Why are (a^2+2ab+b^2) and ((a+b)^2) equal in an area model?
Correct answer: A
One form is the sum of parts and the other is area from the outer side. Exam tip: get both forms from the same diagram.
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 69 · area equality,visual proof,square model
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- Both show the area of the same square
- Both show different perimeters
- Both are always zero
- Both show only lines
Easy · Level 69 · sum difference,rectangle area,identity
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- (x^2+4)
- (x^2-4)
- (x^2-2x)
- (x^2+2x)
Easy · Level 69 · small square,b square,area model
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- From a rectangle of (2a) and (b)
- From the square of (2a)
- From the square of (b)
- From the sum of (a) and (b)
Easy · Level 69 · algebraic identities, distributive property, area model, rectangle area, visual algebra
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- Areas \(ac\) and \(bc\)
- Areas \(ab\) and \(c^2\)
- Areas \(a^2\) and \(b^2\)
- Areas \(a+b\) and \(c\)
Easy · Level 69 · square difference,perfect square,visual model
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- (x+3)
- (x-3)
- (x-6)
- (x+9)
Easy · Level 69 · algebraic identities, visual models, perfect square identity, area model, class 9 mathematics
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- \((p+q)^2 = p^2 + 2pq + q^2\)
- \((p-q)^2 = p^2 - 2pq + q^2\)
- \(p^2-q^2=(p+q)(p-q)\)
- \((p+q)(p-q)=p^2+q^2\)
Easy · Level 69 · difference of squares,area of squares,algebraic identities,visual models,class 9 mathematics
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- 40
- 60
- 84
- 116
Easy · Level 69 · algebraic identities,visual models,binomial square,area model,expansion
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- \(8x\)
- \(16x\)
- \(x^2\)
- \(64\)
Easy · Level 69 · middle term,ab rectangles,visual proof
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- From two (ab) rectangles
- From one (a^2) square
- From one (b^2) square
- From the outer perimeter
Easy · Level 69 · rectangle identity,sum difference,visual model
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- ((x+3)^2=x^2+6x+9)
- ((x+3)(x-3)=x^2-9)
- ((x-3)^2=x^2-6x+9)
- (x^2+9=(x+3)(x-3))
Medium · Level 67 · algebraic identities,visual model,area model
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- (a^2+b^2)
- (a^2+2ab+b^2)
- (a^2-b^2)
- (2a^2+2b^2)
Medium · Level 67 · area tiles,rectangle model,identity
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- (xy)
- (2xy)
- (x^2+y^2)
- (2x+2y)
Medium · Level 67 · difference of squares,visual model,factorisation
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- (p^2+q^2)
- ((p+q)^2)
- ((p-q)(p+q))
- (2pq)
Medium · Level 67 · rectangle model,difference of squares,identity
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- (a^2+2ab+b^2)
- (a^2-2ab+b^2)
- (a^2+b^2)
- (a^2-b^2)
Medium · Level 67 · subtraction area model,square identity,visual algebra
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- ((m-n)^2=m^2-2mn+n^2)
- ((m+n)^2=m^2+2mn+n^2)
- (m^2-n^2=(m-n)(m+n))
- (m^2+n^2=(m+n)^2)
Medium · Level 67 · area addition,perfect square,visual identity
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- (r^2+rs+s^2)
- (r^2+2rs+s^2)
- (r^2-s^2)
- (2r^2+s^2)
Medium · Level 67 · tiles,rectangle area,visual model
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- (u+v)
- (u^2v^2)
- (uv)
- (2uv)
Medium · Level 67 · visual correction,minus square,identity
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- Subtract (d^2)
- Add (cd)
- Subtract (2d^2)
- Add (d^2)
Medium · Level 67 · perfect square,area model,side finding
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- (x+3)
- (x+6)
- (x-3)
- (3x+1)
Medium · Level 67 · minus square,visual model,perfect square
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- (y+4)
- (y-4)
- (y-8)
- (4y-1)
A rectangle model shows ((x+2)(x-2)). What is its simplified area?
Correct answer: B
This is a product of sum and difference, so it gives (x^2-4). Exam tip: identify ((x+a)(x-a)).
View question detailsIn a square of side (2a+b), how is the (b^2) part formed?
Correct answer: C
(b^2) is the area of the small square with side (b). Exam tip: connect a square term with the square of its side.
View question detailsIf the sides of a rectangle are (a+b) and (c), what two parts appear in the area model?
