A square tile model shows (z^2+6z+9). What is the side of the complete square?
Correct answer: A
Since (9=3^2) and the middle term is (6z), the side is (z+3). Exam tip: identify the smaller side from the last term.
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 · perfect square,tile model,side
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- (z+3)
- (z+6)
- (z+9)
- (z-3)
Easy · Level 69 · algebraic identities, difference of squares, rectangle area, visual models, class 9 mathematics
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- \(a^2+25\)
- \(a^2-25\)
- \(a^2-10a+25\)
- \(a^2+10a+25\)
Easy · Level 69 · middle rectangles,area addition,like terms,visual algebra,Visual models of identities,Exploring Algebraic Identities,Mathematics,Class 9 MCQ
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- 7x
- 49x
- 14x
- x²
Easy · Level 69 · algebraic identities,visual models,area model,perfect square,trinomials,class 9 mathematics
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- \((x+y)^2=x^2+2xy+y^2\)
- \((x-y)^2=x^2-2xy+y^2\)
- \(x^2-y^2=(x+y)(x-y)\)
- \((x+y)(x-y)=x^2-y^2\)
Easy · Level 69 · algebraic identities,perfect square trinomial,visual models,area model,class 9 mathematics
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- \(x+6\)
- \(x+12\)
- \(x+36\)
- \(x-6\)
Easy · Level 69 · algebraic identities,rectangle model,area model,binomial multiplication,visual algebra
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- \(2p\)
- \(10\)
- \(5p\)
- \(7p\)
Easy · Level 69 · binomial square,area model,algebraic identity
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- (a^2+2ab+4b^2)
- (a^2+4b^2)
- (a^2+4ab+4b^2)
- (a^2+2ab+2b^2)
Easy · Level 69 · difference of squares,rectangular form,visual proof
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- (c(c-d))
- (d(c+d))
- (c^2+d^2)
- ((c+d)(c-d))
Easy · Level 69 · algebraic identities,visual models,area model,perfect square trinomial,class 9 mathematics
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- \((x-y)^2=x^2-2xy+y^2\)
- \((x+y)(x-y)=x^2-y^2\)
- \((x+y)^2=x^2+2xy+y^2\)
- \((x-y)^2=x^2+2xy-y^2\)
Easy · Level 69 · minus square,last term,area model
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- (-9)
- (9)
- (3t)
- (-6t)
Easy · Level 69 · area model,factor form,rectangle
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- ((u+10)(u+1))
- ((u+2)(u+2))
- ((u+2)(u+5))
- ((u+5)(u+5))
Easy · Level 69 · algebraic identities,perfect square trinomial,visual models,square side,class 9 mathematics
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- \(2x+9\)
- \(4x+3\)
- \(2x+6\)
- \(2x+3\)
Easy · Level 69 · square part,area model,visual identity
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- A square of side (a)
- A square of side (b)
- A rectangle of sides (a) and (b)
- A rectangle of side (a+b)
Easy · Level 69 · four parts,square model,non example
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- (x^2)
- (3x)
- (2x)
- (4)
Easy · Level 69 · difference of squares,visual removal,area
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- Square of sum
- Perfect square trinomial
- Difference of squares
- Sum of two squares
Easy · Level 69 · algebraic identities,rectangle area,distributive property,visual models,polynomial multiplication
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- \(x^2+6\)
- \(6x^2\)
- \(x+6x\)
- \(x^2+6x\)
Easy · Level 69 · equal rectangles,square model,area
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- (2)
- (1)
- (4)
- (8)
Easy · Level 69 · square of difference,visual model,remaining square
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- Square of sum model
- Square of difference model
- Rectangle multiplication model
- Sum of squares model
Easy · Level 69 · algebraic identities, visual models, area model, perfect square, class 9 mathematics
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- \(x^2+y^2+2xy\)
- \(x^2+y^2+xy\)
- \(x^2-y^2\)
- \(x^2+2y^2\)
Easy · Level 69 · outer boundary,square area,visual identity
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- (a^2+2ab+b^2)
- (a^2-b^2)
- (ab)
- ((a+b)^2)
A rectangle has length (a+5) and breadth (a-5). Which area does the visual model show?
Correct answer: B
The area of a rectangle is length × breadth. Therefore, the area is \((a+5)(a-5)\). Using the identity \((x+y)(x-y)=x^2-y^2\), we get \((a+5)(a-5)=a^2-5^2=a^2-25\). Although \(a^2+25\) also has no middle term, 25 is subtracted in this product, not added. Exam tip: the product of the sum and difference of the same terms gives a difference of squares.
View question detailsIf two equal rectangles in an area model have areas 7x and 7x, what is the total middle term?
Correct answer: C
The governing idea is addition of areas and combination of like algebraic terms. In an area model, the two rectangles are separate parts, so their contributions are added rather than multiplied. Their areas are 7x and 7x, and both contain the same variable part x. Therefore, add their coefficients while retaining x: 7x+7x=(7+7)x=14x. Option C is correct. The answer is not 49x because that would incorrectly multiply 7 by 7 while leaving x unchanged; if multiplication were intended, the variable power would also need careful treatment. Option A gives only one rectangle’s area, not the total. Option D would represent a product of two x-length quantities, which is not what the stated area labels require.
