1. If y = 2 and y = 0 are the zeroes of the polynomial f(y) = 2y3 - 5y + ay + b, find the value of a and b.
22. ABCD is a cyclic quadrilateral in which AB || CD. If ZD = 70°, find all the remaining angles.
Answers
Given : y = 2 and y = 0 are the zeroes of the polynomial f(y) = 2y³ - 5y² + ay + b
To find : value of a and b.
Solution:
f(y) = 2y³ - 5y² + ay + b
y = 2 and y = 0 are the zeroes of the polynomial
=> f(2) = f(0) = 0
f(0) = 0 - 0 + 0 + b = 0
=> b = 0
f(y) = 2y³ - 5y² + ay + 0
=> f(y) = 2y³ - 5y² + ay
f(2) = = 0
=> 2(2)³ - 5(2)² + a(2) = 0
=> 16 - 20 + 2a = 0
=> 2a = 4
=> a = 2
a= 2
b = 0
∠D = 70°
=> ∠B = 110° ( sum of opposite angles = 180° in cyclic quadrilateral )
∠A + ∠D = 180° as AB || CD
=> ∠A = 110°
=> ∠C = 70°
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Step-by-step explanation:
Given If y = 2 and y = 0 are the zeroes of the polynomial f(y) = 2y3 - 5y + ay + b, find the value of a and b.
2. ABCD is a cyclic quadrilateral in which AB || CD. If ZD = 70°, find all the remaining angles.
- Now the given equation is
- f(y) = 2y^3 – 5y + ay + b
- Substituting y = 2 in the given equation we get
- f(2) = 2(2)^3 – 5(2) + a(2) + b
- = 2 x 8 – 10 + 2a + b
- = 16 – 10 + 2a + b
- Or 2a + b + 6 = 0----------1
- Now substituting y = 0 in the equation we get
- 0 – 0 + 0 + b = 0
- So b = 0
- Putting value of b in eqn 1 we have
- 2a + 0 + 6 = 0
- 2a = - 6
- a = -6 / 2
- a = - 3
- Now AB is parallel to CD
- So angle A + angle D = 180 degree
- Or angle A + 70 = 180
- Angle A = 180 – 70
- Or angle A = 110 degree
- In a cyclic quadrilateral sum of opposite angles is 180 degree
- So angle A + angle C = 180
- 110 + angle C = 180
- Angle C = 180 – 110
- Angle C = 70 degree
- Angle B + angle D = 180 degree
- Angle B + 70 = 180
- Angle B = 180 – 70
- Angle B = 110 degree
Reference link will be
https://brainly.in/question/2132984