Prove that
Question: Let $A=\{-2,-1,0,2\}$ and $f: A \rightarrow Z: f(x)=x^{2}-2 x-3 .$ Find $f(A)$ Solution: Given : $A=\{-2,-1,0,2\}$ $f: A \rightarrow Z: f(x)=x^{2}-2 x-3$ Finding f(x) for each value of x, $f(-2)=(-2)^{2}-2(-2)-3=4+4-3=5$ $f(-1)=(-1)^{2}-2(-1)-3=1+2-3=0$ $f(0)=(0)^{2}-2(0)-3=0+0-3=-3$ $f(2)=(2)^{2}-2(2)-3=4-4-3=-3$ f in ordered pair is represented as $f=\{(-2,5),(-1,0),(0,-3),(2,-3)\}$...
Read More →Rays from sun converge at a point 15 cm
Question: Rays from sun converge at a point 15 cm in front of a concave mirror. Where should an object be placed so that size of its image is equal to the size of the object ? (a)15 cm in front of the mirror (b)30 cm in front of the mirror (c)between 15 cm and 30 cm in front of the mirror. (d)more than 30 cm in front of the mirror. Solution: (b). Explanation :Here, Focal length of concave mirror, f = -15 cm Radius of curvature of the mirror, R = 2f = -30 cm. In case of concave mirror, size of im...
Read More →vThe value of a car including VAT is Rs 382500.
Question: The value of a car including VAT is Rs 382500. If the basic price of the car be Rs 340000, find the rate of VAT on cars. Solution: Let the rate of VAT be x%. Then, we have: $34000+x \%$ of $34000=382500$ $\Rightarrow\left(\frac{x}{100} \times 340000\right)=382500-340000$ $\Rightarrow 3400 x=42500$ $\Rightarrow x=\frac{42500}{3400}$ $=12.5$ The rate of VAT is 12.5%....
Read More →Magnification produced by a rear view mirror
Question: Magnification produced by a rear view mirror fitted in vehicles. (a)is less than one (b)is more than one (c)is equal to one (d)can be more than or less than one depending upon the position of the object in front of it. Solution: (a). Explanation : $m=\frac{\text { size of image }}{\text { size of object }}$ Rear view mirror is a convex mirror, which always forms an image whose size is less than the size of the object....
Read More →Malti bought a VCR for Rs 19980 including VAT.
Question: Malti bought a VCR for Rs 19980 including VAT. If the original price of VCR be Rs 18500, find the rate of VAT. Solution: Let the rate of VAT bex%. Then, we have: $18500+x \%$ of $18500=19980$ $\Rightarrow\left(\frac{x}{100} \times 18500\right)=19980-18500$ $\Rightarrow 185 x=1480$ $\Rightarrow x=\frac{1480}{185}$ $\quad=8$ The rate of VAT is 8%....
Read More →Solve this
Question: Let $X=\{-1,0,2,5\}$ and $f: X \rightarrow R Z: f(x)=x^{3}+1$. Then, write $f$ as a set of ordered pairs. Solution: Given, $X=\{-1,0,2,5\}$ $f: X \rightarrow R Z: f(x)=x^{3}+1$ Finding f(x) for each value of x, (1) $f(-1)=(-1)^{3}+1=-1+1=0$ (2) $f(0)=(0)^{3}+1=0+1=1$ (3) $f(2)=(2)^{3}+1=8+1=9$ (4) $f(5)=(5)^{3}+1=125+1=126$ f in ordered pair is represented as $f=\{(-1,0),(0,1),(2,9),(5,126)\}$...
Read More →Rohit purchased a pair of shoes for Rs 882 inclusive of VAT.
Question: Rohit purchased a pair of shoes for Rs 882 inclusive of VAT. If the original cost be Rs 840, find the rate of VAT. Solution: Let the rate of VAT bex%. Then, we have: $840+x \%$ of $840=882$ $\Rightarrow\left(\frac{x}{100} \times 840\right)=882-840$ $\Rightarrow \frac{84 x}{10}=42$ $\Rightarrow x=\left(42 \times \frac{10}{84}\right)$ $=5$ $\therefore$ The rate of VAT is $5 \%$...
