# 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, axÂ² + 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 = bÂ² – 4ac, then:

(a) D > 0Â  Â  Â  Â  Â  Â  Â  Â  (b) D < 0

(c) D â‰¥ 0Â  Â  Â  Â  Â  Â  Â  Â  Â (d) D = 0

(C) If Î± and 1/Î± are the zeroes of the quadratic polynomial 2xÂ² – x + 8k, then k is:

(a) 4Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (b) 1/4

(c) -1/4Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (d) 2

(D) The graph of xÂ² + 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)Â  k(-pxÂ² + x/p + 1)

(b)Â  k(pxÂ² – x/p – 1)

(c)Â  k(xÂ² + px – 1/p)

(d)Â  k(xÂ² – px + 1/p)

## Solution:

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

Explatnation: The standard form of the quadratic polynomial is axÂ² + bx + c, where a, b, c are real numbers and a â‰  0

(B) Answer (d) D = 0

Explanation: 2xÂ² – x + 8k

Roots of polynomial = Î±, 1/Î±

Product of the roots = Constant term/ Coefficient of xÂ²

â‡’ Î±Ã— 1/Î± = 8k/2

â‡’ k = 1/4

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

Explanation: xÂ² + 1 = 0

â‡’ xÂ² = -1

It has no real roots

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

(E) Answer (c) k(xÂ² + px – 1/p)

Explanation: sum of the roots = -p

product of the roots = -1/p

= k[xÂ² – (Sum of the roots) x + (Productof the roots)]

= k[xÂ² – (-p)x +(-1/p) ]

= k[xÂ² + 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) Â  Â  Â  Â  Â  (b)

(c) -1Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (d) 0

(B) If Î± and Î² are zeroes of the quadratic polynomial such that Î±Â² + Î²Â² + Î±Î² = 21/4, The value of k is

(a) -2Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (b) 0

(c) Â  Â  Â  (d)

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

(a) 2Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (b) 4

(c) 8Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (d) 12

(D) The zeroes of the quadratic polynomial are:

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

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

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

(a) 10Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (b) -10

(c) -7Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (d) -2

## Solution:

Explanation: Given

Product of the zeroesÂ  =

Explanation:Given

Since,

Given that, Î±Â² + Î²Â² + Î±Î² = 21/4

Î±Â² + Î²Â² + Î±Î² + Î±Î²- Î±Î² = 21/4

â‡’Î±Â² + Î²Â² + 2Î±Î² – Î±Î² = 21/4

â‡’(Î± + Î²)Â²-Î±Î² = 21/4

Explanation: Let zeroes of the equation are Î± and Î²

Then, Î± + Î² = -b/a = -(8)/1 =8

Î±.Î² = c/a = k/1 = k

Given that

Explanation: given that

Hence, zeroesÂ  are = a, 1/a

Let zeroes of the equation are Î± and Î²

Given

and

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