Relations and Functions Test

Result

Relations and Functions Test - 55

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Question 1
1 / -0

Let $$f\left( x \right)=\dfrac { \sin { x } \left( { 2 }^{ x }+{ 2 }^{ -x } \right) \sqrt { \tan ^{ -1 }{ \left( { x }^{ 2 }-x+1 \right) } } }{ { \left( 7{ x }^{ 2 }+3x+1 \right) }^{ 3 } } $$ then $$f'(0)$$ is equal to

A
$$0$$

B
$$\sqrt {\pi}$$

C
$$\pi$$

D
$$Does\ not\ exist$$

Solution
Question 2
1 / -0

Let $$X$$ be a family of sets and $$R$$ be a relation on $$X$$ defined by $$'A$$ is disjoint from $$B ^ { \prime }$$ . Then $$R$$ is _________.

A
Reflexive

B
Symmetric

C
Anti-symmetric

D
Transitive

Solution
Question 3
1 / -0

A function $$ f : R \rightarrow R$$ satisfies x cos y (f (2x+2y)-f(2x-2y))=cos x sin y (f(2x+2y)+f(2x-2y)). If $$f'(0)=\dfrac{1}{2}$$, then

A
f"(x)=f(x)=0

B
4f"(x)+f(x)=0

C
f"(x)+f(x)=0

D
4f"(x)-f(x)=0

Solution
Question 4
1 / -0

If $$f(x) = \left\{ {\begin{array}{lllllllllllllll}{\left[ x \right]\quad \quad \quad ,if\quad - 3 < x \le - 1}\\{\left| x \right|\quad \quad \quad ,if\quad \; - 1 < x < 1}\\{\left| {\left[ { - x} \right]} \right|\quad \quad ,if\quad \;\;1 \le x \le 3}\end{array}} \right.$$, then $$\left\{x: f(x)\ge 0\right\}$$=

A
$$(-1, 3)$$

B
$$[-1, 3)$$

C
$$(-1, 3]$$

D
$$[-1, 3]$$

Solution
Question 5
1 / -0

If $$f(x)=\left| \begin{matrix} sin{ x } & sin{ a } & sin{ b } \\ \cos { x } & \cos { a } & \cos { b } \\ \tan { x } & \tan { a } & \tan { b } \end{matrix} \right| $$, where $$0<a<b<\dfrac{\pi}{2}$$, then the equation $$f(x)=0$$ has in the interval $$(a,b)$$

A
atleast one root

B
atmost one root

C
no root

D
exactiy one root

Solution
Question 6
1 / -0

The sum of infinite terms of the series $$1+2 \left(1-\dfrac{1}{n}\right)+3\left(1-\dfrac{1}{n}\right)^{2}+4\left(1-\dfrac{1}{n}\right)^{3}+........$$ is given by

A
$$n^{2}+1$$

B
$$n(n+1)$$

C
$$n\left(1-\dfrac{1}{n}\right)^{2}$$

D
$$n^{2}$$

Solution
Question 7
1 / -0

The shortest distance between the line $$\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$$ and $$\dfrac{x-2}{3}=\dfrac{y-2}{4}=\dfrac{z-5}{5}$$ is

A
$$\dfrac{1}{\sqrt{6}}$$

B
$$\dfrac{1}{6}$$

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

D
None of these

Solution
Question 8
1 / -0

The complete set of values of $$x$$ for which the function $$f(x)=2\tan^{-1}x+\sin^{-1} \dfrac{2x}{1+x^{2}}$$ behaves like a constant function with positive output is equal to

A
$$x \in [-1,1]$$

B
$$[1,\infty)$$

C
$$(-\infty,1]$$

D
$$(-\infty, -1] \cup [1,\infty)$$

Solution
Question 9
1 / -0

If $$p$$ and $$q$$ are positive real numbers such that $$p^{2}+q^{2}=1$$, then the maximum value of $$(p+q)$$ is-

A
$$2$$

B
$$\dfrac {1}{2}$$

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

D
$$\sqrt {2}$$

Solution

$$\textbf{Step 1: Calculating}$$
$$\text{Given that }p^2+q^2=1$$
$$\text{Applying }A.M\geq G.M$$
$$\Rightarrow \dfrac{p^2+q^2}2\geq \sqrt{p^2.q^2}$$
$$\Rightarrow \dfrac12\geq|p.q|$$
$$\Rightarrow \dfrac12\geq p.q\geq-\dfrac12$$
$$\textbf{Step 2: Calculating p+q}$$
$$\Rightarrow \dfrac12\geq p.q\geq-\dfrac12$$
$$\Rightarrow 1\geq2p.q\geq-1$$
$$\Rightarrow 2\geq1+2p.q\geq0$$
$$\Rightarrow 2\geq p^2+q^2+2pq\geq 0$$
$$\Rightarrow 2\geq (p+q)^2\geq 0$$
$$\Rightarrow \max(p+q)=\sqrt2$$
$$\textbf{Hence , the maximum value of p+q is}\mathbf{\sqrt2.}$$ $$\textbf{Option D is correct.}$$
Question 10
1 / -0

Let $$y$$ be an imlict function of $$x$$ defined by $$x^{2x}-2x^{x}\cot y-1=0$$.Then $$y'(1)$$ equals

A
$$1$$

B
$$\log 2$$

C
$$-\log 2$$

D
$$-1$$

Solution