Relations and Functions Test

Result

Relations and Functions Test - 53

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Weekly Quiz Competition

Question 1
1 / -0

Total number of equivalence relations defined in the set $$S=\left\{a, b, c\right\}$$ is?

A
$$5$$

B
$$3!$$

C
$$2^3$$

D
$$3^3$$

Solution
Question 2
1 / -0

A function $$y=f(x)$$ satisfies $$f"(x)=-\dfrac{1}{x^2}-\pi^2\sin(\pi x);f'(2)=\pi+\dfrac{1}{2}$$ and $$f(1)=0$$, the value of $$f\left(\dfrac{1}{2}\right)$$ is

A
$$ln2$$

B
$$1$$

C
$$\dfrac{\pi}{2}-ln2$$

D
$$1-ln2$$

Question 3

1 / -0

Which of the following is always true

$$(p\Longrightarrow q)\equiv \sim q\Longrightarrow \sim p$$

$$\sim (p\Longrightarrow q)\equiv p\ \wedge \sim q$$

$$\sim (p\ \vee q)\equiv \vee\ p\ \vee \sim q$$

$$\sim (p\ \vee q)\equiv \sim\ p\ \wedge \sim q$$

Solution

Truth table:

$$p$$	$$q$$	$$p\Rightarrow q$$	$$\sim q$$	$$\sim p$$	$$\sim q\Rightarrow \sim p$$
$$T$$	$$T$$	$$T$$	$$F$$	$$F$$	$$T$$
$$T$$	$$F$$	$$F$$	$$T$$	$$F$$	$$F$$
$$F$$	$$T$$	$$T$$	$$F$$	$$T$$	$$T$$
$$F$$	$$F$$	$$T$$	$$T$$	$$T$$	$$T$$

So, $$(p\Rightarrow q)\cong \sim q\Rightarrow \sim p$$

Question 4
1 / -0

$$f:\left( { - \infty ,\infty } \right) \to \left( {0,1} \right]$$ defined by $$f\left( x \right) = \frac{1}{{{x^2} + 1}}$$ is

A
one -one but not onto

B
onto but not one-one

C
bijective

D
neither one-one nor onto

Solution
Question 5
1 / -0

The values of $$a$$ and $$b$$ such that $$f$$ and $$f'$$ are continuous, are

A
$$a = 1, b = \dfrac {1}{\sqrt {2}} + \dfrac {\pi}{6}$$

B
$$a = \dfrac {1}{\sqrt {2}}, b = \dfrac {1}{\sqrt {2}}$$

C
$$a = 1, b = \dfrac {\sqrt {3}}{2} - \dfrac {\pi}{6}$$

D
$$a = \dfrac {\sqrt {3}}{2}, b = \dfrac {\sqrt {3}}{2} + \dfrac {\pi}{6}$$

Solution
Question 6
1 / -0

Let $$X=\left\{ x \epsilon {R};\cos(\sin\ x)=\sin(\cos\ x)\right\}$$. The number of elements in $$X$$ is

A
$$0$$

B
$$2$$

C
$$4$$

D
$$ not\ finite$$

Solution
Question 7
1 / -0

Let f(x) be a polynomial function such that $$f(x)+f(\dfrac{1}{x})=f(x).f(\dfrac{1}{x})$$ and $$ f(3)=82$$, then the value of $$f(5)$$, is

A
126

B
626

C
3126

D
None

Solution
Question 8
1 / -0

If $$f^2(x)\cdot f\left(\dfrac{1-x}{1+x}\right)=x^{3},\ x\neq -1$$ and $$f(x)\neq 0$$ then the value of $$\left\{ f\left( -2 \right) \right\}$$ (the fractional part of $$f(x)$$ is equal to)

A
$$\dfrac{2}{3}$$

B
$$\dfrac{1}{3}$$

C
$$\dfrac{1}{2}$$

D
$$0$$

Solution
Question 9
1 / -0

If $$\phi\left(x\right)=\phi\left(1\right)=2$$, then $$\phi\left(3\right)$$ equals

A
$$e^{2}$$

B
$$5e^{2}$$

C
$$3e^{2}$$

D
$$2e^{2}$$

Solution
Question 10
1 / -0

Let $$g$$ be a function that is differentiable throughout an open interval containing the origin.
Suppose $$g$$ has the following properties
$$g\left( x+y \right) =\left\{ g\left( x \right) +g\left( y \right) \right\} \div \left\{ 1-g\left( x \right) g\left( y \right) \right\}$$ for all real numbers $$x,\ y$$ and $$x+y$$ in the domain of $$g$$.
$$\displaystyle \lim _{ h\rightarrow 0 }{ \dfrac { g\left( h \right) }{ h } =1 }$$.
Then $$g(0)$$ has value equal to

A
$$0$$

B
$$1$$

C
$$2$$

D
$$none\ of\ these$$

Solution