Read More →Which of the following statements is true ?
Question: Which of the following statements is true ? (a)A convex lens has 4 dioptre power having focal length 0.25 m. (b)A convex lens has 4 dioptre power having focal length 0.25 m. (c)A concave lens has 4 dioptre power having focal length 0.25 m. (d)A concave lens has 4 dioptre power having focal length 0.25 m. Solution: (a). Explanation: Convex lens has positive power and positive focal length. $\mathrm{P}=\frac{1}{f} \quad \therefore f=\frac{1}{\mathrm{P}}=\frac{1}{4}=0.25 \mathrm{~m}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $10^{(10 x)}$ Solution: Let $y=10^{10 x}$ Taking log both the sides: $\Rightarrow \log y=\log 10^{10 x}$ $\Rightarrow \log y=10 x \log 10\left\{\log x^{a}=\operatorname{alog} x\right\}$ $\Rightarrow \log y=(10 \log 10) x$ Differentiating with respect to $x$ : $\Rightarrow \frac{d(\log y)}{d x}=\frac{d\{(10 \log 10) x\}}{d x}$ $\Rightarrow \frac{\mathrm{d}(\log \mathrm{y})}{\mathrm{dx}}=10 \times \log (10) \times \frac{\mathrm{...
Read More →A beam of light is incident through the holes
Question: A beam of light is incident through the holes on side A and emerges out of the holes on the other side of the box as shown in figure. Which of the following could be inside the box ? (a)Concave lens (b)Rectangular slab (c)prism (d)Convex lens Solution: (d). Explanation:...
Read More →The sale price of a TV set including VAT is Rs 27000.
Question: The sale price of a TV set including VAT is Rs 27000. If the VAT is charged at 8% of the list price, what is the list price of the TV set? Solution: Let the list price of the IV set be Rs $x$. $\mathrm{VAT}=8 \%$ of Rs. $x$ $=\operatorname{Rs} \cdot\left(x \times \frac{8}{100}\right)$ $=\operatorname{Rs} \cdot \frac{8 x}{100}$ $\therefore$ Price including VAT $=\mathrm{Rs} .\left(x+\frac{8 x}{100}\right)$ $=$ Rs. $\frac{108 x}{100}$ Now, $\frac{108 x}{100}=27000$ $\Rightarrow x=\left(2...
Read More →Find the set of values for which the function
Question: Find the set of values for which the function $f(x)=x+3$ and $g(x)=3 x^{2}-1$ are equal. Solution: $f(x)=x+3, g(x)=3 x^{2}-1$ To find:- Set of values of $x$ for which $f(x)=g(x)$ Consider, $f(x)=g(x)$ $f(x)=g(x)$ $x+3=3^{x^{2}}-1$ $3^{x^{2}}-x-4=0$ $3^{x^{2}}-4 x+3 x-4=0$ $x(3 x-4)+(3 x-4)=0$ $(3 x-4)(x+1)=0$ $x=4 / 3$ or $x=-1$ The set values for which $f(x)$ and $g(x)$ have same value is $\{4 / 3,-1\}$....
Read More →Sajal purchased some car parts for Rs 20776 including VAT at 12%.
Question: Sajal purchased some car parts for Rs 20776 including VAT at 12%. What is the original cost of these spare parts? Solution: Let the original cost of the spare parts be Rs $x$. $\mathrm{VAT}=12 \%$ of Rs. $x$ $=$ Rs. $\left(x \times \frac{12}{100}\right)$ $=$ Rs. $\frac{12 x}{100}$ $\therefore$ Price including VAT $=$ Rs. $\left(x+\frac{12 x}{100}\right)$ $=$ Rs. $\frac{112 x}{100}$ Now, $\frac{112 x}{100}=20776$ $\Rightarrow x=\left(20776 \times \frac{100}{112}\right)$ = 18550 Hence, ...
