\displaystyle \textbf{Question 1: }\text{If }(k+1)x^2+\frac{3}{2}x=7\text{ is a quadratic equation, then }k\text{ cannot be equal to }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle k+1\ne0\Rightarrow k\ne-1
\displaystyle \therefore k\text{ cannot be equal to }-1.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Every quadratic equation has exactly }\underline{\hspace{1cm}}\text{ roots.}
\displaystyle \text{Answer:}
\displaystyle \therefore \text{Every quadratic equation has exactly }2\text{ roots.}
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{The values of }k\text{ for which the quadratic equation } \\ 2x^2-kx+k=0\text{ has equal roots are }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle D=k^2-4(2)(k)=0\Rightarrow k(k-8)=0
\displaystyle \therefore k=0\text{ or }k=8.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{If }\frac{1}{2}\text{ is a root of the equation }x^2+kx-\frac{5}{4}=0,\text{ then the value of }k\text{ is }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle \frac{1}{4}+\frac{k}{2}-\frac{5}{4}=0\Rightarrow \frac{k}{2}=1
\displaystyle \therefore k=2.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{If one root of the quadratic equation } \\ 3x^2-10x+k=0\text{ is reciprocal of the other, then }k=\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle \alpha\beta=1\Rightarrow \frac{k}{3}=1
\displaystyle \therefore k=3.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{The values of }k\text{ for which the quadratic equation } \\ x^2-4kx+k=0\text{ has equal roots are }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle D=(-4k)^2-4(1)(k)=0\Rightarrow 4k(4k-1)=0
\displaystyle \therefore k=0\text{ or }k=\frac{1}{4}.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{If the arithmetic mean of the roots of the equation } \\ x^2-6x+8=0\text{ is }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle \text{Sum of roots}=-\frac{b}{a}=6
\displaystyle \therefore \text{Arithmetic mean}=\frac{6}{2}=3.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{If the arithmetic mean of the roots of the equation } \\ x(x-2)+4ax=5\text{ is }3,\text{ then }a=\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle x^2+(4a-2)x-5=0
\displaystyle \text{Sum of roots}=-(4a-2)=2-4a
\displaystyle \therefore \frac{2-4a}{2}=3\Rightarrow 2-4a=6\Rightarrow a=-1.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{If the equation }mx^2+2x+m=0\text{ is satisfied by only one} \\ \text{real value of }x,\text{ then the values of }m\text{ are }\underline{\hspace{1cm}}\text{ and }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle \text{For equal real roots, }D=0.
\displaystyle 2^2-4(m)(m)=0\Rightarrow4-4m^2=0\Rightarrow m^2=1
\displaystyle \therefore m=1\text{ and }m=-1.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{The quadratic equation with rational coefficients having } \\ \frac{\sqrt{3}}{2}\text{ as a root is }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle \text{The other root is }-\frac{\sqrt{3}}{2}.
\displaystyle \therefore \left(x-\frac{\sqrt{3}}{2}\right)\left(x+\frac{\sqrt{3}}{2}\right)=0
\displaystyle \therefore 4x^2-3=0.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{The total number of values of }x\text{ satisfying } \\ x^2-3|x|+2=0\text{ is }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle \text{Let }y=|x|.\text{ Then }y^2-3y+2=0\Rightarrow(y-1)(y-2)=0.
\displaystyle \Rightarrow |x|=1\text{ or }|x|=2\Rightarrow x=\pm1,\pm2
\displaystyle \therefore \text{Total number of values of }x=4.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{The number of real roots of the equation } \\ x^2+3|x|+2=0\text{ is }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle x^2+3|x|+2>0\text{ for every real }x.
\displaystyle \therefore \text{Number of real roots}=0.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{If the coefficient of }x^2\text{ and the constant term of a quadratic}
\displaystyle \text{equation have opposite signs, then the roots of the quadratic equation are }\underline{\hspace{1cm}}\text{ and }\underline{\hspace{1cm}}.
\displaystyle \text{Answer:}
\displaystyle \alpha\beta=\frac{c}{a}<0
\displaystyle \therefore \text{The roots are real and of opposite signs.}
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{If the coefficient of }x^2\text{ and the constant term of a quadratic equation have the }
\displaystyle \text{same sign and if the coefficient of }x\text{ term is zero, then the quadratic equation has }\underline{\hspace{1cm}}\text{ roots.}
\displaystyle \text{Answer:}
\displaystyle ax^2+c=0\Rightarrow x^2=-\frac{c}{a}<0
\displaystyle \therefore \text{The quadratic equation has no real roots.}
\displaystyle \\

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