Find the value of k for which the system has (i) a unique solution, and (ii) no solution.
Question: Find the value ofkfor which the system has (i) a unique solution (ii) no solution. $k x+2 y=5$ $3 x+y=1$ Solution: GIVEN: $k x+2 y=5$ $3 x+y=1$ To find: To determine for what value ofkthe system of equation has (1) Unique solution (2) No solution We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ (1) For Unique solution $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ Here, $\frac{k}{3} \neq \frac{2}{1}$ $k \neq 6$ Hence for $k \neq 6$ the system of equa...
Read More →In a triangle ABC, AB = 15 cm, BC = 13 cm and AC = 14 cm.
Question: In a triangle ABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of triangle ABC and hence its altitude on AC. Solution: Let the sides of the given triangle be AB = a, BC = b, AC = c respectively. So given, a = 15 cm b = 13 cm c = 14 cm By using Heron's Formula The Area of the triangle $=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ Semi perimeter of a triangle = 2s 2s = a + b + c s =(a + b + c)/2 s =(15 + 13 + 14)/2 s = 21 cm Therefore, Area of the triangle $=\sqrt{s \times(s-a...
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Question: $\sqrt{x^{2}+3 x}$ Solution: Let $I=\int \sqrt{x^{2}+3 x} d x$ $=\int \sqrt{x^{2}+3 x+\frac{9}{4}-\frac{9}{4}} d x$ $=\int \sqrt{\left(x+\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}} d x$ It is known that, $\int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+\mathrm{C}$ $\begin{aligned} \therefore I =\frac{\left(x+\frac{3}{2}\right)}{2} \sqrt{x^{2}+3 x}-\frac{9}{2} \log \left|\left(x+\frac{3}{2}\right)+\sqrt{x^{2}+3 x...
Read More →Find the domain and range of each of the following real valued functions:
Question: Find the domain and range of each of the following real valued functions: (i) $f(x)=\frac{a x+b}{b x-a}$ (ii) $f(x)=\frac{a x-b}{c x-d}$ (iii) $f(x)=\sqrt{x-1}$ (iv) $f(x)=\sqrt{x-3}$ (v) $f(x)=\frac{x-2}{2-x}$ (vi) $f(x)=|x-1|$ (vii) $f(x)=-|x|$ (viii) $f(x)=\sqrt{9-x^{2}}$ (ix) $f(x)=\frac{1}{\sqrt{16-x^{2}}}$ (x) $f(x)=\sqrt{x^{2}-16}$ Solution: (i) Given: $f(x)=\frac{a x+b}{b x-a}$ Domain of $f$ : Clearly, $f(x)$ is a rational function of $x$ as $\frac{a x+b}{b x-a}$ is a rational ...
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Question: $\sqrt{1+3 x-x^{2}}$ Solution: Let $I=\int \sqrt{1+3 x-x^{2}} d x$ $=\int \sqrt{1-\left(x^{2}-3 x+\frac{9}{4}-\frac{9}{4}\right)} d x$ $=\int \sqrt{\left(1+\frac{9}{4}\right)-\left(x-\frac{3}{2}\right)^{2}} d x$ $=\int \sqrt{\left(\frac{\sqrt{13}}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}} d x$ It is known that, $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\mathrm{C}$ $\begin{aligned} \therefore I =\frac{x-\frac{3}{2}}{2} \sqrt{1...
Read More →Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Question: Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm. Solution: Whenever we are given the measurements of all sides of a triangle, we basically look for Heron's formula to find out the area of the triangle. If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by: $A=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ Where, s =(a + b + c)/2 We are given: a = 18 cm b = 10 cm, and peri...
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Question: $\sqrt{x^{2}+4 x-5}$ Solution: Let $I=\int \sqrt{x^{2}+4 x-5} d x$ $=\int \sqrt{\left(x^{2}+4 x+4\right)-9} d x$ $=\int \sqrt{(x+2)^{2}-(3)^{2}} d x$ It is known that, $\int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+\mathrm{C}$ $\therefore I=\frac{(x+2)}{2} \sqrt{x^{2}+4 x-5}-\frac{9}{2} \log \left|(x+2)+\sqrt{x^{2}+4 x-5}\right|+\mathrm{C}$...
