1. The real values of a for which \[\frac{x^{2}-ax-2}{x^{2}-3x+4}>-1\]

for each \[x \epsilon R\] , is

a) (– 1, 2)

b) (0, 7)

c) (-7, 1)

d) (2, 7)

Explanation:

2. If \[a,b,c \epsilon R\] are distinct, then the condition(s) on
a, b, c for which the equation

\[\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}+\frac{1}{x-b-c+a}=0\]

has real roots is

a) \[a+b+c\neq 0,a\neq0\]

b) \[a-b-c\neq 0,a\neq0\]

c) 2a = b + c

d) none of these

Explanation:

3. If \[f\left(x\right)=ax^{2}+bx+c,f\left(-1\right)< 1,f\left(1\right)>-1\]

and \[f\left(-3\right)< -4\]
then

a) a = 0

b) a < 0

c) a > 0

d) sign of a cannot be determined

Explanation:

4. The set of all values of k for which the equation
\[x^{2}+2 \left(k-1\right)x+\left(k-5\right)=0\] has
at least one non-negative root is

a) \[\left[1,\infty\right)\]

b) [-1,1]

c) \[(-\infty,-5\] ]

d) \[\left[5,\infty\right)\]

Explanation:

5.If \[3\pi/4<\alpha<\pi\] , then the set of values of \[\alpha\] for which

\[\left(\sin\alpha\right)x^{2}+\left(2\cos\alpha\right)x+\frac{1}{2}\left(\sin\alpha+\cos\alpha\right)\]

is square of a linear polynomial is

a) \[\left\{\frac{5\pi}{6},\frac{11\pi}{12}\right\}\]

b) \[\left\{\frac{11\pi}{12}\right\}\]

c) \[\left\{\frac{5\pi}{6}\right\}\]

d) \[\phi\]

Explanation:

6. If \[0\leq\phi\leq3\pi\] , then the set of all values of \[\phi\] for which
sum of the squares of the roots of the equation

\[x^{2}+\left(\sin \phi -1\right)x-\frac{1}{2}\cos^{2}\phi=0\]

is greatest is

a) \[\left\{\pi,\frac{3\pi}{2},2\pi,3\pi\right\}\]

b) \[\left\{\pi,3\pi\right\}\]

c) \[\left\{2\pi,\frac{5\pi}{4}\right\}\]

d) \[\left\{\frac{3\pi}{2}\right\}\]

Explanation:

7. The set of values of b for which
\[2\log_{1/36}\left(bx+28\right)=-\log_{6}\left(12-4x-x^{2}\right)\]

has exactly one solution is

a) \[\left(-\infty,-14\right)\cup\] [14/3, \[\infty)\]

b) [14/9, \[\infty)\]

c) \[(-\infty\] ,-14] \[\cup\left\{4\right\}\cup\] [14/3,\[\infty\] ]

d) \[\left\{4\right\}\]

Explanation:

8. If \[2x^{2}+3x+4=0\] and \[ax^{2}+bx+c=0\] , where
\[a,b,c \epsilon N\] have a common root, then the least value
of a + b + c is

a) 10

b) 9

c) 8

d) 6

Explanation:

9. Let p, q, r, s be real numbers such that pr = 2(q +
s). Consider quadratic equations \[x^{2}+px+q=0\] and
\[x^{2}+rx+s=0\] . Then

a) none of these has real roots

b) both have real roots

c) at least one has real roots

d) at most one has real roots

Explanation:

10. Let \[a< 0,a\neq -2\] . The equation \[x^{2}+a\mid x\mid+1=0\] has

a) no real roots

b) at least two real roots

c) exactly four real roots or no real roots

d) none of these

Explanation: