# AP 10th Class Maths Bits Chapter 3 Polynomials with Answers

Practice the AP 10th Class Maths Bits with Answers Chapter 3 Polynomials on a regular basis so that you can attempt exams with utmost confidence.

## AP SSC 10th Class Maths Bits 3rd Lesson Polynomials with Answers

Question 1.
Write the sum of zeroes of
bx2 + ax + c.
$$\frac{-a}{b}$$

Question 2.
Find product of the zeroes of
p(x) = (x-2).(x + 3)
= -6
Explanation:
p(x) = x2 + x – 6,
Product of zeroes = $$\frac{\mathrm{c}}{\mathrm{a}}$$ = – 6.

Question 3.
5x – 3 represents which type of poly¬nomial ?
Linear.

Question 4.
Find the value of ‘x’ which satisfies 2(x – 1) – (1 – x) = 2x + 3 ?
6
Explanation:
2x – 2 – 1 + x = 2x + 3
⇒ 3x – 2x = 3 + 3 = 6
⇒ x = 6.

Question 5.
Write the degree of the polynomial
$$\sqrt{2}$$x2 – 3x + 1.
2

Question 6.
Write order of the polynomial 5x7 – 6x5 + 7x – 6.
7

Question 7.
Find the product of zeroes of 2x2 – 3x + 6.
3
Explanation:
Product of zeroes = $$\frac{c}{a}=\frac{6}{2}$$ = 3.

Question 8.
Find sum of zeroes of a polynomial x3 – 2x2 + 3x – 4.
2
Explanation:
Sum of zeroes = α + β + γ
= $$\frac{-b}{a}=\frac{-(-2)}{1}$$ = 2

Question 9.
Find a quadratic polynomial, the sum of whose zeroes is zero and one zero is 4, is
x2 – 16 = 0
Explanation:
Sum of zeroes = $$\frac{-b}{a}$$ = 0,
α + β = 0, β = 4
⇒ α = -4 x2 – (α + β)x + αβ = 0
⇒ x2 – 0(x) -16 = 0
⇒ x2 – 16 = 0.

Question 10.
If p(x) = x2 – ax – 3 and p(2) = – 3, then find Answer:
2
Explanation:
p(x) = (2)2 – 2a – 3 = – 3
⇒ 1 — 2a = — 3
⇒ 4 – 2a = 0
⇒ 4 = 2a ⇒ a = 2

Question 11.
Write the zero value of polynomial px + q.
$$\frac{-\mathrm{q}}{\mathrm{p}}$$

Question 12.
In a division, if divisor is x + 1, quo¬tient is x and remainder is 4, then find dividend.
x (x + 1) + 4 = x2 + x + 4
Explanation:
Dividend = Divisor x Quotient
+ Remainder
= (x + 1) x + 4 = x2 + x + 4.

Question 13.
Find the zero value of linear polynomial ax – b.
$$\frac{\mathrm{b}}{\mathrm{a}}$$

Question 14.
Find the sum of the zeroes of the poly-nomial x2 + 5x + 6.
– 5

Question 15.
4y2 – 5y + 1 is an example for this type of polynomial.

Question 16.
Write the degree of the polynomial 5x7 – 6x5 + 7x – 4.
7

Question 17.
How much the sum of zeroes of the polynomial 2x2 – 8x + 6 ?
4

Question 18.
f α,β are the zeroes of x2 + x +1, then find $$\frac{1}{\alpha}+\frac{1}{\beta}$$
-1
Explanation:
α + β = -1, αβ = 1
⇒ $$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}=\frac{-1}{1}$$ = -1

Question 19.
Write the number of zeroes of the poly¬nomial in the given graph.

3

Question 20.
Write product of zeroes of the cubic polynomial 3x3 – 5x2 – 11x – 3.
1
Explanation:
Product of zeroes = αβγ
= $$\frac{-d}{a}=\frac{-(-3)}{3}=\frac{3}{3}$$ = 1

Question 21.
The following is the graph of a poly¬nomial. Find the zeroes of the poly¬nomial from the given graph.