Read More →For what value of α, the system of equations will have no solution?
Question: For what value of , the system of equations will have no solution? $\alpha x+3 y=\alpha-3$ $12 x+\alpha y=\alpha$ Solution: GIVEN: $\alpha x+3 y=\alpha-3$ $12 x+\alpha y=\alpha$ To find: To determine for what value ofkthe system of equation has no solution We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For no solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ Here, $\frac{\alpha}{12}=\frac{3}{\alpha} \neq \frac{\alpha-3}{\a...
Read More →Find the area of a triangle whose sides are respectively 9 cm, 12 cm and 15 cm.
Question: Find the area of a triangle whose sides are respectively 9 cm, 12 cm and 15 cm. Solution: Let the sides of the given triangle be a, b, c respectively. So given, a = 9 cm b = 12 cm c = 15 cm By using Heron's Formula The Area of the triangle $=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ Semi perimeter of a triangle = s 2s = a + b + c s =(a + b + c)/2 s =(9 + 12 + 15)/2 s = 18 cm Therefore, Area of the triangle $=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ $=\sqrt{18 \times(18-9) \tim...
Read More →Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm.
Question: Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm. Solution: Let the sides of the given triangle be a, b, c respectively. So given, a = 150 cm b = 120 cm c = 200 cm By using Heron's Formula The Area of the triangle $=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ Semi perimeter of a triangle = s 2s = a + b + c s =(a + b + c)/2 s =(150 + 200 + 120)/2 s = 235 cm Therefore, Area of the triangle $=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ $=\sqrt{235 \ti...
Read More →For what value of k the following system of equations will be inconsistent?
Question: For what value ofkthe following system of equations will be inconsistent? $4 x+6 y=11$ $2 x+k y=7$ Solution: GIVEN: $4 x+6 y=11$ $2 x+k y=7$ To find: To determine for what value ofkthe system of equation will be inconsistent We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For the system of equation to be inconsistent $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ Here, $\frac{4}{2}=\frac{6}{k} \neq \frac{11}{7}$ $\frac{4}{2}=\fra...
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Question: $\sqrt{1-4 x-x^{2}}$ Solution: Let $I=\int \sqrt{1-4 x-x^{2}} d x$ $=\int \sqrt{1-\left(x^{2}+4 x+4-4\right)} d x$ $=\int \sqrt{1+4-(x+2)^{2}} d x$ $=\int \sqrt{(\sqrt{5})^{2}-(x+2)^{2}} d x$ It is known that, $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\mathrm{C}$ $\therefore I=\frac{(x+2)}{2} \sqrt{1-4 x-x^{2}}+\frac{5}{2} \sin ^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+\mathrm{C}$...
Read More →Find the value of k for which each of the following system of equations have no solution :
Question: Find the value ofkfor which each of the following system of equations have no solution : $c x+2 y=3$ $12 x+c y=6$ Solution: GIVEN: $c x+3 y=3$ $12 x+c y=6$ To find: To determine for what value ofcthe system of equation has no solution We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For no solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ Here, $\frac{c}{12}=\frac{3}{c} \neq \frac{3}{6}$ $\frac{c}{12}=\frac{3}{c}$ $c^{2}=12 ...
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Question: $\sqrt{x^{2}+4 x+1}$ Solution: Let $I=\int \sqrt{x^{2}+4 x+1} d x$ $=\int \sqrt{\left(x^{2}+4 x+4\right)-3} d x$ $=\int \sqrt{(x+2)^{2}-(\sqrt{3})^{2}} d x$ It is known that, $\int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+\mathrm{C}$ $\therefore I=\frac{(x+2)}{2} \sqrt{x^{2}+4 x+1}-\frac{3}{2} \log \left|(x+2)+\sqrt{x^{2}+4 x+1}\right|+C$...
