Important Questions Class 10 Maths Chapter 2 Polynomials

Important class 10 maths questions for chapter 2 polynomials are provided here to help the students practice and score well in the CBSE class 10 maths exam. These questions from the polynomials chapter of NCERT class 10 covers most concepts and will help the students to get acquainted with wide variations of questions and thus, develop problem-solving skills to a great extent.

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Important Polynomials Questions for Class 10- Chapter 12 (With Solutions)

A few important class 10 polynomials questions are provided below with solutions. These questions include both short and long answer questions and involve HOTS to let the students get completely acquainted with the in-depth concepts.

1. Find the value of “p” from the equation x2 + 3x + p, if one of the zeroes of the polynomial is 2.

Solution:

As 2 is the zero of the polynomial,

x2 + 3x + p, for x = 2

Now, put x = 2

22 + 3(2) + p = 0

=> 4 + 6 + p = 0

Or, p = -10

2. Does the polynomial a4 + 4a2 + 5 = 0 have real zeroes?

Solution:

In the aforementioned equation, let a2 = x.

Now, the equation becomes,

x2 + 4x2 + 5 = 0

Here, b2 – 4ac will be = 44 – 4(1)(5) = -20

So, D = b2 – 4ac < 0

As the discriminant (D) is negative, the given polynomial does not have real roots or zeroes.

3. Compute the zeroes of the polynomial 4x2 – 4x – 8. Also, establish a relationship between the zeroes and coefficients.

Solution:

Factorise the equation 4x2 – 4x – 8,

4x2 – 4x – 8 = 4x2 – 2x – 2x + 1

= 2x(2x – 1) – 1(2x -1) = (2x – 1) (2x – 1)

So, the roots of 4x2 – 4x – 8 are (½ and ½)

Relation between the sum of zeroes and coefficients:

½ + ½ = 1 = -4/4 i.e. (- coefficient of x/ coefficient of x2)

Relation between the product of zeroes and coefficients:

½ × ½ = ¼ i.e (constant/coefficient of x2)

4. Find the quadratic polynomial if its zeroes are 0, √5.

Solution:

A quadratic polynomial can be written using the sum and product of its zeroes as:

x2 +(α + β)x + αβ = 0

Where α and β are the roots of the equation.

Here, α = 0 and β = √5

So, the equation will be:

x2 +(0 + √5)x + 0(√5) = 0

Or, x2 + √5x = 0

5. Find the value of “x” in the equation 2a2 + 2xa + 5x + 10 if (a + x) is one of its factors.

Solution:

Let f(a) = 2a2 + 2xa + 5x + 10

As (a + x) is a factor of 2a2 + 2xa + 5x + 10, f(-x) = 0

So, f(-x) = 2x2 – 2x2 – 5x + 10 = 0

Or, -5x + 10 = 0

Thus, x = 2

6. How many zeros does the polynomial (x – 3)2 – 4 can have? Also, find its zeroes.

Solution:

Given equation is (x – 3)2 – 4

Now, expand this equation.

=> x2 + 9 – 6x – 4

= x2 – 6x + 5

As the equation has a degree of 2, the number of zeroes it will have is 2.

Now, solve x2 – 6x + 5 = 0 to get the roots.

So, x2 – x – 5x + 5 = 0

=> x(x-1) -5(x-1) = 0

=> (x-1)(x-5)

So, the roots are 1 and 5.

7. α and β are zeroes of the quadratic polynomial x2 – 6x + y. Find the value of ‘y’ if 3α + 2β = 20.

Solution:

Let, f(x) = x² – 6 x + a

From the question,

3α + 2β = 20

From f(x),

α + β = 6———————(ii)

And,

αβ = y———————(iii)

Now, multiply equation (ii) by 2 and subtract from equation (i),

=> α = 20 – 12 = 8

Now, substitute this value in equation (ii),

=> β = 6-8 = -2

By substituting the value of α and β in equation (iii), the value of “y” can be obtained.

y = αβ = -16

8. The zeroes of the polynomial 𝒙𝟑 − 𝟑𝒙𝟐 + 𝒙 + 𝟏 are a – b, a, a + b

Solution:

p(x) = 𝑥3 − 3𝑥2 + 𝑥 + 1

Here, zeroes are given are a – b, a, and a + b

Now, compare the given polynomial equation with general expression.

𝑝𝑥3 + 𝑞𝑥2 + 𝑟𝑥 + 𝑠 = 𝑥3 − 3𝑥2 + 𝑥 + 1

Here, p = 1, q = -3, r = 1 and s = 1

For sum of zeroes:

Sum of zeroes = a – b + a + a + b

-q/p = 3a

Substitute the values q and p.

-(-3)/1 = 3a

Or, a = 1

So, the zeroes are 1-b, 1, 1+b.

For product of zeroes:

Product of zeroes = 1(1-b)(1+b)

-s/p=1-𝑏2

=> -1/1=1-𝑏2

Or, 𝑏2 = 1 + 1 =2

So, b = √2

Thus, 1-√2, 1, 1+√2 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑧𝑒𝑟𝑜𝑒𝑠 𝑜𝑓 the equation 𝑥3 − 3𝑥2 + 𝑥 + 1.

More Topics Related to Class 10 Polynomials

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1 Comment

  1. Very nice app with all solutions of any question .

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