MCQ
If $A = \{a, b, c, d\},$ then a relation $R = \{(a, b), (b, a), (a, a)\}$ on $A$ is:
- ✓Symmetric and transitive only.
- BReflexive and transitive only.
- CSymmetric only.
- DTransitive only.
✓
Answer
Correct option: A.
Symmetric and transitive only.
Given that $A = \{a, b, c, d\}$
then $a$ relation $R = \{(a, b), (b, a), (a, a)\}$ on $A.$
$(a, b), (b, a) \in\text{R}$
$\Rightarrow R$ is symmetric.
Also for $(a, a) R$ is symmetric.
then $a$ relation $R = \{(a, b), (b, a), (a, a)\}$ on $A.$
$(a, b), (b, a) \in\text{R}$
$\Rightarrow R$ is symmetric.
Also for $(a, a) R$ is symmetric.
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Consider the function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ defined by $f(x)=\frac{x^2-a x+1}{x^2+a x+1}, 0 < a < 2 .$
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$(C)$ $f^{\prime}(1) f^{\prime}(-1)=(2-a)^2$
$(D)$ $f^{\prime}(1) f^{\prime}(-1)=-(2+a)^2$
$2.$ Which of the following is true?
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$(C)$ $f(x)$ is increasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
$(D)$ $f(x)$ is decreasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
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$(B)$ $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
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Give the answer question $1,2$ and $3.$
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