Quadratic Equation [BCB Ex6.2]

From the equation whose roots are
1. 3,-2
  
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  Solution :1a
2. -5,4
  
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  Solution :1b
3. $\sqrt{3},-\sqrt{3}$
  
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  Solution :1c
4. $\frac{1}{2} (-1+\sqrt{5}),\frac{1}{2} (-1-\sqrt{5})$
  
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  Solution :1d
5. -3+5i,-i-5i
  
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  Solution :1e
6. a+ib,a-ib
  
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  Solution :1f
1. Find a quadratic equation whose roots are twice the roots of $4x^2+8x-5=0$
  
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  Solution :2a
2. Find a quadratic equation whose roots are reciprocals of the roots of $3x^2-5x-2=0$
  
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  Solution :2(b)
3. Find a quadratic equation whose roots are greater by h than the roots of $x^2-px+q=0$
  
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  Solution :2(c)
4. Find a quadratic equation whose roots are the squares of the roots of $3x^2-5x-2=0$
  
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  Solution :2(d)
Find a quadratic equation with rational coefficients one of whose roots is
1. 4+3i
  
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  Solution :3a
2. $\frac{1}{5+3i}$
  
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  Solution :3b
3. $2+\sqrt{3}$
  
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  Solution :3c
Find the value of k so that the equation
1. $2x^2+kx-15=0$ has one root 3
  
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  Solution :4a
2. $3x^2+kx-2=0$ has roots whose sum is equal to 6
  
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  Solution :4b
3. $2x^2+(4-k)x-17=0$ has roots equal but opposite in sign
  
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  Solution :4c
4. $3x^2+(5+k)x+8=0$ has roots numerically equal but opposite in sign
  
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  Solution :4d
5. $3x^2+7x+6-k=0$ has one root equal to zero
  
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  Solution :4e
6. $4x^2-17x+k=0$ has the reciprocal roots
  
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  Solution :4f
7. $4x^2+kx+5=0$ has roots whose difference is $\frac{1}{4}$
  
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  Solution :4g
Show that -1 is a root of the equation $a+b-2c$ x^2+(2a-b-c)x+(c+a-2b)=0\). Find the other root.

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Solution :5
Find the value of m for which the equation $(m+1)x^2+2(m+3)x+(2m+3)=0$ will have (a) reciprocal roots (b) one root zero.

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Solution :6
If the roots of the equation $x^2+ax+c=0$ differ by 1, prove that $a^2=4c+1$

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Solution :7
If are the roots of the equation , find the equation whose roots are
1. $\alpha ^2 \beta ^{-1}$ and $\beta ^2 \alpha ^{-1}$
  
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  Solution :8a
2. $\alpha + \frac{1}{\beta}$ and $\beta + \frac{1}{\alpha}$
  
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  Solution :8(b)
If are the roots of the equation , find the equation whose roots are
1. $\alpha ^2 \beta ^{-1}$ and $\beta ^2 \alpha ^{-1}$
  
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  Solution :9a
2. $\alpha ^3$ and $\beta ^3$
  
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  Solution :9b
3. $(\alpha-\beta)^2$ and $(\alpha+\beta)^2$
  
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  Solution :9c
4. the reciprocal of the roots of given equation
  
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  Solution :9d
1. If the roots of the equation $ax^2+bx+c=0$ be in the ratio of 3:4, prove that $12b^2=49ac$
  
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  Solution :10a
2. If one root of the equation $ax^2+bx+c=0$ be four times the other root, show that $4b^2=25ac$
  
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  Solution :10b
3. For what values of m, the equation $x^2-mx+m+1=0$ may have its root in the ratio 2:3
  
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  Solution :10c
1. If $\alpha, \beta$ are the roots of the equation $px^2+qx+q=0$ , prove that $\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{q}{p}}=0$
  
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  Solution :11a
2. If roots of the equation $lx^2+nx+n=0$ be in the ratio of p:q, prove that $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$
  
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  Solution :11b
If one root of the equation $ax^2+bx+c=0$ be square of the other root, prove that $b^3+a^2c+ac^2=3abc$

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Solution :12

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