Give the geometrical representation of 2x + 13 = 0 as an equation in

Question: Give the geometrical representation of 2x + 13 = 0 as an equation in (i) one variable (ii) two variables Solution: We are given, 2x + 13 = 0 We get, 2x = -13 x = -13/2 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 (-13/2, 0) is shown below....

If 2 is added to the numerator of a fraction,

Question: If 2 is added to the numerator of a fraction, it reduces to 1/2 and if 1 is subtracted from the denominator, it reduces to 1/3. Find the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ If 2 is added to the numerator of the fraction, it reduces to $\frac{1}{2}$. Thus, we have $\frac{x+2}{y}=\frac{1}{2}$ $\Rightarrow 2(x+2)=y$ $\Rightarrow 2 x+4=y$ $\Rightarrow 2 x-y+4=0$ If 1 is subtracted from the den...

Give the geometric representations of the following equations

Question: Give the geometric representations of the following equations (a) on the number line (b) on the Cartesian plane: (i) x = 2 (ii) y + 3 = 0 (iii) y = 3 (iv) 2x + 9 = 0 (v) 3x - 5 = 0 Solution: (i) We are given, x = 2 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 (2, 0) is shown below (ii) We are given,...

The sum of a numerator and denominator of a fraction is 18.

Question: The sum of a numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. 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 the denominator of the fraction is 18 . Thus, we have $x+y=18$ $\Rightarrow x+y-18=0$ If the denominator is increased by 2 , the fraction reduces to $\frac{1}{3}$. Thus, we have $\frac{x}{y+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...

If f(x)=

Question: If $f(x)=\frac{x}{x-1}=\frac{1}{y}$, then $f(y)=$ _____________ . Solution: $f(x)=\frac{x}{x-1}$ given $\frac{x}{x-1}=\frac{1}{y}$ $x y=x-1 \quad$ i.e $y=\frac{x-1}{x}$ i. e $f(y)=\frac{y}{y-1}$ $=\frac{\frac{x-1}{x}}{\frac{x-1}{x}-1}$ $=\frac{\frac{x-1}{x}}{\frac{x-1-x}{x}}$ $=\frac{x-1}{x} \times \frac{x}{-1}$ $=-(x-1)$ $\therefore f(y)=1-x$...

When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes 1/4.

Question: When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes 1/4. And when 6 is added to numerator and the denominator is multiplied by 3, it becomes 2/3. Find the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ If 3 is added to the denominator and 2 is subtracted from the numerator, the fraction becomes $\frac{1}{4}$. Thus, we have $\frac{x-2}{y+3}=\frac{1}{4}$ $\Right...

Show that

Question: $\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{5} x d x}{\sin ^{5} x+\cos ^{5} x}$ Solution: Let $I=\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{5} x}{\sin ^{5} x+\cos ^{5} x} d x$ ...(1) $\Rightarrow I=\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{5}\left(\frac{\pi}{2}-x\right)}{\sin ^{5}\left(\frac{\pi}{2}-x\right)+\cos ^{5}\left(\frac{\pi}{2}-x\right)} d x$ $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right)$ $\Rightarrow I=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{5} x}{\sin ^{5} x+\cos ^{5} x} d x$ ...

If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes 6/5.

Question: If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes 6/5. And, if the denominator is doubled and the numerator is increased by 8, the fraction becomes 2/5. find the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ If the numerator is multiplied by 2 and the denominator is reduced by 5, the fraction becomes $\frac{6}{5}$. Thus, we have $\frac{2 x}{y-...

Aarushi was driving a car with uniform speed of 60 km/h.

Question: Aarushi was driving a car with uniform speed of 60 km/h. Draw distance-time graph From the graph, find the distance travelled by Aarushi in (i)5/2Hours (ii) 1/2 Hour Solution: Aarushi is driving the car with the uniform speed of 60 km/h. We represent time on X-axis and distance on Y-axis Now, graphically We are given that the car is travelling with a uniform speed 60 km/hr. This means car travels 60 km distance each hour. Thus the graph we get is of a straight line. Also, we know when ...

