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All questions of April Week 3 for Class 10 Exam
If one of the factors of x2 + x – 20 is (x + 5), then other factor is- a)(x – 4)
- b)(x – 5)
- c)(x – 6)
- d)(x – 7)
Correct answer is 'A'. Can you explain this answer?
If one of the factors of x2 + x – 20 is (x + 5), then other factor is
a)
(x – 4)
b)
(x – 5)
c)
(x – 6)
d)
(x – 7)
|
Arun Sharma answered |
Using mid-term splitting,
x2+x-20=x2+5x-4x-20=x(x+5)-4(x+5)
Taking common x+5
(x+5)(x-4) , so the other factor is x-4
x2+x-20=x2+5x-4x-20=x(x+5)-4(x+5)
Taking common x+5
(x+5)(x-4) , so the other factor is x-4
If p and q are the zeroes of the polynomial x2- 5x + k. Such that p - q = 1, find the value of K
- a)6
- b)7
- c)8
- d)9
Correct answer is option 'A'. Can you explain this answer?
If p and q are the zeroes of the polynomial x2- 5x + k. Such that p - q = 1, find the value of K
a)
6
b)
7
c)
8
d)
9
|
Zachary Foster answered |
Given α and β are the zeroes of the polynomial x2 − 5x + k
Also given that α − β = 1 → (1)
Recall that sum of roots (α + β) = −(b/a)
∴ α + β = 5 → (2)
Add (1) and (2), we get
α − β = 1
α + β = 5
2α = 6
∴ α = 3
Put α = 3 in α + β = 5
3 + β = 5
∴ β = 2
Hence 3 and 2 are zeroes of the given polynomial
Put x = 2 in the given polynomial to find the value of k ( Since 2 is a zero of the polynomial, f(2) will be 0 )
x2 − 5x + k = 0
⇒ 22 − 5(2) + k = 0
⇒ 4 − 10 + k = 0
⇒ − 6 + k = 0
∴ k = 6
Also given that α − β = 1 → (1)
Recall that sum of roots (α + β) = −(b/a)
∴ α + β = 5 → (2)
Add (1) and (2), we get
α − β = 1
α + β = 5
2α = 6
∴ α = 3
Put α = 3 in α + β = 5
3 + β = 5
∴ β = 2
Hence 3 and 2 are zeroes of the given polynomial
Put x = 2 in the given polynomial to find the value of k ( Since 2 is a zero of the polynomial, f(2) will be 0 )
x2 − 5x + k = 0
⇒ 22 − 5(2) + k = 0
⇒ 4 − 10 + k = 0
⇒ − 6 + k = 0
∴ k = 6
If α,β be the zeros of the quadratic polynomial 2 – 3x – x2, then α + β =- a)2
- b)3
- c)1
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
If α,β be the zeros of the quadratic polynomial 2 – 3x – x2, then α + β =
a)
2
b)
3
c)
1
d)
None of these
|
Pooja Shah answered |
If α and β are the zeros of the polynomial then
(x−α)(x−β) are the factors of the polynomial
Thus, (x−α)(x−β) is the polynomial.
So, the polynomial =x2 − αx − βx + αβ
=x2 − (α + β)x + αβ....(i)
Now,the quadratic polynomial is
2 − 3x − x2 = x2 + 3x − 2....(ii)
(x−α)(x−β) are the factors of the polynomial
Thus, (x−α)(x−β) is the polynomial.
So, the polynomial =x2 − αx − βx + αβ
=x2 − (α + β)x + αβ....(i)
Now,the quadratic polynomial is
2 − 3x − x2 = x2 + 3x − 2....(ii)
Now, comparing equation (i) and (ii),we get,
−(α + β) = 3
α + β = −3
−(α + β) = 3
α + β = −3
Quadratic polynomial having sum of it's zeros 5 and product of it's zeros – 14 is –- a)x2 – 5x – 14
- b)x2 – 10x – 14
- c)x2 – 5x + 14
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Quadratic polynomial having sum of it's zeros 5 and product of it's zeros – 14 is –
a)
x2 – 5x – 14
b)
x2 – 10x – 14
c)
x2 – 5x + 14
d)
None of these
|
|
Amit Kumar answered |
The quadratic equation is of the form x2 - (sum of zeros) x + (product of zeros)
=x2 - 5x - 14
=x2 - 5x - 14
If x = 2 and x = 3 are zeros of the quadratic polynomial x2 + ax + b, the values of a and b respectively are :- a)5, 6
- b)- 5, - 6
- c)- 5, 6
- d)5, - 6
Correct answer is option 'C'. Can you explain this answer?
