In a rectangle model (x^2+2x-15) must be shown using two binomial sides. Which sides are correct?
Correct answer: A
(5\cdot -3=-15) and (5+(-3)=2) so the sides are (x+5) and (x-3). Exam tip: check both product and sum.
View question detailsMathematics
Visual models of identities
सर्वसमिकाओं के दृश्य मॉडल
In Class 9 Mathematics, under the chapter Exploring Algebraic Identities, students use geometric diagrams and area-based representations to understand algebraic identities visually. They relate squares and rectangles formed from algebraic expressions to expansions such as binomial products, helping them see why both sides of an identity are equal rather than merely memorising formulas.
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Medium · Level 67 · rectangle factorisation,visual algebra,signs
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- (x+5) and (x-3)
- (x-5) and (x+3)
- (x+15) and (x-1)
- (x+2) and (x-15)
Medium · Level 67 · area parts,binomial square,visual model,algebraic identity,Visual models of identities,Exploring Algebraic Identities,Mathematics,Class 9 MCQ
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- 4a², 5a, 25
- 4a², 20a, 25
- 2a², 10a, 25
- 4a², 10a, 5
Medium · Level 67 · middle term,minus square,visual expansion
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- (10x)
- (-10x)
- (-5x)
- (25x)
Medium · Level 67 · algebraic identities,visual models,square of binomial,area model,class 9 mathematics
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- \((a+b)^2=a^2+2ab+b^2\)
- \((a-b)^2=a^2-2ab+b^2\)
- \(a^2-b^2=(a+b)(a-b)\)
- \((a+b)(a-b)=a^2-b^2\)
Medium · Level 67 · area tiles,ab rectangles,perfect square
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- One
- Two
- Three
- Four
Medium · Level 67 · difference of squares,visual explanation,area
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- Removing a square of side (4) from a square of side (x)
- Adding a square of side (4) to a square of side (x)
- Making a rectangle of sides (x) and (16)
- Adding two (4x) rectangles
Medium · Level 67 · rectangle model,binomial product,area
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- (m^2+16m+10)
- (m^2+10m+16)
- (m^2+6m+16)
- (m^2+10m-16)
Medium · Level 67 · correction term,minus square,visual model
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- Because the corner part is not removed by mistake
- Because the corner (b^2) part is subtracted twice and added once
- Because (b^2) is always zero
- Because (a^2) is multiplied by (b^2)
Medium · Level 67 · strip areas,perfect square,visual tiles
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- (7x) and (7x)
- (14x) and (14x)
- (49x) and (49x)
- (x^2) and (49)
Medium · Level 67 · minus square,side finding,area model
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- (x+3)
- (x-6)
- (x-3)
- (x+9)
Medium · Level 67 · algebraic identities,difference of squares,rectangle area,visual models,polynomial multiplication
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- \(4x^2+1\)
- \(4x^2+4x+1\)
- \(4x^2-4x+1\)
- \(4x^2-1\)
Medium · Level 67 · algebraic identities,area model,visual models,perfect square identity,binomial square
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- Two \(pq\) rectangles
- The \(p^2\) square and the \(q^2\) square
- The \(p^2\) square and one \(pq\) rectangle
- The \(q^2\) square and one \(pq\) rectangle
Medium · Level 67 · factor tiles,rectangle model,quadratic
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- (x-2) and (x-10)
- (x-4) and (x-5)
- (x+4) and (x+5)
- (x-1) and (x-20)
Medium · Level 67 · algebraic identities,binomial square,visual models,middle term,class 9 mathematics
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- \(2ab\)
- \(4ab\)
- \(2b^2\)
- \(a^2+4b^2\)
Medium · Level 67 · perfect square,visual model,recognition
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- ((4p+3)^2)
- ((16p-3)^2)
- ((4p-3)^2)
- ((p-3)^2)
Medium · Level 67 · sum difference,area model,identity
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- (9x^2+4)
- (9x^2-12x+4)
- (9x^2+12x+4)
- (9x^2-4)
Medium · Level 67 · algebraic identities,visual models,area model,perfect square identity,class 9 mathematics
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- \(a^2+2ab+b^2\)
- \(a^2-b^2\)
- \(a^2+ab+b^2\)
- \(a^2-2ab+b^2\)
Medium · Level 67 · constant term,rectangle model,signs
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- (12)
- (-12)
- (4)
- (-4)
Medium · Level 67 · perfect square,visual strips,identity
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- ((x+18)^2=x^2+36x+324)
- ((x-9)^2=x^2-18x+81)
- ((x+9)^2=x^2+18x+81)
- (x^2-81=(x-9)(x+9))
Medium · Level 67 · difference of squares,rearranged rectangle,visual identity
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- ((a-b)^2)
- (a^2+b^2)
- (2ab)
- ((a-b)(a+b))
A visual model of a square with side 2a + 5 will show which area parts?
