(Q11 to Q20) (1 – 4 ) The Relation Between the Roots of the Second Degree Equation and the Coefficients of its Terms.

 

1sec ≫ First term  ≫ Algebra ≫ Unit 1 ≫(1 – 4 ) The Relation Between the Roots of the Second Degree Equation and the Coefficients of its Terms.

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(Q11 to Q20)

(11) If \(x=-1\) is one of the two roots of the equation: \(x^{2}-2x+\mathrm{a}=0\)

The correct answer is:

\[ 3, -3 \]

(12) If \((1+\mathrm{i})\) is one of the two roots of the equation: \(x^{2}-2x+\mathrm{a}=0\)

The correct answer is:

\[ a = 2, \text{ other root is } (1-\mathrm{i}) \]

(13) Find the values of \(a\), \(b\) in each of the following equation, if: 2, 5 are the two roots of the equation: \(x^{2} + ax + b = 0\)

The correct answer is:

\(a = -7\), \(b = 10\)

(14) Find the values of \(a\), \(b\) in each of the following equation, if: \(\sqrt{3} i,-\sqrt{3} i\) are the two roots of the equation: \(x^{2} + ax + b = 0\)

The correct answer is:

\(a = 0\), \(b = 3\)

(15) Find the value of \(k\) in each of the following which makes: One of the roots of the equation: \(x^{2} + (k-1)x - 3 = 0\) is the additive inverse of the other roots.

The correct answer is:

\(k = 1\)

(16) Find the value of \(k\) in each of the following which makes: \(4kx^{2} + 7x + k^{2} + 4 = 0\) is the multiplicative inverse of the other.

The correct answer is:

\(k = 2\)

(17) The quadratic equation whose roots are \(\frac{3}{2} \mathrm{i}\) and \(\frac{3}{2} \mathrm{i}^{3}\) is

1. \(4x^{2}-9=0\)
2. \(4x^{2}+9=0\)
3. \(4x^{2}-4=0\)
4. \(9x^{2}+4=0\)

The correct answer is: b

(18) The quadratic equation whose roots are \( -2,3 \) is

1. \( (x+2)(x+3)=0 \)
2. \( x^{2}-4x+6=0 \)
3. \( x^{2}-x=6 \)
4. \( 4x^{2}-2x+3=0 \)

The correct answer is: c

(19) Form the quadratic equation whose two roots are:

• (a) \(\frac{2}{3}, \frac{3}{2}\)
• (b) \(5 \sqrt{3}, -2 \sqrt{3}\)
• (c) \(-5 \mathrm{i}, 5 \mathrm{i}\)
• (d) \(1 - 3 \mathrm{i}, 1 + 3 \mathrm{i}\)
• (e) \(\frac{3}{\mathrm{i}}, \frac{3 + 3i}{1 - \mathrm{i}}\)

The correct answer is:

(a) \(x^2 - \frac{5}{2}x + 1 = 0\)

(b) \(x^2 - 3\sqrt{3}x - 30 = 0\)

(c) \(x^2 + 25 = 0\)

(d) \(x^2 - 2x + 10 = 0\)

(e) \(x^2 + 9 = 0\)

(20) Find the quadratic equation in which each of the two roots exceeds one of the two roots of the equation: \(x^2 - 7x - 9 = 0\)

The correct answer is:

\(x^2 - 9x - 1 = 0\)