Correct answer: A
Splitting the side \(a+b\) into \(a\) and \(b\) divides the rectangle into two smaller rectangles. Each has the other side \(c\), so their areas are \(ac\) and \(bc\), respectively. Hence, the total area is \(c(a+b)=ac+bc\). \(a+b\) and \(c\) are the original side lengths, not the two area parts. Exam tip: in an area model, multiply the side lengths of each small rectangle.
View question detailsA square model shows (x^2-6x+9). It is the area of a square with which side?
Correct answer: B
Since (9=3^2) and the middle term is (-6x), the side is (x-3). Exam tip: identify subtraction from the negative middle term.
View question detailsIf a square of side \(p+q\) is divided into regions of areas \(p^2\), \(q^2\), and two regions of area \(pq\), which algebraic identity does this visual model represent?
Correct answer: A
The square contains regions of area \(p^2\), \(pq\), \(pq\), and \(q^2\). The two equal \(pq\) rectangles combine to make \(2pq\), so A is correct. Exam tip: add the areas of all regions.
View question detailsIf the side of the big square is (10) and the side of the small square is (4), what is the area of the difference of squares?
Correct answer: C
The area of the larger square is \(10^2=100\), and that of the smaller square is \(4^2=16\). Therefore, the difference in their areas is \(100-16=84\). The answer \(60\) results from calculating \(10^2-4\), but the area of the smaller square is \(4^2\), not \(4\). Exam tip: for a difference of squares, square both side lengths first in \(a^2-b^2\).
View question detailsIn a square model, the outer side is (x+8). What will be the last term in the expansion of the area?
Correct answer: D
The area of the square is \((x+8)^2\). Expanding it gives \(x^2+16x+64\), so the last term is \(8^2=64\). The term \(16x\) is the middle term because it comes from \(2\times x\times 8\). Exam tip: in \((a+b)^2\), the last term is always the square of the constant term \(b\).
View question detailsIn the visual model of ((a+b)^2), where does the (2ab) part come from?
Correct answer: A
Two equal rectangles of areas (ab) and (ab) combine to form (2ab). Exam tip: connect the middle term with rectangles.
View question detailsA rectangle with sides (x+3) and (x-3) shows which identity?
Correct answer: B
This is a sum-and-difference rectangle, so the area is (x^2-9). Exam tip: notice the same number (3) to identify difference of squares.
View question detailsA square is divided into two strips with side parts (a) and (b). Which identity is shown by the area of the whole square?
Correct answer: B
The whole square has side (a+b) so its area is (a^2+2ab+b^2). Exam tip: add all four parts in a square model.
View question detailsIf a square of side (x+y) has a small corner square of side (y) then what is the total area of the two equal rectangles?
Correct answer: B
Each rectangle has area (xy) so the total is (2xy). Exam tip: count the number of equal rectangles carefully.
View question detailsA small square of side (q) is removed from a large square of side (p). Which is the factor form of the remaining area?
Correct answer: C
The remaining area is (p^2-q^2) and its factor form is ((p-q)(p+q)). Exam tip: view difference of squares as a rectangle.
View question detailsA rectangle has length (a+b) and breadth (a-b). Its area matches which algebraic identity?
Correct answer: D
The rectangle area is ((a+b)(a-b)) which equals (a^2-b^2). Exam tip: identify factors with opposite signs.
View question detailsIn a visual model two strips of width (n) are subtracted from a square of side (m) and the corner (n^2) part is added back. Which identity is formed?
Correct answer: A
The corner is subtracted twice in the two (mn) strips so (n^2) is added back. Exam tip: list removed and added parts separately.
View question detailsIn an area model a square (r^2) is surrounded by two strips of area (rs) and a small square (s^2). What is the total area?
Correct answer: B
The total area is (r^2+rs+rs+s^2) which becomes (r^2+2rs+s^2). Exam tip: combine like terms.
View question detailsIn a tile model a square of side (u+v) is divided into a square (u^2) a square (v^2) and two rectangles. What is the area of each rectangle?
Correct answer: C
Each rectangle has sides (u) and (v) so its area is (uv). Exam tip: use length times breadth for rectangles.
View question detailsIf the visual model of a square of side (c-d) is made by removing two (cd) strips from (c^2) then what final correction is needed?
Correct answer: D
The corner part (d^2) is subtracted twice so it is added once. Exam tip: identify double subtraction.
View question detailsA square model shows total area (x^2+6x+9). What will be the side of the square?
Correct answer: A
This is (x^2+2\cdot x\cdot 3+3^2) so the side is (x+3). Exam tip: match the middle term with (2ab).
View question detailsA visual model shows area (y^2-8y+16). It is formed from which square side?
Correct answer: B
This is (y^2-2\cdot y\cdot 4+4^2) so the side is (y-4). Exam tip: notice the negative middle term.
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