<\/span><\/h2>\nQuestion 1.<\/strong>
Define a polynomial with real coefficients.
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/3c.png\")
<\/p>\nQuestion 2.<\/strong>
Define the degree of a polynomial.
Solution:<\/strong>
The exponent of the highest degree term in a polynomial is known as its degree. A polynomial of degree O is called a constant polynomial.<\/p>\nQuestion 3.<\/strong>
Write the standard form of a linear polynomial with real coefficients.
Solution:<\/strong>
ax + b is the standard form of a linear polynomial with real coefficients and a \u2260 0<\/p>\nQuestion 4.<\/strong>
Write the standard form of a quadratic polynomial with real coefficients.
Solution:<\/strong>
ax2<\/sup> + bx + c is a standard form of a quadratic polynomial with real coefficients and a \u2260 0.<\/p>\nQuestion 5.<\/strong>
Write the standard form of a cubic polynomial with real coefficients.
Solution:<\/strong>
ax3<\/sup> + bx2<\/sup> + cx + d is a standard form of the cubic polynomial with real coefficients and a \u2260 0.<\/p>\nQuestion 6.<\/strong>
Define the value of a polynomial at a point.
Solution:<\/strong>
If f(x) is a polynomial and a is any real number then the real number obtained by replacing x by \u03b1 in f(x) is called the value of f(x) at x = \u03b1 and is denoted by f(\u03b1).<\/p>\nQuestion 7.<\/strong>
Define zero of a polynomial.
Solution:<\/strong>
A real number a is a zero of a polynomial f(x) if f(\u03b1) = 0.<\/p>\nQuestion 8.<\/strong>
The sum and product of the zeros of a quadratic polynomial are \u2013 12 and -3 respectively. What is the quadratic polynomial?
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/3b.png\")
<\/p>\nQuestion 9.<\/strong>
Write the family of quadratic polynomials having \u2013 14 and 1 as its zeros.
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/3a.png\")
<\/p>\nQuestion 10.<\/strong>
If the product of zeros of the quadratic polynomial f(x) = x2<\/sup> \u2013 4x + k is 3, find the value of k.
Solution:<\/strong>
We know that a quadratic polynomial x2<\/sup> \u2013 (sum of zeros) x + product of zeros
In the given polynomial f(x) = x2<\/sup> \u2013 4x + k is the product of zeros which is equal to 3
k = 3<\/p>\nQuestion 11.<\/strong>
If the sum of the zeros of a quadratic polynomial f(x) = kx2<\/sup> \u2013 3x + 5 is 1, write the value of k.
Solution:<\/strong>
f (x) = kx2<\/sup> \u2013 3x + 5
Here a = k, b = -3, c = 5
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/2e.png\")
<\/p>\nQuestion 12.<\/strong>
In the figure, the graph of a polynomial p (x) is given. Find the zeros of the polynomial.
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/2d.png\")
Solution:<\/strong>
The graph of the given polynomial meets the x-axis at -1 and -3
Zero will be -1 and -3
Zero of a polynomial is 3<\/p>\nQuestion 13.<\/strong>
The graph of a polynomial y = f(x) is given below. Find the number of real zeros of f (x).
![\"2c\"](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/2c.png\")
Solution:<\/strong>
The curve touches the x-axis at one point and also intersects at one point So the number of zeros will be 3, two equal, and one distinct<\/p>\nQuestion 14.<\/strong>
The graph of the polynomial f(x) = ax2<\/sup> + bx + c is as shown below (in the figure) write the signs of \u2018a\u2019 and b2<\/sup> \u2013 4ac.
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/2b.png\")
Solution:<\/strong>
The shape of the parabola is up to word a > 0
and b2<\/sup> \u2013 4ac >0 i.e., both are positive.<\/p>\nQuestion 15.<\/strong>
The graph of the polynomial f(x) = ax2<\/sup> + bx + c is as shown in the figure write the value of b2<\/sup> \u2013 4ac and the number of real zeros of f(x).
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/2a.png\")
Solution:<\/strong>
The curve parabola touches the x-axis at one point
It has two equal zeros
b2<\/sup> \u2013 4ac = 0<\/p>\nQuestion 16.<\/strong>
In Q. No. 14<\/strong>, write the sign of c
Solution:<\/strong>
The mouth of a parabola is upward and intersect the y-axis above the x-axis
c > 0<\/p>\nQuestion 17.<\/strong>
In Q. No. 15<\/strong>, write the sign of c.