View question detailsA square visual model is divided into one large square, a square of side x, a square of side y, and two equal rectangles of dimensions xy. Which algebraic identity does this model represent?
Correct answer: A
The large square has area \((x+y)^2\). Its parts have areas \(x^2\), \(y^2\), and \(xy+xy=2xy\), so option A is correct. A subtraction model has a negative middle term. Exam tip: add the areas of all visual pieces.
View question detailsA square model shows parts \(x^2\), (12x), and (36). What side of the square does it indicate?
Correct answer: A
The total area is \(x^2+12x+36\). Here, \(36=6^2\) and the middle term is \(12x=2\cdot x\cdot 6\). Therefore, \(x^2+12x+36=(x+6)^2\), so the side of the square is \(x+6\). Squaring \(x-6\) would give a middle term of \(-12x\). Exam tip: for a perfect-square trinomial, check the square root of the constant term against the middle term.
View question detailsIn the rectangle model of ((p+2)(p+5)), what is the area of the small corner rectangle?
Correct answer: B
In the rectangle model, the small corner rectangle has side lengths \(2\) and \(5\). Therefore, its area is \(2\times5=10\). The terms \(2p\) and \(5p\) represent rectangles with one side \(p\), so they are not the corner rectangle’s area. Exam tip: in an area model, identify both side lengths of each small region before multiplying.
View question detailsWhich expansion is given by the area model of a square with side (a+2b)?
Correct answer: C
In ((a+2b)^2), the middle term is (2\cdot a\cdot 2b=4ab). Exam tip: write the square of the second term as (4b^2).
View question detailsIn a visual model, the rectangular form of (c^2-d^2) is equal to what?
Correct answer: D
(c^2-d^2) can be rearranged into a ((c+d)(c-d)) rectangle. Exam tip: identify the big and small squares.
View question detailsEach side of a square model is divided into parts x and y. The model contains one square of area x², one square of area y², and two rectangles of area xy. Which algebraic identity does this model represent?
Correct answer: C
The whole square has side \(x+y\), so its area is \((x+y)^2\). Adding the four regions gives \(x^2+xy+xy+y^2=x^2+2xy+y^2\). Option A has a negative middle term. Exam tip: always count both \(xy\) rectangles.
View question detailsIn the model of a square with side (t-3), what will be the last term?
Correct answer: B
((-3)^2=9), so the last term is positive. Exam tip: the square of a negative number is positive.
View question detailsIn an area model, the four parts of a rectangle are (u^2), (2u), (5u), and (10). Which product does the whole rectangle show?
Correct answer: C
The parts (2u), (5u), and (10) show the numbers (2) and (5). Exam tip: match the numbers using the corner product.
View question detailsA square model shows \(4x^2+12x+9\). What will be its side?
Correct answer: D
The given area \(4x^2+12x+9\) is a perfect-square trinomial: \(4x^2+12x+9=(2x)^2+2(2x)(3)+3^2=(2x+3)^2\). Hence, the side of the square is \(2x+3\). Squaring \(2x+6\) gives a middle term of \(24x\), not \(12x\). Exam tip: take the square roots of the first and last terms, then check whether the middle term equals \(2ab\).
View question detailsIn a square with side (a+b), what shape is the (a^2) part?
Correct answer: A
(a^2) means the area of a square of side (a). Exam tip: keep the difference between area and side clear.
View question detailsIf the side of a square is (x+2), which part will not appear among its four regions?
Correct answer: B
The parts of ((x+2)^2) are (x^2), (2x), (2x), and (4). Exam tip: identify every part of the model separately.
View question detailsIn a visual model, when a square of side (b) is removed from a square of side (a), what is the remaining area called?
Correct answer: C
The remaining area is (a^2-b^2), called difference of squares. Exam tip: connect removal with subtraction.
View question detailsIn a rectangle model, the length is (x+6) and the breadth is (x). What is the area?
Correct answer: D
The area of a rectangle is length × breadth. Thus, the area is \(x(x+6)\). Using the distributive property, \(x(x+6)=x^2+6x\), so option D is correct. In \(x^2+6\), the 6 has not been multiplied by \(x\). Exam tip: When multiplying algebraic dimensions, multiply each term by the other factor.
View question detailsWhile making a square model of side (s+4), how many (4s) rectangles appear besides (s^2)?
Correct answer: A
In ((s+4)^2), there are two equal rectangles (4s) and (4s). Exam tip: count both rectangles using symmetry.
View question detailsWhich area model has the whole square (a^2) and the inner remaining square ((a-b)^2)?
Correct answer: B
This visually shows ((a-b)^2). Exam tip: observe the side (a-b) of the remaining inner square.
View question detailsIn a tile model of a square with side \(x+y\), there is one \(x^2\) square, one \(y^2\) square, and two \(xy\) rectangles. Which algebraic expression does this model represent?
Correct answer: A
The two \(xy\) rectangles together contribute \(2xy\). Thus, the total tile area is \(x^2+2xy+y^2\), which equals \((x+y)^2\). Option B misses one \(xy\) rectangle. Exam tip: add the areas of all tiles before selecting the identity.
View question detailsIn a square model with side (a+b), in which form is the total area directly obtained from the outer boundary?
Correct answer: D
The outer boundary has side (a+b), so the area is ((a+b)^2). Exam tip: write the total area from the outside side.
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