– 2,1

Question 22.
Find the value of p(x) = 4x2 + 3x + 1 at x = -1.
2

Question 23.
If α, β, γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d and (a ≠ 0), then find αβγ

Question 24.
4x + 6y = 18 doesn’t pass through ori-gin, then its graph indicates
A straight line

Question 25.
If ‘3’ is one of the zeroes of
p(x) = x2 + kx – 9, then find the value of k.
0
Explanation:
p(3) = 32 + 3k – 9 = 0
⇒ 3k = 0 ⇒ k = 0

Question 26.
When p(x) = x2 – 8x + k leaves a re-mainder when it is divided by (x – 1), then find k.
k > 7
Explanation:
p(1) = 1 – 8 + k = 0
⇒ k > 7

Question 27.
If α, β, γ are zeroes of x3 + 3x2 – x + 2, then find the value of αβγ.
-2

Question 28.
Make a quadratic polynomial having 2, 3 as zeroes.
x2 – 5x + 6 = 0
Explanation:
x2 – (2 + 3)x + 2 . 3 = 0
⇒ x2 – 5x + 6 = 0.

Question 29.
Write a quadratic polynomial, whose zeroes are $$\sqrt{2}$$ and –$$\sqrt{2}$$.
x2 – 2 = 0
Explanation:
x2 – $$(\sqrt{2}-\sqrt{2}) x+\sqrt{2}(-\sqrt{2})$$ = 0
⇒ x2 – 0(x) -2 = 0
⇒ x2 – 2 = 0.

Question 30.
Find the coefficient of x7 in the poly¬nomial 7x17 – 17x11 + 27x5 – 7.
0

Question 31.
If α, β are the zeroes of the polyno¬mial x2 – x – 6, then find α2β2
36
Explanation:
αβ = $$\frac{\mathrm{c}}{\mathrm{a}}$$ = – 6 ⇒ (αβ)2 = (-6)2 = 36

Question 32.
If ‘4’ is one of the zeroes of p(x) = x2 + kx – 8, then the value of k.
-2

Question 33.
If the polynomial p(x) = x3 – x2 + 3x + k is divided by (x – 1), the remainder obtained is 3, then find the value of k.
0
Explanation:
p(1) = 1-1 + 3 + k = 3 ⇒ k = 0

Question 34.
If one zero of the polynomial f(x) = 5x2 + 13x + k is reciprocal of the other, then find the value of k.
5
Explanation:
α = x, β = 1/x
⇒ α . β = x . 1/x = k/5 ⇒ k = 5

Question 35.
Find the number of zeroes of the po!y: nomial, whose graph is given below.

Question 36.
Number of zeroes that can be identi¬fied by the given figure.

Question 37.
Observe the given rectangular figure, then write its area in polynomial func-tion.

x2 – 7x – 30 = 0
Explanation:
(10 – x)(x + 3) = 10x + 30 – x2 – 3x = 0
⇒ x2 – 7x – 30 = 0

Question 38.
f(x) = x + 3, then find zero of f(x).
-3

Question 39.
One zero of the polynomial 2x2 + 3x + k is 1/2, then find k.
-2
Explanation:
im 19
α + β + γ = $$\frac{-b}{a}$$ = -3
$$\sqrt{5}-\sqrt{5}+\gamma$$ = -3
γ = -3

Question 40.
Find factors of x2 + x(a + b) + ab.
x + a and x + b

Question 41.
f(x) = 4x2 + 4x – 3, then find f($$\frac{-3}{2}$$)
0

Question 42.
If $$\sqrt{5}$$ and $$-\sqrt{5}$$ are two zeroes of the polynomial x3 + 3x2 – 5x – 15, then find its third zero.
-3
Explanation:

Question 43.
If $$\sqrt{3}$$ and – $$\sqrt{3}$$ are the zeroes of a polynomial p(x), then find p(x).
x2 – 3

