \((x+6)^2\). Note that \((x-6)^2=x^2-12x+36\), so it is not correct. Exam tip: use the sign of the middle term to decide whether the binomial has \(+\) or \(-\).
View question detailsUsing an identity, what is the value of \( 96\times104 \)?
Correct answer: A
Here, \(96=100-4\) and \(104=100+4\). Therefore, \((100-4)(100+4)=100^2-4^2=10000-16=9984\). The option \(10016\) results from adding \(4^2\) instead of subtracting it. Exam tip: when two numbers are equally spaced around a middle number, use \((a-b)(a+b)=a^2-b^2\).
View question detailsWhich of the following expressions represents the expansion of \((p-q)^2\)?
Correct answer: B
Using \((a-b)^2=a^2-2ab+b^2\), put \(a=p\) and \(b=q\) to get \(p^2-2pq+q^2\). Option C is the difference of squares. Exam tip: the middle term must be \(-2pq\).
View question detailsWhich of the following expressions represents the square of the sum of \(x\) and \(y\)?
Correct answer: A
\((x+y)^2=(x+y)(x+y)\). On multiplying, the middle terms give \(xy+xy=2xy\), so the expression is \(x^2+2xy+y^2\). Exam tip: check the sign of the middle term carefully.
View question detailsWhat is the correct factorisation of (36-x^2)?
Correct answer: B
This is a difference of squares: \(36-x^2=6^2-x^2\). Using \(a^2-b^2=(a+b)(a-b)\), we get \((6+x)(6-x)\). Option A expands to \(x^2-36\), which is the negative of the given expression. Exam tip: first identify the square roots of both terms, then write one sum factor and one difference factor.
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