A lending library has a fixed charge for the first three days and an additional charge for each day thereafter.
Question: A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Aarushi paid Rs 27 for a book kept for seven days. If fixed charges are Rs x and per day charges are Rs y. Write the linear equation representing the above information. Solution: Total charges of Rs 27 of which Rs x for first three days and Rs y per day for 4 more days is given by x + y (7 3) = 27 x + 4y = 27 Here, (7 - 3) is taken as the charges for the first three days are ...
Read More →A three-wheeler scooter charges Rs 15 for first kilometer and Rs 8 each for every subsequent kilometer.
Question: A three-wheeler scooter charges Rs 15 for first kilometer and Rs 8 each for every subsequent kilometer. For a distance of x km, an amount of Rs y is paid. Write the linear equation representing the above information. Solution: Total fare of Rs y for covering the distance of x km is given by y = 15 + 8(x - 1) y = 15 + 8x - 8 y = 8x + 7 Where, Rs y is the total fare (x - 1) is taken as the cost of first kilometer is already given Rs 15 and 1 has to subtracted from the total distance trav...
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Question: $\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^{2} x} d x$ Solution: $\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^{2} x} d x$ Let $\cos x=t \Rightarrow-\sin x d x=d t$ When $x=0, t=1$ and when $x=\frac{\pi}{2}, t=0$ $\Rightarrow \int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^{2} x} d x=-\int_{1}^{0} \frac{d t}{1+t^{2}}$ $=-\left[\tan ^{-1} t\right]_{1}^{0}$ $=-\left[\tan ^{-1} 0-\tan ^{-1} 1\right]$ $=-\left[-\frac{\pi}{4}\right]$ $=\frac{\pi}{4}$...
Read More →The domain of definition of f(x) =
Question: The domain of definition of $f(x)=\sqrt{4 x-x^{2}}$ is (a) R [0, 4] (b) R (0, 4) (c) (0, 4) (d) [0, 4] Solution: (d) [0, 4] Given: $f(x)=\sqrt{4 x-x^{2}}$ Clearly,f(x) assumes real values if $4 x-x^{2} \geq 0$ $\Rightarrow x(4-x) \geq 0$ $\Rightarrow-x(x-4) \geq 0$ $\Rightarrow x(x-4) \leq 0$ $\Rightarrow x \in[0,4]$ Hence, domain (f)= [0, 4]....
Read More →The domain of definition of f(x) =
Question: The domain of definition of $f(x)=\sqrt{4 x-x^{2}}$ is (a) R [0, 4] (b) R (0, 4) (c) (0, 4) (d) [0, 4] Solution: (d) [0, 4] Given: $f(x)=\sqrt{4 x-x^{2}}$ Clearly,f(x) assumes real values if $4 x-x^{2} \geq 0$ $\Rightarrow x(4-x) \geq 0$ $\Rightarrow-x(x-4) \geq 0$ $\Rightarrow x(x-4) \leq 0$ $\Rightarrow x \in[0,4]$ Hence, domain (f)= [0, 4]....
Read More →The sum of a two-digit number and the number formed by reversing the order of digit is 66.
Question: The sum of a two-digit number and the number formed by reversing the order of digit is 66. If the two digits differ by 2, find the number. How many such numbers are there? Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The two digits of the number are differing by 2 . Thus, we have $x-y=\pm 2$ After interchanging the digits, the number becomes $10 x+y$. The sum of the numbers obtained by interchanging the ...
Read More →Give the equations of two lines passing through
Question: Give the equations of two lines passing through (3, 12). How many more such lines are there, and why? Solution: We observe that x = 3 and y = 12 is the solution of the following equations 4x y = 0 and 3x y + 3 = 0 So, we get the equations of two lines passing through (3, 12) are, 4x - y = 0 and 3x - y + 3 = 0. We know that passing through the given point infinitely many lines can be drawn. So, there are infinitely many lines passing through (3, 12)...
