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 State Syllabus 10th Class Maths Bits 3rd Lesson Polynomials with Answers

Question 1.

Write the sum of zeroes of

bx^{2} + ax + c.

Answer:

\(\frac{-a}{b}\)

Question 2.

Find product of the zeroes of

p(x) = (x-2).(x + 3)

Answer:

= -6

Explanation:

p(x) = x^{2} + x – 6,

Product of zeroes = \(\frac{\mathrm{c}}{\mathrm{a}}\) = – 6.

Question 3.

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

Answer:

Linear.

Question 4.

Find the value of ‘x’ which satisfies 2(x – 1) – (1 – x) = 2x + 3 ?

Answer:

6

Explanation:

2x – 2 – 1 + x = 2x + 3

⇒ 3x – 2x = 3 + 3 = 6

⇒ x = 6.

Question 5.

Write the degree of the polynomial

\(\sqrt{2}\)x^{2} – 3x + 1.

Answer:

2

Question 6.

Write order of the polynomial 5x^{7} – 6x^{5} + 7x – 6.

Answer:

7

Question 7.

Find the product of zeroes of 2x^{2} – 3x + 6.

Answer:

3

Explanation:

Product of zeroes = \(\frac{c}{a}=\frac{6}{2}\) = 3.

Question 8.

Find sum of zeroes of a polynomial x^{3} – 2x^{2} + 3x – 4.

Answer:

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

Answer:

x^{2} – 16 = 0

Explanation:

Sum of zeroes = \(\frac{-b}{a}\) = 0,

α + β = 0, β = 4

⇒ α = -4 x2 – (α + β)x + αβ = 0

⇒ x^{2} – 0(x) -16 = 0

⇒ x^{2} – 16 = 0.

Question 10.

If p(x) = x^{2} – ax – 3 and p(2) = – 3, then find Answer:

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.

Answer:

\(\frac{-\mathrm{q}}{\mathrm{p}}\)

Question 12.

In a division, if divisor is x + 1, quo¬tient is x and remainder is 4, then find dividend.

Answer:

x (x + 1) + 4 = x^{2} + x + 4

Explanation:

Dividend = Divisor x Quotient

+ Remainder

= (x + 1) x + 4 = x^{2} + x + 4.

Question 13.

Find the zero value of linear polynomial ax – b.

Answer:

\(\frac{\mathrm{b}}{\mathrm{a}}\)

Question 14.

Find the sum of the zeroes of the poly-nomial x^{2} + 5x + 6.

Answer:

– 5

Question 15.

4y^{2} – 5y + 1 is an example for this type of polynomial.

Answer:

Quadratic polynomial.

Question 16.

Write the degree of the polynomial 5x^{7} – 6x^{5} + 7x – 4.

Answer:

7

Question 17.

How much the sum of zeroes of the polynomial 2x^{2} – 8x + 6 ?

Answer:

4

Question 18.

f α,β are the zeroes of x^{2} + x +1, then find \(\frac{1}{\alpha}+\frac{1}{\beta}\)

Answer:

-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.

Answer:

3

Question 20.

Write product of zeroes of the cubic polynomial 3x^{3} – 5x^{2} – 11x – 3.

Answer:

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.

Answer:

– 2,1

Question 22.

Find the value of p(x) = 4x^{2} + 3x + 1 at x = -1.

Answer:

2

Question 23.

If α, β, γ are the zeroes of the cubic polynomial ax^{3} + bx^{2} + cx + d and (a ≠ 0), then find αβγ

Question 24.

4x + 6y = 18 doesn’t pass through ori-gin, then its graph indicates

Answer:

A straight line

Question 25.

If ‘3’ is one of the zeroes of

p(x) = x^{2} + kx – 9, then find the value of k.

Answer:

0

Explanation:

p(3) = 3^{2} + 3k – 9 = 0

⇒ 3k = 0 ⇒ k = 0

Question 26.

