Find the range of the function f
Question: Find the range of the function f(x) = sin x. Solution: The graph of sin(x) is Sin(x) is a periodic function whose values always lies between -1 to +1. The maximum value is attained at $n^{\frac{\pi}{2}}$ where $n$ is odd and minimum when $n$ is even. Hence, Range is $[-1,+1]$....
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Question: Tick (✓) the correct answer: On selling 100 pencils a man gains the selling price of 20 pencils. His gain per cent is (a) 20% (b) 25% (c) $22 \frac{1}{2} \%$ (d) $16 \frac{2}{3} \%$ Solution: (b) 25% Let the SP of 100 pens be Rs $x$. SP of 1 pen $=$ Rs $\frac{x}{100}$ Profit $=$ Rs $\frac{20 x}{100}$ $=\mathrm{Rs} \frac{x}{5}$ Now, $\mathrm{CP}=x-\frac{x}{5}$ $=\frac{4 x}{5}$ $\therefore$ Gain percentage $=\frac{\frac{x}{5}}{\frac{5 x}{5}} \times 100$ = 25%...
Read More →Convex mirror always forms a diminished
Question: Convex mirror always forms a diminished image of the object irrespective of the position of the object from the mirror. In which of the following, the image of an object placed at infinity will be highly diminished and point sized ? (a)Concave mirror only (b)Convex mirror only (c)Convex lens only (d)Concave mirror, convex mirror, concave lens and convex lens. Solution: (d).Explanation :...
Read More →Let A = {6, 10, 11, 15, 12} and let f
Question: Let $A=\{6,10,11,15,12\}$ and let $f: A \rightarrow N: f(n)$ is the highest prime factor of n. Find range (f). Solution: Given, A = {6, 10, 11, 15, 12} $f: A \rightarrow N: f(n)$ is the highest prime factor of $n$ (1) When $n=6$, the highest prime factor of 6 is 3 . Hence, f(6) = 3. (2) When n = 10, the highest prime factor of 10 is 5. Hence, f(10) = 5. (3) When $n=11$, the highest prime factor of 11 is 11 as 11 itself is a prime number. Hence, $f(11)=11$. (4) When n = 15, the highest ...
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Question: Tick (✓) the correct answer: If the selling price of 10 pens is the same as the cost price of 12 pens then gain per cent is (a) 2% (b) 12% (c) 20% (d) 25% Solution: (c) 20% Let Rs $x$ be the SP of each pen. SP of 10 pens $=$ CP of 12 pens $=$ Rs $12 x$ CP of 10 pens $=$ Rs $10 x$ Now, gain $=$ Rs $(12 \mathrm{x}-10 \mathrm{x})$ = Rs 12x $\therefore$ Gain percentage $=\left(\frac{\text { gain }}{\mathrm{CP}} \times 100\right) \%$ $=\left(\frac{2 x}{10 x} \times 100\right) \%$ $=20 \%$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $x^{\sin ^{-1} x}$ Solution: Let $y=x^{\sin ^{-1} x}$ Taking log both the sides: $\Rightarrow \log y=\log x^{\sin ^{-1} x}$ $\Rightarrow \log y=\sin ^{-1} x \log x\left\{\log x^{a}=\operatorname{alog} x\right\}$ Differentiating with respect to $x$ : $\Rightarrow \frac{\mathrm{d}(\log \mathrm{y})}{\mathrm{dx}}=\frac{\mathrm{d}\left(\sin ^{-1} \mathrm{x} \log \mathrm{x}\right)}{\mathrm{dx}}$ $\Rightarrow \frac{\mathrm{d}(\log \m...
Read More →A girl is standing in front of a magic mirror.
Question: A girl is standing in front of a magic mirror. She finds the image of her head bigger, the middle portion of her body of the same size and that of the legs smaller. The order of combinations for the magic mirror from the top is : (a)Convex, plane and concave (b)Plane, convex and concave (c)Concave, plane and convex (d)Convex, concave and plane. Solution: (c). Explanation :Concave mirror forms a magnified (enlarged) image of the object if the object is placed close to the concave mirror...
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Question: Tick (✓) the correct answer: Bananas are bought at 3 for Rs 2 and sold at 2 for Rs 3. The gain per cent is (a) 25% (b) 50% (c) 75% (d) 125% Solution: (d) 125% Cost price of a banana $=\mathrm{Rs} \frac{2}{3}$ Selling price of a banana $=\mathrm{Rs} \frac{3}{2}$ Now, profit $=\mathrm{Rs}\left(\frac{3}{2}-\frac{2}{3}\right)=\mathrm{Rs} \frac{9-4}{6}=\mathrm{Rs} \frac{5}{6}$ $\therefore$ Gain percentage $=\frac{\text { gain }}{\mathrm{CP}} \times 100$ $=\frac{\left(\frac{5}{6}\right)}{\le...