Question 44.
If ‘m’ and ‘n’ are zeroes of the polyno-mial 3x2 + 11x – 4, then find the value $$\frac{\mathbf{m}}{\mathbf{n}}+\frac{\mathbf{n}}{\mathbf{m}}$$
$$\frac{-145}{12}$$
Explanation:
3x2 + 11x – 4 = 3x2 + 12x – x – 4 = 0
⇒(x + 4)(3x – 1)= 0
⇒x = -4, $$\frac { 1 }{ 2 }$$

Question 45.
In the product (x + 4) (x + 2). Write the constant term.
8

Question 46.
Find the polynomial whose zeroes are – 5 and 4.
x2 + x – 20

Question 47.
Flnd product of zeroes of 3x2 = 1
$$-\frac{1}{3}$$

Question 48.
Find the sum of the zeroes of x2 + 7x + 10.
-7

Question 49.
If one root of the polynomial
fix) = 5x2 + 13x + k is reciprocal of the other, then find ‘k’.
5

Question 50.
f(x) = 3x – 2. then find zero of f(x).
$$\frac{2}{3}$$

Question 51.
Write the zeroes from the below graph.

-2, 0 and 2

Question 52.
Make a quadratic polynomial whose zeroes are 5 and -2.
x2 – 3x – 10.

Question 53.
If α, β are (1w zero ai polynomial
f(x) = x2 – p(x + 1) – c then find (α + 1)(β + 1).
1 – c
Explanation:
α + β = $$\frac{-b}{a}=\frac{+p}{1}$$ = +p
αβ = $$\frac{c}{a}=\frac{-p-c}{1}$$ = -(p+c)
(α + 1) (β + 1) = αβ + α + β + 1
= -p – c + p + 1
= 1 – c

Question 54.
Write number of constant polynomial x2 + 7x – 7.
1

Question 55.
Write the quadratic polynomial, whose sum and product of zeroes are 1 and – 2 respectively.
x2 – x – 2

Question 56.
Find the product of the zeroes of x3 + 4x2 + x – 6.
6

Question 57.
p(x) = $$\frac{x+1}{1-x}$$ , then find p(0).
1

Question 58.
Write the maximum number of zeroes that a polynomial of degree 3 can have.
Three

Question 59.
If x + 2 is a factor of x2 + ax + 2 b and a 4 b = 4, then find Answer:
3
Explanation:
(- 2)2 – 2a + 2b = 0
⇒ 4 – 2a + 2b = 0
⇒ -a + b = -2

⇒ b = 1,a = 4- b = 4 – 1 = 3.

Question 60.
The graph of ax + b represents which type of polynomial ?
linear polynomial.

Question 61.
If ‘1’ is the zero of the quadratic poly- nomlal x2 + kx – 5, then find the value ofk.
4
Explanation:
1 + k – 5 = 0 ⇒ k = 4.

Question 62.
Find a quadratic polynomial, the sum of whose zeroes is W and product of zero is 3.
x2 – 9

Question 63.
If y = p(x) is represented by the given pqih, then find die number of zeroes.

4

Question 64.
If α + β = 0, αβ = $$\sqrt{3}$$, then write the quadratic polynomial.
x2 + $$\sqrt{3}$$

Question 65.
Find the degree of the polynomial ax4 + bx3 + cx2 + dx + e.
4.

Question 66.
If α, β, γ are the zeroes of the polyno- mid f(x) = x3 – px2 + qx – r, then find
$$\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}$$
$$\frac{\mathbf{p}}{\mathrm{r}}$$
Explanation:
$$\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}=\frac{\alpha+\beta+\gamma}{\alpha \beta \gamma}=\frac{\frac{p}{1}}{\frac{r}{1}}=\frac{p}{r}$$

Question 67.
If α and β are the zeroes of the poly¬nomial p(x) = 2x2 + 5x + k satisfying the relation α2 + β2 + αβ = $$\frac{21}{4}$$, then find k.
2
Explanation:
α2 + β2 + αβ

Question 68.
If p(t) = t3 , then find p(- 2).
– 9

Question 69.
If the polynomial f(x) = ax3 – bx – a is divisible % the polynomial g(x) = x2 + bx + c, then find ah.
1

Question 70.
Write degree of a linear polynomial.
1

Question 71.
If the sum off the zeroes off die polyno¬mial fix) = 2x3 – 3kx2 + 4x – 5 is 6, then find k.
4.

