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Question: $\left(x^{2}-y^{2}\right) d x+2 x y d y=0$ Solution: The given differential equation is: $\left(x^{2}-y^{2}\right) d x+2 x y d y=0$ $\Rightarrow \frac{d y}{d x}=\frac{-\left(x^{2}-y^{2}\right)}{2 x y}$ ...(1) Let $F(x, y)=\frac{-\left(x^{2}-y^{2}\right)}{2 x y}$. $\therefore F(\lambda x, \lambda y)=\left[\frac{(\lambda x)^{2}-(\lambda y)^{2}}{2(\lambda x)(\lambda y)}\right]=\frac{-\left(x^{2}-y^{2}\right)}{2 x y}=\lambda^{0} \cdot F(x, y)$ Therefore, the given differential equation is ...
Read More →There are two cones .The curved surface area of one is twice that of the other.
Question: There are two cones .The curved surface area of one is twice that of the other. The slant height of the later is twice that of the former. Find the ratio of their radii. Solution: Let the curved surface area of $1^{5 t}$ cone $=2 x$ C.S.A of $2^{\text {nd }}$ cone $=x$ Slant height of $1^{\text {st }}$ cone $=h$ Slant height of $2^{\text {nd }}$ cone $=2 \mathrm{~h}$ Therefore $\frac{\text { C.S. A of } 1^{\text {st }} \text { cone }}{\text { C. S. A of } 2^{\text {nd }} \text { cone }...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions:(i)f(x) = 2 sinx, 0 x (ii) $g(x)=3 \sin \left(x-\frac{\pi}{4}\right), 0 \leq x \leq \frac{5 \pi}{4}$ (iii) $h(x)=2 \sin 3 x, 0 \leq x \leq \frac{2 \pi}{3}$ (iv) $\phi(x)=2 \sin \left(2 x-\frac{\pi}{3}\right), 0 \leq x \leq \frac{7 \pi}{5}$ (v) $\psi(x)=4 \sin 3\left(x-\frac{\pi}{4}\right), 0 \leq x \leq 2 \pi$ (vi) $\theta(x)=\sin \left(\frac{x}{2}-\frac{\pi}{4}\right), 0 \leq x \leq 4 \pi$ (vii) $u(x)=\sin ^{2} x, 0 \leq x \leq 2 \pi v(x)=|...
Read More →Find the ratio of the curved surface area of two cones if their diameter
Question: Find the ratio of the curved surface area of two cones if their diameter of the bases are equal and slant heights are in the ratio 4: 3. Solution: Given that, Diameter of two coins are equal. Therefore their radius are equal. Letr1= r2= r Let ratio be x Therefore slant height $\left.\right|_{1}$ of $1^{5 t}$ cone $=4 x$ Similarly slant height $I_{2}$ of $2^{\text {nd }}$ cone $=3 x$ Therefore $C . S . A_{1} / C . S . A_{2}$ $=\frac{\pi * r_{1} * l_{1}}{\pi * r_{2} * l_{2}}$ $=\frac{\pi...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions:(i)f(x) = 2 sinx, 0 x (ii) $g(x)=3 \sin \left(x-\frac{\pi}{4}\right), 0 \leq x \leq \frac{5 \pi}{4}$ (iii) $h(x)=2 \sin 3 x, 0 \leq x \leq \frac{2 \pi}{3}$ (iv) $\phi(x)=2 \sin \left(2 x-\frac{\pi}{3}\right), 0 \leq x \leq \frac{7 \pi}{5}$ (v) $\psi(x)=4 \sin 3\left(x-\frac{\pi}{4}\right), 0 \leq x \leq 2 \pi$ (vi) $\theta(x)=\sin \left(\frac{x}{2}-\frac{\pi}{4}\right), 0 \leq x \leq 4 \pi$ (vii) $u(x)=\sin ^{2} x, 0 \leq x \leq 2 \pi v(x)=|...
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Question: $(x-y) d y-(x+y) d x=0$ Solution: The given differential equation is: $(x-y) d y-(x+y) d x=0$ $\Rightarrow \frac{d y}{d x}=\frac{x+y}{x-y}$ ..(1) $\operatorname{Let} F(x, y)=\frac{x+y}{x-y}$ $\therefore F(\lambda x, \lambda y)=\frac{\lambda x+\lambda y}{\lambda x-\lambda y}=\frac{x+y}{x-y}=\lambda^{0} \cdot F(x, y)$ Thus, the given differential equation is a homogeneous equation. To solve it, we make the substitution as: y=vx $\Rightarrow \frac{d}{d x}(y)=\frac{d}{d x}(v x)$ $\Rightarr...
Read More →In a ∆ABC, D and E are points on the sides AB and AC respectively such that DE || BC.