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Question: $\sqrt{x^{2}+4 x+6}$ Solution: Let $I=\int \sqrt{x^{2}+4 x+6} d x$ $=\int \sqrt{x^{2}+4 x+4+2} d x$ $=\int \sqrt{\left(x^{2}+4 x+4\right)+2} d x$ $=\int \sqrt{(x+2)^{2}+(\sqrt{2})^{2}} d x$ It is known that, $\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+\mathrm{C}$ $\begin{aligned} \therefore I =\frac{(x+2)}{2} \sqrt{x^{2}+4 x+6}+\frac{2}{2} \log \left|(x+2)+\sqrt{x^{2}+4 x+6}\right|+\mathrm{C} \\ =\frac{(x+2)}{2} \sq...
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Question: $\sqrt{x^{2}+4 x+6}$ Solution: Let $I=\int \sqrt{x^{2}+4 x+6} d x$ $=\int \sqrt{x^{2}+4 x+4+2} d x$ $=\int \sqrt{\left(x^{2}+4 x+4\right)+2} d x$ $=\int \sqrt{(x+2)^{2}+(\sqrt{2})^{2}} d x$ It is known that, $\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+\mathrm{C}$ $\begin{aligned} \therefore I =\frac{(x+2)}{2} \sqrt{x^{2}+4 x+6}+\frac{2}{2} \log \left|(x+2)+\sqrt{x^{2}+4 x+6}\right|+\mathrm{C} \\ =\frac{(x+2)}{2} \sq...
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Question: $\sqrt{1-4 x^{2}}$ Solution: Let $I=\int \sqrt{1-4 x^{2}} d x=\int \sqrt{(1)^{2}-(2 x)^{2}} d x$ Let $2 x=t \Rightarrow 2 d x=d t$ $\therefore I=\frac{1}{2} \int \sqrt{(1)^{2}-(t)^{2}} d t$ It is known that, $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\mathrm{C}$ $\begin{aligned} \Rightarrow I =\frac{1}{2}\left[\frac{t}{2} \sqrt{1-t^{2}}+\frac{1}{2} \sin ^{-1} t\right]+\mathrm{C} \\ =\frac{t}{4} \sqrt{1-t^{2}}+\frac{1}{4} \sin ^{-1...
Read More →Write the coordinates of each of the following points marked in the graph paper.
Question: Write the coordinates of each of the following points marked in the graph paper. Solution: A (3, 1), B (6, 0), C (0, 6), D (-3, 0), E (- 4, 3), F (- 2, - 4), G (0, - 5), H (3,- 6), P (7, - 3), Q (7, 6) 1) The distance of point A from y-axis is 3 units and that of from x-axis is 1 units. Since A lies in the first quadrant, so its coordinates are (3, 1). 2) The distance of point B from y-axis is 6 units and that of from x-axis is 0 units. Since B lies on x-axis, so its coordinates are (6...
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Question: $\sqrt{4-x^{2}}$ Solution: Let $I=\int \sqrt{4-x^{2}} d x=\int \sqrt{(2)^{2}-(x)^{2}} d x$ It is known that, $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\mathrm{C}$ $\begin{aligned} \therefore I =\frac{x}{2} \sqrt{4-x^{2}}+\frac{4}{2} \sin ^{-1} \frac{x}{2}+\mathrm{C} \\ =\frac{x}{2} \sqrt{4-x^{2}}+2 \sin ^{-1} \frac{x}{2}+\mathrm{C} \end{aligned}$...
Read More →Find the value of k for which each of the following system of equations have no solution :
Question: Find the value ofkfor which each of the following system of equations have no solution : $2 x-k y+3=0$ $3 x+2 y-1=0$ Solution: GIVEN: $2 x-k y+3=0$ $3 x+2 y-1=0$ To find: To determine for what value ofkthe system of equation has no solution We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For no solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ Here, $\frac{2}{3}=\frac{-k}{2} \neq \frac{-3}{1}$ $\frac{2}{3}=\frac{-k}{2}$ $k=...
