The domain of the function f(x)

Question: The domain of the function $f(x)=\frac{x^{2}+1}{x^{2}-3 x+2}$ is Solution: $f(x)=\frac{x^{2}+1}{x^{2}-3 x+2}$ f(x) is defined ifx2 3x+ 2 0 i.ef(x) is not defined forx2 3x+ 2 = 0 i.ex2 2xx+ 2 = 0 (x2 2x) 1 (x 2) = 0 x(x 2) 1 (x 2) = 0 i.e (x 1) (x 2) = 0 i.ex= 1 orx= 2 i.e forx= 1 orx= 2,f(x) is not defined Domain off(x) =R~ {1,2}...

The angles of a quadrilateral are in the ratio 3:5:9:13.

Question: The angles of a quadrilateral are in the ratio 3:5:9:13. Find all the angles of the quadrilateral. Solution: Let the common ratio between the angles is 't' So the angles will be 3t, 5t, 9t and 13t respectively. Since the sum of all interior angles of a quadrilateral is 360 Therefore, 3t + 5t + 9t + 13t = 360 ⟹ 30t = 360 ⟹ t = 12 Hence, the angles are 3t = 3*12 = 36 5t = 5*12 = 60 9t = 9*12 = 108 13t = 13*12 = 156...

The domain for which the functions f(x)

Question: The domain for which the functionsf(x) = 3x21 andg(x) = 3 +xare equal is __________ . Solution: f(x) = 3x2 1 g(x) = 3 +x Given:- f(x) =g(x) i.e 3x2 1 = 3 +x 3x2x 1 3 = 0 3x2x 4 = 0 3x2+ 3x 4x 4 = 0 3x(x+ 1) 4 (x+ 1) = 0 (3x 4) (x+ 1) = 0 i.e $x=\frac{4}{3}$ or $x=-1$ $\therefore$ Domain for which $f(x)$ and $g(x)$ are equal is $\left\{-1, \frac{4}{3}\right\}$...

A father is three times as old as his son. After twelve years,

Question: A father is three times as old as his son. After twelve years, his age will be twice as that of his son then. Find the their present ages. Solution: Let the present age of father bexyears and the present age of son beyyears. Father is three times as old as his son. Thus, we have $x=3 y$ $\Rightarrow x-3 y=0$ After 12 years, father's age will be $(x+12)$ years and son's age will be $(y+12)$ years. Thus using the given information, we have $x+12=2(y+12)$ $\Rightarrow x+12=2 y+24$ $\Right...

In a quadrilateral ABCD, CO and Do are the bisectors of ∠C and ∠D respectively.

Question: In a quadrilateral ABCD, CO and Do are the bisectors of C and D respectively. Prove that COD = 1/2(A and B). Solution: In ΔDOC 1 + COD + 2 = 180 [Angle sum property of a triangle] ⟹COD = 180 (1 2) ⟹COD = 180 1 + 2 ⟹COD = 180 [1/2 LC + 1/2 LD] [∵ OC and OD are bisectors of LC and LD respectively] ⟹COD = 180 1/2(LC + LD) ... (i) In quadrilateral ABCD A + B + C + D = 360 [Angle sum property of quadrilateral] C + D = 360 (A + B) .... (ii) Substituting (ii) in (i) ⟹ COD = 180 1/2(360 (A + B...

Let f and g be two real functions given by f =

Question: Letfandgbe two real functions given byf= {(10, 1), (2, 0), (3, 4), (4, 2), (5, 1)} andg= {(1, 0), (2, 2), (3, 1), (4, 4), (5, 3)}. Then the domainfgis given by __________ . Solution: f= {(10, 1),(2, 0), (3, 4), (4, 2), (5, 1)} g= {(1, 0), (2, 2), (3, 1), (4, 4), (5, 3)} Since (fg) (x) =f(x)g(x) Domain offgis (Domain off) (Domain ofg) Since Domain off= {10, 2, 3, 4, 5} Domain of g= {1, 2, 3, 4, 5} ⇒ Domain off gis {2, 3, 4, 5}...

Show that

Question: $\int_{0}^{1} x(1-x)^{n} d x$ Solution: Let $I=\int_{0}^{1} x(1-x)^{n} d x$ $\therefore I=\int_{0}^{1}(1-x)(1-(1-x))^{n} d x$ $=\int_{0}^{1}(1-x)(x)^{n} d x$ $=\int_{0}^{d}\left(x^{n}-x^{n+1}\right) d x$ $=\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_{0}^{1}$ $\left(\int_{0}^{\infty} f(x) d x=\int_{0}^{0} f(a-x) d x\right)$ $=\left[\frac{1}{n+1}-\frac{1}{n+2}\right]$ $=\frac{(n+2)-(n+1)}{(n+1)(n+2)}$ $=\frac{1}{(n+1)(n+2)}$...

