In a square of side \(x+7\), the small corner square is \(7^2\). What is its area?
Correct answer: C
The small square has side \(7\), so its area is \(7^2=49\). Exam tip: treat the square of the number as the constant 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 68 · constant square,area model,visual identity
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- \(7\)
- \(14\)
- \(49\)
- \(x+49\)
Easy · Level 68 · algebraic identities,area model,perfect square trinomial,expansion,grade 9 mathematics
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- \(x^2+5x+25\)
- \(x^2+10x+25\)
- \(x^2+25x+10\)
- \(x^2+10x+5\)
Easy · Level 68 · algebraic identities,area model,square expansion,visual models,binomial square
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- \(x^2+4\)
- \(x^2+2x+4\)
- \(x^2+4x+4\)
- \(x^2-4x+4\)
Easy · Level 68 · minus square,area model,visual identity
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- \(x^2-8x+16\)
- \(x^2+8x+16\)
- \(x^2-4x+16\)
- \(x^2-16\)
Easy · Level 68 · numeric square,big square,area model
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- \(100\)
- \(2500\)
- \(2600\)
- \(2704\)
Easy · Level 68 · algebraic identities,visual models,binomial square,area model,class 9 mathematics
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- \(50^2\)
- \(49^2\)
- \(1^2\)
- \(2 \times 50 \times 1\)
Easy · Level 68 · algebraic identities,area model,binomial square,polynomial expansion,class 9 mathematics
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- \(x^2+3x+9\)
- \(x^2+9x+9\)
- \(x^2+6x+9\)
- \(x^2+6x+3\)
Easy · Level 68 · area model,visual identity,basic concept
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- Area model
- Number line model
- Clock model
- Only verbal model
Easy · Level 68 · plus minus models,visual difference,identity
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- In the first parts are added and in the second some parts are subtracted
- Both have no area
- The first has only a line
- The second always has side \(a+b\)
Easy · Level 68 · algebraic identities,perfect square trinomial,visual models,square area,class 9 mathematics
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- \(p-q\)
- \(p+q\)
- \(2pq\)
- \(p^2+q^2\)
Easy · Level 70 · algebraic identities,area model,square
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- \( (a+b)^2=a^2+2ab+b^2 \)
- \( (a-b)^2=a^2-2ab+b^2 \)
- \( a^2-b^2=(a+b)(a-b) \)
- \( a+b=ab \)
Easy · Level 70 · visual model,identity expansion,class 9 math
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- \(x^2+3x+9\)
- \(x^2+6x+9\)
- \(x^2-6x+9\)
- \(x^2+9x+6\)
Easy · Level 70 · difference of squares,area model,visual identity
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- \(a^2+b^2\)
- \(2ab\)
- \(a^2-b^2\)
- \(a-b\)
Easy · Level 70 · rectangle model,difference identity,algebra
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- \(a^2+2ab+b^2\)
- \(a^2-2ab+b^2\)
- \(2a+2b\)
- \(a^2-b^2\)
Easy · Level 70 · square model,minus identity,easy algebra
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- \(x^2-4x+4\)
- \(x^2+4x+4\)
- \(x^2-2x+4\)
- \(x^2+2x-4\)
Easy · Level 70 · area partition,square identity,visual algebra
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- \(p^2+q^2\)
- \(p^2+2pq+q^2\)
- \(p^2-2pq+q^2\)
- \(2p+2q\)
Easy · Level 70 · visual parts,square identity,area tiles
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- \(mn\)
- \(2mn\)
- \(m^2\)
- \(m+n\)
Easy · Level 70 · algebraic identities,algebra tiles,visual models,square expansion,binomial square
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- \((x-5)^2\)
- \(x^2-25\)
- \(x^2+5x+25\)
- \((x+5)^2\)
Easy · Level 70 · algebraic identities,visual models,perfect square trinomial,square of a difference,class 9 mathematics
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- \((y-4)^2=y^2-8y+16\)
- \((y+4)^2=y^2+8y+16\)
- \(y^2-16=(y-4)(y+4)\)
- \((y-4)(y+4)=y^2-8y+16\)
Easy · Level 70 · numeric area model,square identity,easy
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- \(36+4\)
- \(36+24+4\)
- \(12+8\)
- \(64-4\)
If a square area model has parts \(x^2\), \(5x\), \(5x\), and \(25\), what is the whole area?
Correct answer: B
The whole area is the sum of all four regions: \(x^2+5x+5x+25\). Combining like terms, \(5x+5x=10x\), gives \(x^2+10x+25\). This is also \((x+5)^2\). Option A misses one \(5x\) region. Exam tip: In an area model, add every small region and then combine like terms.
View question detailsWhich option correctly writes the visual area of \((x+2)^2\)?
Correct answer: C
A square of side \(x+2\) can be divided into one \(x\times x\) square, two \(x\times2\) rectangles, and one \(2\times2\) small square. Therefore, its total area is \(x^2+2x+2x+4=x^2+4x+4\). Option B includes the area of only one \(2x\) rectangle. Exam tip: use \((a+b)^2=a^2+2ab+b^2\).
View question detailsWhich option correctly gives the visual area of \((x-4)^2\)?
Correct answer: A
\((x-4)^2=x^2-2 \times x \times 4+16=x^2-8x+16\). Exam tip: take care of the negative middle term.
View question detailsIf \(52^2\) is solved using the area model of \((50+2)^2\), what is the large part \(50^2\)?
Correct answer: B
The big square has side \(50\), so its area is \(50^2=2500\). Exam tip: start with the large square.