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Question: $\int e^{x} \sec x(1+\tan x) d x$ equals (A) $e^{x} \cos x+\mathrm{C}$ (B) $e^{x} \sec x+\mathrm{C}$ (C) $e^{x} \sin x+\mathrm{C}$ (D) $e^{x} \tan x+\mathrm{C}$ Solution: $\int e^{x} \sec x(1+\tan x) d x$ Let $I=\int e^{x} \sec x(1+\tan x) d x=\int e^{x}(\sec x+\sec x \tan x) d x$ Also, let $\sec x=f(x) \Rightarrow \sec x \tan x=f^{\prime}(x)$ It is known that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+\mathrm{C}$ $\therefore I=e^{x} \sec x+\mathrm{C}$ Hence, the corr...
Read More →Find the domain of each of the following real valued functions of real variable:
Question: Find the domain of each of the following real valued functions of real variable: (i) $f(x)=\sqrt{x-2}$ (ii) $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ (iii) $f(x)=\sqrt{9-x^{2}}$ (iv) $f(x)=\frac{\sqrt{x-2}}{3-x}$ Solution: (i) Given: $f(x)=\sqrt{x-2}$ Clearly,f(x) assumes real values ifx2 0. $\Rightarrow x \geq 2$ $\Rightarrow x \in[2, \infty)$ Hence, domain $(f)=[2, \infty)$. (ii) Given: $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ Clearly, $f(x)$ is defined for $x^{2}-10$. $(x+1)(x-1)0 \quad\left[\right.$ ...
Read More →Find the domain of each of the following real valued functions of real variable:
Question: Find the domain of each of the following real valued functions of real variable: (i) $f(x)=\sqrt{x-2}$ (ii) $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ (iii) $f(x)=\sqrt{9-x^{2}}$ (iv) $f(x)=\frac{\sqrt{x-2}}{3-x}$ Solution: (i) Given: $f(x)=\sqrt{x-2}$ Clearly,f(x) assumes real values ifx2 0. $\Rightarrow x \geq 2$ $\Rightarrow x \in[2, \infty)$ Hence, domain $(f)=[2, \infty)$. (ii) Given: $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ Clearly, $f(x)$ is defined for $x^{2}-10$. $(x+1)(x-1)0 \quad\left[\right.$ ...
Read More →Find the value of k for which each of the following system of equations have no solution :
Question: Find the value ofkfor which each of the following system of equations have no solution : $3 x-4 y+7=0$ $k x+3 y-5=0$ Solution: GIVEN: $3 x-4 y+7=0$ $k x+3 y-5=0$ To find: To determine for what value ofkthe system of equation has no solution We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For no solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ Here, $\frac{3}{k}=\frac{-4}{3} \neq \frac{-7}{5}$ $\frac{3}{k}=\frac{-4}{3}$ $k=...
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Question: $\int x^{2} e^{x^{3}} d x$ equals (A) $\frac{1}{3} e^{x^{2}}+\mathrm{C}$ (B) $\frac{1}{3} e^{x^{2}}+\mathrm{C}$ (C) $\frac{1}{2} e^{x^{3}}+\mathrm{C}$ (D) $\frac{1}{3} e^{x^{2}}+\mathrm{C}$ Solution: Let $I=\int x^{2} e^{x^{3}} d x$ Also, let $x^{3}=t \Rightarrow 3 x^{2} d x=d t$ $\begin{aligned} \Rightarrow I =\frac{1}{3} \int e^{t} d t \\ =\frac{1}{3}\left(e^{t}\right)+\mathrm{C} \\ =\frac{1}{3} e^{x^{3}}+\mathrm{C} \end{aligned}$ Hence, the correct answer is A....
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