Use of Monotonicity for Proving Inequalities

Q. The number of solutions of the equation ${\log }_{(x+1)}(2x^2+7x+5)+{\log }_{(2x+5)}(x+1{)}^2-4=0,x>0,$ is

Q. Number of solutions of the equation $z^3+\left[3{\left(\overline{z}\right)}^2\right]/\left|z\right|=0$ where z is a complex number is

1. 2
2. 6
3. 5
4. 3

Q. Match the following

	Column I		Column II
(A)	If $lo{g}_{45}75=x$ and $lo{g}_{135}375=y$ then $xy+5(x–y)$ equals	(p)	3
(B)	The number of real solutions of the equation $X^{(lo{g}_{3}X{)}^2-\frac{9}{2}(lo{g}_{3}X-5)}=3^{3/2}$	(q)	1
(C)	The number of real solutions (x, y, z) of the system of equations log(2xy) = logx logy, logyz = logy logz log(2zx) = logz logx is	(r)	4
(D)	If $lo{g}_{12}27=x$ and $lo{g}_{6}16=ythen\frac{y(3+x)}{(3-x)}$ equals	(s)	2

Q.

x $\ge$ sin(x) for $\forall$ x $\ge$ 0

1. True

2. False

Q.

Debye-Huckel-Onsager equation is represented as ${\wedge }_c={\wedge }_o-b\sqrt{C}$. From the given options, identify the equation which fits the above equation?

1. $\frac{82.4}{(DT{)}_{\eta }^{1/2}}+\frac{8.20\times {10}^5}{(DT{)}^{1/2}}{\wedge }_o$
2. $\frac{82.4}{(DT{)}_{\eta }^{1/2}}+\frac{8.20\times {10}^5}{(DT{)}^{3/2}}{\wedge }_o$
3. $\frac{82.4}{(DT{)}_{\eta }^{1/2}}+\frac{8.20\times {10}^5}{(DT{)}^{1/2}}$
4. $\frac{8.24}{(DT{)}_{\eta }^{1/2}}+\frac{8.20\times {10}^5}{(DT{)}^{1/2}}{\wedge }_o$

Q.

Given $h\left(x\right)=f\left(x\right)-g\left(x\right).If{h}^{\prime }\left(x\right)\ge 0,f\left(x\right)\ge g\left(x\right)$.

1. True

2. False

Q. $\underset{x\to -1^{+}}{lim}\frac{\sqrt{\pi }-\sqrt{{\cos }^{-1}x}}{\sqrt{x+1}}$ is equal to

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{\sqrt{2\pi }}$
3. none of these
4. $\frac{1}{\sqrt{\pi }}$

Q. If $f\left(x\right)=x+\sin x;g\left(x\right)=e^{-x};u=\sqrt{c+1}-\sqrt{c};v=\sqrt{c}-\sqrt{c-1};(c>1),$ then

1. $f\circ g\left(u\right)<f\circ g\left(v\right)$
2. $g\circ f\left(u\right)<g\circ f\left(v\right)$
3. $g\circ f\left(u\right)>g\circ f\left(v\right)$
4. $f\circ g\left(u\right)>f\circ g\left(v\right)$

Q. $\int \frac{x}{\sqrt{(9+8x-x^2)}}dx.$

1. $\sqrt{9+8x-x^2}-{\sin }^{-1}\frac{x-4}{5}$
2. $-\sqrt{9+8x+x^2}+{\sin }^{-1}\frac{x-4}{5}$
3. $-\sqrt{9+8x-x^2}+{\sin }^{-1}\frac{x-4}{5}$
4. $-\sqrt{9-8x+x^2}+{\sin }^{-1}\frac{x-4}{5}$

Q. $\underset{x\to 0}{lim}\frac{{27}^x-9^x-3^x+1}{\sqrt{2}-\sqrt{1+\cos x}}=$

1. $0$
2. $8\sqrt{2}(\log 3{)}^2$
3. $8(\log 3{)}^2$
4. $1$

Q.

x $\ge$ sin(x) for $\forall$ x $\ge$ 0

1. True

2. False

Q. Which of the following statements is/are correct in stating the number of solutions of the equation
$x\log x+x=\lambda ,$ where $\lambda \in R.$

1. If $\lambda =0$, then the equation has exactly two solutions in $x$.
2. If $\lambda \in (0,\frac{1}{e^2})$, then the equation has exactly one solution in $x$.
3. If $\lambda \in (\frac{-1}{e^2},0)$, then equation has at least one solution in $x$.
4. the equation has at least one solution in $[1,\lambda ]$

Q. lf $y=\frac{1}{2}$ cosech-1 $(\frac{1}{2_X\sqrt{1+x^2}})$ , then $x=$

1. $\sinh y$
2. $\cosh y$
3. $\tanh y$
4. $\coth y$

Q.

