Domain

Q. The domain of f(x) is (0, 1). Then the domain of $f\left(e^x\right)+f\left(ln|x|\right)$ is

1. $(-e,1)$
2. $(-1,e)$
3. $(1,e)$
4. $(-e,-1)$

Q.

Let f and g be real functions, defined by $f\left(x\right)=\sqrt{x+2}$ and $g\left(x\right)=\sqrt{4-x^2}$. Find

(i) $(f+g)\left(x\right)$

(ii) $(f-g)\left(x\right)$

(iii) $\left(fg\right)\left(x\right)$

(iv) $\left(ff\right)\left(x\right)$

(v) $\left(gg\right)\left(x\right)$

(vi) $\left(\frac{f}{g}\right)\left(x\right)$

Q. 43. The value of tan15^° /2-sec15^° is equal to (1) 1/23 (2) 3/2 (3) -1/23 (4) 1/83

Q.

Find the domain of f(2x - 1) if the domain of f(x) is [-1, 1]

1. [-1, 1]

2. [0, 1]

3. [-1, 0]

4. [1, 2]

Q. The domain of the function $f\left(x\right)={\log }_{1/2}(x-\frac{1}{2})+{\log }_2\sqrt{4x^2-4x+5}$ is

1. $[\frac{1}{2},\infty )$
2. $(\frac{1}{2},\infty )$
3. $(-\infty ,\infty )$
4. $(-\frac{1}{2},\infty )$

Q. If I=∫\operatorname{sin(6-4x)dx, then / is

Q.

If 3sin$\beta$ = sin(2$\alpha$ + $\beta$), then tan ($\alpha$ + $\beta$) is equal to

1. 2tan

2. 2tan

3. tan + tan

4. None of these

Q. Let R be the relation on Z defined by R = {( a , b ): a , b ∈ Z , a – b is an integer}. Find the domain and range of R.

Q. The values of $x$ for which the function $f\left(x\right)=\frac{x^2+x+1}{x^2-7x+12}$ is not defined, is

1. $\{3,4\}$
2. $\{3,-4\}$
3. $\{-3,-4\}$
4. the function is defined everywhere

Q.

Write the complex number $z=\frac{i-1}{cos\frac{\pi }{3}+isin\frac{\pi }{3}}$ in the polar form.

Q.

Solve the following systems of inequalities graphically:

$3x+2y\le 12,x\ge 1,y\ge 2$

Q.

If $\frac{|x-3|}{x-3}>0$, then

1. $xϵ(-3,\infty )$

2. $xϵ(3,\infty )$

3. $xϵ[-3,\infty )$

4. $xϵ[3,\infty )$

Q.

Draw the graph of the greatest integer function:

$f:R\to R:f\left(x\right)=\left[x\right]$ for all $xϵR$

Q.

$\underset{x\to 0}{lim}\frac{2^x-1}{(1+x{)}^{\frac{1}{2}}-1}$

1. 

2. 

3. 

4. 

Q. If $f\left(x\right)=\frac{1}{|x-1|-x^2},$ then the domain of the function is

1. $R-\left\{\frac{-1\pm \sqrt{5}}{2}\right\}$
2. $R-\left\{\frac{-1\pm \sqrt{3}}{2}\right\}$
3. $R-\left\{\frac{-1\pm \sqrt{7}}{2}\right\}$
4. $R-\left\{\frac{-1\pm \sqrt{6}}{2}\right\}$

Q.

Identify the function based on the given information.

1. Domain of the function is $R^{+}$

2. $F(x_1.x_2)=F\left(x_1\right)+F\left(x_2\right)$

3. $F\left(\frac{x_1}{x_2}\right)=F\left(x_1\right)-F\left(x_2\right)$

4. $F\left(1\right)=0$

5. $F\left(2^m\right)$ is proportional to m.

6. Range of function is R

1. Logarithm Function

2. Exponential Function

3. Absolute value function

4. 

Q. Find the domain and the range of the real function f defined by .

Q. Consider the rational function $f\left(x\right)=\frac{x^3-4x^2+15x-8}{x^2-2}$. Then

1. domain of $f$ is $R-\left\{\sqrt{2}\right\}$
2. there are three pre-images of $4$
3. there is only one pre-image of $4$
4. domain of $f$ is $R-\{\pm \sqrt{2}\}$

Q. 71.WHAT IS THE PRINCIPAL VALUE OF SIN x = -1/2

Q. Find the number of sets of 5-tuples (A1, A2, A3, A4, A5) so that A1 + A2(sinx)+ A3(cosx) + A4(2cosx) + A5(2sinx) = 0 is true for all values of X? A) 0 B) 1 C) 2 D) Infinite

Q. Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = { ( x , y ): 3 x – y = 0, where x , y ∈ A}. Write down its domain, codomain and range.

Q. integrate b/w the limits 0 to 2 pi cos^2x dx

Q. Find the domain and range of the real function $f$ defined by $f\left(x\right)=|x-1|$

Q.

$f\left(x\right)=\log (\sqrt{x-4}+\sqrt{6-x})$ is defined for

1. $\left[6,8\right]$

2. $\left[2,4\right]$

3. $\left[4,6\right]$

4. None of these

Q. Find the domain and range of the following real function: (i) f ( x ) = –| x | (ii)

Q. The principal value of sin^-1( √3 - 1/2√2)

Q. 7. Solve for x : log(x-1)+log(x+1) = log of 1 to the base 2

Q.

If f(x) = sin log $\left(\frac{\sqrt{4-x^2}}{1-x}\right)$, then the domain of f(x) is ___

1. $(1,2)$

2. $(-2,1)$

3. $(0,2)$

4. $(-1,2)$

Q. Let f(x) = x, $g\left(x\right)=\frac{1}{x}$ and h(x) = f(x) g(x). Then, h(x) = 1
(a) x ∈ R
(b) x ∈ Q
(c) x ∈ R − Q
(d) x ∈ R, x ≠ 0

Q.

If $a{x}^2+bx+c,a,b,cϵR$, a < 0 has no real zeros and a - b+c<0, then the

value of ac

1. > 0

2. < 0

3. = 0

4. < -1