Mix Questions - Worksheet 1

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Chapter 2 Class 9 Polynomials - Mix Questions Worksheet 1 by teachoo Chapter: Chapter 2 Class 9 Maths – Polynomials Name: _____________________________ School: _____________________________ Roll Number: _____________________________ (2 Marks) A box has a volume given by V(y)=2y^3+y^2-2y-1. If one of its dimensions is (y-1), is it possible for the box to be a cube? Justify your answer conceptually. (2 Marks) The area of a rectangle is given by A(a)=25a^2-35a+12. Without finding the exact dimensions, determine if a rectangle with a=1 unit is possible. (2 Marks) The path of a diver from a diving board is given by p(x)=-x^2+2x+3, where x is the horizontal distance. Does the diver ever reach a height of 5 units? Explain using the concept of the vertex of a parabola. (2 Marks) If the side of a square is (x+y), its area is (x+y)^2. If the side is (x-y), its area is (x-y)^2. What is the geometric meaning of the term 2xy in the expansion of these identities? (2 Marks) A farmer has a rectangular field of area A(x)=x^2+5x+6. He wants to fence it. Find a polynomial expression for the perimeter of the field. (2 Marks) The cost of producing x toys is given by a linear polynomial C(x)=10x+500. What do the coefficient ' 10 ' and the constant '500' represent in this real-world scenario? (2 Marks) The polynomial x-a is a factor of x^10-a^10. Without performing division, explain why this must be true using the Factor Theorem. (2 Marks) Give a real-world example of a situation that can be modelled by a cubic polynomial and one by a constant polynomial. (3 Marks) A rectangular garden has a length of (2x+3) meters and a width of (x-1) meters. A path of uniform width of 1 meter is built around it. Find a polynomial expression for the area of the path. (3 Marks) The volume of a cuboid is V(x)=x^3+13x^2+32x+20. If one of its dimensions is (x+1), find the other two dimensions. Could this cuboid be a cube? Important links Answer of this worksheet - https://www.teachoo.com/25598/5365/Mix-Questions---Worksheet-1/category/Teachoo-Questions---Mix/ Full Chapter with Explanation, Activity, Worksheets and more – https://www.teachoo.com/subjects/cbse-maths/class-9th/ch2-9th-polynomials/ Science Class 9 – https://www.teachoo.com/subjects/science/class-9/ Maths Class 9- https://www.teachoo.com/subjects/cbse-maths/class-9th/ For more worksheets, ad-free videos and Sample Papers – subscribe to Teachoo Black here - https://www.teachoo.com/black/   Answer Key to Mix Questions Worksheet 1 No. Factoring the volume gives V(y)=(y-1)(2y+1)(y+1). For a box to be a cube, all three dimensions (factors) must be identical. These are not. Yes. For a shape to be possible, its area must be positive. At a=1, the area is A(1)=25(1)^2-35(1)+12=2. Since the area is positive, the rectangle is possible. No. The vertex of the parabola p(x)=-x^2+2x+3 gives the maximum height. The x-coordinate of the vertex is x=-b/(2a)= -2/(2(-1))=1. The maximum height is p(1)=-(1)^2+2(1)+3=4. Since the maximum height is 4 units, the diver cannot reach 5 units. The term 2xy represents the combined area of two rectangles, each with dimensions x by y, that are part of the larger square of side (x+y). Area A(x)=x^2+5x+6=(x+2)(x+3). The dimensions are (x+2) and (x+3). The perimeter is P(x)=2×((x+2)+ (x+3))=2(2x+5)=4x+10. ' 10 ' is the variable cost per toy (e.g., cost of materials). '500' is the fixed cost that must be paid regardless of production (e.g., rent, machine setup). Let p(x)=x^10-a^10. According to the Factor Theorem, if p(a)=0, then (x-a) is a factor. Here, p(a)=a^10-a^10=0. Therefore, (x-a) must be a factor. Cubic: The volume of a balloon as it's being inflated (volume changes with the cube of the radius). Constant: The price of a movie ticket (it doesn't change based on how many people are watching). New length =(2x+3)+2=2x+5. New width =(x-1)+2=x+1. New Area =(2x+5)(x+1)=2x^2+7x+5. Original Area =(2x+3)(x-1)=2x^2+x-3. Path Area = New Area - Original Area =(2x^2+7x+5)-(2x^2+x-3)=6x+8. Dividing V(x) by (x+1) gives the quadratic x^2+12x+20. Factoring this gives (x+2)(x+10). The other two dimensions are (x+2) and (x+10). It cannot be a cube as the three dimensions are different.