When a square of side (a+b+c) is divided into (3) parts in both directions, how many small regions are formed?
Correct answer: C
Three rows and three columns make (3\cdot3=9) regions. Exam tip: multiply rows and columns.
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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Easy · Level 71 · three term square,grid model,visual identity
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- (3)
- (6)
- (9)
- (12)
Easy · Level 71 · square terms,three term model,area grid
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- Rectangular parts
- Square parts
- Line parts
- Perimeter parts
Easy · Level 71 · cross rectangles,commutative property,visual model
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- Equal to (ab)
- Equal to (a^2)
- Equal to (b^2)
- Equal to (c^2)
Easy · Level 71 · algebraic identities,area model,rectangle model,factorisation,binomial products,class 9 mathematics
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- \((x+2)\) and \((x+3)\)
- \((x+1)\) and \((x+6)\)
- \((x+2)\) and \((x+6)\)
- \((x+3)\) and \((x+3)\)
Easy · Level 71 · minus square,numeric model,small area
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- (2)
- (4)
- (40)
- (80)
Easy · Level 71 · algebraic identities,rectangle model,like terms,binomial multiplication,linear terms
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- \(2x\)
- \(5x\)
- \(7x\)
- \(10x\)
Easy · Level 71 · algebraic identities,area model,visual models,binomial square,class 9 mathematics
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- A square of side \(a+b\) divided into \(a^2\), \(b^2\), and two \(ab\) rectangles
- A rectangle with length \(a+b\) and width \(a-b\)
- Two separate squares of side \(a\) and one square of side \(b\)
- A square of side \(a-b\) with a square of side \(b\) removed
Easy · Level 71 · algebraic identities,perfect square,trinomials,visual models,area model
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- \(x+12\)
- \(x+6\)
- \(x+3\)
- \(x-6\)
Easy · Level 71 · algebraic identities, visual models, area model, square identity, class 9 mathematics
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- \(a^2+ab+ab+b^2\)
- \(a^2+b^2\)
- \(a^2+ab+b^2\)
- \(a^2+2b^2\)
Easy · Level 71 · algebraic identities,visual models,perfect square expansion,area model,class 9 mathematics
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- 7
- 14
- 49
- x
Easy · Level 71 · difference of squares,rearrangement,visual proof
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- Square addition model
- Square removal and rearrangement model
- Three-term square model
- Only perimeter model
Easy · Level 71 · algebraic identities,difference of squares,mental multiplication,grade 9 mathematics,numerical identities
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- 624
- 625
- 626
- 650
Easy · Level 71 · coefficient,binomial square,area model
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- (6a)
- (12a)
- (4a^2)
- (9)
Easy · Level 71 · visual expansion,area model,binomial square
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- (4x^2+12x+9)
- (2x^2+12x+9)
- (4x^2+6x+9)
- (4x^2+12x+3)
Easy · Level 71 · algebraic identities, visual models, area model, square identity, class 9 mathematics
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- \((a+b)^2=a^2+2ab+b^2\)
- \((a-b)^2=a^2-2ab+b^2\)
- \(a^2-b^2=(a+b)(a-b)\)
- \((a+b)(a-b)=a^2+b^2\)
Easy · Level 71 · algebraic identities,area model,binomial multiplication,like terms,linear term
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- \(5x\)
- \(6x\)
- \(7x\)
- \(8x\)
Easy · Level 71 · algebraic identities,area model,constant term,binomial multiplication,class 9 mathematics
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- 1
- 6
- 7
- 6a
Easy · Level 71 · quadratic model,rectangle factors,visual algebra
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- ((x+1)(x+6))
- ((x+2)(x+3))
- ((x+5)(x+6))
- ((x+1)(x+5))
Easy · Level 71 · numeric square,area model,quick calculation
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- (900)
- (960)
- (1024)
- (1064)
Easy · Level 71 · algebraic identities,visual models,area model,binomial square,class 9 mathematics
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- \((a+b)^2=a^2+2ab+b^2\)
- \((a-b)^2=a^2-2ab+b^2\)
- \((a+b)(a-b)=a^2-b^2\)
- \((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
In the grid model of ((a+b+c)^2), what type of parts are (a^2), (b^2), and (c^2)?
Correct answer: B
(a^2), (b^2), and (c^2) are areas of separate squares. Exam tip: identify square terms and rectangle terms separately.
View question detailsIf (ab) is one rectangle in the (a+b+c) model, what area will (ba) give?
Correct answer: A
(ab) and (ba) are two rectangles with equal area. Exam tip: the order of multiplication does not change area.
View question detailsIn the area model of a rectangle, the four regions have areas \(x^2\), \(3x\), \(2x\), and \(6\). Which two expressions represent the sides of the rectangle?
Correct answer: A
One side can be split into \(x\) and \(2\), and the other into \(x\) and \(3\). The four products are \(x\cdot x, x\cdot3, 2\cdot x, 2\cdot3\), giving the stated regions. Hence the sides are \((x+2)\) and \((x+3)\). Exam tip: use the corner constant to identify the two numbers.
View question detailsIf (18^2) is found using ((20-2)^2), what is the small square added at the end?
Correct answer: B
The identity \\( (20-2)^2 \\) can be shown by starting with a square of side \\(20\\), removing two rectangular strips of width \\(2\\), and then accounting for the small corner. The corner has side lengths \\(2\\) and \\(2\\), so its area is \\(2^2=4\\). In the subtraction model, this small corner is added back because it was removed twice when the two strips were subtracted.
Algebra confirms the same result: \\( (20-2)^2=20^2-2(20)(2)+2^2\\). The last term is the small square, namely \\(4\\). The number \\(2\\) is only its side length, while \\(40\\) and \\(80\\) come from other parts of the calculation and are not the corner area. Therefore option B is correct.