Question 72.
If p and q are the zeroes of the poly¬nomial t2 – 4t + 3, then find $$\frac{1}{p}+\frac{1}{9}-2 p q+\frac{14}{3}$$
0
Explanation:
t2 – 3t – t + 3
= t(t – 3) – 1(t-3)
= (t – 3) (t – 1)
t = 3,1
p = 3, q = 1

Question 73.
If the inudnct of zeroes rf9x2+3x + p is 7, then find p”.
63

Question 74.
If α, β, γ are the zeroes of the polyno-mial f(x) = ax3 + hx2 + cx + d, then find α2 + β2 + γ2
$$\frac{b^{2}-2 a c}{a^{2}}$$

Question 75.
p(x) = x2 + 5x + 6, then find zeroes of p(x).
-2,-3.

Question 76.
If one of the zeroes of the quadratic polynomial ax2 + bx + c is ‘0’, then find the other zero.
-b

Question 77.
α = a – b, β = a 4 b, then make the quadratic polynomial.
x2 – 2ax + a2 – b2

Question 78.
Find the quadratic polynomial whose zeroes are $$4+\sqrt{5}$$ and $$4-\sqrt{5}$$
x2 – 8x + 11

Question 79.
Find die remainder when
3x3 + x2 + 2x + 5 is divided by x2 + 2x + 1.
9x + 10

Question 80.
What must be subtracted or added to p(x) = 8x4 + 14x3 – 2x2 + 8x – 12, so that 4x2 + 3x – 2 is a factor off p(x) ?
15x – 14

Question 81.
Prepare a quadratic polynomial whose zeroes are $$\sqrt{15}$$ and $$-\sqrt{15}$$.
x2 – 15

Question 82.
Divide(x3 – 6x2 + 11x – 12) by (x2 – x + 2)r then find quotient.
x – 5

Question 83.
If -1 is a zero of the polynomial f(x) = x2 – 7x – 8, then find the other zero.
8

Question 84.
If a > 0, dim draw the shape off ax2 + bx + c = 0.

Question 85.
If 2x + 3 is a factor of 2x3 – x – b + 9x2, then find the value erf b.
-15

Question 86.
If the order erf ax5 + 3x4 + 4x3 + 3x2 + 2x + 1 is 4, then find Answer:
0
Explanation:
If a = 0, then given equation order becomes 4.

Question 87.
Find the zeroes erf the polynomial p(x) = 4x2 – 12x + 9.
$$\frac{3}{2}, \frac{3}{2}$$

Question 88.
If α, β, γ are roots erf a cubic polynomial ax3 + bx2 + cx + d, then find αβ + βγ + γα
$$\frac{\mathbf{c}}{\mathbf{a}}$$

Question 89.
If one of the zeroes of the quadratic polynomial f(x) = 14x2 – 42k2x – 9 is negative of the other, then find k.
0
Explanation:
Let α = x, β = – x
α + β = $$-\frac{b}{a}$$
x – x = $$\frac{-\left(-42 \mathrm{k}^{2}\right)}{14}$$
⇒ + 3k2 = 0
⇒ k2 = $$\frac{0}{3}$$ = 0 ⇒ k = 0

Question 90.
p(x) = 4x2 + 3x – 1, then find P($$\frac{1}{4}$$)
0

Question 91.
Find the sum of zeroes of the polynomial 3x2 – 8x + 1 = 0.
$$\frac{8}{3}$$
Explanation:
3x2 – 8x + 1 = 0
Sum of zeroes = $$-\frac{b}{a}$$
= $$\frac{-(-8)}{3}=\frac{8}{3}$$

