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Question: Solution: Let $I=\int_{1}^{4}\left(x^{2}-x\right) d x$ $=\int_{1}^{4} x^{2} d x-\int_{1}^{4} x d x$ Let $I=I_{1}-I_{2}$, where $I_{1}=\int_{1}^{4} x^{2} d x$ and $I_{2}=\int_{1}^{4} x d x$ ...(1) It is known that, $\int_{a}^{b} f(x) d x=(b-a) \lim _{n \rightarrow \infty} \frac{1}{n}[f(a)+f(a+h)+f(a+(n-1) h)]$, where $h=\frac{b-a}{n}$ For $I_{1}=\int_{1}^{4} x^{2} d x$ $a=1, b=4$, and $f(x)=x^{2}$ $\therefore h=\frac{4-1}{n}=\frac{3}{n}$ $I_{1}=\int_{1}^{4} x^{2} d x=(4-1) \lim _{n \rig...
Read More →Find the values of a and b for which the following system of linear equations has infinite number of solutions :
Question: Find the values ofaandbfor which the following system of linear equations has infinite number of solutions : $2 x-3 y=7$ $(a+b) x-(a+b-3) y=4 a+b$ Solution: GIVEN: $2 x-3 y=7$ $(a+b) x-(a+b-3) y=4 a+b$ To find: To determine for what value ofkthe system of equation has infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Here $\frac{2}{(a+b...
Read More →If f(x) be defined on [−2, 2] and is given by f(x)
Question: If $f(x)$ be defined on $[-2,2]$ and is given by $f(x)=\left\{\begin{array}{rr}-1, -2 \leq x \leq 0 \\ x-1, 0x \leq 2\end{array}\right.$ and $g(x)=f(|x|)+|f(x)|$, find $g(x)$. Solution: Given: $f(x)= \begin{cases}-1, -2 \leqslant x \leqslant 0 \\ x-1, 0x \leqslant 2\end{cases}$ Thus, $g(x)=f(|x|)+|f(x)|$ $= \begin{cases}x-1+1, -2 \leqslant x \leqslant 0 \\ x-1+(-x+1), 0x1 \\ x-1+x-1, 1 \leq x \leq 2\end{cases}$ $= \begin{cases}x, -2 \leqslant x \leqslant 0 \\ 0, 0x1 \\ 2 x-2, 1 \leq x ...
Read More →Let f(x) = 2x + 5
Question: Let $f(x)=2 x+5$ and $g(x)=x^{2}+x$. Describe (i) $f+g$ (ii) $f-g$ (iii) $f g$ (iv) $f / g$. Find the domain in each case. Solution: Given: f(x) = 2x+ 5 andg(x) =x2+x Clearly,f(x) andg(x) assume real values for allx. Hence, domain (f) =Rand domain (g) =R. $\therefore \quad D(f) \cap D(g)=R$ Now, (i) (f+g) :RRis given by (f+g) (x) =f(x) +g(x) = 2x+ 5 +x2+x=x2+ 3x + 5. Hence, domain (f+g) =R. (ii) $(f-g): R \rightarrow R$ is given by $(f-g)(x)=f(x)-g(x)=(2 x+5)-\left(x^{2}+x\right)=5+x-x...
Read More →Let f(x) = 2x + 5
Question: Let $f(x)=2 x+5$ and $g(x)=x^{2}+x$. Describe (i) $f+g$ (ii) $f-g$ (iii) $f g$ (iv) $f / g$. Find the domain in each case. Solution: Given: f(x) = 2x+ 5 andg(x) =x2+x Clearly,f(x) andg(x) assume real values for allx. Hence, domain (f) =Rand domain (g) =R. $\therefore \quad D(f) \cap D(g)=R$ Now, (i) (f+g) :RRis given by (f+g) (x) =f(x) +g(x) = 2x+ 5 +x2+x=x2+ 3x + 5. Hence, domain (f+g) =R. (ii) $(f-g): R \rightarrow R$ is given by $(f-g)(x)=f(x)-g(x)=(2 x+5)-\left(x^{2}+x\right)=5+x-x...
