###### Computer-Organization

October 13, 2023###### Nielit Scientist-B IT 22-07-2017

October 14, 2023# Digital-Logic-Design

Question 22 |

Consider a quadratic equation *x ^{2} – 13x + 36 = 0* with coefficients in a base

*b*. The solutions of this equation in the same base b are

*x = 5*and

*x = 6*. Then

*b*=________.

8 | |

9 | |

10 | |

11 |

Question 22 Explanation:

x

Generally if a, b are roots.

(x – a)(x – b) = 0

x

Given that x=5, x=6 are roots of (1)

So, a + b = 13

ab=36 (with same base ‘b’)

i.e., (5)

Convert them into decimal value

5

6

13

11 = b+3

b = 8

Now check with ab = 36

5

Convert them into decimals

5

30 = b × 3 + 6

24 = b × 3

b = 8

∴ The required base = 8

^{2}– 13x + 36 = 0 ⇾(1)Generally if a, b are roots.

(x – a)(x – b) = 0

x

^{2}– (a + b)x + ab = 0Given that x=5, x=6 are roots of (1)

So, a + b = 13

ab=36 (with same base ‘b’)

i.e., (5)

_{b}+ (6)_{b}= (13)_{b}Convert them into decimal value

5

_{b}= 5_{10}6

_{10}= 6_{10}13

_{b}= b+311 = b+3

b = 8

Now check with ab = 36

5

_{b}× 6_{b}= 36_{b}Convert them into decimals

5

_{b}× 6_{b}= (b×3) + 6_{10}30 = b × 3 + 6

24 = b × 3

b = 8

∴ The required base = 8

Correct Answer: A

Question 22 Explanation:

x

Generally if a, b are roots.

(x – a)(x – b) = 0

x

Given that x=5, x=6 are roots of (1)

So, a + b = 13

ab=36 (with same base ‘b’)

i.e., (5)

Convert them into decimal value

5

6

13

11 = b+3

b = 8

Now check with ab = 36

5

Convert them into decimals

5

30 = b × 3 + 6

24 = b × 3

b = 8

∴ The required base = 8

^{2}– 13x + 36 = 0 ⇾(1)Generally if a, b are roots.

(x – a)(x – b) = 0

x

^{2}– (a + b)x + ab = 0Given that x=5, x=6 are roots of (1)

So, a + b = 13

ab=36 (with same base ‘b’)

i.e., (5)

_{b}+ (6)_{b}= (13)_{b}Convert them into decimal value

5

_{b}= 5_{10}6

_{10}= 6_{10}13

_{b}= b+311 = b+3

b = 8

Now check with ab = 36

5

_{b}× 6_{b}= 36_{b}Convert them into decimals

5

_{b}× 6_{b}= (b×3) + 6_{10}30 = b × 3 + 6

24 = b × 3

b = 8

∴ The required base = 8

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