Question 92.
If the product of zeroes of the polynomial f(x) = ax3 – 6x2 + 11x – 6 is 4, then find Answer:
$$\frac{3}{2}$$
Explanation:
αβγ = $$\frac{-\mathrm{d}}{\mathrm{a}}$$ = 4
$$\frac{-(-6)}{a}$$ = 4 ⇒ 4a = 6 ⇒ a = $$\frac{6}{4}=\frac{3}{2}$$

Question 93.
Name a polynomial of degree ‘3’.
Cubic polynomial.

Question 94.
Is the below graph represents a polynomial ?

No, it is not a polynomial.

Question 95.
Find the quotient when x4 + x3 + x2 – 2x – 3 is divided by x2 – 2.
x2 + x + 3

Question 96.
If α, β, γ are roots of a cubic polyno-mial ax3 + bx2 + cx + d, then find α + β + γ
$$\frac{-b}{a}$$

Question 97.
If one zero of the quadratic polynomial 2x2 + kx – 15 is 3, then find the other zero.
$$\frac{-5}{2}$$

Question 98.
If α, β, γ are the zeroes of the polynomial f(x) = ax3 + bx2 + cx + d, then
find $$\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}$$
$$-\frac{c}{d}$$

Question 99.
The graph

represents, which type of polynomial ?
Cubic polynomial.

Question 100.
Write the product and sum of the zeroes of the quadratic polynomial ax2 + bx + c respectively.
$$\frac{c}{a}, \frac{-b}{a}$$

Question 101.
Write the number of zeroes of the poly¬nomial in the given graph.

0

Question 102.
If f(x) = ax2 + bx + c has no real zeroes and a + b + c < 0, then write the condition.
c < 0

Question 103.
Which type of polynomial ax2 + bx + c?

Question 104.
If one zero of the polynomial
f(x) = (k2 + 4)x2 + 13x + 4k is recip¬rocal of the other, then find k.
2
Explanation:
$$x \cdot \frac{1}{x}=\frac{4 k}{k^{2}+4}$$
⇒ k2 + 4 = 4k
=» k2 – 4k + 4 = 0
⇒ (k – 2)2 = 0
⇒ k = 2

Question 105.
Write the number of zeroes of the poly-nomial function p(x), whose graph is given below.

3

Question 106.
If two zeroes of x3 + x2 – 5x – 5 are $$\sqrt{5}$$ and $$-\sqrt{5}$$ then find its third zero.
-1

Question 107.
If one zero of the quadratic polynomial x2 – 5x – 6 is 6, then find the other zero.
– 1

Question 108.
Find the degree of the polynomial $$a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\ldots .+a_{n} x^{n}$$
n

Question 109.
If α and β are the zeros of the polyno-mial f(x) = x2 + px + q, then find a
polynomial having $$\frac{1}{\alpha}$$ and $$\frac{1}{\beta}$$ as its a p
zeroes.
qx2 + px + 1

Question 110.
If both the zeroes of a quadratic poly-nomial ax2 + bx + c are equal and opposite in sign, then find ‘b’.
0

Question 111.
Write the number of zeroes of the poly-nomial in the given graph.

1

Question 112.
Find the sum and product of the ze¬roes of polynomial x2 – 3 respectively.
0,-3

Question 113.
If a < 0, then draw the shape of ax2 + bx + c = 0.

Question 114.
From the graph write the number of zeroes of the polynomial.

2

Question 115.
If α and β are zeroes of the polynomial p(x) = x2 – 5x + 6, then find the value of α + β – 3αβ.
– 13
Explanation:
α + β = $$\frac{-b}{a}=\frac{-(-5)}{1}$$ = 5
αβ = $$\frac{\mathrm{c}}{\mathrm{a}}$$ = 6
= α + β – 3αβ = 5 – 3(6)
= 5- 18 = – 13.