In a quadrilateral ABCD, the angles A, B, C and D are in the ratio of 1: 2: 4: 5

Question: In a quadrilateral ABCD, the angles A, B, C and D are in the ratio of 1: 2: 4: 5. Find the measure of each angles of the quadrilateral Solution: Let the angles of the quadrilaterals be A = x, B = 2x, C = 4x and D = 5x Then, A + B + C + D = 360 ⟹ x + 2x + 4x + 5x = 360 ⟹ 12x = 360 ⟹x = 360/12 ⟹ x = 30 Therefore, A = x = 30 B = 2x = 60 C = 4x = 120 D = 5x = 150...

Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angle.

Question: Three angles of a quadrilateral are respectively equal to 110, 50 and 40. Find its fourth angle. Solution: Given, Three angles are 110, 50and 40 Let the fourth angle be 'x' We have, Sum of all angles of a quadrilateral = 360 110+ 50+ 40= 360 ⟹ x = 360- 200 ⟹ x = 160 Therefore, the required fourth angle is 160....

Let f and g be two functions given by f = {(2, 4), (5, 6), (8, −1), (10, −3)}

Question: Letfandgbe two functions given byf= {(2, 4), (5, 6), (8, 1), (10, 3)} andg= {(2, 5), (7, 1) (8, 4), (10, 13), (11, 5). Then, domain off+gis __________ . Solution: f= {(2, 4), (5, 6), (8, 1), (10, 3)} g= {(2, 5), (7, 1), (8, 4), (10, 13), (11, 5)} Domain offis {2, 5, 8, 10} and Domain ofg= {2, 7, 8, 10, 11} Since (f +g) (x) =f(x) +g(x) xshould lie in Both Domain offand Domain ofg x Domain (f) Domain (g) i.e Domain off +gis {2, 8, 10}...

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3,

Question: The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ The sum of the numerator and denominator of the fraction is 12. Thus, we have $x+y=12$ $\Rightarrow x+y-12=0$ If the denominator is increased by 3, the fraction becomes $\frac{1}{2}$. Thus, we have $\frac{x}{y+3}=\frac{1}...

The range of the function f(x)

Question: The range of the functionf(x) = logax,a 0 is _________ Solution: f(x) = logax;a 0 $f(x)=\frac{\log _{10}(x)}{\log _{10}(a)}$ Sincea 0⇒f(x) is defined forR+{0} Range set off(x) isR...

Show that

Question: $\int_{2}^{8}|x-5| d x$ Solution: Let $I=\int_{2}^{8}|x-5| d x$ It can be seen that $(x-5) \leq 0$ on $[2,5]$ and $(x-5) \geq 0$ on $[5,8]$. $I=\int_{2}^{5}-(x-5) d x+\int_{2}^{8}(x-5) d x$ $\left(\int_{a}^{b} f(x)=\int_{a}^{c} f(x)+\int_{c}^{b} f(x)\right)$ $=-\left[\frac{x^{2}}{2}-5 x\right]_{2}^{5}+\left[\frac{x^{2}}{2}-5 x\right]_{5}^{8}$ $=-\left[\frac{25}{2}-25-2+10\right]+\left[32-40-\frac{25}{2}+25\right]$ $=9$...

The range of the function

Question: The range of the function $f(x)=\frac{x+2}{|x+2|}$ is _______ . Solution: $f(x)=\frac{x+2}{|x+2|}$ $= \begin{cases}\frac{x+2}{x+2} ; x \geq-2 \\ \frac{x+2}{-(x+2)} ; x-2\end{cases}$ i. e $f(x)= \begin{cases}1 ; x \geq-2 \\ -1 ; x-2\end{cases}$ $\therefore$ Range of function $f(x)$ is $\{-1,1\}$...

The sum of the numerator and denominator of a fraction is 3 less than twice the denominator.

Question: The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ The sum of the numerator and denominator of the fraction is 3 less than twice the denominator. Thus, we have $x+y=2 y-3$ $\Rightarrow x+y-2 y+3=0$ $\Rig...

Write the equation of the line that is parallel to y-axis and passing through the Points

Question: Write the equation of the line that is parallel to y-axis and passing through the Points (i) (4, 0) (ii) (- 2, 0) (iii) (3, 5) (iv) (- 4, - 3) Solution: (i) We are given the coordinates of the Cartesian plane at (4, 0) For the equation of the line parallel to y axis, we assume the equation as a one variable equation independent of y containing x equal to 4 We get the equation as y = 3 (ii) We are given the coordinates of the Cartesian plane at (-2, 0) For the equation of the line paral...

The range of the function f(x)

Question: The range of the functionf(x) = [x] xis __________ . Solution: f(x) = [x] x Sincex [x] Every number is greater than or equal to its greatest integral value i.ex [x] = {x} fractional part ofx. [x]x= {x}fraction part only. also [x] =xfor integral value ofx hence, for non-integral valuesf(x) = {x}(1, 0) and for integral valuesf(x) = 0 Hence, Range off(x) is (1, 0]....

The domain of the function

Question: The domain of the function $f(x)=\frac{1}{\sqrt{|x|-x}}$ is _________ . Solution: $f(x)=\frac{1}{\sqrt{|x|-x}}$ f(x) is defined if |x|x 0 i.e |x| x i.ex |x| which is possible for negative real numbers $\therefore$ Domain for $f(x)$ is $\mid R^{-} \sim\{0\}$...

