Question

$\begin{array}{cccc}& \text{Column - I}& & \text{Column - II}\\ \left(a\right)& \text{The number of roots of}{x}^2-2=\left[\sin x\right]& \left(p\right)& 0\\ & \text{is (where [x] is greatest interer)}\le x& & \\ \left(b\right)& \text{Let f: D}\to R\text{and}& \left(q\right)& 1\\ & f\left(x\right)=\ln \left\{\underset{\text{n times}}{\underset{}{\ln \ln \cdots \ln }}(x^2-\frac{x}{2}+\frac{33}{16})\right\},\text{then the}& & \\ & \text{value (s) of n for which f(x) is onto is (are)}& & \\ \left(c\right)& \text{The number of common points on the graphs of y}& \left(r\right)& 2\\ & =|\sin |x||andy=x+|x|\text{for}xϵ[-2\pi ,2\pi ],\text{is}& & \\ \left(d\right)& \text{If f(x) is differentiable function for}\forall xϵ\text{R and}& \left(s\right)& 3\\ & \text{exactly two of its tangents are parallel to}& & \\ & \text{x-axis, then number of real values of c for which}& & \\ & f\left(c\right)=0\text{could be}& & \end{array}$

• A. $\left(a\right)\to \left(r\right);\left(b\right)\to (r,s);\left(c\right)\to \left(s\right);\left(d\right)\to (p,q,r,s)$
• B. $\left(a\right)\to \left(r\right);\left(b\right)\to (r,s);\left(c\right)\to \left(r\right);\left(d\right)\to (p,q)$
• C. $\left(a\right)\to \left(r\right);\left(b\right)\to (p,q);\left(c\right)\to \left(s\right);\left(d\right)\to (p,s)$
• D. $\left(a\right)\to \left(r\right);\left(b\right)\to (p,q);\left(c\right)\to \left(q\right);\left(d\right)\to (r,s)$

Answer 1

The correct option is A $\left(a\right)\to \left(r\right);\left(b\right)\to (r,s);\left(c\right)\to \left(s\right);\left(d\right)\to (p,q,r,s)$

(a)

(b) Let $g\left(x\right)=x^2-\frac{x}{2}+\frac{33}{16}={(x-\frac{1}{4})}^2+\frac{32}{16}$

so $g\left(x\right)ϵ[2,\infty ),\text{so}\ln \left(g\right(x\left)\right)ϵ[ln2,\infty )$

$\text{so}\ln \left(\ln \right(g\left(x\right)\left)\right)ϵ\left[\ln \right(\ln 2),\infty )$

where $ln\left(ln2\right)<0$, hence $lnln\left(ln2\right)ϵ$ R for the domain, hence here $n\ge 2$
For n=1, the range is not entire Real numbers and for $n\ge 2$ the domain is not entire Real Numbers and gives all Real numbers as the range.

(c)

(d) For a cubic polynomial, we can have 1, 2 or 3 values of c. And in a case of asymptotes at infinity and negative infinity above X-axis, we have c=0.