What is the coefficient of (x^2) in ( (x-1)(x-2)(x-3) )?
Correct answer: B
The sum of constants is (1+2+3=6) and the sign is negative. Exam tip: stepwise multiplication also gives the same result.
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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Expert · Level 63 · coefficient,cubic expansion,expert
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- (6)
- (-6)
- (11)
- (-11)
Expert · Level 63 · cube identity,advanced simplification,expert
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- (2b(3a^2+b^2))
- (2a(a^2+3b^2))
- (6ab)
- (a^3-b^3)
Expert · Level 63 · cube identity,sum of cubes,binomial cube
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- (2a(a^2+3b^2))
- (2b(3a^2+b^2))
- (6a^2b)
- (2a^3+2b^3)
Expert · Level 63 · algebraic identities, difference of cubes, factorisation, class 9 mathematics, algebra
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- a^3-b^3
- a^2-b^2
- a^3+b^3
- a^2+b^2
Expert · Level 63 · algebraic identities,perfect square trinomial,quadratic expressions,completing the square,class 9 mathematics
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- \(q=\frac{p^2}{4}\)
- \(q=2p\)
- \(q=p^2\)
- \(q=\frac{p}{2}\)
Expert · Level 63 · three variable identity,cube relation,expert
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- (ab+bc+ca)
- (abc)
- (3abc)
- (0)
Expert · Level 63 · three variable identity,zero condition,cube
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- (xyz)
- (3xyz)
- (x^2+y^2+z^2)
- (0)
Expert · Level 63 · algebraic identities, factorisation, difference of squares, divisibility, class 9 mathematics
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- \(a^2-b^2\)
- \(a^2+b^2\)
- \(a^2+ab+b^2\)
- \((a+b)^2\)
Expert · Level 63 · three term square,mixed terms,identity
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- (xy+yz+zx)
- (2xy+2yz+2zx)
- (xyz)
- (x+y+z)
Expert · Level 63 · common factor,combined identity,simplification
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- (12x+18)
- (6x+9)
- (18)
- (4x^2+9)
Expert · Level 63 · cube expansion,term identification,expert
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- (12x^2)
- (24x^2)
- (48x^2)
- (96x^2)
Expert · Level 63 · cube identity,symmetric expression,simplification
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- (2x^3+24x)
- (2x^3+12x)
- (x^3+8)
- (2x^3+16)
Expert · Level 63 · algebraic identities,polynomial simplification,sign handling,binomial square,grade 9 mathematics
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- \(2ab+2b^2\)
- \(2ab\)
- \(a^2+2ab\)
- \(2b^2\)
Expert · Level 63 · minus square,middle term,identity
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- (2ab)
- (-2ab)
- (a^2-b^2)
- (-b^2)
Expert · Level 63 · algebraic identities, perfect square, non-negative expressions, binomial square, class 9 mathematics
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- \(x^2-y^2\)
- \(x^2+2xy+y^2\)
- \(x^2-2xy-y^2\)
- \(x^2+xy+y^2\)
Expert · Level 63 · reciprocal expression,minus square,identity
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- (34)
- (36)
- (38)
- (40)
Expert · Level 63 · reciprocal identity,difference,expert
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- (2)
- (4)
- (x^2+\frac{1}{x^2})
- (0)
Expert · Level 63 · nested identity,difference of squares,quartic
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- (x^4+16)
- (x^4-8)
- (x^4-16)
- (x^4+8x^2+16)
Expert · Level 63 · algebraic identities,difference of squares,polynomials,class 9 mathematics,binomial products
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- \((a+b)^2=a^2+2ab+b^2\)
- \((a-b)^2=a^2-2ab+b^2\)
- \((a+b)(a-b)=a^2-b^2\)
- \((a+b)^2-(a-b)^2=4ab\)
Expert · Level 63 · algebraic identities,difference of squares,factorisation,perfect squares,class 9 mathematics
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- \(p^4-81q^4\)
- \(p^4+81q^4\)
- \(p^4-18p^2q^2+81q^4\)
- \(p^4+9p^2q^2+81q^4\)
What is the simplified form of ( (a+b)^3-(a-b)^3 )?
Correct answer: A
On expansion, (6a^2b+2b^3=2b(3a^2+b^2)). Exam tip: handle signs carefully in cube expansions.
View question detailsWhat is ( (a+b)^3+(a-b)^3 ) equal to?
Correct answer: A
On adding, terms with odd powers of (b) cancel and (2a^3+6ab^2) remains. Exam tip: use symmetry.
View question detailsWhich of the following expressions can be written using the identity for the difference of two cubes?
Correct answer: A
A difference of two cubes has the form \(a^3-b^3\) and factorises as \((a-b)(a^2+ab+b^2)\). \(a^3+b^3\) is a sum of cubes, so it uses a different identity. Exam tip: for a difference, the first binomial is \(a-b\).
