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question_answer1) If the solution of the equation, \[{{4}^{x}}-{{3}^{x-1/2}}={{3}^{x+1/2}}-{{2}^{2x-1}},x\in R\,is\,\frac{k}{2}\]then find k.
question_answer2) Let \[p,q\in \left\{ 1,2,3,4 \right\}\]. The number of equation of form \[p{{x}^{2}}+qx+1=0\] having real roots is k. Then find k.
question_answer3) If \[{{x}^{2}}+3x+5=0\] and \[a{{x}^{2}}+bx+c=0\] have common root/roots and \[a,\text{ }b,\text{ }c\text{ }\in \text{ }N\]. Find the minimum value of \[a+b+c\].
question_answer4) Find the product of real roots of equation, \[{{x}^{2}}+18x+30=2\sqrt{{{x}^{2}}+18x+45}\].
question_answer5) If the roots of the equation \[12{{x}^{2}}-mx+5=0\] are in the ratio \[2\text{ }:\text{ }3\] and the value of m is \[5\sqrt{k}\] then find k.
question_answer6) Find the least integral value of k for which, \[(k-2){{x}^{2}}+8x+k+4>0\] for all \[x\in R\].
question_answer7) If \[{{x}^{2}}+px+q=0\] is the quadratic equation whose roots are \[a-2\] and \[b-2\] where a and b are the roots of \[{{x}^{2}}-3x+1=0\], then find value of \[p+q\].
question_answer8) If sum of the non-real roots of \[\left( {{x}^{2}}+x-2 \right)\left( {{x}^{2}}+x-3 \right)=12\] is \[100-k\] then find k.
question_answer9) If the product of roots of equation, \[{{x}^{2}}-5kx+2{{e}^{4\,\,ln\,\,k}}-1\] is 31, then find sum of roots.
question_answer10) If \[x=2+{{2}^{2/3}}+{{2}^{1/3}}\], then find the value of \[{{x}^{3}}-6{{x}^{2}}+6x\].
question_answer11) The coefficient of x in the equation \[{{x}^{2}}+px+q=0\] was taken as 17 in place of 13 and its roots were found to be -2 and -15. Then find product of roots of the original equation.
question_answer12) If one root of the equations \[a{{x}^{2}}+bx+c=0\] and \[b{{x}^{2}}+cx+a=0\] \[\left( a,b,c\in R \right)\] is common, then find the value\[\left( \frac{{{a}^{3}}+{{b}^{3}}+{{c}^{3}}}{abc} \right)\].
question_answer13) If \[\alpha ,\beta \] and \[\gamma \] are the roots of the equation \[5{{x}^{3}}-qx-1=0\], \[\left( q\in R \right)\] then find the value of\[\frac{{{\alpha }^{2}}-3}{\beta \gamma }+\frac{{{\beta }^{2}}-3}{\gamma \alpha }+\frac{{{\gamma }^{2}}-3}{\alpha \beta }\].
question_answer14) If the difference of the roots of the equation \[{{x}^{2}}+ax+b=0\] is equal to the difference of the roots of the equation \[{{x}^{2}}+bx+a=0\], if \[a\ne b\] then find \[a+b+4\].
question_answer15) Find the value of a for which the sum of the squares of the roots of the equation \[{{x}^{2}}-\left( a-2 \right)x-a-1=0\] assume the least value.
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