Continuity in an Interval

Q. If f(x+10)+f(x+4)=0, there f(x) is a periodic function with period 1. 2 2. 4 3. 6 4. 1

Q.

Given $f\left(x\right)=\sqrt{9-x^2}$, then the function is continuous at [-3, 3].

1. True

2. False

Q.

What is the difference between continuous and differentiable?

Q. 10. If f(a-x) = f(a+x) and f(b-x) = f(b+x) for all real x, where a, b (b less than a) are constants, then prove that f(x) is a periodic function.

Q. 23. If f(x)=|x+2|+|2x-p|+|x-2| attains its minimum value in the interval (-1, 1)then sum of all possible integral value of p is (A)0 , (B)1, (C)3 , (D) 4.

Q. If $f\left(x\right)=\frac{\sin 3x}{\sin x},x\ne n\pi$ then the range of values of f(x) for real values of x is

1. [-1, 3)
2. [-1, 3]
3. (-, -1)
4. (3, )

Q. 16. Let f(x) = ax2 + bx+ c where a b c are rational and f: Z--->Z where Z is the set of integer . Then a+b =

Q. If f is continuous on its domain D, then | f | is also continuous on D.

1. True
2. False

Q.

The value of $\sum _{k=1}^{10}$$(sin\frac{2k\pi }{11}-icos\frac{2k\pi }{11})$ is

1. -i

2. i

3. 1

4. -1

Q. The function f (x) = [x], where [x] denotes the greatest integer function, is continuous at

1. 4
2. -2
3. 1
4. 1.5
5. -2.5

Q. $Letf\left(x\right)=\{\begin{array}{cc}(1+|cosx|{)}^{\frac{ab}{|cosx|}},& n\pi <x<(2n+1)\frac{\pi }{2}\\ e^a.e^b,& x=(2n+1)\frac{\pi }{2}\\ e^{\frac{cot2x}{cot8x}},& (2n+1)\frac{\pi }{2}<x<(n+1)\pi \end{array}Iff\left(x\right)iscontinuousin\left(\right(n\pi ),(n+1)\pi ,then)$

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

Q.

$\underset{x\to \frac{\pi }{2}}{lim}\frac{(1-tan\left(\frac{x}{2}\right))(1-sinx)}{(1+tan\left(\frac{x}{2}\right))(\pi -2x{)}^3}is$

1. 1/8

2. 0

3. 1/32

4. ∞

Q.

so the least integral value of n is

1. -3

2. 4

3. 3

4. -4

Q. If the function $f$ defined on $(-\frac{1}{3},\frac{1}{3})$ by $f\left(x\right)=\{\begin{array}{c}\frac{1}{x}{\log }_e\left(\frac{1+3x}{1-2x}\right),\text{when}x\ne 0\\ k,\text{when}x=0\end{array}$
is continuous, then $k$ is equal to

Q. If $f\left(x\right)=1-\frac{1}{x}$, then write the value of $f\left(f\left(\frac{1}{x}\right)\right)$.

Q. 8. If (x-|x|-12)/(x-3) ≥0

Q. If the function $f\left(x\right)=\{\begin{array}{cc}x+a^2\sqrt{2}sinx,& 0\le x<\frac{\pi }{4}\\ xcotx+b,& \frac{\pi }{4}\le x<\frac{\pi }{2}\\ bsin2x-acos2x,& \frac{\pi }{2}\le x\le \pi \end{array}$ is continuous in the interval $[0,\pi ]$ , then the values of (a, b) are

1. (–1, –1)
2. (0, 0)
3. (–1, 1)
4. (1, -1)

Q. If the function f(x) given by $f\left(x\right)=(\begin{array}{c}4\mathrm{ax}+b,\mathrm{if}x>1\\ 11,\mathrm{if}x=1\\ 3\mathrm{ax}-2b,\mathrm{if}x<1\end{array}$ is continuous at x = 1, then the value of a + b is

1. 3
2. 2
3. 1
4. 4

Q. State and explain fouriers theorem determine the coefficient of fouriers series derive the formula

Q. Let $A=\{a,b,c\}$ and $B=\{1,2,3,4\}$. Then the number of elements in the set $C=\{f:A\to B|2\in f(A\left)\text{and}f\text{is not one–one}\right\}$ is