View question detailsIn the visual model of \(49^2\), using \((50-1)^2\), which is the small square?
Correct answer: C
In the visual model of \((50-1)^2\), each side of the small square is \(1\), so its area is \(1^2\). \(2 \times 50 \times 1\) represents the total area of two rectangular strips, not the small square. Exam tip: in the model of \((a-b)^2\), the small square always has area \(b^2\).
View question detailsIn an area model, what expansion is obtained by adding the parts of \((x+3)(x+3)\)?
Correct answer: C
The area model has four parts with areas \(x^2\), \(3x\), \(3x\), and \(9\). Combining like terms gives \(3x+3x=6x\), so the expansion is \(x^2+6x+9\). Option A misses one \(3x\) part. Exam tip: verify using \((a+b)^2=a^2+2ab+b^2\).
View question detailsIn which model is an identity explained by only adding areas?
Correct answer: A
In an area model, areas of smaller parts are added to get total area. Exam tip: add the parts in such questions.
View question detailsWhat is the main visual difference between \((a+b)^2\) and \((a-b)^2\) in square models?
Correct answer: A
\((a+b)^2\) is an addition model while \((a-b)^2\) is a subtraction model. Exam tip: identify the model from the sign.
View question detailsIf a visual model has total area \(p^2+2pq+q^2\), what is the side of the whole square?
Correct answer: B
The given area is \(p^2+2pq+q^2=(p+q)^2\). Since the area of a square equals the square of its side, the side of the whole square is \(p+q\). Note that \((p-q)^2=p^2-2pq+q^2\), which has a negative middle term. Exam tip: recognise the identity \(a^2+2ab+b^2=(a+b)^2\) quickly.
View question detailsWhich algebraic identity is shown by the area model of a square with side \(a+b\)?
Correct answer: A
Splitting the square into two smaller squares and two rectangles gives \(a^2+2ab+b^2\). Exam tip: connect square area with the identity.
View question detailsBy observing a square of side \(x+3\), what is the expansion of \((x+3)^2\)?
Correct answer: B
The area model has \(x^2\), two \(3x\) rectangles, and \(9\). Exam tip: the middle term comes from two equal rectangles.
View question detailsA large square has side \(a\), and a square of side \(b\) is removed from it. How is the remaining area expressed?
Correct answer: C
The smaller area \(b^2\) is subtracted from the larger area \(a^2\). Exam tip: treat the removed part as subtraction.
View question detailsThe area of a rectangle with dimensions \(a+b\) and \(a-b\) is linked with which identity?
Correct answer: D
The rectangle area is \((a+b)(a-b)\), which equals \(a^2-b^2\). Exam tip: multiply length and breadth.
View question detailsWhich expansion is correctly shown by the area model of a square with side \(x-2\)?
Correct answer: A
In \((x-2)^2\), there is \(x^2\), two subtracted \(2x\) parts, and \(4\) added. Exam tip: watch the negative sign.
View question detailsIf a square is divided into parts \(p^2\), \(pq\), \(pq\), and \(q^2\), what is its total area?
Correct answer: B
The total area of a divided square is found by adding the areas of all its non-overlapping parts. The four given parts are one region of area \\(p^2\\), two regions each of area \\(pq\\), and one region of area \\(q^2\\). Since the two \\(pq\\) regions have equal areas, together they contribute \\(pq+pq=2pq\\). Adding every part gives \\(p^2+pq+pq+q^2=p^2+2pq+q^2\\). Therefore, option B is correct. This is also the familiar identity \\( (p+q)^2=p^2+2pq+q^2\\), representing the area of a square whose side is \\(p+q\\).
The important point is that a diagram must be counted completely, including both middle rectangular parts. Option A leaves out the two \\(pq\\) regions, so it is incomplete. Option C has the signs used in the identity for \\( (p-q)^2\\), which does not match the stated positive parts. Option D, \\(2p+2q\\), represents a length such as a perimeter expression in a suitable setting, not an area made from square units. Thus the sum of the four areas leads unambiguously to option B.
In a square of side \(m+n\), which can be the largest square part among its four parts?
Correct answer: C
The square parts are \(m^2\) and \(n^2\), and if \(m>n\), then \(m^2\) is larger. Exam tip: distinguish square and rectangular parts.
View question detailsAn \(x\) by \(x\) square, two \(x\) by \(5\) rectangles, and one \(5\) by \(5\) square are combined. This is a model of what?
Correct answer: D
The \(x\) by \(x\) square has area \(x^2\). The two \(x\) by \(5\) rectangles together have area \(5x+5x=10x\), and the \(5\) by \(5\) square has area \(25\). Thus, the total area is \(x^2+10x+25=(x+5)^2\). In contrast, \((x-5)^2\) has the middle term \(-10x\), so it does not match. Exam tip: add the areas of the two equal rectangles first to identify the middle term.
View question detailsIf a visual model subtracts two \(4y\) rectangles from a \(y^2\) square and adds a \(16\) square, what is the identity?
Correct answer: A
Using \((a-b)^2=a^2-2ab+b^2\), put \(a=y\) and \(b=4\): \((y-4)^2=y^2-2(y)(4)+4^2=y^2-8y+16\). The two \(4y\) rectangles together give \(-8y\), and the added small square is \(4^2=16\). Option B has a positive middle term, so it represents \((y+4)^2\). Exam tip: in the square of a difference, the middle term is negative.
View question detailsIn the area model of a square with side \(6+2\), how can the total area be written?
Correct answer: B
In \((6+2)^2\), \(6^2\), two \(6\times2\) parts, and \(2^2\) are added. Exam tip: understand the identity using numerical models.
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