51). When a polynomial is divided by a linear polynomial, then the remainder is
| A). a linear polynomial |
| B). constant |
| C). zero |
| D). either constant or zero |
|
52). Consider the following statements :
I. Every rational function is a polynomial .
II. A rational function may be a polynomial
III. A rational function cannot be a polynomial
IV. A polynomial is always a rational function.
Which of these are correct?
| A). II and IV |
| B). I and IV |
| C). III and IV |
| D). I and II |
|
53). The value of \( \Large \left(x-y\right)^{3}+ \left(y-z\right)^{3}+ \left(z-x\right)^{3}-3 \left(x-y\right) \left(y-z\right) \left(z-x\right) \) is equal to
|
54). If \( \Large a+b+c=0,\ then\ a^{2}+ab+b^{2} \) is equal to
| A). \( \Large b^{2}-bc+c^{2} \) |
| B). \( \Large c^{2}-ab+b^{2} \) |
| C). \( \Large b^{2}+bc+c^{2} \) |
| D). zero |
|
55). If \( \Large pqr = 1 \), then the value of \( \Large \frac{1}{ \left(1+p+q^{-1}\right) }+\frac{1}{ \left(1+p+r^{-1}\right)}+\frac{1}{ \left(1+r+p^{-1}\right) } \) will be
| A). 1 |
| B). 0 |
| C). -1 |
| D). -2 |
|
56). If \( \Large 2^{x-1}=4^{x-3} \), then x is equal to
|
57). If x e R, then maximum value of \( \Large \left(\sqrt{3}-x+5\right) \left(\sqrt{3}+x-5\right) \) is
| A). \( \Large \sqrt{3} \) |
| B). 5 |
| C). 3 |
| D). \( \Large \sqrt{3+5} \) |
|
58). Factors of \( \Large 2x^{3}+5x^{2}-11x+4 \) are
| A). \( \Large \left(x-1\right) \left(2x-1\right) \left(x+4\right) \) |
| B). \( \Large \left(x+1\right) \left(2x-1\right) \left(x+4\right) \) |
| C). \( \Large \left(x-1\right) \left(2x+1\right) \left(x+4\right) \) |
| D). \( \Large \left(x-1\right) \left(2x-1\right) \left(x-4\right) \) |
|
59). If x e R, then the value of \( \Large -4x^{2}+12x-2 \) is maximum at x equal to
| A). \( \Large \frac{2}{3} \) |
| B). \( \Large \frac{3}{2} \) |
| C). 7 |
| D). 0 |
|
60). When an \( \Large x (x \ne -1) \) is replaced by \( \Large \left(x+1\right) \) is a polynomial of the form \( \Large x^{2}-ax+1 \), then the resulting polynomial is \( \Large x^{2}+2x+2 \). The value of a is
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