Exercise 2.1 & 2.2

Exercise 2.2 - Class 10 Math Solutions

Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x² - 2x - 8

Zeroes by factorization: (x - 4)(x + 2)

Zeroes: x = 4, -2

Sum = 2, Product = -8 (Verified)

(ii) 4s² - 4s + 1 = (2s - 1)²

Zeroes: s = 1/2, 1/2

Sum = 1, Product = 1/4 (Verified)

(iii) 6x² - 3 - 7x = 6x² - 7x - 3

Use quadratic formula

x = [7 ± sqrt(49 + 72)]/12 = [7 ± sqrt(121)]/12 = [7 ± 11]/12

x = 3/2, -1/3

Sum = 7/6, Product = -1/2 (Verified)

(iv) 4u² + 8u = 4u(u + 2)

Zeroes: u = 0, -2

Sum = -2, Product = 0 (Verified)

(v) t² - 15

Zeroes: ±sqrt(15)

Sum = 0, Product = -15 (Verified)

(vi) 3x² - x - 4

Use quadratic formula: x = [1 ± sqrt(1 + 48)]/6 = [1 ± 7]/6

x = 4/3, -1

Sum = 1/3, Product = -4 (Verified)

Q2. Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.

(i) Zeroes: 1/4 and -1

Sum = -3/4, Product = -1/4

Polynomial: x² + (3/4)x - 1/4

(ii) Zeroes: √2 and 1/3

Sum = √2 + 1/3, Product = √2/3

Polynomial: x² - (√2 + 1/3)x + √2/3

(iii) Zeroes: 0 and √5

Sum = √5, Product = 0

Polynomial: x² - √5x

(iv) Zeroes: 1 and 1

Sum = 2, Product = 1

Polynomial: x² - 2x + 1

(v) Zeroes: -1/4 and 1/4

Sum = 0, Product = -1/16

Polynomial: x² + 1/16

(vi) Zeroes: 4 and 1

Sum = 5, Product = 4

Polynomial: x² - 5x + 4

अभ्यास 2.2 - कक्षा 10 गणित समाधान

प्रश्न 1: निम्नलिखित द्विघात बहुपदों के शून्य खोजिए और गुणांक एवं शून्यों के बीच संबंध सत्यापित कीजिए।

(i) x² - 2x - 8

गुणनखंड: (x - 4)(x + 2)

शून्य: x = 4, -2

योग = 2, गुणनफल = -8 (सत्यापित)

(ii) 4s² - 4s + 1 = (2s - 1)²

शून्य: s = 1/2, 1/2

योग = 1, गुणनफल = 1/4 (सत्यापित)

(iii) 6x² - 3 - 7x = 6x² - 7x - 3

समीकरण हल करें: x = [7 ± √(121)]/12 = [7 ± 11]/12

x = 3/2, -1/3

योग = 7/6, गुणनफल = -1/2 (सत्यापित)

(iv) 4u² + 8u = 4u(u + 2)

शून्य: u = 0, -2

योग = -2, गुणनफल = 0 (सत्यापित)

(v) t² - 15

शून्य: ±√15

योग = 0, गुणनफल = -15 (सत्यापित)

(vi) 3x² - x - 4

x = [1 ± √(49)]/6 = [1 ± 7]/6

x = 4/3, -1

योग = 1/3, गुणनफल = -4 (सत्यापित)

प्रश्न 2: निम्नलिखित शून्यों के योग और गुणनफल के अनुसार द्विघात बहुपद बनाइए।

(i) शून्य: 1/4 और -1

योग = -3/4, गुणनफल = -1/4

बहुपद: x² + (3/4)x - 1/4

(ii) शून्य: √2 और 1/3

योग = √2 + 1/3, गुणनफल = √2/3

बहुपद: x² - (√2 + 1/3)x + √2/3

(iii) शून्य: 0 और √5

योग = √5, गुणनफल = 0

बहुपद: x² - √5x

(iv) शून्य: 1 और 1

योग = 2, गुणनफल = 1

बहुपद: x² - 2x + 1

(v) शून्य: -1/4 और 1/4

योग = 0, गुणनफल = -1/16

बहुपद: x² + 1/16

(vi) शून्य: 4 और 1

योग = 5, गुणनफल = 4

बहुपद: x² - 5x + 4