When p(x) = x^{2} – 8x + k leaves a re-mainder when it is divided by (x – 1), then find k.

Answer:

k > 7

Explanation:

p(1) = 1 – 8 + k = 0

⇒ k > 7

Question 27.

If α, β, γ are zeroes of x^{3} + 3x^{2} – x + 2, then find the value of αβγ.

Answer:

-2

Question 28.

Make a quadratic polynomial having 2, 3 as zeroes.

Answer:

x^{2} – 5x + 6 = 0

Explanation:

x^{2} – (2 + 3)x + 2 . 3 = 0

⇒ x^{2} – 5x + 6 = 0.

Question 29.

Write a quadratic polynomial, whose zeroes are \(\sqrt{2}\) and –\(\sqrt{2}\).

Answer:

x^{2} – 2 = 0

Explanation:

x^{2} – \((\sqrt{2}-\sqrt{2}) x+\sqrt{2}(-\sqrt{2})\) = 0

⇒ x^{2} – 0(x) -2 = 0

⇒ x^{2} – 2 = 0.

Question 30.

Find the coefficient of x7 in the poly¬nomial 7x^{17} – 17x^{11} + 27x^{5} – 7.

Answer:

0

Question 31.

If α, β are the zeroes of the polyno¬mial x^{2} – x – 6, then find α^{2}β^{2}

Answer:

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) = x^{2} + kx – 8, then the value of k.

Answer:

-2

Question 33.

If the polynomial p(x) = x^{3} – x^{2} + 3x + k is divided by (x – 1), the remainder obtained is 3, then find the value of k.

Answer:

0

Explanation:

p(1) = 1-1 + 3 + k = 3 ⇒ k = 0

Question 34.

If one zero of the polynomial f(x) = 5x^{2} + 13x + k is reciprocal of the other, then find the value of k.

Answer:

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.

Answer:

Question 36.

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

Answer:

Question 37.

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

Answer:

x^{2} – 7x – 30 = 0

Explanation:

(10 – x)(x + 3) = 10x + 30 – x^{2} – 3x = 0

⇒ x^{2} – 7x – 30 = 0

Question 38.

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

Answer:

-3

Question 39.

One zero of the polynomial 2x^{2} + 3x + k is 1/2, then find k.

Answer:

-2

Explanation:

im 19

α + β + γ = \(\frac{-b}{a}\) = -3

\(\sqrt{5}-\sqrt{5}+\gamma\) = -3

γ = -3

Question 40.

Find factors of x^{2} + x(a + b) + ab.

Answer:

x + a and x + b

Question 41.

f(x) = 4x^{2} + 4x – 3, then find f(\(\frac{-3}{2}\))

Answer:

0

Question 42.

If \(\sqrt{5}\) and \(-\sqrt{5}\) are two zeroes of the polynomial x^{3} + 3x^{2} – 5x – 15, then find its third zero.

Answer:

-3

Explanation:

Question 43.

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

Answer:

x^{2} – 3

Question 44.

If ‘m’ and ‘n’ are zeroes of the polyno-mial 3x^{2} + 11x – 4, then find the value \(\frac{\mathbf{m}}{\mathbf{n}}+\frac{\mathbf{n}}{\mathbf{m}}\)

Answer:

\(\frac{-145}{12}\)

Explanation:

3x^{2} + 11x – 4 = 3x^{2} + 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.

Answer:

8

Question 46.

Find the polynomial whose zeroes are – 5 and 4.

Answer:

x^{2} + x – 20

Question 47.

Flnd product of zeroes of 3x^{2} = 1

Answer:

\(-\frac{1}{3}\)

Question 48.

Find the sum of the zeroes of x^{2} + 7x + 10.

Answer:

-7

Question 49.

If one root of the polynomial

fix) = 5x^{2} + 13x + k is reciprocal of the other, then find ‘k’.

Answer:

5

Question 50.

f(x) = 3x – 2. then find zero of f(x).

Answer:

\(\frac{2}{3}\)

Question 51.