Solution:<\/strong>
The mouth of the parabola is downward and intersects the y-axis below the x-axis
c < 0<\/p>\nQuestion 18.<\/strong>
The graph of a polynomial f (x) is as shown in the figure. Write the number of real zeros of f (x).
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1j.png\")
Solution:<\/strong>
The curves touch the x-axis at two distinct point
It has a pair of two equal zeros i.e., it has 4 real zeros<\/p>\nQuestion 19.<\/strong>
If x = 1, is a zero of the polynomial f(x) = x3<\/sup> \u2013 2x2<\/sup> + 4x + k, write the value of k.
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1i.png\")
<\/p>\nQuestion 20.<\/strong>
State division algorithm for polynomials.
Solution:<\/strong>
If f(x) is a polynomial and g (x) is a non zero polynomial, there exist two polynomials q (x) and r (x) such that
f(x) = g (x) x q (x) + r (x)
where r (x) = 0 or degree r (x) < degree g (x)
This is called division algorithm<\/p>\nQuestion 21.<\/strong>
Give an example of polynomials f(x), g (x), q (x) and r (x) satisfying f(x) = g (x) . q (x) + r (x), where degree r (x) = 0.
Solution:<\/strong>
f (x) = x3<\/sup> + x2<\/sup> + x + 4
g (x) = x + 1
q (x) = x2<\/sup> + 1
r (x) = 3
is an example of f (x) = g (x) x q (x) + r (x)
where degree of r (x) is zero.<\/p>\nQuestion 22.<\/strong>
Write a quadratic polynomial, a sum of whose zeros is 2\u221a3 and their product is 2.
Solution:<\/strong>
Sum of zeros = 2 \u221a3
and product of zeros = 2
Quadratic polynomial will be f (x) = x2 \u2013 (sum of zeros) x + product of zeros
= x2<\/sup> \u2013 2 \u221a3 x + 2<\/p>\nQuestion 23.<\/strong>
If the fourth-degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.
Solution:<\/strong>
Degree of the given polynomial = 4
and degree of divisor = 2
The degree of quotient will be 4 \u2013 2 = 2
and degree of the remainder will be less than 2 In other words equal to or less than one degree<\/p>\nQuestion 24.<\/strong>
If f(x) = x3<\/sup> + x2<\/sup> \u2013 ax + b is divisible by x2<\/sup> \u2013 x, write the value of a and b.
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1h.png\")
<\/p>\nQuestion 25.<\/strong>
If a \u2013 b, a and a + b are zeros of the polynomial f(x) = 2x3<\/sup> \u2013 6x2<\/sup> + 5x \u2013 7, write the value of a.
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1g.png\")
<\/p>\nQuestion 26.<\/strong>
Write the coefficients of the polynomial p (z) = z5<\/sup> \u2013 2z2<\/sup> + 4.
Solution:<\/strong>
p (z) = z5<\/sup> + oz4<\/sup> + oz3<\/sup> \u2013 2z2<\/sup> + oz + 4
Coefficient of z5<\/sup> = 1
Coefficient of z4<\/sup> = 0
Coefficient of z3<\/sup> = 0
Coefficient of z2<\/sup> = \u2013 2
Coefficient of z = 0
Constant = 4<\/p>\nQuestion 27.<\/strong>
Write the zeros of the polynomial x2<\/sup> \u2013 x \u2013 6. (C.B.S.E. 2008)<\/strong>
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1f.png\")
<\/p>\nQuestion 28.<\/strong>
If (x + a) is a factor of 2x2<\/sup> + 2ax + 5x + 10, find a. (C.B.S.E. 2008)<\/strong>
Solution:<\/strong>
x + a is a factor of
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1e.png\")
<\/p>\nQuestion 29.<\/strong>
For what value of k, -4 is a zero of the polynomial x2<\/sup> \u2013 x \u2013 (2k + 2)? (CBSE 2009)<\/strong>
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1d.png\")
<\/p>\nQuestion 30.<\/strong>
If 1 is a zero of the polynomial p (x) = ax2<\/sup> \u2013 3 (a \u2013 1) x \u2013 1, then find the value of a.