Read More →Determine the values of a and b so that the following system of linear equations have infinitely many solutions :
Question: Determine the values ofaandbso that the following system of linear equations have infinitely many solutions : $(2 a-1) x+3 y-5=0$ $3 x+(b-1) y-2=0$ Solution: GIVEN: $(2 a-1) x+3 y-5=0$ $3 x+(b-1) y-2=0$ To find: To determine for what value ofkthe system of equation has infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Here $\frac{(2 a-1...
Read More →The perimeter of an isosceles triangle is 42 cm and its base is 3/2 times each of the equal side.
Question: The perimeter of an isosceles triangle is 42 cm and its base is 3/2 times each of the equal side. Find the length of each of the triangle, area of the triangle and the height of the triangle. Solution: Let 'x' be the length of two equal sides, Therefore the base =1/2 x Let the sides a, b, c of a triangle be1/2 x, x and x respectively So, the perimeter = 2s = a + b + c 42 = a + b + c 42 =3/2 x+ x + x Therefore, x = 12 cm So, the respective sides are a = 12 cm b = 12 cm c = 18 cm Now, se...
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Question: $\int_{2}^{3} x^{2} d x$ Solution: It is known that, $\int_{a}^{b} f(x) d x=(b-a) \lim _{n \rightarrow \infty} \frac{1}{n}[f(a)+f(a+h)+f(a+2 h) \ldots f\{a+(n-1) h\}]$, where $h=\frac{b-a}{n}$ Here, $a=2, b=3$, and $f(x)=x^{2}$ $\Rightarrow h=\frac{3-2}{n}=\frac{1}{n}$ $\therefore \int_{2}^{3} x^{2} d x=(3-2) \lim _{n \rightarrow \infty} \frac{1}{n}\left[f(2)+f\left(2+\frac{1}{n}\right)+f\left(2+\frac{2}{n}\right) \ldots f\left\{2+(n-1) \frac{1}{n}\right\}\right]$ $=\lim _{n \rightarro...
Read More →Find f + g, f − g,
Question: Find $1+g, 1-g, c f(c \in \mathrm{R}, c \neq 0), f g, \frac{1}{f}$ and $\frac{f}{g}$ in each of the following: (a) If $f(x)=x^{3}+1$ and $g(x)=x+1$ (b) If $f(x)=\sqrt{x-1}$ and $g(x)=\sqrt{x+1}$ Solution: (a) Given: $f(x)=x^{3}+1$ and $g(x)=x+1$ Thus, $(f+g)(x): R \rightarrow R$ is given by $(f+g)(x)=f(x)+g(x)=x^{3}+1+x+1=x^{3}+x+2$ $(f-g)(x): R \rightarrow R$ is given by $(f-g)(x)=f(x)-g(x)=\left(x^{3}+1\right)-(x+1)=x^{3}+1-x-1=x^{3}-x$ $c f: R \rightarrow R$ is given by $(c f)(x)=c\...
Read More →Obtain the condition for the following system of linear equations to have a unique solution
Question: Obtain the condition for the following system of linear equations to have a unique solution $a x+b y=c$ $b x+m y=n$ Solution: GIVEN: $a x+b y=c$ $l x+m y=n$ To find: To determine the condition for the system of equation to have a unique equation We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For unique solution $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ Here $\frac{a}{l} \neq \frac{b}{m}$ $a m \neq b l$ Hence for $a m \neq b l$ the system of eq...
Read More →The lengths of the sides of a triangle are in a ratio of 3: 4: 5 and its perimeter is 144 cm.
Question: The lengths of the sides of a triangle are in a ratio of 3: 4: 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side? Solution: Given the perimeter of a triangle is 160m and the sides are in a ratio of3: 4: 5 Let the sides a, b, c of a triangle be 3x, 4x, 5x respectively So, the perimeter = 2s = a + b + c 144 = a + b + c 144 = 3x + 4x + 5x Therefore, x = 12 cm So, the respective sides are a = 36 cm b = 48 cm c = 60 cm Now, semi pe...