Write the zeroes from the below graph.

Answer:

-2, 0 and 2

Question 52.

Make a quadratic polynomial whose zeroes are 5 and -2.

Answer:

x^{2} – 3x – 10.

Question 53.

If α, β are (1w zero ai polynomial

f(x) = x^{2} – p(x + 1) – c then find (α + 1)(β + 1).

Answer:

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 x^{2} + 7x – 7.

Answer:

1

Question 55.

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

Answer:

x^{2} – x – 2

Question 56.

Find the product of the zeroes of x^{3} + 4x^{2} + x – 6.

Answer:

6

Question 57.

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

Answer:

1

Question 58.

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

Answer:

Three

Question 59.

If x + 2 is a factor of x^{2} + ax + 2 b and a 4 b = 4, then find Answer:

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 ?

Answer:

linear polynomial.

Question 61.

If ‘1’ is the zero of the quadratic poly- nomlal x^{2} + kx – 5, then find the value ofk.

Answer:

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.

Answer:

x^{2} – 9

Question 63.

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

Answer:

4

Question 64.

If α + β = 0, αβ = \(\sqrt{3}\), then write the quadratic polynomial.

Answer:

x^{2} + \(\sqrt{3}\)

Question 65.

Find the degree of the polynomial ax^{4} + bx^{3} + cx^{2} + dx + e.

Answer:

4.

Question 66.

If α, β, γ are the zeroes of the polyno- mid f(x) = x^{3} – px^{2} + qx – r, then find

\(\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}\)

Answer:

\(\frac{\mathbf{p}}{\mathrm{r}}\)

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) = 2x^{2} + 5x + k satisfying the relation α^{2} + β^{2} + αβ = \(\frac{21}{4}\), then find k.

Answer:

2

Explanation:

α^{2} + β^{2} + αβ

Question 68.

If p(t) = t^{3} , then find p(- 2).

Answer:

– 9

Question 69.

If the polynomial f(x) = ax^{3} – bx – a is divisible % the polynomial g(x) = x^{2} + bx + c, then find ah.

Answer:

1

Question 70.

Write degree of a linear polynomial.

Answer:

1

Question 71.

If the sum off the zeroes off die polyno¬mial fix) = 2x^{3} – 3kx^{2} + 4x – 5 is 6, then find k.

Answer:

4.

Question 72.

If p and q are the zeroes of the poly¬nomial t^{2} – 4t + 3, then find \(\frac{1}{p}+\frac{1}{9}-2 p q+\frac{14}{3}\)

Answer:

0

Explanation:

t^{2} – 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”.

Answer:

63

Question 74.

If α, β, γ are the zeroes of the polyno-mial f(x) = ax^{3} + hx^{2} + cx + d, then find α^{2} + β^{2} + γ^{2}

Answer:

\(\frac{b^{2}-2 a c}{a^{2}}\)

Question 75.

p(x) = x^{2} + 5x + 6, then find zeroes of p(x).

Answer:

-2,-3.

Question 76.

If one of the zeroes of the quadratic polynomial ax^{2} + bx + c is ‘0’, then find the other zero.

Answer:

-b

Question 77.

α = a – b, β = a 4 b, then make the quadratic polynomial.

Answer:

x^{2} – 2ax + a^{2} – b^{2}

Question 78.

Find the quadratic polynomial whose zeroes are \(4+\sqrt{5}\) and \(4-\sqrt{5}\)

Answer:

x^{2} – 8x + 11

Question 79.

Find die remainder when

3x^{3} + x^{2} + 2x + 5 is divided by x^{2} + 2x + 1.

Answer:

9x + 10

Question 80.

What must be subtracted or added to p(x) = 8x^{4} + 14x^{3} – 2x^{2} + 8x – 12, so that 4x^{2} + 3x – 2 is a factor off p(x) ?

Answer:

15x – 14

Question 81.

