1. Let $f\left(x\right)=x^{3}+3x^{2}+6x+2009$      and
$g\left(x\right)=\frac{1}{x-f\left(1\right)}+\frac{2}{x-f\left(2\right)}+\frac{3}{x-f\left(3\right)}$
The number of real solutions of g(x) = 0 is
a) 0
b) 1
c) 2
d) infinite

Explanation:

2. Let $\alpha,\beta \epsilon R ,\beta\neq 0$     and $\alpha+i\beta$  be a root of $x^{3}+qx+r=0$    where $q, r \epsilon R$ . A cubic equation with real coefficients one of whose roots is $\alpha$ , is
a) $x^{3}-qx+r=0$
b) $x^{3}+2qx+r=0$
c) $x^{3}+4qx-8r=0$
d) none of these

Explanation:

3. Suppose a, b are complex numbers and $x^{3}+ax+b=0$    has a pair of complex conjugate roots, then
a) a is real and b is imaginary
b) a is imaginary and b is real
c) both a and b are real
d) none of these

Explanation:

4. The number of solution of $10^{2/x}+25^{1/x}=\left(\frac{65}{8}\right)\left(50^{1/x}\right)$
is
a) 0
b) 1
c) 2
d) infinite

Explanation:

5. The equation $\left(\frac{10}{9}\right)^{x}= - 3x^{2}+2x-\frac{9}{11}$
has
a) no solution
b) exactly one solution
c) exactly two solutions
d) none of these

Explanation:

6. Let $a,b,c \epsilon R$   and a $\neq$ 0 be such that $\left(a+c\right)^{2}< b^{2}$   , then the quadratic equation $ax^{2}+bx+c=0$    has
a) imaginary roots
b) real roots
c) two real roots lying between (– 1, 1)
d) none of these

Explanation:

7. The number of real solution of $4^{x+1.5}+9^{x+0.5}=\left(10\right)\left(6^{x}\right)$
is
a) zero
b) one
c) two
d) infinite

Explanation:

8. If roots of the equation $x^{2}-2mx +m^{2}-1=0$      lie in the interval (– 2, 4), then
a) – 1 < m < 3
b) 1 < m < 5
c) 1 < m < 3
d) -1 < m < 5

Explanation:

9. The number of solutions of the equation $\sin \left(\frac{\pi x}{2\sqrt{3}}\right)=x^{2}-2\sqrt{3}x+4$
is
a) 1
b) 2
c) 0
d) infinite

10. If x is real, then the least value of the expression $\frac{x^{2}-6x+5}{x^{2}+2x+2}$