Case Study 1: Understanding Polynomials
Ravi wrote some expressions: 2x+3, x²+2x+1, 5, 7x³+4x+2.
Questions:
- A polynomial of degree 1 is called:
(a) linear (b) quadratic (c) cubic (d) constant
Answer: (a) - Degree of x²+2x+1 is:
(a) 1 (b) 2 (c) 3 (d) 0
Answer: (b) - The polynomial 5 is:
(a) constant (b) linear (c) quadratic (d) cubic
Answer: (a) - In 7x³+4x+2, the degree is:
(a) 1 (b) 2 (c) 3 (d) 0
Answer: (c) - Which is not a polynomial?
(a) 2x²+3x (b) √x + 2 (c) x³−7x+6 (d) 5
Answer: (b)
Case Study 2: Zeros of a Polynomial
Teacher wrote f(x)=x²−4.
Questions:
- Zeros are solutions of:
(a) f(x)=0 (b) f(x)=1 (c) f(x)=x (d) none
Answer: (a) - Zeros of x²−4 are:
(a) 2,−2 (b) 4,0 (c) 1,−1 (d) none
Answer: (a) - Zeros of linear polynomial ax+b are:
(a) a/b (b) −a/b (c) −b/a (d) b/a
Answer: (c) - How many zeros can quadratic polynomial have?
(a) 0 (b) 1 (c) 2 (d) up to 2
Answer: (d) - Zeros of f(x)=x²+1 are:
(a) real numbers (b) imaginary numbers (c) rationals (d) integers
Answer: (b)
Case Study 3: Remainder Theorem
For p(x)=x²+3x+2 divided by (x+1).
Questions:
- Remainder theorem states: remainder = ?
(a) p(0) (b) p(−1) (c) p(1) (d) none
Answer: (b) - p(−1) = (−1)²+3(−1)+2 = ?
(a) 0 (b) 1 (c) −2 (d) 4
Answer: (a) - Hence, (x+1) is:
(a) factor (b) not a factor (c) remainder (d) degree
Answer: (a) - If remainder = 0, divisor is called:
(a) factor (b) quotient (c) polynomial (d) term
Answer: (a) - Division algorithm: p(x)=d(x)×q(x)+r(x), where r(x) = ?
(a) divisor (b) remainder (c) factor (d) zero
Answer: (b)
Case Study 4: Factor Theorem
For polynomial f(x)=x²−5x+6.
Questions:
- If x=2 is zero, then (x−2) is:
(a) remainder (b) factor (c) divisor (d) term
Answer: (b) - f(2) = 0, so x−2 is:
(a) factor (b) not factor (c) constant (d) degree
Answer: (a) - Zeros of f(x)=x²−5x+6 are:
(a) 2,3 (b) −2,3 (c) 1,6 (d) none
Answer: (a) - Factorised form:
(a) (x−2)(x−3) (b) (x+2)(x−3) (c) (x−1)(x−6) (d) none
Answer: (a) - Which identity helps factorisation?
(a) a²−b² (b) (a+b)² (c) (a−b)² (d) all
Answer: (d)
Case Study 5: Identities in Factorisation
A teacher wrote a²+2ab+b².
Questions:
- This is:
(a) (a+b)² (b) (a−b)² (c) a²−b² (d) none
Answer: (a) - x²−y² factors into:
(a) (x+y)(x−y) (b) (x−y)² (c) (x+y)² (d) none
Answer: (a) - (x+a)(x+b)=?
(a) x²+ax+b (b) x²+(a+b)x+ab (c) x²+(a−b)x (d) none
Answer: (b) - Factorise x²+7x+10.
(a) (x+2)(x+5) (b) (x+1)(x+10) (c) (x+3)(x+4) (d) none
Answer: (a) - Identity: (a−b)²=?
(a) a²+b²−2ab (b) a²+b²+2ab (c) a²−b² (d) none
Answer: (a)
Case Study 6: Types of Polynomials
Students listed: 7, 4x+3, x²+5, 3x³−2x+1.
Questions:
- Polynomial 7 is:
(a) constant (b) linear (c) quadratic (d) cubic
Answer: (a) - Polynomial 4x+3 is:
(a) linear (b) quadratic (c) cubic (d) constant
Answer: (a) - Polynomial x²+5 is:
(a) linear (b) quadratic (c) cubic (d) none
Answer: (b) - Polynomial 3x³−2x+1 is:
(a) linear (b) quadratic (c) cubic (d) constant
Answer: (c) - Degree of constant polynomial is:
(a) 0 (b) 1 (c) −1 (d) undefined
Answer: (c) (i.e. −∞ or not defined, CBSE treats it as –∞)
Case Study 7: Division Algorithm
For p(x)=x³−3x²+4, divided by (x−2).
Questions:
- Division Algorithm: p(x)=d(x)q(x)+r(x). Here d(x) is:
(a) divisor (b) quotient (c) remainder (d) factor
Answer: (a) - Degree of remainder < degree of:
(a) quotient (b) divisor (c) p(x) (d) zero
Answer: (b) - Remainder theorem: remainder = ?
(a) p(0) (b) p(2) (c) p(−2) (d) none
Answer: (b) - If remainder = 0, then divisor is:
(a) not a factor (b) a factor (c) quotient (d) none
Answer: (b) - Quotient degree = ?
(a) same as divisor (b) degree of p(x)−degree of divisor (c) 0 (d) none
Answer: (b)
Case Study 8: Standard Form of Polynomial
Ramesh wrote 4x³−2x+7+5x². Teacher asked him to rearrange.
Questions:
- Standard form is:
(a) 4x³+5x²−2x+7 (b) 7+5x²+4x³−2x (c) −2x+4x³+5x²+7 (d) any order
Answer: (a) - Leading coefficient = ?
(a) 7 (b) 5 (c) 4 (d) −2
Answer: (c) - Constant term = ?
(a) 4 (b) 5 (c) 7 (d) −2
Answer: (c) - Degree = ?
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (c) - Number of terms = ?
(a) 2 (b) 3 (c) 4 (d) 5
Answer: (c)
Case Study 9: Graphs of Polynomials
Teacher drew graphs of y=x, y=x², y=x³.
Questions:
- Graph of y=x is:
(a) straight line (b) parabola (c) cubic curve (d) circle
Answer: (a) - Graph of y=x² is:
(a) straight line (b) parabola (c) cubic curve (d) circle
Answer: (b) - Graph of y=x³ passes through:
(a) origin (b) (0,1) (c) (1,0) (d) none
Answer: (a) - Zeros of polynomial are x-intercepts of graph:
(a) true (b) false (c) sometimes (d) none
Answer: (a) - A linear polynomial has graph as:
(a) line (b) parabola (c) cubic (d) hyperbola
Answer: (a)
Case Study 10: Applications of Polynomials
A rectangle has sides (x+3) and (x+2).
Questions:
- Area = ?
(a) x²+5x+6 (b) x²+6x+5 (c) x²+3x+2 (d) none
Answer: (a) - Area is a polynomial of degree:
(a) 1 (b) 2 (c) 3 (d) 0
Answer: (b) - Perimeter = ?
(a) 2x+10 (b) 2x+12 (c) 2x+8 (d) 2x+14
Answer: (a) - Area factorised form:
(a) (x+3)(x+2) (b) (x+5)(x+1) (c) (x+4)(x+2) (d) none
Answer: (a) - Zeros of x²+5x+6=?
(a) −2,−3 (b) 2,3 (c) 1,6 (d) none
Answer: (a)