Prepare a quadratic polynomial whose zeroes are \(\sqrt{15}\) and \(-\sqrt{15}\).

Answer:

x^{2} – 15

Question 82.

Divide(x^{3} – 6x^{2} + 11x – 12) by (x^{2} – x + 2)r then find quotient.

Answer:

x – 5

Question 83.

If -1 is a zero of the polynomial f(x) = x^{2} – 7x – 8, then find the other zero.

Answer:

8

Question 84.

If a > 0, dim draw the shape off ax^{2} + bx + c = 0.

Answer:

Question 85.

If 2x + 3 is a factor of 2x^{3} – x – b + 9x^{2}, then find the value erf b.

Answer:

-15

Question 86.

If the order erf ax^{5} + 3x^{4} + 4x^{3} + 3x^{2} + 2x + 1 is 4, then find Answer:

Answer:

0

Explanation:

If a = 0, then given equation order becomes 4.

Question 87.

Find the zeroes erf the polynomial p(x) = 4x^{2} – 12x + 9.

Answer:

\(\frac{3}{2}, \frac{3}{2}\)

Question 88.

If α, β, γ are roots erf a cubic polynomial ax^{3} + bx^{2} + cx + d, then find αβ + βγ + γα

Answer:

\(\frac{\mathbf{c}}{\mathbf{a}}\)

Question 89.

If one of the zeroes of the quadratic polynomial f(x) = 14x^{2} – 42k^{2}x – 9 is negative of the other, then find k.

Answer:

0

Explanation:

Let α = x, β = – x

α + β = \(-\frac{b}{a}\)

x – x = \(\frac{-\left(-42 \mathrm{k}^{2}\right)}{14}\)

⇒ + 3k^{2} = 0

⇒ k^{2} = \(\frac{0}{3}\) = 0 ⇒ k = 0

Question 90.

p(x) = 4x^{2} + 3x – 1, then find P(\(\frac{1}{4}\))

Answer:

0

Question 91.

Find the sum of zeroes of the polynomial 3x^{2} – 8x + 1 = 0.

Answer:

\(\frac{8}{3}\)

Explanation:

3x^{2} – 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) = ax^{3} – 6x^{2} + 11x – 6 is 4, then find Answer:

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’.

Answer:

Cubic polynomial.

Question 94.

Is the below graph represents a polynomial ?

Answer:

No, it is not a polynomial.

Question 95.

Find the quotient when x^{4} + x^{3} + x^{2} – 2x – 3 is divided by x^{2} – 2.

Answer:

x^{2} + x + 3

Question 96.

If α, β, γ are roots of a cubic polyno-mial ax^{3} + bx^{2} + cx + d, then find α + β + γ

Answer:

\(\frac{-b}{a}\)

Question 97.

If one zero of the quadratic polynomial 2x^{2} + kx – 15 is 3, then find the other zero.

Answer:

\(\frac{-5}{2}\)

Question 98.

If α, β, γ are the zeroes of the polynomial f(x) = ax^{3} + bx^{2} + cx + d, then

find \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\)

Answer:

\(-\frac{c}{d}\)

Question 99.

The graph

represents, which type of polynomial ?

Answer:

Cubic polynomial.

Question 100.

Write the product and sum of the zeroes of the quadratic polynomial ax^{2} + bx + c respectively.

Answer:

\(\frac{c}{a}, \frac{-b}{a}\)

Question 101.

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

Answer:

0

Question 102.

If f(x) = ax^{2} + bx + c has no real zeroes and a + b + c < 0, then write the condition.

Answer:

c < 0

Question 103.

Which type of polynomial ax^{2} + bx + c?

Answer:

Quadratic polynomial.

Question 104.

If one zero of the polynomial

f(x) = (k^{2} + 4)x2 + 13x + 4k is recip¬rocal of the other, then find k.

Answer:

2

Explanation:

\(x \cdot \frac{1}{x}=\frac{4 k}{k^{2}+4}\)

⇒ k^{2} + 4 = 4k

=» k^{2} – 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.

