University Maths Solution

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Maths Question
Question 1

$\begin{align}  & \text{Find the equation whose roots are numerically equal to those of }2{{x}^{2}}+2x-12=0 \\ & \text{but of opposite signs} \\\end{align}$ 

Question 2

$\begin{align}  & \text{If the roots of the equation }{{x}^{2}}+7x+8=0\text{ are whose roots are }\alpha \text{ and }\beta ,\text{ } \\ & \text{Find the equation whose roots are }{{\alpha }^{2}}\text{ and }{{\beta }^{2}} \\\end{align}$ 

Question 3

$\begin{align}  & \text{If the roots of the equation }2{{x}^{2}}+x+1=0\text{ are }\alpha \text{ and }\beta .\text{ Form the equation whose roots are } \\ & \tfrac{1}{{{\alpha }^{2}}}\text{ and }\tfrac{1}{{{\beta }^{2}}} \\\end{align}$

Question 4

$\begin{align}  & \text{If }\alpha \text{ and }\beta \text{ are the roots of the equation }7+12x-7{{x}^{2}}=0,\text{ } \\ & \text{Find the equation whose roots are }\tfrac{\alpha }{\beta }\text{ and }\tfrac{\beta }{\alpha } \\\end{align}$

Question 5

$\begin{align}
& \text{If the roots of the equation }{{x}^{2}}+2{{x}^{2}}-1=0\text{ are }\alpha \text{ and }\beta \text{, } \\
& \text{form the equation whose roots are (i) }{{\alpha }^{2}},{{\beta }^{2}}\text{ (ii) }{{\alpha }^{2}}+{{\beta }^{2}},4\alpha \beta \\
\end{align}$

Question 6

$\begin{align} & \text{If }\alpha \text{ and }\beta \text{ are roots of the equation }6{{x}^{2}}-10x+3=0,\text{ } \\ & \text{Find the value of } \\ & \text{(}i\text{) }{{(\alpha -\beta )}^{2}}\text{  } \\ & \text{(}ii\text{) }{{\alpha }^{2}}+{{\beta }^{2}}\text{  } \\ & \text{(}iii\text{) }({{\alpha }^{3}}+{{\beta }^{3}})\text{  } \\ & \text{(}iv\text{) }({{\alpha }^{3}}-{{\beta }^{3}})\text{  } \\ & \text{(}v\text{) }\tfrac{1}{\alpha }+\tfrac{1}{\beta } \\ & (vi)\text{ }\tfrac{1}{{{\alpha }^{2}}}+\tfrac{1}{{{B}^{2}}}\text{  } \\ & (vii)\text{ }\tfrac{\alpha }{\beta }+\tfrac{\beta }{\alpha } \\\end{align}$

Question 7

$\begin{align}
& \text{The coefficient of }x\text{ in quadratic equation }{{x}^{2}}+px+q\text{ was taken as 17 in place of 13 } \\
& \text{and its roots were found to }-2\text{ and }-15.\text{ Find the roots of the original equation} \\
\end{align}$

Question 8

$\begin{align}
& \text{The roots of the quadratic equation }a{{x}^{2}}+bx+c\text{ are }\alpha \text{ and }\beta ,\text{ those of }{{a}^{1}}{{x}^{2}}+{{b}^{1}}x+{{c}^{1}}=0\text{ are } \\
& \zeta \text{ and }\delta \text{. Find the equation whose roots }\left( \frac{\alpha }{\zeta }+\frac{\beta }{\beta } \right)\text{ and }\left( \frac{\alpha }{\delta }+\frac{\beta }{\zeta } \right) \\
\end{align}$

Question 9

$\text{Show that }2{{x}^{2}}+3x+7\text{ is always for real value  of }x.\text{ Find its minimum value}$ 

Question 10

$\begin{align}  & \text{Express the following in terms of }\alpha \beta \text{ and }\alpha +\beta  \\ & (i){{\alpha }^{2}}-{{\beta }^{2}} \\ & (ii){{\alpha }^{2}}+{{\beta }^{2}} \\ & (iii){{\alpha }^{3}}-{{\beta }^{3}} \\ & (iv){{\alpha }^{3}}+{{\beta }^{3}} \\ & (v){{\alpha }^{4}}-{{\beta }^{4}} \\ & (vi)\frac{1}{\alpha }+\frac{1}{\beta } \\ & (vii)\frac{1}{{{\alpha }^{2}}}+\frac{1}{{{\beta }^{2}}} \\\end{align}$