If x = 2 and x = 3 are zeros of the quadratic polynomial x2 + ax + b, the values of a and b respectively are :
a)
5, 6
b)
- 5, - 6
c)
- 5, 6
d)
5, - 6
|
Gaurav Kumar answered |
Zeros of the polynomials are the values which gives zero when their value is substituted in the polynomial
When x=2,
x2+ax+b =(2)2+a*2+b=0
4+2a+b=0
b=-4-2a ….1
When x=3,
(3)2+ 3a + b=0
9 + 3a + b=0
Substituting
9 + 3a - 4 - 2a =0
5 + a =0
a = -5
b = 6
When x=2,
x2+ax+b =(2)2+a*2+b=0
4+2a+b=0
b=-4-2a ….1
When x=3,
(3)2+ 3a + b=0
9 + 3a + b=0
Substituting
9 + 3a - 4 - 2a =0
5 + a =0
a = -5
b = 6
The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is –2 is :- a) x2 + 3x – 2
- b)x2 – 2x + 3
- c)x2 – 3x + 2
- d)x2 – 3x – 2
Correct answer is 'C'. Can you explain this answer?
The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is –2 is :
a)
x2 + 3x – 2
b)
x2 – 2x + 3
c)
x2 – 3x + 2
d)
x2 – 3x – 2
|
|
Pooja Shah answered |
Sum of zeros = 3/1
-b/a = 3/1
Product of zeros = 2/1
c/a = 2/1
This gives
a = 1
b = -3
c = -2,
The required quadratic equation is
ax2+bx+c
So, x2-3x+2
-b/a = 3/1
Product of zeros = 2/1
c/a = 2/1
This gives
a = 1
b = -3
c = -2,
The required quadratic equation is
ax2+bx+c
So, x2-3x+2
If α,β be the zeros of the quadratic polynomial 2x2 + 5x + 1, then value of α + β + αβ =- a)- 2
- b)- 1
- c)1
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
If α,β be the zeros of the quadratic polynomial 2x2 + 5x + 1, then value of α + β + αβ =
a)
- 2
b)
- 1
c)
1
d)
None of these
|
Naina Sharma answered |
P(x) = 2x² + 5x + 1
Sum of roots = -5/2
Product of roots = 1/2
Therefore substituting these values,
α + β +αβ
=(α + β) + αβ
= -5/2 + 1/2
= -4/2
= -2
Sum of roots = -5/2
Product of roots = 1/2
Therefore substituting these values,
α + β +αβ
=(α + β) + αβ
= -5/2 + 1/2
= -4/2
= -2
The sum and product of zeros of the quadratic polynomial are – 5 and 3 respectively the quadratic polynomial is equal to –- a)x2 + 2x + 3
- b)x2 – 5x + 3
- c)x2 + 5x + 3
- d)x2 + 3x – 5
Correct answer is option 'C'. Can you explain this answer?
The sum and product of zeros of the quadratic polynomial are – 5 and 3 respectively the quadratic polynomial is equal to –
a)
x2 + 2x + 3
b)
x2 – 5x + 3
c)
x2 + 5x + 3
d)
x2 + 3x – 5
|
Bhaskar Dasgupta answered |
If α, β be the zeros of the quadratic polynomial ,then
(x−α)(x−β) is the quadratic polynomial.
Thus, (x−α)(x−β) is the polynomial.