View question detailsIf, for real constants p and q, the expression \(x^2+px+q\) can be written as the perfect square of a binomial involving \(x\) and a constant, which relation between \(p\) and \(q\) is necessary?
Correct answer: A
For a perfect square, \(x^2+px+q=(x+\frac p2)^2\). Squaring gives the constant term \(\frac{p^2}{4}\), so \(q=\frac{p^2}{4}\). The choice \(q=p^2\) wrongly ignores halving the coefficient of \(x\). Exam tip: halve the middle coefficient, then square it.
View question detailsIf (a+b+c=0), what is (a^3+b^3+c^3) equal to?
Correct answer: C
By identity, (a^3+b^3+c^3-3abc=(a+b+c)(\cdots)). Exam tip: if (a+b+c=0), remember (a^3+b^3+c^3=3abc).
View question detailsIf (x+y+z=0), what is the value of ( x^3+y^3+z^3-3xyz )?
Correct answer: D
In the identity, this expression is multiplied by (x+y+z), so the value is (0). Exam tip: check the condition first.
View question detailsWhich of the following expressions is divisible by \,a-b\, for all values of the variables?
Correct answer: A
\(a^2-b^2=(a-b)(a+b)\), so \(a-b\) is a factor for every value of the variables. \(a^2+b^2\) does not generally contain this factor. Exam tip: factorise using identities before testing divisibility.
View question detailsWhat remains when (x^2+y^2+z^2) is removed from ( (x+y+z)^2 )?
Correct answer: B
The extra mixed terms in the square of three terms are (2xy+2yz+2zx). Exam tip: write all three mixed terms.
View question detailsWhat is the simplified form of ( (2x+3)^2-(2x+3)(2x-3) )?
Correct answer: A
((2x+3)) is common, giving ((2x+3)[(2x+3)-(2x-3)]). Exam tip: taking common factors makes work easier.
View question detailsWhat will be the term containing (x^2) in ( (x+4)^3-(x-4)^3 )?
Correct answer: B
The difference gives (6x^2\cdot4=24x^2). Exam tip: identify the (x^2) term quickly in cube differences.
View question detailsWhat is the simplified form of ( (x+2)^3+(x-2)^3 )?
Correct answer: A
In the sum, constant cube terms cancel and (2x^3+6x\cdot4=2x^3+24x) remains. Exam tip: understand symmetric cube expansion.
View question detailsWhat is the simplified form of ( (a+b)^2-(a^2-b^2) )?
Correct answer: A
Using \((a+b)^2=a^2+2ab+b^2\), subtracting \(a^2-b^2\) changes the signs of both terms inside it. Thus, \(a^2+2ab+b^2-a^2+b^2=2ab+2b^2\). Option B misses the remaining \(2b^2\) term. Exam tip: when a bracket is preceded by a minus sign, change the sign of every term in that bracket.
View question detailsWhat is ( (a-b)^2-(a^2+b^2) ) equal to?
Correct answer: B
( (a-b)^2=a^2-2ab+b^2 ), so subtracting leaves (-2ab). Exam tip: remember the negative middle term.
View question detailsWhich of the following expressions is always non-negative for all real values of \(x\) and \(y\)?
Correct answer: B
\(x^2+2xy+y^2=(x+y)^2\), a perfect square. The square of every real number is never negative, so this expression is always non-negative. Exam tip: recognise the middle term \(2xy\).
View question detailsIf (t-\frac{1}{t}=6), what is (t^2+\frac{1}{t^2})?
Correct answer: C
On squaring, (t^2-2+\frac{1}{t^2}=36), so the value is (38). Exam tip: the square of a difference gives (-2).
View question detailsWhat is the value of ( (x+\frac{1}{x})^2-(x-\frac{1}{x})^2 )?
Correct answer: B
This is (4AB) where (A=x) and (B=\frac{1}{x}), so the value is (4). Exam tip: reciprocal terms multiply to (1).
View question detailsWhat is the simplified form of ( (x-2)(x+2)(x^2+4) )?
Correct answer: C
First ( (x-2)(x+2)=x^2-4 ), then ((x^2-4)(x^2+4)=x^4-16). Exam tip: apply difference of squares twice.
View question detailsWhich of the following algebraic identities expresses the product of the sum and difference of two terms as the difference of their squares?
Correct answer: C
In \((a+b)(a-b)\), the cross terms \(+ab\) and \(-ab\) cancel, leaving \(a^2-b^2\). Option A is the square of a sum, not this product. Exam tip: look for opposite signs.
View question detailsWhich of the following expressions can be factorised directly using the difference of squares identity \(A^2-B^2=(A-B)(A+B)\)?
Correct answer: A
Since \(p^4=(p^2)^2\) and \(81q^4=(9q^2)^2\), option A is a difference of two squares. It factorises as \((p^2-9q^2)(p^2+9q^2)\). Exam tip: first rewrite each term as a perfect square.
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