# Class 10 Case based problem of Chapter 2 Polynomials 3

## Class 10 Chapter 2(Poloynomials)

### Case based 4:

The pictures are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures,their curve represents an efficient method of carrying load and so can be found in bridges and in architecture in a variety of forms.(Class 10 Case based problem of Chapter 2 Polynomials 3)

(A) In the standard form of quadratic polynomial, ax² + bx + c, a, b and c are :

(a) All are real numbers.

(b) All are rational numbers.

(c) ‘a’ is a non-zero real number and, b and c are any real numbers.

(d) All are integers.

(B) If the root of quadratic polynomialare equal, where the discrimnant D = b² – 4ac, then:

(a) D > 0                (b) D < 0

(c) D ≥ 0                 (d) D = 0

(C) If α and 1/α are the zeroes of the quadratic polynomial 2x² – x + 8k, then k is:

(a) 4                          (b) 1/4

(c) -1/4                      (d) 2

(D) The graph of x² + 1 = 0:

(a) Intersects x-axis at two distinct points

(b) Touches x – axis at a point.

(c) Neither touches nor intersects x – axis

(d) Either touches or intersects x- axis.

(E) If the sum of the roots is -p and product of the roots is -1/p, then then the quadratic polynomial is

(a)  k(-px² + x/p + 1)

(b)  k(px² – x/p – 1)

(c)  k(x² + px – 1/p)

(d)  k(x² – px + 1/p)

## Solution:

(A) Answer  (c) ‘a’ is non-zero real number and b and c are any real numbers.

Explatnation: The standard form of the quadratic polynomial is ax² + bx + c, where a, b, c are real numbers and a ≠ 0

(B) Answer (d) D = 0

Explanation: 2x² – x + 8k

Roots of polynomial = α, 1/α

Product of the roots = Constant term/ Coefficient of x²

⇒ α× 1/α = 8k/2

⇒ k = 1/4

(D) Answer (c) Neither touches nor intersects x – axis,

Explanation: x² + 1 = 0

⇒ x² = -1

It has no real roots

Hence the graph of equation never touches and intersect the x- axis

(E) Answer (c) k(x² + px – 1/p)

Explanation: sum of the roots = -p

product of the roots = -1/p

= k[x² – (Sum of the roots) x + (Productof the roots)]

= k[x² – (-p)x +(-1/p) ]

= k[x² + px -1/p]

# Case Based 5:

Government of India allotted some funds to the National Disaster Relief Fund to help the families of flood affected village. The fund alloted is represented by . The fund is equally divided between each of the families of that village. Each family received an amount of . After distribution, the amount left was . Which was used to construct a primary health centre in each village.

(A) If the product of the zeros of the polynomial is 4, then then the value of a is:

(a)           (b)

(c) -1                                  (d) 0

(B) If α and β are zeroes of the quadratic polynomial such that α² + β² + αβ = 21/4, The value of k is

(a) -2                             (b) 0

(c)       (d)

(C) If the sum of squares of zeroes of the polynomial is 40, the value of k is

(a) 2                             (b) 4

(c) 8                              (d) 12

(D) The zeroes of the quadratic polynomial are:

(a) -a, -1/a                   (b) a, 1/a

(c) -1, a                         (d) 1, -a

(E) If one of the zeroes of the quadratic polynomial is 2, then the value of a is:

(a) 10                            (b) -10

(c) -7                              (d) -2

### Solution:

Explanation: Given

Product of the zeroes  =

Explanation:Given

Since,

Given that, α² + β² + αβ = 21/4

α² + β² + αβ + αβ- αβ = 21/4

⇒α² + β² + 2αβ – αβ = 21/4

⇒(α + β)²-αβ = 21/4

Explanation: Let zeroes of the equation are α and β

Then, α + β = -b/a = -(8)/1 =8

α.β = c/a = k/1 = k

Given that

Explanation: given that

Hence, zeroes  are = a, 1/a