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SubjectsMathematics
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.
TOPIC PRACTICE
Up to 10 questions from this page. Select your focus, then start.
When a square of side \(a+b\) is partitioned, each middle rectangle has length \(a\) and breadth \(b\). Therefore, its area is \(a\times b=ab\). The terms \(a^2\) and \(b^2\) represent the areas of the corner squares, not the rectangles. Exam tip: the two equal rectangles together have area \(2ab\).
View question detailsThe full side of the square is \(x+y\), so its area is \((x+y)^2\). The sum of the four regions is \(x^2+xy+xy+y^2=x^2+2xy+y^2\). Hence, option A is correct. In option B, the middle term is \(-2xy\), which does not match the two positive \(xy\) regions in the diagram. Exam tip: combine the two equal \(xy\) rectangles to obtain \(2xy\).
View question detailsBoth have (x^2+6x) common, and the difference of constants is (4). In exams compare after removing common regions.
View question detailsThe expression \(9x^2+30x+25\) has the pattern of a square identity. The first term is \(9x^2=(3x)^2\), and the last term is \(25=5^2\). For the square of a sum, the middle term must be twice the product of these two quantities: \(2\cdot3x\cdot5=30x\). This exactly matches the given middle term.
Therefore, the expression is \((3x+5)^2\), because expanding it gives \((3x+5)^2=9x^2+30x+25\). A side of \(9x+25\) would produce a different expansion, and \((3x-5)^2\) would have a negative middle term, namely \(-30x\). Hence the square model with side \(3x+5\), option A, is correct.
The larger square has area \((x+6)^2\), and the smaller square has area \(x^2\). Thus, the remaining area is \((x+6)^2-x^2\). Expanding gives \(x^2+12x+36-x^2=12x+36\). In \(6x+36\), the middle term is incorrect because \(2\times x\times 6=12x\). Exam tip: you can also use \(a^2-b^2=(a-b)(a+b)\) for such questions.
View question detailsThe middle strips are (9x) and (-3x), whose sum is (6x). In exams subtract the negative strip.
View question detailsThis is a difference of squares and the area becomes (x(x+2y)). In exams rearrange the remaining part into a rectangle after removing the small square.
View question detailsThe area of a rectangle is the product of its side lengths. \((2x+5)(x+2)=2x^2+4x+5x+10=2x^2+9x+10\), so the correct sides are \((2x+5)\) and \((x+2)\). In option D, \((2x+10)(x+1)=2x^2+12x+10\), whose middle term is not \(9x\). Exam tip: to check factors, add the two cross-product \(x\)-terms.
View question detailsThe larger square has area \((x+1)^2\), and the smaller square has area \((x-1)^2\). Thus, the difference in their areas is \((x+1)^2-(x-1)^2\). On expanding, \((x^2+2x+1)-(x^2-2x+1)=4x\). The expression \(x^2-1\) is the product \((x-1)(x+1)\), not the difference of the two square areas. Exam tip: write the larger area first and subtract the smaller area.
View question detailsThe remaining two parts are (xy) and (xy), whose sum is (2xy). In exams see the middle term as two equal rectangles.
View question detailsQUIZ COMPLETE