Correct answer: B
The governing concept is the visual form of the identity (p + q)² = p² + 2pq + q². A square whose side is divided into lengths p and q contains one p-by-p square, two p-by-q rectangles, and one q-by-q square. Here p = 2a and q = 5. The first square has area (2a)² = 4a². Each rectangle has area (2a)(5) = 10a, so the two rectangles together have area 20a. The final small square has area 5² = 25. Therefore the model shows 4a², 20a, and 25, so option B is correct. Option A counts only one rectangle; C gives an incorrect first area; and D gives an incorrect constant and middle contribution.
View question detailsWhat is the middle term in the visual expansion of a square with side (5x-1)?
Correct answer: B
The area of a square is found by multiplying its side by itself. For a side of \\(5x-1\\), use the identity \\((a-b)^2=a^2-2ab+b^2\\), with \\(a=5x\\) and \\(b=1\\). The expansion is \\(25x^2-10x+1\\). The middle term is therefore \\(-10x\\), which comes from the two cross-products of the large part and the one-unit strip.
Option B is correct. The negative sign appears because the side is \\(5x-1\\), so the strips are removed rather than added. Option A would be the middle term for \\((5x+1)^2\\), not for the given side. The terms \\(-5x\\) and \\(25x\\) do not represent the required cross-term. Hence the visual model and the algebraic identity give the same answer.
Each side of a square is divided into two parts, \(a\) and \(b\). The model shows regions of areas \(a^2\), \(b^2\), and two rectangles each of area \(ab\). Which algebraic identity does this model represent?
Correct answer: A
The large square has side \(a+b\), so its area is \((a+b)^2\). Adding its parts gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Exam tip: always count both \(ab\) rectangles.
View question detailsA rectangle has sides (a+b) and (a+b). How many (ab) rectangles appear in the area model?
Correct answer: B
The ((a+b)^2) model has two equal (ab) rectangles. Exam tip: understand (2ab) as two separate rectangles.
View question detailsWhich visual model can directly explain (x^2-16)?
Correct answer: A
(16=4^2) so (x^2-16) is a difference of squares model. Exam tip: view the constant term as a square.
View question detailsA rectangle with sides (m+2) and (m+8) shows which expression in an area model?
Correct answer: B
The four parts are (m^2) (2m) (8m) and (16) so the total is (m^2+10m+16). Exam tip: add all parts.
View question detailsIf a square of side (a-b) has final area (a^2-2ab+b^2) then why does the (+b^2) part appear?
Correct answer: B
When two strips are removed the corner is removed twice so it must be added back. Exam tip: treat (+b^2) as a correction term.
View question detailsSquare tiles form (x^2+14x+49). Which two equal strip areas must have been added?
Correct answer: A
(14x) is formed by two equal strips (7x) and (7x). Exam tip: split the middle term into two equal parts.
View question detailsA visual model has area (x^2-6x+9). It is the area of a square with which side?
Correct answer: C
(9=3^2) and (6x=2\cdot x\cdot 3) so the side is (x-3). Exam tip: the negative sign indicates a removed model.
View question detailsA rectangle has sides (2x+1) and (2x-1). What area is obtained from the visual model?