Question 116.
What should be subtracted from the polynomial x2 – 16x + 30 so that 15 is the zero of the resulting polynomial ?
15

Question 117.
Find the number of zeroes lying be-tween – 2 and 2 of the polynomial f(x) whose graph is given below.

2

Question 118.
If the zeroes of a quadratic polynomial are equal in magnitude but opposite in sign, then write the relation be¬tween zeroes.
Sum of its zeroes is 0.

Question 119.
Find where the graph of the polyno¬mial f(x) = 2x – 5 is a straight line which intersects the x – axis.
[$$\frac{5}{2}$$,0]

Question 120.
If α,β are the zeroes of the polyno¬mial f(x) = ax2 + bx + c, then find $$\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}=$$
$$\frac{b^{2}-2 a c}{c^{2}}$$

Choose the correct answer satis¬fying the following statements.

Question 121.
Statement (A): (2 – $$\sqrt{3}$$)is °ne zero of the quadratic polynomial, then other zero will be (2 + $$\sqrt{3}$$).
Statement (B) : Irrational zeroes (roots) always occurs in pairs.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(i)
Explanation:
As irrational roots / zeroes always oc¬curs in pairs therefore, when one, zero
is 2 – √3 , then other will be 2 + √3
So, both A and B are correct and B explains A.
Hence, (i) is the correct option.

Question 122.
Statement (A) : If both zeroes of the quadratic polynomial x2 – 2kx + 2 are equal in magnitude but opposite in sign, then value of k is $$\frac { 1 }{ 2 }$$.
Statement (B) : Sum of zeroes of a quadratic polynomial ax2 + bx + c is $$\frac{-b}{a}$$
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(iii)
Explanation:
As the polynomial is x2 – 2kx + 2 and
its zeros are equal but opposition sign.
∴ Sum of zeroes = 0 = $$\frac{-(-2 \mathrm{k})}{1}$$ = 0
⇒ 2k = 0 ⇒ k = 0
So, A is incorrect but B is correct.
Hence, (iii) is the correct option.

Question 123.
Statement (A): p(x) = 14x3 – 2x2 + 8x4 + 7x – 8 is a polynomial of degree 3. Statement (B) : The highest power of x in any polynomial p(x) is the degree of the polynomial.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(iii)
Explanation:
The highest power of x in the polyno¬mial p(x) = 14x3 – 2x2 + 8x4 + 7x – 8 is 4.
∴ Degree of p(x) is 4.
So, A is incorrect but B is correct.
Hence, (iii) is the correct option.

Question 124.

Statement (A) : The graph y = f(x) is shown in figure, for the polynomial f(x). The number of zeroes of f(x) is 4.
Statement (B): The number of zero of polynomial f(x) is the number of point of which f(x) cuts or touches the axes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(ii)
Explanation:
As the number of zero of polynomial f(x) is the number of {mints at which f(x) cuts (intersects) the x-axis and number of zero in the given figure is 4. So A is correct but B is incorrect Hence, (ii) is the correct option.

Question 125.
Statement (A) : The sum and product of the zeroes of a quadratic polynomial
are – $$\frac { 1 }{ 4 }$$ and $$\frac { 1 }{ 4 }$$ respectively. Then the
quadratic polynomial is 4x2 + x + 1.
Statement (B): The quadratic polyno-mial whose sum and product of zeroes are given is x2 – (sum of zeroes)x + product of zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false,
(i)
Explanation:
Sum of zeroes = –$$\frac { 1 }{ 4 }$$ and
Product of zeroes = $$\frac { 1 }{ 4 }$$

∴ Quadratic polynomial be 4x2 + x + 4.
So, both A and B are correct and B explains A. Hence, (i) is the correct option.