Read More →For what value of k, the following system of equations will represent the coincident lines?
Question: For what value ofk, the following system of equations will represent the coincident lines? $x+2 y+7=0$ $2 x+k y+14=0$ Solution: GIVEN: $x+2 y+7=0$ $2 x+k y+14=0$ To find: To determine for what value ofkthe system of equation will represents coincident lines 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 represent coincident lines we have the following relation $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}...
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Question: $\int_{0}^{5}(x+1) d x$ Solution: $\operatorname{Let} I=\int_{0}^{5}(x+1) d x$ It is known that, $\int_{a}^{b} f(x) d x=(b-a) \lim _{n \rightarrow \infty} \frac{1}{n}[f(a)+f(a+h) \ldots f(a+(n-1) h)]$, where $h=\frac{b-a}{n}$ Here, $a=0, b=5$, and $f(x)=(x+1)$ $\Rightarrow h=\frac{5-0}{n}=\frac{5}{n}$ $\therefore \int_{0}^{5}(x+1) d x=(5-0) \lim _{n \rightarrow \infty} \frac{1}{n}\left[f(0)+f\left(\frac{5}{n}\right)+\ldots+f\left((n-1) \frac{5}{n}\right)\right]$ $=5 \lim _{n \rightarro...
Read More →Find the values of k for which the system will have (i) a unique solution, and (ii) no solution. Is there a value of k for which the system has infinitely many solutions?
Question: Find the values ofkfor which the system will have (i) a unique solution (ii) no solution. Is there a value ofkfor which the system has infinitely many solutions? $2 x+k y=1$ $3 x-5 y=7$ Solution: GIVEN: $2 x+k y=1$ $3 x-5 y=7$ To find: To determine for what value ofkthe system of equation has (1) Unique solution (2) No solution (3) Infinitely many 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}} \...
Read More →A triangle has sides 35 cm, 54 cm, 61 cm long. Find its area. Also, find the smallest of its altitudes?
Question: A triangle has sides 35 cm, 54 cm, 61 cm long. Find its area. Also, find the smallest of its altitudes? Solution: Given, The sides of the triangle are a = 35 cm b = 54 cm c = 61 cm Perimeter 2s = a + b + c 2s = 35 + 54 + 61 cm Semi perimeter s = 75 cm By using Heron's Formula, Area of the triangle $=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ $=\sqrt{75 \times(75-35) \times(75-54) \times(75-61)}$ $=939.14 \mathrm{~cm}^{2}$ The altitude will be smallest provided the side corresponding...
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Question: $\int_{0}^{b} x d x$ Solution: It is known that, $\int_{a}^{b} f(x) d x=(b-a) \lim _{n \rightarrow \infty} \frac{1}{n}[f(a)+f(a+h)+\ldots+f(a+(n-1) h)]$, where $h=\frac{b-a}{n}$ Here, $a=a, b=b$, and $f(x)=x$ $\therefore \int_{a}^{b} x d x=(b-a) \lim _{n \rightarrow \infty} \frac{1}{n}[a+(a+h) \ldots(a+2 h) \ldots a+(n-1) h]$ $=(b-a) \lim _{n \rightarrow \infty} \frac{1}{n}[(a+a+a+\ldots+a)+(h+2 h+3 h+\ldots+(n-1) h)]$ $=(b-a) \lim _{n \rightarrow \infty} \frac{1}{n}[n a+h(1+2+3+\ldots...
Read More →The perimeter of a triangular field is 240 dm. If two of its sides are 78 dm and 50 dm,
Question: The perimeter of a triangular field is 240 dm. If two of its sides are 78 dm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex. Solution: Given, In a triangle ABC, a = 78 dm = AB, b = 50 dm = BC Now, Perimeter = 240 dm Then, AB + BC + AC = 240 dm 78 + 50 + AC = 240 AC = 240 - (78 + 50) AC = 112 dm = c Now, 2s = a + b + c 2s = 78 + 50 + 112 s = 120 dm Area of a triangle $\mathrm{ABC}=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ $=\sqrt...
