1. If \[\alpha\] and \[\beta\] \[\left(\alpha<\beta\right)\] are the roots of the equation
\[x^{2}+bx+c\] , where c < 0 < b, then

a) \[\alpha < 0<\beta<\mid\alpha\mid\]

b) \[\beta>\mid\alpha\mid\]

c) \[\mid\alpha\mid +\mid\beta\mid<2\]

d) none of these

Explanation:

2. If x is real, and y = \[\frac{x^{2}+2x+c}{x^{2}+4x+3}\]
takes all real values
then

a) 0 < c < 2

b) \[0\leq c\leq 1\]

c) – 1 < c < 1

d) \[-3\leq c\leq 1\]

Explanation:

3. The smallest integer x for which the inequality \[\frac{x-5 }{x^{2}+5x-14}>0\] is satisfied is given by

a) -6

b) -5

c) -4

d) -3

Explanation:

4. If \[\mid a\pm b \mid>c\] and \[a \neq0\] , then the quadratic equation
\[a^{2}x^{2}+\left(b^{2}+a^{2}-c^{2}\right)x+b^{2}=0\]

a) has two real roots

b) both positive roots

c) cannot have real roots

d) none of these

Explanation:

5. If abc < 0, then the equation
\[ax^{2}+2\left(b+c-a\right)x+bc=0\] , has

a) real roots

b) one positive and one negative root

c) both positive roots

d) both a and b

Explanation:

6. The equation \[1+\sqrt{1-\sqrt{x^{4}-x^{2}}}=x,x \epsilon R\]

has

a) only positive solutions

b) exactly one solution

c) at least two solutions

d) both a and b

Explanation:

7. The equation \[x+\sqrt{x}+\sqrt{x+2}+\sqrt{x^{2}+2x}=3\]

has

a) no solution

b) at least one solution

c) only positive solutions

d) both b and c

Explanation:

8.Let a, b, c be the sides of an obtuse angled triangle
with \[\angle C >\pi/2\] . The equation
\[a^{2}x^{2}+\left(b^{2}+a^{2}-c^{2}\right)x+b^{2}=0\]

has

a) two positive roots

b) one positive and one negative root

c) two real roots

d) two imaginary roots

Explanation: It has two imaginary roots

9. The equation
\[\sqrt{x-1}+\sqrt{x+3}+2\sqrt{\left(x-1\right)\left(x+3\right)}=4-2x\]

has

a) exactly one integral solution

b) all its solutions in [1, 2]

c) sum of all the solutions is 1

d) All of the above

Explanation:

10. The equation \[x^{3}+1=2 \left(2x-1\right)^{1/3}\] has

a) one rational solution

b) two irrational solutions

c) sum of roots equal to zero

d) All of the above

Explanation: