Problem 1
If for the equation \(x^{3}-3 x^{2}+h x+3=0\), one root is the negative of another, then the value of \(k\) is (a) 3 (b) \(-3\) (c) 1 \((\) ch \()-1 .\)
Problem 2
If the roots of the equation \(x^{n}-1=0\) are \(1, a_{1}, \alpha_{2}, \ldots \ldots, \alpha_{m-1}\), then \(\left(1-\alpha_{1}\right)\left(1-\alpha_{2}\right) \ldots \ldots\left(1-\alpha_{n-1}\right)\) is equal to (a) 0 (b) 1 (c) \(n\) (d) \(n+1\).
Problem 4
\(x+2\) is a factor of (a) \(x^{4}+2\) (b) \(x^{4}-x^{2}+12\) (c) \(x^{4}-2 x^{3}-x+2\) (d) \(x^{4}+2 x^{3}-x-2\)
Problem 6
If one of the roots of the equation \(x^{3}-6 x^{2}+11 x-6=0\) is 2, then the other two roots are (a) 1 and 3 (b) 0 and 4 (c) \(-1\) and 5 (d) \(-2\) and 6 .
Problem 8
If 1 and 2 are two roots of the equation \(x^{4}-x^{3}-19 x^{2}+49 x-30=0\), then the remaining two roets are (a) \(-3\) and 5 (b) 3 and \(-5\) (c) \(-6\) and 5 (d) 6 and \(-5\).
Problem 9
If the roots of \(x^{3}-3 x^{2}+p x+1=0\), are in arithmetic progression, then the sum of squares of the langest and the smallest roots is (a) 3 (b) 5 (c) 6 (d) 10 .
Problem 10
A root of \(x^{3}-8 x^{2}+p x+q=0\) where \(p\) and \(q\) are real numbers is \(3+i \sqrt{3}\). The real mot is \(\begin{array}{lll}\text { (a) } 2 & \text { (b) } 6 & \text { (c) } 9\end{array}\) (d) 12 .
Problem 12
If a real root of \(f(x)=0\) lies in \(\mid a, b]\), then the sign of \(f(a) \cdot f(b)\) is \(\ldots .\)
Problem 14
If \(\alpha, \beta, Y\) are the roots of the equation \(x^{3}-p x+q=0\), then \(\Sigma 1 / \alpha=\ldots\)
Problem 20
If ene root in double of another in \(x^{3}-7 x^{2}+36=0\), then its roots are .....