Read More →Which of the following ray diagrams
Question: Which of the following ray diagrams is correct for the ray of light incident on a lens shown in figure ? Solution: (a). Explanation :A ray of light passing through the focus of a lens travels parallel to the principal axis after refracting through the lens....
Read More →Solve this
Question: Let $f: R^{+} \rightarrow R: f(x)=\log _{e} x .$ Find $\{x: f(x)=-2\}$ Solution: Given, $f: R^{+} \rightarrow R: f(x)=\log _{e} x$ $f(x)=-2$ $\log _{e} x=-2$ Taking antilog on both sides $x=e^{-2}$ Hence, the value of $x$ for which $f(x)=-2$ is $e^{-2}$....
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Question: Tick (✓) the correct answer: On selling an article at a certain price a man gains 10%. On selling the same article at double the price, gain per cent is (a) 20% (b) 100% (c) 120% (d) 140% Solution: (c) 120% Let the SP and CP of the article be Rsxandy,respectively. Gain percentage = 10% $\Rightarrow 10=\frac{x-y}{y} \times 100$ $\Rightarrow \mathrm{y}=\frac{10 x}{11}$ According to the question, we have: SP = Rs 2x $\therefore$ Gain percentage $=\frac{\text { gain }}{\mathrm{CP}} \times ...
Read More →Which of the following ray diagrams
Question: Which of the following ray diagrams is correct for the ray of light incident on a concave mirror as shown in figure ? Solution: (d). Explanation :Any ray of light parallel to the principal axis passes through the focus (F) after reflecting from the concave mirror....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\left(\sin ^{-1} x\right)^{x}$ Solution: Let $y=\left(\sin ^{-1} x\right)^{x}$ Taking log both the sides: $\Rightarrow \log y=\log \left(\sin ^{-1} x\right)^{x}$ $\Rightarrow \log y=x \log \left(\sin ^{-1} x\right)\left\{\log x^{a}=\operatorname{alog} x\right\}$ Differentiating with respect to $x$ : $\Rightarrow \frac{\mathrm{d}(\log \mathrm{y})}{\mathrm{dx}}=\frac{\mathrm{d}\left(\mathrm{x} \log \left(\sin ^{-1} \mathrm{x}\r...
Read More →On selling an article for Rs 48, a shopkeeper loses 20%.
Question: On selling an article for Rs 48, a shopkeeper loses 20%. In order to gain 20%, what would be the selling price? (a) Rs 52 (b) Rs 56 (c) Rs 68 (d) Rs 72 Solution: (d) Rs 72 $\mathrm{SP}=\mathrm{Rs} 48$ Loss $=20 \%$ Now, $\mathrm{CP}=\frac{100}{100-\text { loss } \%} \times \mathrm{SP}$ $=\operatorname{Rs}\left(\frac{100}{(100-\text { loss } \%)} \times \mathrm{SP}\right)$ $=\operatorname{Rs}\left(\frac{100}{(100-20)} \times 48\right)$ $=\operatorname{Rs}\left(\frac{100}{80} \times 48\r...
Read More →Prove that
Question: Let $f: R \rightarrow R: f(x)=x^{2}+1$. Find $f^{-1}\{10\}$ Solution: Given: $f: R \rightarrow R: f(x)=x^{2}+1$ To find inverse of $f(x)$ Let $y=f(x)$ $y=x^{2}+1$ $y-1=x^{2}$ $\mathrm{x}=\sqrt{y-1}$ $f^{-1}(x)=\sqrt{x-1}$ Substituting x = 10, $f^{-1}(10)=\sqrt{10-1}=\sqrt{9}=3$...
Read More →You are given water, mustard oil,
Question: You are given water, mustard oil, glycerine and kerosene. In which of these media, a ray of light incident obliquely at same angle would bend the most ? (a)Kerosene (b)Water (c)Mustard Oil (d)Glycerine Solution: (d). Explanation :The ray would bend the most, when it goes from rarer medium (say air) to the most denser medium. Since refractive index of glycerine is the highest among all these medium, so glycerine is the most denser medium....