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1c.png\")
<\/p>\nQuestion 31.<\/strong>
If \u03b1, \u03b2 are the zeros of a polynomial such that \u03b1 + \u03b2 = -6 and \u03b1 \u03b2 = -4, then write the polynomial. [CBSE 2010]<\/strong>
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1b.png\")
<\/p>\nQuestion 32.<\/strong>
If \u03b1, \u03b2 are the zeros of the polynomial 2y2<\/sup> + 7y + 5, write the value of \u03b1 + \u03b2 + \u03b1\u03b2. [CBSE 2010]<\/strong>
Solution:<\/strong>
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/1a.png\")
<\/p>\nQuestion 33.<\/strong>
For what value of k, is 3 a zero of the polynomial 2x2<\/sup> + x + k ? [CBSE 2010]<\/strong>
Solution:<\/strong>
3 is a zero of f(x) = 2x2<\/sup> + x + k
It will satisfy the polynomial
f(x) = 0 \u21d2 f(3) = 0
Now 2x2<\/sup> + x + k = 0
=> 2 (3)2<\/sup> + 3 + k = 0
=> 18 + 3 + k = 0
=> 21 + k = 0
=> k = -21<\/p>\nQuestion 34.<\/strong>
For what value of k, is -3 a zero of the polynomial x2<\/sup> + 11x + k ? [CBSE 2010]<\/strong>
Solution:<\/strong>
-3 is a zero of polynomial f(x) = x2<\/sup> + 11x + k
It will satisfy the polynomial
f (x) = 0 => f(-3) = 0
Now x2<\/sup> + 11x + k = 0
=> (-3)2<\/sup>+ 11 x (-3) + k = 0
\u21d2 9 \u2013 33 + k = 0
\u21d2 -24 + k = 0
\u21d2 k = 24<\/p>\nQuestion 35.<\/strong>
For what value of k, is -2 a zero of the polynomial 3x2<\/sup> + 4x + 2k ? [CBSE 2010]<\/strong>
Solution:<\/strong>
-2 is a zero of the polynomial
f(x) = 3x2<\/sup> + 4x + 2k
f(-2) = 0
=> 3 (-2)2<\/sup> + 4 (-2) + 2k = 0
=> 12 \u2013 8 + 2k = 0
=> 4 + 2k = 0
=> 2k = -4
=> k = -2<\/p>\nQuestion 36.<\/strong>
If a quadratic polynomial f(x) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of f (x)?
Solution:<\/strong>
In a quadratic polynomial f(x) its degree is 2 and it can be factorized into two distinct linear factors.
f(x) has two distinct zeros<\/p>\nQuestion 37.<\/strong>
If a quadratic polynomial f(x) is a square of a linear polynomial, then its two zeros are coincident. (True \/ False)
Solution:<\/strong>
In a quadratic polynomial f(x), it is the square of a linear polynomial It has two zeros that are equal i.e. coincident
It is true<\/p>\nQuestion 38.<\/strong>
If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real zero. (True \/ False)
Solution:<\/strong>
A quadratic polynomial f(x) is not factorized into linear factors It has no real zeros It is true<\/p>\nQuestion 39.<\/strong>
If f(x) is a polynomial such that f(a) f(b) < 0, then what is the number of zeros lying between a and b?
Solution:<\/strong>
f(x) is a polynomial such that f(a) f(b) < 0
At least one of its zeros will be between a and b<\/p>\nQuestion 40.<\/strong>
If a graph of quadratic polynomial ax2<\/sup> + bx + c cuts the positive direction of the y-axis, then what is the sign of c?
Solution:<\/strong>
The graph of quadratic polynomial ax2<\/sup> + bx + c cuts positive direction of y-axis Then sign of constant term c will be also positive.<\/p>\nQuestion 41.<\/strong>
If the graph of quadratic polynomial ax2<\/sup> + bx + c cuts the negative direction of the y-axis, then what is the sign of c?
Solution:<\/strong>
The graph of quadratic polynomial ax2<\/sup> + bx + c cuts the negative side of the y-axis
Then the sign of constant term c will be negative.<\/p>\nWe have provided complete details of RD Sharma Class 10 Solutions Chapter 2 VSAQs. If you have any queries related to CBSE<\/strong><\/a> Class 10, feel free to ask us in the comment section below.<\/p>\n
![\"RD](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-2-VSAQS.jpg\")
<\/p>\n