Answer:

3

Question 106.

If two zeroes of x^{3} + x^{2} – 5x – 5 are \(\sqrt{5}\) and \(-\sqrt{5}\) then find its third zero.

Answer:

-1

Question 107.

If one zero of the quadratic polynomial x^{2} – 5x – 6 is 6, then find the other zero.

Answer:

– 1

Question 108.

Find the degree of the polynomial \(a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\ldots .+a_{n} x^{n}\)

Answer:

n

Question 109.

If α and β are the zeros of the polyno-mial f(x) = x^{2} + px + q, then find a

polynomial having \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) as its a p

zeroes.

Answer:

qx^{2} + px + 1

Question 110.

If both the zeroes of a quadratic poly-nomial ax^{2} + bx + c are equal and opposite in sign, then find ‘b’.

Answer:

0

Question 111.

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

Answer:

1

Question 112.

Find the sum and product of the ze¬roes of polynomial x^{2} – 3 respectively.

Answer:

0,-3

Question 113.

If a < 0, then draw the shape of ax^{2} + bx + c = 0.

Answer:

Question 114.

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

Answer:

2

Question 115.

If α and β are zeroes of the polynomial p(x) = x^{2} – 5x + 6, then find the value of α + β – 3αβ.

Answer:

– 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 x^{2} – 16x + 30 so that 15 is the zero of the resulting polynomial ?

Answer:

15

Question 117.

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

Answer:

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.

Answer:

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.

Answer:

[\(\frac{5}{2}\),0]

Question 120.

If α,β are the zeroes of the polyno¬mial f(x) = ax^{2} + bx + c, then find \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}=\)

Answer:

\(\frac{b^{2}-2 a c}{c^{2}}\)

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.

Answer:

(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 x^{2} – 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 ax^{2} + 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.

Answer:

(iii)

Explanation:

As the polynomial is x^{2} – 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) = 14x^{3} – 2x^{2} + 8x^{4} + 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.

Answer:

(iii)

Explanation:

The highest power of x in the polyno¬mial p(x) = 14x^{3} – 2x^{2} + 8x^{4} + 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.

Answer:

(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 4x^{2} + x + 1.

Statement (B): The quadratic polyno-mial whose sum and product of zeroes are given is x^{2} – (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,

Answer:

(i)

Explanation:

Sum of zeroes = –\(\frac { 1 }{ 4 }\) and

Product of zeroes = \(\frac { 1 }{ 4 }\)

∴ Quadratic polynomial be

∴ Quadratic polynomial be 4x^{2} + 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 x^{2} – 3x + p and 2α + 3β = 15, then p – 54.

Statement (B) : If α, β are the zeroes of the polynomial ax^{2} + 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.

Answer:

(iii)

Question 127.

Statement (A) : A quadratic polyno¬mial having 4 and – 2 as zeroes is 2x^{2} – 4x – 16.

Statement (B): The quadratic polyno-mial having a and (3 as zeroes is given by p(x) = x^{2} – (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.

Answer:

(i)

Question 128.

Statement (A) : The polynomial p(x) = x^{3} + 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.

Answer:

(ii)

Question 129.

Statement (A): If α, β, γ are the zeroes of x^{3} – 2x^{2} – qx – r and α + β = 0, then 2q= r.

Statement (B) : If α, β, γ are the ze¬roes of ax^{3} + bx^{2} + 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.

Answer:

(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) = (k^{2} + 4)x^{2} + 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.

Answer:

(i)

Explanation:

(B) is true

Let α, 1/α be the zeroes of p(x), then

\(\alpha \cdot \frac{1}{\alpha}=\frac{4 \mathbf{k}}{\mathbf{k}^{2}+4} \Rightarrow \mathbf{1}=\frac{4 \mathbf{k}}{\mathbf{k}^{2}+4}\)

∴ k^{2} -4k + 4 = 0 ⇒ (k- 2)^{2} = 0

∴ k = 2

∴ (A) is true. So, (i) is correct option.