Correct answer: D
The area of a rectangle is the product of its side lengths: \((2x+1)(2x-1)\). This matches \((a+b)(a-b)=a^2-b^2\), where \(a=2x\) and \(b=1\). Hence, the area is \((2x)^2-1^2=4x^2-1\). The expression \(4x^2+1\) does not apply the difference-of-squares identity correctly. Exam tip: when two binomials differ only in the middle sign, look for the difference of squares.
View question detailsIn the area model of \((p+q)^2\), which two parts are always rectangles of equal area?
Correct answer: A
The model for \((p+q)^2\) has squares of areas \(p^2\) and \(q^2\), along with two rectangles, each measuring \(p\times q\). Hence the two \(pq\) rectangles have equal area. The squares need not be equal. Exam tip: identify the middle term as \(2pq\).
View question detailsA rectangle model must show (x^2-9x+20) in factors. Which pair is correct?
Correct answer: B
The product of (-4) and (-5) is (20) and the sum is (-9). Exam tip: two negative terms give a positive constant term.
View question detailsWhat is the middle term in the visual expansion of a square with side (a+2b)?
Correct answer: B
The area of the square is \((a+2b)^2\). Using \((x+y)^2=x^2+2xy+y^2\), take \(x=a\) and \(y=2b\). The middle term is \(2\times a\times 2b=4ab\). \(2ab\) is incorrect because it does not fully include the coefficient 2 in \(2b\). Exam tip: the middle term in a binomial square is twice the product of the two terms.
View question detailsA square model shows (16p^2-24p+9). It is equal to which square?
Correct answer: C
(16p^2=(4p)^2) and (9=3^2) with middle term (-24p) so ((4p-3)^2) is correct. Exam tip: identify the first and last squares.
View question detailsIf a rectangle is made with sides (3x+2) and (3x-2) then which area is obtained?
Correct answer: D
This is a product of sum and difference so ((3x)^2-2^2=9x^2-4). Exam tip: look for the same first part and opposite second part.
View question detailsWhich algebraic expression is best represented by an area model that divides a large square into four parts—two squares of different sizes and two congruent rectangles?
Correct answer: A
Splitting a square of side \(a+b\) gives areas \(a^2\), \(ab\), \(ab\), and \(b^2\). Their sum is \(a^2+2ab+b^2\). \(a^2-b^2\) represents a difference, not this complete square model. Exam tip: count both \(ab\) rectangles.
View question detailsIn a rectangle model with sides (x-2) and (x+6) what will be the constant part?
Correct answer: B
The constant part of a product comes from multiplying the terms that do not contain the variable. The rectangle has sides \\(x-2\\) and \\(x+6\\). Its area is \\( (x-2)(x+6)\\). The constant terms are \\(-2\\) and \\(6\\), so their product is \\((-2)(6)=-12\\). Therefore, the constant part is \\(-12\\), and option B is correct. The negative sign is essential because the first side contains \\(-2\\), not \\(2\\).
For a complete check, distribute the factors: \\(x\cdot x=x^2\\), \\(x\cdot6=6x\\), \\((-2)\cdot x=-2x\\), and \\((-2)\cdot6=-12\\). Thus the full product is \\(x^2+4x-12\\), whose constant term is indeed \\(-12\\). Option A would result from incorrectly treating \\(-2\\) as positive. Options C and D do not come from multiplying the two constant terms. In an area model, the bottom-right constant rectangle has signed area \\(-12\\), so its sign must be retained.
In which visual identity do two equal rectangles give middle term (18x) and the small square is (81)?
Correct answer: C
The small square is (81=9^2) and the two strips give (9x+9x=18x). Exam tip: identify the small side from half of the middle term.
View question detailsIn a square model the large square is (a^2) and the removed small square has area (b^2). If the remaining part is made into a rectangle then how is the area written?
Correct answer: D
The remaining area is (a^2-b^2) and after rearrangement it becomes ((a-b)(a+b)). Exam tip: connect the removed square with difference of squares.
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