Modulus Function

Q. If $a$ and $b$ are arbitary positive real numbers, then the least possible value of $\frac{6a}{5b}+\frac{10b}{3a}$ is

1. $4$
2. $\frac{6}{5}$
3. $\frac{10}{3}$
4. $\frac{68}{15}$

Q. Let $f\left(x\right)=|x^2-9|-|x-a|,$ then number of integers in the range of $a$ so that $f\left(x\right)=0$ has 4 distinct real roots, is

Q. Find the domain of F(×)= 【√log0.5(-x^2 + x + 6)】+ 1/x^2 +2x

Q.

the set values of'a 'for which the $f\left(x\right)=a{x}^2+2x(1-a)-4$ is negative for exactly three integral value of $x$, is

Q. The values of $x$ satisfying the inequality $[{\tan }^{-1}x{]}^2-[{\tan }^{-1}x]-2\le 0$, where $[.]$ denotes integer part function, is

1. $[-\tan 1,\infty )$
2. $[-\frac{\pi }{4},\tan 2]$
3. $[-\tan 1,\tan 3]$
4. $[\tan 1,\tan 3]$

Q. If $|ln|x||=|k-1|-4$ has exactly two distinct real roots for $x\in R-\left\{0\right\}$, then the sum of all possible integral values of k is

1. -5
2. -3
3. 2
4. Infinite

Q. The integral value(s) of $x$ satisfying $|x^2-9|+|x^2-4|=5$ is/are

1. $-3$
2. $2$
3. $-2$
4. $3$

Q. Which of the following is/are true for function $f\left(x\right)=|{\log }_{e}x|$

1. $f\left(x\right)$ is one-one function.
2. $f\left(x\right)$ is many-one function.
3. Domain of $f\left(x\right)$ is $(0,\infty )$
4. Range of $f\left(x\right)$ is $[0,\infty )$

Q. Number of solution(s) of $2^{\sin |x|}=4^{|\cos x|}$ in $[-\pi ,\pi ]$ is equal to :

1. $2$
2. $4$
3. $6$
4. $8$

Q.

Let f: R → R be defined as f(x) = x⁴. Choose the correct answer.

(A) f is one-one onto (B) f is many-one onto

(C) f is one-one but not onto (D) f is neither one-one nor onto

Q. The range of the function $f\left(x\right)=|x-1|+|x-8|$ is

1. $[7,\infty )$
2. $R-\{1,8\}$
3. $[0,\infty )$
4. $[8,\infty )$

Q. Let $\left[x\right]$ denote the greatest integer less than or equal to $x.$ If the domain of the function $\frac{1}{[x{]}^2-7[x]+12}$ is $R-[a,b),$ then the value of $a+b$ is

1. $9$
2. $8$
3. $5$
4. $15$

Q. The least value of $f\left(x\right)=|x-a|+|x-b|+|x-c|+|x-d|$, where $a<b<c<d$ are fixed real numbers, is

1. $a+b+c-d$
2. $a+b-c+d$
3. $a-b-c+d$
4. $-a-b+c+d$

Q. Let S be the set of values of $x$ for which the tangent to the curve $y=f\left(x\right)=x^3-x^2-2x$ at $(x,y)$ is parallel to the line segment joining the points $(1,f(1\left)\right)\text{and}(-1,f(-1\left)\right)$, then S is equal to:

1. $\{\frac{1}{3},1\}$
2. $\{\frac{1}{3},-1\}$
3. $\{-\frac{1}{3},-1\}$
4. $\{-\frac{1}{3},1\}$

Q. Pointing to Abhay, Shilpa said, "His mother's brother is the father of my son Ashish. How is Abhay related to Shilpa?

1. Brother
2. Cousin
3. Nephew
4. Brother-in-law

Q. Let f : [0, ∞) → [–4, ∞) be defined by f(x) = x^2 + x – 4, then f^–1(2) is

Q.

A point c in the domain of a function f is called a critical point of f if

1. Either f' (c) = 0 or f is not differentiable

2. f' (x) = 0 at x = c

3. f" (x) = 0, at x = c

4. f is not differentiable at x = c

Q.

Find the number of integers satisfying the condition $|x-3|$$>$ 5 and $|x+1|$ $\le$ 4

___

Q. The number of value(s) of $x$ satisfying $\sin \left(\frac{1}{3}{\cos }^{-1}x\right)=1$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q.

If f(x) =x²+kx+1, for all x and if it is an even function, find k.

Q.

If$a=5$ and $b=7$, then $|a-b|$ is

1. $-2$

2. $2$

3. $5$

4. $12$

Q. Pointing to an old man Satya said, "His son is my son's uncle". How is the old man related to Satya?

1. Brother
2. Uncle
3. Father
4. Grand-father

Q.

What is the sign of modulus?

Q.

Prove that the greatest integer function defined byis not

differentiable at x = 1 and x = 2.

Q.

Find the value of $\left|4-9\right|$

Q. If $f\left(x\right)=|||x|-2|+p|$ has more than $3$ points of non differentiability, then the value of $p$ can be:

1. $4$
2. $3$
3. $-1$
4. $0$

Q. Write the order and degree of the differential equation $\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^{\frac{1}{4}}+x^{\frac{1}{5}}=0$

Q.

Verify Rolle’s Theorem for the function

Q. If $f\left(x\right)=\text{sgn}(x^2-ax+1)$ has maximum number of points of non differentiability, then

1. $a\in (-\infty ,-2)\cup (2,\infty )$
2. $a\in (-2,2)$
3. $a\in (-1,1)$
4. $a\in R$

Q. The number of solution pairs $(x,y)$ of the simultaneous equations
${\log }_{1/3}(x+y)+{\log }_3(x-y)=2$
$2^{y^2}={512}^{x+1}$ is

1. $0$
2. $1$
3. $2$
4. $3$