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KC Sinha: Exercise 7.1 - Mathematics Solution Class 10 Chapter 7 Quadratic Equations

[mathjax] Question 1 A Which of the following is a quadratic polynomial? (i) 2 - 13x 2 (ii) x+ 1x√ (iii) x+ 1x (iv) x 2 +3 x−−√+2 Sol : (i) On solving the equations, 2−13x2 Re-writing in the format...

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Question 1 A

Which of the following is a quadratic polynomial?

(i) 2 -13x²

(ii) x+1x√

(iii) x+1x

(iv) x² +3x−−√+2

Sol :
(i) On solving the equations,
2−13x2
Re-writing in the format of ax² + bx + c = 0
(−13)x2+x(0)+2=0
∵ a=−13, b = 0 & c = 2
So, following the ideal pattern of a quadratic polynomial 2−13x2 is a quadratic polynomial.

(ii) On solving the equations,
x+1x√
∵ it can’t be re-written in the format of ax² + bx + c = 0
So, following the ideal pattern of a quadratic polynomial, x+1x√ is not a quadratic polynomial.

(iii) On solving the equations,
x+1x
∵ it can’t be re-written in the format of ax² + bx + c = 0
So, following the ideal pattern of a quadratic polynomial, x+1xis not a quadratic polynomial.

(iv) On solving the equations,
x² + 3√x + 2
∵ it can’t be re-written in the format of ax² + bx + c = 0
So, following the ideal pattern of a quadratic polynomial, x² + 3√x + 2 is not a quadratic polynomial.

Question 1 B

Which of the following is a quadratic polynomial?
(i) 2 x²+1

(ii) x2+1x√

(iii) x2+1−−−−−√+1x√

(iv) 3x2+1−−−−−√+x

Sol :
(i) On solving the equations,
2x² + 1 = 0
Re-writing in the format of ax² + bx + c = 0

(2)x² + (0)x + 1 = 0
∵ a = 2 b = 0 & c = 1.
So, following the ideal pattern of a quadratic polynomial 2x² + 1 is a quadratic polynomial.

(ii) On solving the equations,
x2+1x√
∵ it can’t be re-written in the format of ax² + bx + c = 0
So, following the ideal pattern of a quadratic polynomial, x2+1x√ is not a quadratic polynomial.

(iii) On solving the equations,
x2+1−−−−−√+1x√
∵ it can’t be re-written in the format of ax² + bx + c = 0
So, following the ideal pattern of a quadratic polynomial, x2+1−−−−−√+1x√ is not a quadratic polynomial.

(iv) On solving the equations,
3√(x²+ 1) + x = 0
∵ it can’t be re-written in the format of ax² + bx + c = 0
So, following the ideal pattern of a quadratic polynomial, 3√(x²+ 1) + x = 0 is not a quadratic polynomial.

Question 2

Which of the following is a polynomial?

(i) 2x+13x2

(ii)3√2+x2

(iii) y² + y^-3

(iv) 3x−−√+7

Sol :
(i) On solving the equations,
2x+13x2=0
2x+3x−2=0
∵ After simplifying the equation, one of the term has a negative (-2) exponent.
So, following the ideal pattern of a polynomial, 2x+13x2=0 is not a polynomial.

(ii) On solving the equations,
3√2+x2=0
∵ After simplifying the equation, as it has a positive (2) exponent.
So, following the ideal pattern of a polynomial, 3√2+x2=0 is a polynomial.

(iii) On solving the equations,
y2+1y3=0
y2+y−3=0
∵ After simplifying the equation, as the one of the term has a negative (-3) exponent.
So, following the ideal pattern of a polynomial, y2+1y3=0 is not a polynomial.

(iv) On solving the equations,
3x−−√+7=0
3x12+7=0
∵ After simplifying the equation, as the expression has a degree of 12.
So, following the ideal pattern of a polynomial, 3x12+7=0 is not a polynomial.

Question 3

Fill in the blanks:

(i) x² + x + 3 is a ....... polynomial.
(ii) axⁿ + bx + c is a quadratic polynomial if n = ......
(iii) The value of the quadratic polynomial x² — 5x + 4 for x = — 1 is ......
(iv) The degree of the polynomial 2x² + 4x — x³is ..........
(v) A real number a will be called the zero of the quadratic polynomial ax² + bx + c if ........ is equal to zero.
Sol :
(i) Quadratic, because it is in the form of ax² + bx² + c = 0
(ii) n = 2, and also a≠0, as it will make the polynomial 0.
(iii) Putting the value of x = -1, in x² — 5x + 4
(-1)² - 5(-1) + 4
1 + 5 + 4
10
The value is 10.
(iv) ∵ The degree is the highest power of the term in the expression, so it is 3.
(v) ∵ The zeroes of a polynomial are α & β.
∴ to be zero, αx²+b α+c=0 & βx²+b β+c=0

Question 4 A

Find the zeroes of the quadratic polynomial 9 — x².