Read More →Prove that there is a value of c(≠ 0) for which the system has infinitely many solutions. Find this value.
Question: Prove that there is a value ofc( 0) for which the system has infinitely many solutions. Find this value. $6 x+3 y=c-3$ $12 x+c y=c$ Solution: GIVEN: $6 x+3 y=c-3$ $12 x+c y=c$ To find: To determine for what value ofcthe system of equation has infinitely many 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 infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Here $\frac{6}{12}=\frac{3}{c}=\frac{c-3}{c}$ ...
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Question: $\int \sqrt{x^{2}-8 x+7} d x$ is equal to A. $\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}+9 \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|+C$ B. $\frac{1}{2}(x+4) \sqrt{x^{2}-8 x+7}+9 \log \left|x+4+\sqrt{x^{2}-8 x+7}\right|+C$ C. $\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}-3 \sqrt{2} \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|+C$ D. $\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}-\frac{9}{2} \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|+C$ Solution: Let $I=\int \sqrt{x^{2}-8 x+7} d x$ $=\int \sqrt{\left(x^{2}-8 x+16\right)-...
Read More →The perimeter of a triangle is 300 m. If its sides are in the ratio of 3: 5: 7. Find the area of the triangle.
Question: The perimeter of a triangle is 300 m. If its sides are in the ratio of 3: 5: 7. Find the area of the triangle. Solution: Given the perimeter of a triangle is 300 m and the sides are in a ratio of 3: 5: 7 Let the sides a, b, c of a triangle be 3x, 5x, 7x respectively So, the perimeter = 2s = a + b + c 200 = a + b + c 300 = 3x + 5x + 7x 300 = 15x Therefore, x = 20 m So, the respective sides are a = 60 m b = 100 m c = 140 m Now, semi perimeter $s=\frac{a+b+c}{2}$ $=\frac{60+100+140}{2}$ =...
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Question: $\int \sqrt{1+x^{2}} d x$ is equal to A. $\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log \left|x+\sqrt{1+x^{2}}\right|+\mathrm{C}$ B. $\frac{2}{3}\left(1+x^{2}\right)^{\frac{2}{3}}+\mathrm{C}$ C. $\frac{2}{3} x\left(1+x^{2}\right)^{\frac{3}{2}}+\mathrm{C}$ D. $\frac{x^{2}}{2} \sqrt{1+x^{2}}+\frac{1}{2} x^{2} \log \left|x+\sqrt{1+x^{2}}\right|+\mathrm{C}$ Solution: It is known that, $\int \sqrt{a^{2}+x^{2}} d x=\frac{x}{2} \sqrt{a^{2}+x^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\...
Read More →The perimeter of a triangular field is 540 m and its sides are in the ratio
Question: The perimeter of a triangular field is 540 m and its sides are in the ratio 25: 17: 12. Find the area of triangle. Solution: Let the sides of the given triangle be a = 25x, b = 17x, c = 12x respectively, So, a = 25x cm b = 17x cm c = 12x cm Given Perimeter = 540 cm 2s = a + b + c a + b + c = 540 cm 25x + 17x + 12x = 540 cm 54x = 540 cm x = 10 cm Therefore, the sides of a triangle are a = 250 cm b = 170 cm c = 120 cm Now, Semi perimeter s =(a + b + c)/2 =540/2 = 270 cm By using Heron's ...
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: $\sqrt{1+\frac{x^{2}}{9}}$ Solution: Let $I=\int \sqrt{1+\frac{x^{2}}{9}} d x=\frac{1}{3} \int \sqrt{9+x^{2}} d x=\frac{1}{3} \int \sqrt{(3)^{2}+x^{2}} d x$ 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{1}{3}\left[\frac{x}{2} \sqrt{x^{2}+9}+\frac{9}{2} \log \left|x+\sqrt{x^{2}+9}\right|\right]+\mathrm{C} \\ =\frac{x}{6} \sqrt{x^{2}+9}+\frac{3}{2} \log ...
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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