Quadratic Equation [BCB Ex6.1]

  1. Determine the nature of roots of each of the following equations
    1. x^2-6x+5=0

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    2. x^2-4x-3=0

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    3. x^2-6x+9=0

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    4. 4x^2-4x+1=0

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    5. 2x^2-9x+35=0

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    6. 4x^2+8x-5=0

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  2. For what values of p will the equation 5x^2-px+45=0?

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  3. if the equation x^2+2(k+2)x+9k=0 has equal roots, find k.

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  4. For what value of a will the equation x^2-(3a-1)x+2(a^2-1)=0 have equal roots?

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  5. If the roots of the equation (a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0 are equal, then show that \frac{a}{b}=\frac{c}{d}

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  6. Show that the roots of the equation (a^2-bc)x^2+2(b^2-ca)x+(c^2-ab)=0 will be equal, if either b=0 or a^3+b^3+c^3-3abc=0

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  7. If a,b,c are rational and a+b+c=0, show that the roots (b+c-a)x^2+(c+a-b)x+(a+b-c)=0 are rational.

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  8. Prove that the roots of the equation (x-a)(x-b)=k^2 are real for all values of k.

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  9. Show that the roots of the equation x^2-4abx+(a^2+2b^2)^2=0 are imaginary.

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  10. If the roots of the quadratic equation qx^2+2px+2q=0 are real and unequal, prove that the roots of the equation (p+q)x^2+2qx+(p-q)=0 are imaginary

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Quadratic Equation [BCB Ex6.2]

  1. From the equation whose roots are
    1. 3,-2

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    2. -5,4

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    3. \sqrt{3},-\sqrt{3}

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    4. \frac{1}{2} (-1+\sqrt{5}),\frac{1}{2} (-1-\sqrt{5})

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    5. -3+5i,-i-5i

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    6. a+ib,a-ib

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    1. Find a quadratic equation whose roots are twice the roots of 4x^2+8x-5=0

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    2. Find a quadratic equation whose roots are reciprocals of the roots of 3x^2-5x-2=0

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    3. Find a quadratic equation whose roots are greater by h than the roots of x^2-px+q=0

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    4. Find a quadratic equation whose roots are the squares of the roots of 3x^2-5x-2=0

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  2. Find a quadratic equation with rational coefficients one of whose roots is
    1. 4+3i

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    2. \frac{1}{5+3i}

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    3. 2+\sqrt{3}

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  3. Find the value of k so that the equation
    1. 2x^2+kx-15=0 has one root 3

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    2. 3x^2+kx-2=0 has roots whose sum is equal to 6

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    3. 2x^2+(4-k)x-17=0 has roots equal but opposite in sign

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    4. 3x^2+(5+k)x+8=0 has roots numerically equal but opposite in sign

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    5. 3x^2+7x+6-k=0 has one root equal to zero

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    6. 4x^2-17x+k=0 has the reciprocal roots

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    7. 4x^2+kx+5=0 has roots whose difference is \frac{1}{4}

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  4. Show that -1 is a root of the equation a+b-2cx^2+(2a-b-c)x+(c+a-2b)=0\). Find the other root.

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  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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  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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  7. If \alpha, \beta are the roots of the equation x^2-x-6=0, find the equation whose roots are
    1. \alpha ^2 \beta ^{-1}and \beta ^2 \alpha ^{-1}

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    2. \alpha + \frac{1}{\beta} and \beta + \frac{1}{\alpha}

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  8. If \alpha, \beta are the roots of the equation ax^2+bx+c=0, find the equation whose roots are
    1. \alpha ^2 \beta ^{-1}and \beta ^2 \alpha ^{-1}

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    2. \alpha ^3 and \beta ^3

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    3. (\alpha-\beta)^2 and (\alpha+\beta)^2

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    4. the reciprocal of the roots of given equation

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    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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    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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    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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    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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    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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  9. 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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Quadratic Equation[BCB Ex6.3]

  1. Show that each pair of following equations has a common root
    1. x^2-8x+15=0 and 2x^2-x-15 =0

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    2. 3x^2-8x+4=0 and 4x^2-7x-2 =0

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  2. Find the value of p so that each pair of the equations may have one root common
    1. 4x^2+px-12=0 and 4x^2+3px-4 =0

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    2. 2x^2+px-1=0 and 3x^2-2x-5 =0

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  3. If the quadratic equations x^2+px+q=0 and x^2+p'x+q'=0 have common roots show that it must be either \frac{pq'-p'q}{q-q'} or \frac{q-q'}{p'-p}

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  4. If the quadratic equations x^2+px+q=0 and x^2+qx+p=0 have common roots show that it must be either p=q or p+q+1=0

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  5. If the quadratic equations ax^2+bx+c=0 and bx^2+cx+a=0 have common roots show that it must be either a=b=c or a+b+c=0

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  6. Prove that if the equations x^2+bx+ca=0 and x^2+cx+ab=0 have a common root, their other root will satisfy x^2+ax+bc=0

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