Let A and B be any two sets such that n(A) = p and n(B) = q,

Question: LetAandBbe any two sets such thatn(A) =pandn(B) =q, then the total functions fromAtoBis equal to __________ . Solution: n(A) =p, n(B) =q. here any element of setA, can be connected with elements ofBinqways.and there arepsuch elements inA. $\therefore$ Total function possible is $\frac{q \times q \times q \ldots \ldots \times q}{p \text { times }}$ i.e $q^{p}$ $\therefore$ Total functions from $A$ to $B q^{p}$ i.e $n(B)^{n(A)}$....

Ravish tells his daughter Aarushi, "Seven years ago, I was seven times as old as you were then.

Question: Ravish tells his daughter Aarushi, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be". If present ages of Aarushi and Ravish are x and y years respectively, represent this situation algebraically as well as graphically. Solution: We are given the present age of Ravish as y years and Aarushi as x years. Age of Ravish seven years ago = y - 7 Age of Aarushi seven years ago = x 7 It has already been said by...

The domain of the function f defined by

Question: The domain of the function $f$ defined by $f(x)=\frac{1}{\sqrt{x-|x|}}$ is (a) R0 (b) R+ (c) R (d) none of these Solution: $f(x)=f(x)=\frac{1}{\sqrt{x-|x|}}$ f(x) is defined ifx |x| 0 i.ex |x| i.e |x| x Since no such real numberxexist such that |x| x $\therefore$ Domain of $f(x)$ is empty set. Hence, the correct answer is option D....

Show that

Question: $\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x d x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}$ Solution: Let $I=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x$ ...(1) $\Rightarrow I=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}}\left(\frac{\pi}{2}-x\right)}{\sin ^{\frac{3}{2}}\left(\frac{\pi}{2}-x\right)+\cos ^{\frac{3}{2}}\left(\frac{\pi}{2}-x\right)} d x$ $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right)$...

If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1.

Question: If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1. It also becomes 1/2 if we only add 1 to the denominator. What is the fraction? Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ If 1 is added to the numerator and 1 is subtracted from the denominator, the fraction becomes. Thus, we have $\frac{x+1}{y-1}=1$ $\Rightarrow x+1=y-1$ $\Rightarrow x+1-y+1=0$ $\Rightarrow x-y+2=0$ If...

If f(x)=

Question: If $f(x)=x^{3}-\frac{1}{x^{3}}$, then $f(x)+f\left(\frac{1}{x}\right)$ is equal to (a) $2 x^{3}$ (b) $\frac{2}{x^{3}}$ (c) 0 (d) 1 Solution: $f(x)=x^{3}-\frac{1}{x^{3}}$ $\Rightarrow f\left(\frac{1}{x}\right)=\left(\frac{1}{x}\right)^{3}-\left(\frac{1}{\frac{1}{x}}\right)^{3}$ $f\left(\frac{1}{x}\right)=\frac{1}{x^{3}}-x^{3}$ $\therefore f(x)+f\left(\frac{1}{x}\right)=x^{3}-\frac{1}{x^{3}}+\frac{1}{x^{3}}-x^{3}$ i. e $f(x)+f\left(\frac{1}{x}\right)=0$ Hence, the correct answer is optio...

The path of a train A is given by the equation 3x + 4y -12 = 0

Question: The path of a train A is given by the equation 3x + 4y -12 = 0 and the path of another train B is given by the equation 6x + 8y - 48 = 0. Represent this situation graphically. Solution: We are given the path of train A, 3x + 4y - 12 = 0 We get, $y=\frac{12-3 x}{4}$ Now, substituting x = 0 in $y=\frac{12-3 x}{4}$ we get y = 3 Substituting x = 4 in $y=\frac{12-3 x}{4}$ we get y = 0 Thus, we have the following table exhibiting the abscissa and ordinates of points on the line represented b...

A fraction becomes 1/3 if 1 is subtracted from both its numerator and denominator.