Question 126.
Statement (A) : If α, β are zeroes of x2 – 3x + p and 2α + 3β = 15, then p – 54.
Statement (B) : If α, β are the zeroes of the polynomial ax2 + bx + c, then
α + β = $$\frac{-b}{a}$$ and αβ = $$\frac{c}{a}$$
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(iii)

Question 127.
Statement (A) : A quadratic polyno¬mial having 4 and – 2 as zeroes is 2x2 – 4x – 16.
Statement (B): The quadratic polyno-mial having a and (3 as zeroes is given by p(x) = x2 – (α + β)x + αβ
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(i)

Question 128.
Statement (A) : The polynomial p(x) = x3 + x has one real zero.
Statement (B) : A polynomial of nth degree has at most n zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(ii)

Question 129.
Statement (A): If α, β, γ are the zeroes of x3 – 2x2 – qx – r and α + β = 0, then 2q= r.
Statement (B) : If α, β, γ are the ze¬roes of ax3 + bx2 + cx + d, then
α + β + γ = $$\frac{-b}{a}$$, αβ, βγ, γα = $$\frac{c}{a}$$, αβγ = $$\frac{-d}{a}$$
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(iii)
Explanation:
Clearly, (B) is true, [standard result] cc + p + y = -(-2) = 2 ⇒ 0 + y = 2 ‘ Y – 2 ‘
α + β + γ = – (- 2) = 2=
o + γ = 2
γ = 2
αβγ = -(-r) = r
∴ αβ(2) = r
⇒ αβ = $$\frac{r}{2}$$
⇒ αβ + βγ + γα = -q
⇒$$\frac{r}{2}$$ + γ(α+β) = -q
⇒$$\frac{r}{2}$$ + 2(0) = -q
⇒ r = 2q
∴ (A) is false
Hence, (iii) is the correct option.

Question 130.
Statement (A) : If one zero of polyno¬mial p(x) = (k2 + 4)x2 + 13x + 4k is reciprocal of other, then k = 2.
Statement (B) : If x – a is a factor of p(x), then p(α) = 0 i.e., a is a zero of p(x).
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(i)
Explanation:
(B) is true
Let α, 1/α be the zeroes of p(x), then
$$\alpha \cdot \frac{1}{\alpha}=\frac{4 \mathbf{k}}{\mathbf{k}^{2}+4} \Rightarrow \mathbf{1}=\frac{4 \mathbf{k}}{\mathbf{k}^{2}+4}$$
∴ k2 -4k + 4 = 0 ⇒ (k- 2)2 = 0
∴ k = 2
∴ (A) is true. So, (i) is correct option.

Question 131.
Statement (A) : The polynomial x4 + 4x2 + 5 has four zeroes. Statement (B) : If p(x) is divided by (x – k), then the remainder = p(k).
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(iii)
Explanation:
(B) is true by remainder theorem. Again, x4 + 4x2 + 5
= (x2 + 2)2 + 1 > 0 for all x.
∴ Given polynomial has no zeroes.
∴ (A) is not true.
Hence, (iii) is the correct option.

Question 132.
Statement (A) : x3 + x has only one real zero.
Statement (B) : A polynomial of nth degree must have ‘n’ real zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(ii)
Explanation:
(B) is false [v a polynomial of n<sup<th degree has at most x zeroes]
Again, x3 + x = x (x2 + 1) which has only one real zero (x = 0)
[∵ x2 + 1 ≠ 0 for all x ∈ R]
(A) is true.
Hence, (ii) is the correct option.

Question 133.
Statement (A) : If 2, 3 are the zeroes of a quadratic polynomial, then poly¬nomial is x2 – 5x + 6.
Statement (B) : If a, P are the zeroes of a monic quadratic polynomial, then polynomial is x2 – (a + p)x + ap.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(i)

Question 134.
Statement (A): Degree of a zero poly-nomial is not defined.
Statement (B) : Degree of a non-zero constant polynomial is ‘0’.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(ii)

Question 135.
Statement (A) : Zeroes of f(x) = x3 – 4x – 5 are 5, – 1.
Statement (B): The polynomial whose zeroes are $$2+\sqrt{3}, 2-\sqrt{3}$$ is x2 – 4x + 7.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(i)

Question 136.
Statement (A) : x2 + 4x + 5 has two zeroes.
Statement (B) : A quadratic polyno¬mial can have at the most two zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
(i)

Read the below passages and an¬swer to the following questions.