In a rectangle model, length is (x+5) and breadth is (x+2). What will be the total of the (x)-terms?
Correct answer: C
Expanding \((x+5)(x+2)\) in the rectangle model gives \(x^2+2x+5x+10\). The \(x\)-terms are \(2x\) and \(5x\), so their total is \(2x+5x=7x\). The term \(10\) is a constant, so it is not included among the \(x\)-terms. Exam tip: add coefficients only for like terms with the same variable and exponent.
View question detailsWhich area model represents the identity \((a+b)^2=a^2+2ab+b^2\)?
Correct answer: A
A square with side \(a+b\) contains squares of areas \(a^2\) and \(b^2\), plus two rectangles of area \(ab\). Thus its area is \(a^2+2ab+b^2\). Option B represents \(a^2-b^2\). Exam tip: always count both \(ab\) regions.
View question detailsIf \(x^2+12x+36\) is a perfect square area model, what is the whole side?
Correct answer: B
\(x^2+12x+36=x^2+2\cdot x\cdot6+6^2=(x+6)^2\). Thus, the square’s area is \((x+6)^2\), so its whole side is \(x+6\). If the side were \(x+12\), the middle term would be \(24x\), not \(12x\). Exam tip: use the identity \(x^2+2ax+a^2=(x+a)^2\).
View question detailsIn an area-based visual model of a square, which regions are needed to represent \((a+b)^2\)?
Correct answer: A
A square of side \(a+b\) splits into an \(a^2\) square, a \(b^2\) square, and two \(ab\) rectangles. Thus A is correct; C misses one \(ab\) rectangle. Exam tip: count all four regions in the area model.
View question detailsIf in a square of side (x+7), the total area of two equal rectangles is (14x), what is the side of the small square?
Correct answer: A
For a square of side \(x+7\), the visual model contains one large \(x\times x\) square, two equal \(x\times 7\) rectangles, and one small square. The two rectangles have total area \(2\times 7x=14x\), so each side of the small square is \(7\), and its area is \(49\). Thus, \(49\) is the area, not the side length. Exam tip: In the model of \((a+b)^2\), the side of the small square is always \(b\).
View question detailsIn which model is (a^2-b^2) shown as ((a-b)(a+b))?
Correct answer: B
The smaller square is removed from the larger square and the remaining area is arranged as a rectangle. This gives ((a-b)(a+b)).
View question detailsIf \(24\cdot26\) is viewed as ((25-1)(25+1)), what result will be obtained?
Correct answer: A
Here, \(24=25-1\) and \(26=25+1\). Using \((a-b)(a+b)=a^2-b^2\), we get \((25-1)(25+1)=25^2-1^2=625-1=624\). Therefore, 624 is correct. 625 is only the square of the middle number 25; \(1^2\) must still be subtracted. Exam tip: for two numbers equally spaced from a middle number, square the middle number and subtract the square of the distance.
View question detailsIn the area model of a square with side (2a+3), what is the total area of the two rectangles?
Correct answer: B
A square with side \(2a+3\) has area \((2a+3)^2\). In its area model, the two mixed rectangles each use one side of length \(2a\) and one side of length \(3\). Thus each rectangle has area \((2a)(3)=6a\). Adding both equal rectangles gives \(6a+6a=12a\).
The identity \((u+v)^2=u^2+2uv+v^2\) gives the same result when \(u=2a\) and \(v=3\): the mixed term is \(2(2a)(3)=12a\). Therefore option B is correct. Option A counts only one of the two rectangles, option C is the area of the \(2a\) square, and option D is the area of the constant square. The coefficient 2 in the variable part must be included in both rectangle calculations.
If ((2x+3)^2) is expanded, which area is correct?
Correct answer: A
The larger square area is (4x^2), the middle part is (12x), and the small square is (9). Exam tip: add all three parts.
View question detailsIn a square area model, there are regions of areas a^2, b^2, and two congruent rectangular regions each of area ab. Which algebraic identity does this model represent?
Correct answer: A
A square with side \(a+b\) has area \((a+b)^2\). Its parts are \(a^2\), \(b^2\), and two \(ab\) rectangles, giving \(a^2+2ab+b^2\). Option B has \(-2ab\), so it cannot match this added-area model. Exam tip: count both \(ab\) rectangles.
View question detailsIf the rectangle sides are (3x+1) and (x+2), what will be the total (x)-term in the area model?
Correct answer: C
In the area model, the linear terms come from two products: \(3x\times 2=6x\) and \(1\times x=x\). Therefore, the total \(x\)-term is \(6x+x=7x\). Choosing only \(6x\) misses the \(1\times x\) part. Exam tip: write the products of all four small regions and then combine like terms.
View question detailsWhat is the constant part in the area model of ((a+1)(a+6))?
Correct answer: B
In the area model, the side lengths are \(a+1\) and \(a+6\). The constant part comes from multiplying the two constants: \(1\times 6=6\). Therefore, the correct answer is \(6\). The value \(7\) is \(1+6\), which relates to the coefficient of the middle term, not the constant term. Exam tip: identify the small rectangle formed by number × number.
View question detailsIn a visual model, (x^2+5x+6) can be formed by which rectangle?
Correct answer: B
((x+2)(x+3)=x^2+3x+2x+6=x^2+5x+6). Exam tip: check that the two factors add to (5) and multiply to (6).
View question detailsIf the side of a big square is (30+2), the whole area is equal to which number?
Correct answer: C
(30+2=32), and the area is (32^2=1024). Exam tip: you can also add the parts of the area model.
View question detailsA 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 whole 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: two \(ab\) rectangles indicate the \(+2ab\) term.
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