Given $h\left(x\right)=f\left(x\right)-g\left(x\right).If{h}^{\prime }\left(x\right)\ge 0,f\left(x\right)\ge g\left(x\right)$.

1. True

2. False

Q.

The number of values of x where the function f(x) = 2 (cos 3x + cos $\sqrt{3x}$ attains its maximum, is

1. 1

2. 2

3. 0

4. Infinite

Q. Let $(x_0,y_0)$ be the solution of the following equations
$(2x{)}^{\ln 2}=(3y{)}^{\ln 3}$
$3^{\ln x}=2^{\ln y}$
Then $x_0$is

1. $\frac{1}{6}$
2. $6$
3. $\frac{1}{3}$
4. $\frac{1}{2}$

Q. Let $F\left(x\right)=af\left(x\right)+bg\left(x\right)+ch\left(x\right)$. To find out the maximum and minimum value of trigonometric expression containing $\sin x$ and $\cos x$ we have $\sin x\in [-1,1],\cos x\in [-1,1]\forall x\in R$ and $a\cos x+b\sin x+c$ always lie in $[c-\sqrt{a^2+b^2},c+\sqrt{a^2+b^2}]$

1. Option c
2. Option b
3. Option d
4. Option a

Q. Let $\lambda \in \{-2,-1,0,1,2\}$ then the chance that the equation $|x^2+3x|+\lambda -\lambda x=0$ possesses exactly three solutions, given that it possesses atleast $2$ solutions is ?

1. $\frac{1}{3}$
2. $\frac{1}{5}$
3. $\frac{2}{3}$
4. $0$

Q. If$\int \frac{1}{\sqrt{x^2+x+1}}dx=a{\sinh }^{-1}(bx+c)+d$, then descending order of a, b, c is

1. a, b, c
2. b, c, a
3. b, a, c
4. c, a, b

Q.

x $\ge$ sin(x) for $\forall$ x $\ge$ 0

1. True

2. False

Q. Which of the following statements is/are correct in stating the number of solutions of the equation
$x\log x+x=\lambda ,$ where $\lambda \in R.$

1. If $\lambda =0$, then the equation has exactly two solutions in $x$.
2. If $\lambda \in (0,\frac{1}{e^2})$, then the equation has exactly one solution in $x$.
3. If $\lambda \in (\frac{-1}{e^2},0)$, then equation has at least one solution in $x$.
4. the equation has at least one solution in $[1,\lambda ]$

Q.

x $\ge$ sin(x) for $\forall$ x $\ge$ 0

1. False

2. True

Q.

The number of values of x where the function f(x) = 2 (cos 3x + cos $\sqrt{3x}$ attains its maximum, is

1. 0

2. Infinite

3. 1

4. 2

Q.

The number of values of x where the function f(x) = 2 (cos 3x + cos $\sqrt{3x}$ attains its maximum, is

1. 2

2. 0

3. 1

4. Infinite

Q.

Given $h\left(x\right)=f\left(x\right)-g\left(x\right).If{h}^{\prime }\left(x\right)\ge 0,f\left(x\right)\ge g\left(x\right)$.

1. True

2. False

Q. If $f\left(x\right)=x+\sin x;g\left(x\right)=e^{-x};u=\sqrt{c+1}-\sqrt{c};v=\sqrt{c}-\sqrt{c-1};(c>1),$ then

1. $f\circ g\left(u\right)<f\circ g\left(v\right)$
2. $g\circ f\left(u\right)<g\circ f\left(v\right)$
3. $g\circ f\left(u\right)>g\circ f\left(v\right)$
4. $f\circ g\left(u\right)>f\circ g\left(v\right)$

Q.

The value of x for which $co{s}^{-1}\left(cos4\right)>3x^2-4x$ is

1. 
   

2. 
   

3. 
   

4. (-2, 2)