Question: A fraction becomes 1/3 if 1 is subtracted from both its numerator and denominator. It 1 is added to both the numerator and denominator, it becomes 1/2. Find the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ If 1 is subtracted from both numerator and the denominator, the fraction becomes $\frac{1}{3}$. Thus, we have $\frac{x-1}{y-1}=\frac{1}{3}$ $\Rightarrow 3(x-1)=y-1$ $\Rightarrow 3 x-3=y-1$ $\Righ...

The domain and range of the function f given by f(x)

Question: The domain and range of the functionfgiven byf(x) = 2 |x 5|, is (a) Domain =R+, Range = (, 1] (b) Domain =R, Range = (, 2] (c) Domain =R, Range = (, 2) (d) Domain =R+, Range = (, 2] Solution: f(x) = 2 |x 5| Sincef(x) is defined for everyxR $\therefore$ Domain is $R$ also |x 5| 0 i.e |x 5| 0 i.e 2 |x 5| 2 $\therefore f(x) \leq 2$ i.e Range forf(x) is (, 2] Hence, the correct answer is option B....

The domain and range of the function f given by f(x)

Question: The domain and range of the functionfgiven byf(x) = 2 |x 5|, is (a) Domain =R+, Range = (, 1] (b) Domain =R, Range = (, 2] (c) Domain =R, Range = (, 2) (d) Domain =R+, Range = (, 2] Solution: f(x) = 2 |x 5| Sincef(x) is defined for everyxR $\therefore$ Domain is $R$ also |x 5| 0 i.e |x 5| 0 i.e 2 |x 5| 2 $\therefore f(x) \leq 2$ i.e Range forf(x) is (, 2] Hence, the correct answer is option B....

Show that

Question: $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ Solution: $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ Let $I=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ ...(1) $\Rightarrow I=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}}{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}+\sqrt{\cos \left(\frac{\pi}{2}-x\right)}} d x$ $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x)...

The domain of the function f given by

Question: The domain of the function $f$ given by $f(x)=\frac{x^{2}+2 x+1}{x^{2}-x-6}$ (a)R{2, 3} (b)R{3, 2} (c)R[2, 3] (d)R(2, 3) Solution: $f(x)=\frac{x^{2}+2 x+1}{x^{2}-x-6}$ f(x) is not defined forx2x 6 = 0 i.e x2 3x+ 2x 6 = 0 i.e x(x 3) + 2 (x 3) = 0 i.e (x 3) (x+ 2) = 0 i.e x= 3 orx= 2 $\therefore$ Domain for $f(x)$ is $R-\{-2,3\}$ Hence the correct answer is option A....

A fraction becomes 9/11 if 2 is added to both numerator and the denominator.

Question: A fraction becomes 9/11 if 2 is added to both numerator and the denominator. If 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction. Solution: Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$ If 2 is added to both numerator and the denominator, the fraction becomes $\frac{9}{11}$. Thus, we have $\frac{x+2}{y+2}=\frac{9}{11}$ $\Rightarrow 11(x+2)=9(y+2)$ $\Rightarrow 11 x+22=9 y+18$ $\Rig...

Draw the graphs of the linear equations

Question: Draw the graphs of the linear equations 4x - 3y + 4 = 0 and 4x + 3y - 20 = 0. Find the area bounded by these lines and x-axis. Solution: We are given, 4x - 3y + 4 = 0 We get, $y=\frac{4 x+4}{3}$ Now, substituting x = 0 in $y=\frac{4 x+4}{3}$ we get Substituting x = -1 in $y=\frac{4 x+4}{3}$ we get y = 0 Thus, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation Plotting E(0, 4/3) and A (-1, 0) on the graph and by join...

The domain and range of real function f defined by

Question: The domain and range of real function $t$ defined by $f(x)=\sqrt{x-1}$ is given by (a) Domain = (1, ), Range = (0, ) (b) Domain = [1, ), Range =(0, ) (c) Domain = [1, ), Range = [0, ) (d) Domain = [1, ), Range = [0, ) Solution: $f(x)=\sqrt{x-1}$ Sincex1 0 i.ex 1 $\therefore$ Domain of $f(x)$ is $[1, \infty)$ and forx [1,) f(x) 0 ⇒ Range off(x) is[0,) Hence, the correct answer is option C....