Sol :
-x² + 9 = 0
-x² = -9
x² = 9
∴ x = ±3
∴ The zeroes of the given polynomial are 3 & -3.

Question 4 B

Find the zeroes of the quadratic polynomial 4x²-1.

Sol :
4x² - 1 = 0
4x² = 1
x2=14
∴x=±12
∴ The zeroes of the given polynomial are 12 & −12.

Question 4 C

Which of the following are the zeroes of the quadratic polynomial 9 — 4 x² ?

(a) 4

(b) 9
(c) 32

(d) 23
Sol :
9 - 4x² = 0
4x² = 9
x2=94
∴x=±32
∴ The zeroes of the given polynomial are 32 & −32 and the option (c) is correct.

Question 4 D

Find the zeroes of the polynomial 4−12x2

(a) 2 (b) 22–√ (c) 0 (d) 4
Sol :
4−12x2=0
12x2=4
x² =8
∴ x = ±2√2
∴ The zeroes of the given polynomial 2√2 and the option (b) is correct.

Question 5 A

Is — 2 a zero of the quadratic polynomial 3x² + x — 10?

Sol :
Putting the value of -2, in the given polynomial,
3(-2)² + (-2) – 10
3(4) – 2 – 10
12 – 12
0
∵ the value comes out to be 0.
∴ -2 is one of the zeroes and, yes 3x² + x — 10 is a quadratic polynomial.

Question 5 B

Is — 1 a zero of the quadratic polynomial x² + 2x — 3?

Sol :
Putting the value of -1, in the given polynomial,
(-1)² + 2(-1) – 3
1 – 2 – 3
3 – 3
0
∵ the value comes out to be 0.
∴ -1 is one of the zeroes of the given polynomial.

Question 6 A

Which of the following is a polynomial? Find its degree and the zeroes.
2−12x2

Sol :
∵ The highest power is 2, so the degree is also 2.
Equating the expression with 0,
2−12x2=0
2=12x2
x² = 4
∴ x = ±2
Yes, the above expression is a polynomial, as it has no negative powers in any of the terms and its zeroes are 2 & -2.

Question 6 B

Which of the following is a polynomial? Find its degree and the zeroes.
x+1x√

Sol :
∵ the power of a term is in negative (−12).
∴ The above given expression is not a polynomial.

Question 7 A

Which of the following is a polynomial '? Find its zeroes.

(i) x2+x−−√+2

(ii) x+1x

(iii) 4−14x2
Sol :
In the above expressions, only the thirdone has the positive power unlike others.
∴ It is the only polynomial.
Equating the expression with 0,
4−14x2=0
4=14x2
x² = 16
∴ x = ±4
The zeroes of the polynomial 4−14x2 are 4 & -4.

Question 7 B

Which of the following expressions is a polynomial? Find the degree and zeroes of the polynomial.

(i) x2+2x

(ii) x² +2x

Sol :
In the above expressions, only the secondone has a positive power, unlike others.
∴ It is the only polynomial.
Equating the expression with 0,
x² + 2x = 0
x ( x+2) = 0
∴ x = 0 or x + 2 = 0
X = 0 Or x = -2
The zeroes of the polynomial x² +2x are 2 & -2, having a degree of 2, being the highest power of the terms in the same expression.

Question 7 C

Which among the expressions 1-116z² and z² + z + 1 is a polynomial in z? Find its zeroes and degree.

Sol :
∵ The highest power is 2, so the degree is also 2, in both the expressions.
Equating the expression with 0,
1−116z2=0
1=116z2
x² =16
∴ x = ±4
Yes, the above expression (1−116z2 ⁾ is a polynomial, and its zeroes are 4 & -4.
Equating the expression with 0,
z² + z + 1 =0
Using Sreedharacharya formula, −b±b2−4ac√2a
ax²+bx+c = 0
x=(−(1))±(1)2−4(1)(1)√2(1)
x=−1±1−4√2
x=−1±−3√2
∵ it does not have real values.
∴ The zeroes of z² + z + 1 are complex numbers, though it is a polynomial having the degree 2.

Question 8

Find the zeroes of the quadratic polynomial x² — 6x + 8.

Sol :
Equating the expression with 0,
x² — 6x + 8 = 0

On factorising it further,
x² - 4x - 2x + 8 = 0
x(x - 4) – 2(x - 4) = 0
(x - 4) (x - 2) = 0

∴ x = 4 or x = 2

∴ The zeroes of x² — 6x + 8 are 4 & 2.

Question 9 A

Find the zeroes of the quadratic polynomial:
2x² + x — 1

Sol :
Equating the expression with 0,
2x² +x -1 = 0

On factorising it further,
2x² -x + 2x - 1 = 0
x(2x - 1) +1(2x - 1) = 0
(2x - 1) (x + 1) = 0
∴ x = 12 or x = -1
∴ The zeroes of 2x² + x — 1 are 12 and -1.