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Question: $\int_{0}^{2} x \sqrt{x+2}\left(\right.$ Put $\left.x+2=t^{2}\right)$ Solution: $\int_{0}^{2} x \sqrt{x+2} d x$ Let $x+2=t^{2} \Rightarrow d x=2 t d t$ When $x=0, t=\sqrt{2}$ and when $x=2, t=2$ $\therefore \int_{0}^{2} x \sqrt{x+2} d x=\int_{\sqrt{2}}^{2}\left(t^{2}-2\right) \sqrt{t^{2}} 2 t d t$ $=2 \int_{\sqrt{2}}^{2}\left(t^{2}-2\right) t^{2} d t$ $=2 \int_{\sqrt{2}}^{2}\left(t^{4}-2 t^{2}\right) d t$ $=2\left[\frac{t^{5}}{5}-\frac{2 t^{3}}{3}\right]_{\sqrt{2}}^{2}$ $=2\left[\frac{...
Read More →The domain of definition of the function f(x) =
Question: The domain of definition of the functionf(x) = log |x| is (a) R (b) (, 0) (c) (0, ) (d) R {0 Solution: (d) R {0} f(x) = log |x| For $f(x)$ to be defined, $|\mathrm{x}|0$, which is always true. But $|x| \neq 0$ $\Rightarrow x \neq 0$ Thus, $\operatorname{dom}(\mathrm{f})=\mathrm{R}-\{0\}$....
Read More →The domain of definition of the function f(x)
Question: The domain of definition of the function $f(x)=\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{1+x}}$ is (a) (, 2] [2, ) (b) [1, 1] (c) ϕ (d) None of these Solution: (c) ϕ $f(x)=\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{1+x}}$ For $\mathrm{f}(\mathrm{x})$ to be defined, $x+2 \neq 0$ $\Rightarrow x \neq-2 \ldots(1)$ And $1+x \neq 0$ $\Rightarrow \mathrm{x} \neq-1 \ldots(2)$ Also, $\frac{x-2}{x+2} \geq 0$ $\Rightarrow \frac{(x-2)(x+2)}{(x+2)^{2}} \geq 0$ $\Rightarrow(x-2)(x+2) \geq 0$ $\Rightarrow...
Read More →The domain of definition of the function f(x)
Question: The domain of definition of the function $f(x)=\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{1+x}}$ is (a) (, 2] [2, ) (b) [1, 1] (c) ϕ (d) None of these Solution: (c) ϕ $f(x)=\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{1+x}}$ For $\mathrm{f}(\mathrm{x})$ to be defined, $x+2 \neq 0$ $\Rightarrow x \neq-2 \ldots(1)$ And $1+x \neq 0$ $\Rightarrow \mathrm{x} \neq-1 \ldots(2)$ Also, $\frac{x-2}{x+2} \geq 0$ $\Rightarrow \frac{(x-2)(x+2)}{(x+2)^{2}} \geq 0$ $\Rightarrow(x-2)(x+2) \geq 0$ $\Rightarrow...
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Question: $\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$ Solution: Let $I=\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$ Also, let $x=\tan \theta \Rightarrow d x=\sec ^{2} \theta d \theta$ When $x=0, \theta=0$ and when $x=1, \theta=\frac{\pi}{4}$ $\begin{aligned} I =\int_{0}^{\pi} \sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right) \sec ^{2} \theta d \theta \\ =\int_{0}^{\pi} \sin ^{-1}(\sin 2 \theta) \sec ^{2} \theta d \theta \\ =\int_{0}^{\pi} 2 \theta ...
Read More →The domain of definition of the function f(x) =
Question: The domain of definition of the function $f(x)=\sqrt{x-1}+\sqrt{3-x}$ is (a) [1, ) (b) (, 3) (c) (1, 3) (d) [1, 3] Solution: (d) [1, 3] $f(x)=\sqrt{x-1}+\sqrt{3-x}$ For $\mathrm{f}(\mathrm{x})$ to be defined, $(x-1) \geq 0$ $\Rightarrow x \geq 1 \quad \ldots(1)$ and $(3-x) \geq 0$ $\Rightarrow 3 \leq x \quad \ldots(2)$ From $(1)$ and $(2)$, $x \in[1,3]$...