If α, β are the zeroes of the quadratic polynomial f(x) = ax2 + bx + c, then
α + β = $$\frac{-b}{a}$$, αβ = $$\frac{c}{a}$$

If α, β are the zeroes of the quadratic polynomial f(x) = x2 – px + q, then find $$\frac{1}{\alpha}+\frac{1}{\beta}$$
$$\frac{p}{q}$$
Explanation:
α + β = p, αβ = q
∴ $$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}=\frac{p}{q}$$

Question 138.
If α, β are the zeroes of the quadratic polynomial fix) = x2 + x – 2, then find $$\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}$$
$$\frac{9}{4}$$
Explanation:
α + β = -1, αβ = -2

Question 139.
If α, β are the zeroes of the quadratic polynomial fix) = x2 – 5x + 4, then
find $$\frac{1}{\alpha}+\frac{1}{\beta}-2 \alpha \beta$$
–$$\frac{27}{4}$$

If α, β, γ are the zeroes of ax3 + bx2 + cx + d, then $$\Sigma \alpha=\frac{-b}{a}$$, $$\Sigma \alpha \beta=\frac{\mathbf{c}}{\mathbf{a}}, \Sigma \alpha \beta \gamma=\frac{-\mathbf{d}}{\mathbf{a}}$$
Explanation:
α + β = 5, αβ = 4

Question 140.
If α, β, γ are the zeroes of x3 – 5x2 – 2x + 24 and ap = 12, then
find αβ = 12, then find ‘γ’
-2

Question 141.
If a – b, a, a + b are the roots of x3 – 3x2 + x + 1, then find a + b2.
3

Question 142.
If two zeroes of the polynomial x3 – 5x2 – 16x + 80 are equal in magni¬tude but opposite in sign, then find ze¬roes.
4,-4, 5.

Manow says that the order of the polynomial (x2 – 5) (x3 + 1) is 6.

Question 143.
Do you agree with Manow ?
No.

Question 144.
Which mathematical concept is used to judge Manow ?
Polynomial.

Question 145.
How much the actual order of given problem ?
Degree is 5.

The length of a rectangle is ’5’ more than its breadth.

Question 146.
Express the information in the form of polynomial.
(x + 5 + x) = 2x + 5.

Question 147.
Find the perimeter of the rectangle given above.
(4x + 10)m

Question 148.
To solve this given problem which mathematical concept was used by you?
Polynomial.

Question 149.
Write the correct matching option.

A – (ii), B – (iv).

Question 150.
Write the correct matching option.

A – (iii), B – (i).

Question 151.
Write the correct matching option.

A – (iv), B – (iii).

Question 152.
Write the correct matching option.

A – (ii), B – (i).

Question 153.
Write the correct matching option.

A – (iii), B – (i).

Question 154.
Write the correct matching option.

A – (iii), B – (v).

Question 155.
Write the correct matching option.

A – (iv), B – (i).

Question 156.
If α, β, γ are the zeroes of the polyno-mial px3 + qx2 + rx + s then, which of the following matching is correct ?

a) A(i), B(ii), C(iii)
b) A(ii), B(iii), C(i)
c) A(iii), B(i), C(ii)
d) A(ii), B(i), C(iii)
(b)

Question 157.
What is the zero of the polynomial 3x – 2 ?
Solution:
f(x) = 3x – 2; f(x) = 0
3x – 2 = 0 ⇒ 3x = 2
⇒ x = 2/3

Question 158.
Write the polynomial in variable ‘x’ whose zero is $$\frac{-k}{a}$$.
Solution:
x – $$\frac{-k}{a}$$ = 0 ⇒ x + $$\frac{k}{a}$$ = 0
⇒ ax + k = 0
∴ ax + k = 0 is a polynomial with degree T in variable ‘x’ whose zero is $$\frac{-k}{a}$$.