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Question: Tick (✓) the correct answer: A bookseller sells a book for Rs 100, gaining Rs 20. His gain per cent is (a) 20% (b) 25% (c) 22% (d) none of these Solution: (b) 25% $\mathrm{CP}=\mathrm{SP}-\mathrm{Gain}$ $=\mathrm{Rs}(100-20)$ $=\mathrm{Rs} 80$ $\therefore$ Gain percentage $=\left(\frac{\text { gain }}{\mathrm{CP}} \times 100\right) \%$ $=\left(\frac{20}{80} \times 100\right) \%$ $=25 \%$...
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Question: Let $f: R \rightarrow R: f(x)=x^{2}$ Determine (i) range (f) (ii) $\{x: f(x)=4\}$ Solution: Given: $f(x)=x^{2}$ The graph for the given function is (i) Range(f): For finding the range of the given function, let y = f(x) Therefore, $y=x^{2}$ $x=\sqrt{y}$ The value of $y \geq 0$. Hence, Range(f) is $[0, \infty)$. (ii) Let $y=f(x)=x^{2}$ Given $y=4$ Therefore, $x^{2}=4$ $x=2$ or $x=-2$ The set of values for which $y=4$ is $x=\{2,-2\}$....
Read More →The path of a ray of light coming from air passing
Question: The path of a ray of light coming from air passing through a rectangular glass slab traced by four students are shown as A, B, C and D in the figure. Which one of them is correct ? Solution: (B) Explanation :Glass slab causes the lateral displacement of a ray of light falling on it. However, incident ray and emergent ray are parallel to each other....
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Question: Tick (✓) the correct answer: A bat is bought for Rs 120 and sold for Rs 105. The loss per cent is (a) 15% (b) $12 \frac{1}{2} \%$ (c) $16 \frac{2}{3} \%$ (d) $14 \frac{1}{5} \%$ Solution: (b) $12 \frac{1}{2} \%$ $\mathrm{CP}=\mathrm{Rs} 120$ $\mathrm{SP}=\mathrm{Rs} 105$ Loss $=\mathrm{Rs}(120-105)$ = Rs15 $\therefore$ Loss percentage $=\left(\frac{\text { loss }}{\mathrm{CP}} \times 100\right)$ $=\left(\frac{15}{120} \times 100\right)$ $=12 \frac{1}{2} \%$...
Read More →The laws of reflection hold good for
Question: The laws of reflection hold good for (a)plane mirror only (b)concave mirror only (c)convex mirror only (d)all mirrors irrespective of their shape Solution: (d)all mirrors irrespective of their shape...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin \left(x^{x}\right)$ Solution: Let $y=\sin \left(x^{x}\right)$ Take sin inverse both sides: $\Rightarrow \sin ^{-1} y=\sin ^{-1}\left(\sin x^{x}\right)$ $\Rightarrow \sin ^{-1} y=x^{x}$ Taking log both the sides: $\Rightarrow \log \left(\sin ^{-1} y\right)=\log x^{x}$ $\Rightarrow \log \left(\sin ^{-1} y\right)=x \log x\left\{\log x^{a}=a \log x\right\}$ Differentiating with respect to $\mathrm{x}$ : $\Rightarrow \frac{\m...
Read More →In torches, search lights and head lights of vehicles,
Question: In torches, search lights and head lights of vehicles, the bulb is placed (a)between the pole and the focus of the reflector (b)very near to the focus of the reflector (c)between the focus and centre of curvature of the reflector (d)at the centre of curvature of the reflector. Solution: (b). Explanation :...
Read More →A full length image of a distance tall building
Question: A full length image of a distance tall building can definitely be seen by using (a)concave mirror (b)convex mirror (c)plane mirror (d)both concave as well as plane mirror. Solution: (b). Explanation :Convex mirror forms full length of a distant tall object irrespective of the position of the object. However, plane mirror forms full size image of the object if the size of the plane mirror is half the size of the object. Concave mirror forms full size image of the object if the object is...
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Question: Tick (✓) the correct answer: Rajan buys a toy for Rs 75 and sells it for Rs 100. His gain per cent is (a) 25% (b) 20% (c) $33 \frac{1}{3} \%$ (d) $37 \frac{1}{2} \%$ Solution: (c) $33 \frac{1}{3} \%$ $\mathrm{SP}=\mathrm{Rs} 100$ Gain $=\mathrm{Rs}(100-75)$ = RS 25 $\therefore$ Gain percentage $=\left(\frac{\text { gain }}{\mathrm{CP}} \times 100\right) \%$ $\quad=\left(\frac{25}{75} \times 100\right) \%$ $\quad=33 \frac{1}{3} \%$...
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