Read More →Draw the graph of each of the following linear equations in two variables:
Question: Draw the graph of each of the following linear equations in two variables: (i) x + y = 4 (ii) x - y = 2 (iii) -x + y = 6 (iv) y = 2x (v) 3x + 5y = 15 (vi) $\frac{x}{2}-\frac{y}{3}=2$ (vii) $\frac{x-2}{3}=y-3$ (viii) 2y = -x +1 Solution: (i) We are given, x + y = 4 We get, y = 4 - x, Now, substituting x = 0 in y = 4 - x, We get y = 4 Substituting x = 4 in y = 4 - x, we get y = 0 Thus, we have the following table exhibiting the abscissa and ordinates of points on the line represented by ...
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Question: $\int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^{5} \phi d \phi$ Solution: Let $I=\int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^{5} \phi d \phi=\int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^{4} \phi \cos \phi d \phi$ Also, let $\sin \phi=t \Rightarrow \cos \phi d \phi=d t$ When $\phi=0, t=0$ and when $\phi=\frac{\pi}{2}, t=1$ $\therefore I=\int_{0}^{t} \sqrt{t}\left(1-t^{2}\right)^{2} d t$ $=\int_{0}^{1} t^{\frac{1}{2}}\left(1+t^{4}-2 t^{2}\right) d t$ $=\int_{0}^{1}\left[t^{\frac...
Read More →The domain of the function f(x)
Question: The domain of the function $f(x)=\sqrt{\frac{(x+1)(x-3)}{x-2}}$ is (a) [1, 2) [3, ) (b) (1, 2) [3, ) (c) [1, 2] [3, ) (d) None of these Solution: (a) [1, 2) [3, ) $f(x)=\sqrt{\frac{(x+1)(x-3)}{x-2}}$ For $f(x)$ to be defined, $(x-2) \neq 0$ $\Rightarrow \mathrm{x} \neq 2$ ....(1) Also, $\frac{(x+1)(x-3)}{(x-2)} \geq 0$ $\Rightarrow \frac{(x+1)(x-3)(x-2)}{(x-2)^{2}} \geq 0$ $\Rightarrow(\mathrm{x}+1)(\mathrm{x}-3)(\mathrm{x}-2) \geq 0$ $\Rightarrow \mathrm{x} \in[-1,2) \cup[3, \infty) \...
Read More →The domain of the function f(x)
Question: The domain of the function $f(x)=\sqrt{\frac{(x+1)(x-3)}{x-2}}$ is (a) [1, 2) [3, ) (b) (1, 2) [3, ) (c) [1, 2] [3, ) (d) None of these Solution: (a) [1, 2) [3, ) $f(x)=\sqrt{\frac{(x+1)(x-3)}{x-2}}$ For $f(x)$ to be defined, $(x-2) \neq 0$ $\Rightarrow \mathrm{x} \neq 2$ ....(1) Also, $\frac{(x+1)(x-3)}{(x-2)} \geq 0$ $\Rightarrow \frac{(x+1)(x-3)(x-2)}{(x-2)^{2}} \geq 0$ $\Rightarrow(\mathrm{x}+1)(\mathrm{x}-3)(\mathrm{x}-2) \geq 0$ $\Rightarrow \mathrm{x} \in[-1,2) \cup[3, \infty) \...
Read More →The domain of definition of f(x)
Question: The domain of definition of $f(x)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$ is (a) (, 3] (2, 5) (b) (, 3) (2, 5) (c) (, 3) [2, 5] (d) None of these Solution: (a) (, 3] (2, 5) $f(x)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$ For $\mathrm{f}(\mathrm{x})$ to be defined, $(2-x)(x-5) \neq 0$ $\Rightarrow x \neq 2,5$ ....(1) Also, $\frac{(x+3)}{(2-x)(x-5)} \geq 0$ $\Rightarrow \frac{(x+3)(2-x)(x-5)}{(2-x)^{2}(x-5)^{2}} \geq 0$ $\Rightarrow(x+3)(x-2)(x-5) \leq 0$ $\Rightarrow x \in(-\infty,-3] \cup(2,5) \quad \ldot...
Read More →The domain of definition of f(x)
Question: The domain of definition of $f(x)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$ is (a) (, 3] (2, 5) (b) (, 3) (2, 5) (c) (, 3) [2, 5] (d) None of these Solution: (a) (, 3] (2, 5) $f(x)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$ For $\mathrm{f}(\mathrm{x})$ to be defined, $(2-x)(x-5) \neq 0$ $\Rightarrow x \neq 2,5$ ....(1) Also, $\frac{(x+3)}{(2-x)(x-5)} \geq 0$ $\Rightarrow \frac{(x+3)(2-x)(x-5)}{(2-x)^{2}(x-5)^{2}} \geq 0$ $\Rightarrow(x+3)(x-2)(x-5) \leq 0$ $\Rightarrow x \in(-\infty,-3] \cup(2,5) \quad \ldot...