Question 131.

Statement (A) : The polynomial x^{4} + 4x^{2} + 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.

Answer:

(iii)

Explanation:

(B) is true by remainder theorem. Again, x^{4} + 4x^{2} + 5

= (x^{2} + 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) : x^{3} + 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.

Answer:

(ii)

Explanation:

(B) is false [v a polynomial of n<sup<th degree has at most x zeroes]

Again, x^{3} + x = x (x^{2} + 1) which has only one real zero (x = 0)

[∵ x^{2} + 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 x^{2} – 5x + 6.

Statement (B) : If a, P are the zeroes of a monic quadratic polynomial, then polynomial is x^{2} – (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.

Answer:

(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.

Answer:

(ii)

Question 135.

Statement (A) : Zeroes of f(x) = x^{3} – 4x – 5 are 5, – 1.

Statement (B): The polynomial whose zeroes are \(2+\sqrt{3}, 2-\sqrt{3}\) is x^{2} – 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.

Answer:

(i)

Question 136.

Statement (A) : x^{2} + 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.

Answer:

(i)

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

If α, β are the zeroes of the quadratic polynomial f(x) = ax^{2} + bx + c, then

α + β = \(\frac{-b}{a}\), αβ = \(\frac{c}{a}\)

If α, β are the zeroes of the quadratic polynomial f(x) = x^{2} – px + q, then find \(\frac{1}{\alpha}+\frac{1}{\beta}\)

Answer:

\(\frac{p}{q}\)

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) = x^{2} + x – 2, then find \(\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}\)

Answer:

\(\frac{9}{4}\)

Explanation:

α + β = -1, αβ = -2

Question 139.

If α, β are the zeroes of the quadratic polynomial fix) = x^{2} – 5x + 4, then

find \(\frac{1}{\alpha}+\frac{1}{\beta}-2 \alpha \beta\)

Answer:

–\(\frac{27}{4}\)

If α, β, γ are the zeroes of ax^{3} + bx^{2} + 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 x^{3} – 5x^{2} – 2x + 24 and ap = 12, then

find αβ = 12, then find ‘γ’

Answer:

-2

Question 141.

If a – b, a, a + b are the roots of x^{3} – 3x^{2} + x + 1, then find a + b^{2}.

Answer:

3

Question 142.

If two zeroes of the polynomial x^{3} – 5x^{2} – 16x + 80 are equal in magni¬tude but opposite in sign, then find ze¬roes.

Answer:

4,-4, 5.

Manow says that the order of the polynomial (x^{2} – 5) (x^{3} + 1) is 6.

Question 143.

Do you agree with Manow ?

Answer:

No.

Question 144.

Which mathematical concept is used to judge Manow ?

Answer:

Polynomial.

Question 145.

How much the actual order of given problem ?

Answer:

Degree is 5.

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

Question 146.

Express the information in the form of polynomial.

Answer:

(x + 5 + x) = 2x + 5.

Question 147.

Find the perimeter of the rectangle given above.

Answer:

(4x + 10)m

Question 148.

To solve this given problem which mathematical concept was used by you?

Answer:

Polynomial.

Question 149.

Write the correct matching option.

Answer:

A – (ii), B – (iv).

Question 150.

Write the correct matching option.

A – (iii), B – (i).

Question 151.

Write the correct matching option.

Answer:

A – (iv), B – (iii).

Question 152.

Write the correct matching option.

Answer:

A – (ii), B – (i).

Question 153.

Write the correct matching option.

Answer:

A – (iii), B – (i).

Question 154.

Write the correct matching option.

Answer:

A – (iii), B – (v).

Question 155.

Write the correct matching option.

Answer:

A – (iv), B – (i).

Question 156.

If α, β, γ are the zeroes of the polyno-mial px^{3} + qx^{2} + 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)

Answer:

(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 \(\frac{-k}{a}\).