Read More →Beams of light are incident through the holes A and B
Question: Beams of light are incident through the holes A and B and emerge out of a box through the holes C and D respectively as shown in figure. Which of the following could be inside the box ? (a)A rectangular glass slab (b)A convex lens (c)A concave lens (d)A prism. Solution: (a). Explanation :Rectangular glass slab causes the lateral displacement of a ray of light following on it. However, incident ray and emergent ray are paralled to each other....
Read More →Find the set of values for which the function
Question: Find the set of values for which the function $f(x)=1-3 x$ and $g(x)=2 x^{2}-1$ are equal. Solution: $f(x)=1-3 x, g(x)=2 x^{2}-1$ To find:- Set of values of $x$ for which $f(x)=g(x)$ Consider, $f(x)=g(x)$ $1-3 x=2^{x^{2}}-1$ $2^{x^{2}}+3 x-2=0$ $2^{x^{2}}+4 x-x-2=0$ $2 x(x+2)-(x+2)=0$ $(x+2)(2 x-1)=0$ $x=-2$ or $x=$ \gg The set values for which $f(x)$ and $g(x)$ have same value is $\{-2$, \diamond $\}$....
Read More →A light ray enters from medium A to medium B
Question: A light ray enters from medium A to medium B as shown in figure. The refractive index of medium B relative to medium A will be (a)greater than unity (b)less than unity (c)equal to unity (d)zero Solution: (a). Explanation: Explanation : $n_{\mathrm{BA}}=\frac{v_{1}}{v_{2}}$ where v1= speed of light in medium A V2= speed of light in medium B. Since ray of light bends towards the normal, when it goes from medium A to rrtedium B, therefore, medium A is rarer and medium B is denser medium. ...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $(\log x)^{\log x}$ Solution: Let $y=(\log x)^{\log x}$ Taking log both the sides: $\Rightarrow \log y=\log (\log x)^{\log x}$ $\Rightarrow \log y=\log x \log (\log x)\left\{\log x^{a}=\operatorname{alog} x\right\}$ Differentiating with respect to $\mathrm{x}$ : $\Rightarrow \frac{\mathrm{d}(\log y)}{\mathrm{dx}}=\frac{\mathrm{d}(\log x \log (\log x))}{d x}$ $\Rightarrow \frac{d(\log y)}{d x}=\log x \times \frac{d(\log (\log x...
Read More →Mohini purchased a computer for Rs 37960 including VAT at 4%.
Question: Mohini purchased a computer for Rs 37960 including VAT at 4%. What is the original price of the computer? Solution: Let the original price of the computer be Rs $x$. $\mathrm{VAT}=4 \%$ of Rs. $x$ $=$ Rs. $\left(x \times \frac{4}{100}\right)$ $=$ Rs. $\frac{4 x}{100}$ $\therefore$ Price including VAT $=$ Rs. $\left(x+\frac{4 x}{100}\right)$ $=$ Rs. $\frac{104 x}{100}$ Now, $\frac{104 x}{100}=37960$ $\Rightarrow x=\left(37960 \times \frac{100}{104}\right)$ = 36500 The original price of ...
Read More →Figure shows a ray of light as it travels
Question: Figure shows a ray of light as it travels from medium A to medium B. Refractive index of the medium B relative to medium A is (a) $\sqrt{3} / \sqrt{2}$ (b) $\sqrt{2} / \sqrt{3}$ (c) $1 / \sqrt{2}$ (d) $\sqrt{2}$ Solution: (a). Explanation. $\mathrm{n}_{\mathrm{BA}}=\frac{\sin i}{\sin r}=\frac{\sin 60^{\circ}}{\sin 45^{\circ}}=\frac{\sqrt{3} / 2}{1 / \sqrt{2}}=\frac{\sqrt{3}}{\sqrt{2}}$...
Read More →Under which of the following conditions
Question: Under which of the following conditions a concave mirror can form a real image larger than the actual object ? (a)When object is kept at a distance equal to its radius of curvature. (b)When object is placed between the focus and centre of curvature. (c)When object is kept at a distance less than its focal length. (d)When object is kept at a distance greater than its radius of curvature. Solution: (b).Explanation :...