Read More →The sum of digits of a two number is 15.
Question: The sum of digits of a two number is 15. The number obtained by reversing the order of digits of the given number exceeds the given number by 9. Find the given number. Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The sum of the digits of the number is 15 . Thus, we have $x+y=15$ After interchanging the digits, the number becomes $10 x+y$. The number obtained by interchanging the digits is exceeding by 9 ...
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Question: $\int_{0}^{1} \frac{x}{x^{2}+1} d x$ Solution: $\int_{0}^{1} \frac{x}{x^{2}+1} d x$ Let $x^{2}+1=t \Rightarrow 2 x d x=d t$ When $x=0, t=1$ and when $x=1, t=2$ $\therefore \int_{0}^{1} \frac{x}{x^{2}+1} d x=\frac{1}{2} \int_{1}^{2} \frac{d t}{t}$ $=\frac{1}{2}[\log |t|]_{1}^{2}$ $=\frac{1}{2}[\log 2-\log 1]$ $=\frac{1}{2} \log 2$...
Read More →The domain of the function f(x)
Question: The domain of the function $f(x)=\sqrt{2-2 x-x^{2}}$ is (a) $[-\sqrt{3}, \sqrt{3}]$ (b) $[-1-\sqrt{3},-1+\sqrt{3}]$ (c) $[-2,2]$ (d) $[-2-\sqrt{3},-2+\sqrt{3}]$ Solution: (b) $[-1-\sqrt{3},-1+\sqrt{3}]$ $f(x)=\sqrt{2-2 x-x^{2}}$ Since, $2-2 x-x^{2} \geq 0$ $x^{2}+2 x-2 \leq 0$ $\Rightarrow x^{2}-2 x-2+1-1 \leq 0$ $\Rightarrow(x-1)^{2}-(\sqrt{3})^{2} \leq 0$ $\Rightarrow[x-(-1-\sqrt{3})][x-(-1+\sqrt{3})] \leq 0$ $\Rightarrow(-1-\sqrt{3}) \leq x \leq(-1+\sqrt{3})$ Thus, $\operatorname{do...
Read More →The domain of the function f(x)
Question: The domain of the function $f(x)=\sqrt{2-2 x-x^{2}}$ is (a) $[-\sqrt{3}, \sqrt{3}]$ (b) $[-1-\sqrt{3},-1+\sqrt{3}]$ (c) $[-2,2]$ (d) $[-2-\sqrt{3},-2+\sqrt{3}]$ Solution: (b) $[-1-\sqrt{3},-1+\sqrt{3}]$ $f(x)=\sqrt{2-2 x-x^{2}}$ Since, $2-2 x-x^{2} \geq 0$ $x^{2}+2 x-2 \leq 0$ $\Rightarrow x^{2}-2 x-2+1-1 \leq 0$ $\Rightarrow(x-1)^{2}-(\sqrt{3})^{2} \leq 0$ $\Rightarrow[x-(-1-\sqrt{3})][x-(-1+\sqrt{3})] \leq 0$ $\Rightarrow(-1-\sqrt{3}) \leq x \leq(-1+\sqrt{3})$ Thus, $\operatorname{do...
Read More →A number consist of two digits whose sum is five.
Question: A number consist of two digits whose sum is five. When the digits are reversed, the number becomes greater by nine. Find the number. Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The sum of the digits of the number is 5 . Thus, we have $x+y=5$ After interchanging the digits, the number becomes $10 x+y$. The number obtained by interchanging the digits is greater by 9 from the original number. Thus, we have...
Read More →A number consist of two digits whose sum is five.
Question: A number consist of two digits whose sum is five. When the digits are reversed, the number becomes greater by nine. Find the number. Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The sum of the digits of the number is 5 . Thus, we have $x+y=5$ After interchanging the digits, the number becomes $10 x+y$. The number obtained by interchanging the digits is greater by 9 from the original number. Thus, we have...
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