=x^2−αx−βx+αβ
=x^2−x(α+β)+αβ(i)
(α+β)=−5αβ=3
Now putting the value of (α+β),αβ in equation (i) we get,
x^2−x(−5)+3
=x^2+5x+3
Let p(x) = ax2 + bx + c be a quadratic polynomial. It can have at most –- a)One zero
- b)Two zeros
- c)Three zeros
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Let p(x) = ax2 + bx + c be a quadratic polynomial. It can have at most –
a)
One zero
b)
Two zeros
c)
Three zeros
d)
None of these
|
Uday Datta answered |
The polynomial ax^2+bx+c has three terms. The first one is ax^2, the second is bx, and the third is c.
The exponent of the first term is 2.
The exponent of the second term is 1 because bx=bx^1.
The exponent of the third term is 0 because c=cx^0.
Since the highest exponent is 2, therefore, the degree of ax^2+bx+c is 2.
Since, the degree of the polynomial is 2, hence, the polynomial ax^2+bx+c can have zero, one or two zeroes.
Hence, the polynomial ax^2+bx+c can have at most two zereos.
If 2 and (- 1/2) as the sum and product of its zeros respectively then the quadratic polynomial f(x) is –- a)x2 – 2x – 4
- b)4x2 – 2x + 1
- c)2x2 + 4x – 1
- d)2x2 – 4x – 1
Correct answer is option 'D'. Can you explain this answer?
If 2 and (- 1/2) as the sum and product of its zeros respectively then the quadratic polynomial f(x) is –
a)
x2 – 2x – 4
b)
4x2 – 2x + 1
c)
2x2 + 4x – 1
d)
2x2 – 4x – 1
|
Archana gupta answered |
Understanding Quadratic Polynomials
A quadratic polynomial can be expressed in the form f(x) = ax² + bx + c, where the zeros (roots) of the polynomial can be derived from its coefficients.
Given Conditions
- Sum of Zeros (α + β): 2
- Product of Zeros (αβ): -1/2
Using the Relationships
For a quadratic polynomial f(x) = ax² + bx + c:
- The sum of the zeros is given by -b/a.
- The product of the zeros is given by c/a.
Using the conditions:
1. Sum of Zeros:
- -b/a = 2
- Therefore, b = -2a
2. Product of Zeros:
- c/a = -1/2
- Therefore, c = -a/2
Forming the Polynomial
Substituting b and c into the polynomial form:
f(x) = ax² + (-2a)x + (-a/2)
To eliminate the fraction, multiply through by 2:
f(x) = 2ax² - 4ax - a
Assuming a = 1 for simplicity:
f(x) = 2x² - 4x - 1
Thus, the polynomial becomes:
Final Polynomial
- f(x) = 2x² - 4x - 1
This corresponds to option D.
Conclusion
The correct quadratic polynomial based on the provided sum and product of its zeros is:
- 2x² - 4x - 1 (Option D)
A quadratic polynomial can be expressed in the form f(x) = ax² + bx + c, where the zeros (roots) of the polynomial can be derived from its coefficients.
Given Conditions
- Sum of Zeros (α + β): 2
- Product of Zeros (αβ): -1/2
Using the Relationships
For a quadratic polynomial f(x) = ax² + bx + c:
- The sum of the zeros is given by -b/a.
- The product of the zeros is given by c/a.
Using the conditions:
1. Sum of Zeros:
- -b/a = 2
- Therefore, b = -2a
2. Product of Zeros:
- c/a = -1/2
- Therefore, c = -a/2
Forming the Polynomial
Substituting b and c into the polynomial form:
f(x) = ax² + (-2a)x + (-a/2)
To eliminate the fraction, multiply through by 2:
f(x) = 2ax² - 4ax - a
Assuming a = 1 for simplicity:
f(x) = 2x² - 4x - 1
Thus, the polynomial becomes:
Final Polynomial
- f(x) = 2x² - 4x - 1
This corresponds to option D.
Conclusion
The correct quadratic polynomial based on the provided sum and product of its zeros is:
- 2x² - 4x - 1 (Option D)
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