Read More →Solve this
Question: Let $\mathrm{f}:(2, \infty) \rightarrow \mathrm{R}: \mathrm{f}(\mathrm{x})=\sqrt{x-2}$, and $\mathrm{g}:(2, \infty) \rightarrow \mathrm{R}: \mathrm{g}(\mathrm{x})=\sqrt{x+2}$ Find: (i) $(f+g)(x)$ (ii) $(f-g)(x)$ (iii) $(f g)(x)$ Solution: Given: $\mathrm{f}(\mathrm{x})=\sqrt{x-2}: \mathrm{x}2$ and $\mathrm{g}(\mathrm{x})=\sqrt{x+2}: \mathrm{x}2$ (i) To find: $(f+g)(x)$ Domain $(f)=(2, \infty)$ Range $(f)=(0, \infty)$ Domain $(g)=(2, \infty)$ Range $(\mathrm{g})=(2, \infty)$ $(f+g)(x)=f...
Read More →Karuna bought 10 g of gold for Rs 15756 including VAT at 1%.
Question: Karuna bought 10 g of gold for Rs 15756 including VAT at 1%. What is the rate of gold per 10 g? Solution: Let the price of $10 \mathrm{~g}$ of gold be Rs $x$. $\mathrm{VAT}=1 \%$ of Rs $x$ $=\operatorname{Rs}\left(x \times \frac{1}{100}\right)$ $=\operatorname{Rs} \frac{x}{100}$ $\therefore$ Price including VAT $=$ Rs. $\left(x+\frac{x}{100}\right)$ $=\mathrm{Rs} \frac{101 x}{100}$ Now, $\frac{101 x}{100}=15756$ $\Rightarrow x=\operatorname{Rs}\left(15756 \times \frac{100}{101}\right)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $10^{\log \sin x}$ Solution: Let $y=10^{\log \sin x}$ Taking log both the sides: $\Rightarrow \log y=\log 10^{\log \sin x}$ $\Rightarrow \log y=\log \sin x \log 10\left\{\log x^{a}=\operatorname{alog} x\right\}$ Differentiating with respect to $\mathrm{x}$ :\ $\Rightarrow \frac{d(\log y)}{d x}=\frac{d(\log 10 \log \sin x)}{d x}$ $\Rightarrow \frac{d(\log y)}{d x}=\log 10 \times \frac{d(\log \sin x)}{d x}$ $\left\{\right.$ Usin...
Read More →A 10 mm long owl pin is placed vertically in front of a concave mirror.
Question: A 10 mm long owl pin is placed vertically in front of a concave mirror. A 5 mm long image of the owl pin is formed at 30 cm in front of the mirror. The focal length of this mirror is (a) 30 cm (b) 20 cm (c) 40 cm (d) 60 cm Solution: (b). Explanation : $\mathrm{m}=\frac{h^{\prime}}{h}=\frac{-v}{u}$ or $u=\frac{-0 \times b}{h^{\prime}}=\frac{-(-30 \mathrm{~cm}) \times(10 \mathrm{~mm})}{-5 \mathrm{~mm}}$ $=-60 \mathrm{~cm}$ $\therefore \frac{1}{f}=\frac{1}{u}+\frac{1}{v}=-\frac{1}{60}-\fr...
Read More →Mohit bought a shirt for Rs 1337.50 including VAT at 7%.
Question: Mohit bought a shirt for Rs 1337.50 including VAT at 7%. Find the original price of the shirt. Solution: Let the original price of the shirt be Rsx. VAT = 7% of Rsx $\therefore$ Price including VAT $=$ Rs. $\left(x+\frac{7 x}{100}\right)$ $=$ Rs. $\frac{107 x}{100}$ Now, $\frac{107 x}{100}=1337.50$ $\Rightarrow x=\operatorname{Rs}\left(1337.50 \times \frac{100}{107}\right)$ = 1250 Hence, the original price of the shirt is Rs 1,250....
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