Show that

Question: $\int_{-5}^{5}|x+2| d x$ Solution: Let $I=\int_{-5}^{5}|x+2| d x$ It can be seen that $(x+2) \leq 0$ on $[-5,-2]$ and $(x+2) \geq 0$ on $[-2,5]$. $\therefore I=\int_{-5}^{-2}-(x+2) d x+\int_{-2}^{5}(x+2) d x$ $\left(\int_{a}^{b} f(x)=\int_{a}^{c} f(x)+\int_{c}^{b} f(x)\right)$ $I=-\left[\frac{x^{2}}{2}+2 x\right]_{-5}^{-2}+\left[\frac{x^{2}}{2}+2 x\right]_{-2}^{5}$ $=-\left[\frac{(-2)^{2}}{2}+2(-2)-\frac{(-5)^{2}}{2}-2(-5)\right]+\left[\frac{(5)^{2}}{2}+2(5)-\frac{(-2)^{2}}{2}-2(-2)\ri...

Write the equation of the line that is parallel to x-axis and passing through the points

Question: Write the equation of the line that is parallel to x-axis and passing through the points (i) (0, 3) (ii) (0, - 4) (iii) (2, 5) (iv) (3, 4) Solution: (i) We are given the co-ordinates of the Cartesian plane at (0, 3). For the equation of the line parallel to x axis, we assume the equation as a one variable equation independent of x containing y equal to 3. We get the equation as y = 3 (ii) We are given the co-ordinates of the Cartesian plane at (0,- 4). For the equation of the line para...

The sum of the numerator and denominator of a fraction is 4 more than twice the numerator.

Question: The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2 : 3. Determine the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ The sum of the numerator and denominator of the fraction is 4 more than twice the numerator. Thus, we have $x+y=2 x+4$ $\Rightarrow 2 x+4-x-y=0$ $\Rightarrow x-y+4=0$ If...

If y=f(x)

Question: If $y=f(x)=\frac{a x+b}{c x-d}$, then $f(y)=$ ________ . Solution: If $y=f(x)=\frac{a x+b}{c x-d}$ $f(y)=\frac{a y+b}{c y-d}$ $=\frac{a\left(\frac{a x+b}{c x-d}\right)+b}{c\left(\frac{a x+b}{c x-d}\right)-d}$ $f(y)=\frac{a(a x+b)+b(c x-d)}{\frac{c x-d}{\frac{c(a x+b)-d(c x-d)}{c x-d}}}$ $=\frac{a^{2} x+a b+b c x-d b}{a c x+b c-c d x+d^{2}}$ $f(y)=\frac{\left(a^{2}+b c\right) x+a b-b d}{(a c-c d) x+b c-d^{2}}$...

Question: $\int_{-5}^{5}|x+2| d x$ Solution: Let $I=\int_{-5}^{5}|x+2| d x$ It can be seen that $(x+2) \leq 0$ on $[-5,-2]$ and $(x+2) \geq 0$ on $[-2,5]$. $\therefore I=\int_{-5}^{-2}-(x+2) d x+\int_{-2}^{5}(x+2) d x$ $\left(\int_{a}^{b} f(x)=\int_{a}^{c} f(x)+\int_{c}^{b} f(x)\right)$ $I=-\left[\frac{x^{2}}{2}+2 x\right]_{-5}^{-2}+\left[\frac{x^{2}}{2}+2 x\right]_{-2}^{5}$ $=-\left[\frac{(-2)^{2}}{2}+2(-2)-\frac{(-5)^{2}}{2}-2(-5)\right]+\left[\frac{(5)^{2}}{2}+2(5)-\frac{(-2)^{2}}{2}-2(-2)\ri...

Solve the equation 3x + 2 = x - 8, and represent the solution on

Question: Solve the equation 3x + 2 = x - 8, and represent the solution on (i) the number line (ii) the Cartesian plane. Solution: We are given, 3x + 2 = x - 8 we get, 3x - x =- 8 - 2 2x = - 10 x = - 5 The representation of the solution on the number line, when given equation is treated as an equation in one variable. The representation of the solution on the Cartesian plane, it is a line parallel to y axis passing through the point (- 5, 0) is shown below...

If y=f(x)

Question: If $y=f(x)=\frac{a x+b}{c x-d}$, then $f(y)=$ ________ . Solution: If $y=f(x)=\frac{a x+b}{c x-d}$ $f(y)=\frac{a y+b}{c y-d}$ $=\frac{a\left(\frac{a x+b}{c x-d}\right)+b}{c\left(\frac{a x+b}{c x-d}\right)-d}$$f(y)=\frac{a(a x+b)+b(c x-d)}{\frac{c x-d}{\frac{c(a x+b)-d(c x-d)}{c x-d}}}$ $=\frac{a^{2} x+a b+b c x-d b}{a c x+b c-c d x+d^{2}}$ $f(y)=\frac{\left(a^{2}+b c\right) x+a b-b d}{(a c-c d) x+b c-d^{2}}$...