Question: In a ∆ABC, D and E are points on the sides AB and AC respectively such that DE || BC. (i) If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.(ii) IfADDB=34and AC = 15 cm, find AE.(iii) IfADDB=23and AC = 18 cm, find AE.(iv) If AD = 4, AE = 8, DB =x 4, and EC = 3x 19, findx.(v) If AD = 8 cm, AB = 12 cm and AE = 12 cm, find CE.(vi) if AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC.(vii) If AD = 2 cm, AB = 6 cm, and AC = 9 cm, find AE.(viii) IfADBD=45and EC = 2.5 cm, find AE.(ix) If AD =x, DB =...
Read More →A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm.
Question: A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps. Solution: Given that, Radius of conical cap(r) = 7 cm Height of the conical cap (h) = 24 cm Slant height (l) of conical cap $=\sqrt{\mathrm{r}^{2}+\mathrm{h}^{2}}$ $=\sqrt{7^{2}+24^{2}}$ = 25 cm C.S.A of 1 conical cap = rl = 22/7 7 25 $=550 \mathrm{~cm}^{2}$ Curved surface area of 0 such conical caps $=5500 \mathrm{~cm}^{2}$ Thus, $5500 \...
Read More →The radius and slant height of a cone are in the ratio 4: 7.
Question: The radius and slant height of a cone are in the ratio 4: 7.If its curved surface area is $792 \mathrm{~cm}^{2}$, find its radius. Solution: It is given that Curved surface area = rl= 792 Let the radius (r) = 4x Height (h) = 7x Now, C. S. A = 792 22/7 4x 7x = 792 $\Rightarrow 88 x^{2}=792$ $\Rightarrow x^{2}=792 / 88=9$ ⟹ x = 3 Therefore Radius = 4x = 4 * 3 = 12 cm...
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Question: $y^{\prime}=\frac{x+y}{x}$ Solution: The given differential equation is: $y^{\prime}=\frac{x+y}{x}$ $\Rightarrow \frac{d y}{d x}=\frac{x+y}{x}$ ...(1) Let $F(x, y)=\frac{x+y}{x}$. Now, $F(\lambda x, \lambda y)=\frac{\lambda x+\lambda y}{\lambda x}=\frac{x+y}{x}=\lambda^{0} F(x, y)$ Thus, the given equation is a homogeneous equation. To solve it, we make the substitution as: y=vx Differentiating both sides with respect tox, we get: $\frac{d y}{d x}=v+x \frac{d v}{d x}$ Substituting the ...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = tan 2x Solution: Step I- We find the value ofcandaby comparingy= 2 tan 2xwithy=ctanax, i.e.c= 1 anda= 2. Step II- Then, we draw the graph ofy= tanxand mark the point where it crosses thex-axis. Step III- Divide thex-coordinates of the points wherey= tanxcrossesx-axis by 2(i.e.a= 2) and mark the maximum value (i.e.c= 1) and minimum value (i.e.-c=-1). Then , we obtain the following graph:...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = cosec2x Solution: f(x) = cosec2x...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = sec2x Solution: f(x) = sec2x...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: $f(x)=\cot \frac{\pi x}{2}$ Solution: $f(x)=\cot \frac{\pi x}{2}$...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = cot2x Solution: f(x) = cot2x...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = tan2x Solution: f(x) = tan2x...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = 2 sec x Solution: f(x) = 2 sec x...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = 3 secx Solution: f(x) = 3 secx...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = cot 2x Solution: f(x) = cot 2x...
Read More →Sketch the graphs of the following functions:
Question: Sketch the graphs of the following functions: f(x) = 2 cosec x Solution: f(x) = 2 cosec x...
Read More →The curved surface area of a cone is 4070 cm2
Question: The curved surface area of a cone is $4070 \mathrm{~cm}^{2}$ and diameter is $70 \mathrm{~cm}$. What is its slant height? Solution: Diameter of the cone (d) = 70 cm Radius of the cone $(\mathrm{r})=\mathrm{d}^{2}=35 \mathrm{~cm}$ Slant height of the cone (l) = ? Now, Curved Surface Area $=4070 \mathrm{~cm}^{2}$ $\Rightarrow \pi r \mid=4070$ Where, r = Radius of the cone l = Slant height of the cone Thereforerl= 4070 ⟹ 22/7 35 l = 4070 $\Rightarrow \mathrm{l}=\frac{4070 * 7}{22 * 35}=37...
Read More →The area of the curved surface of a cone is 60 π cm2.
Question: The area of the curved surface of a cone is $60 \mathrm{~m} \mathrm{~cm}^{2}$. If the slant height of the cone be $8 \mathrm{~cm}$, find the radius of the base. Solution: It is given that: Curved surface area (C. S. A) $=60 \pi \mathrm{cm}^{2}$ Slant height of the cone (l) = 8 cm Radius of the cone(r) = ? Now we know, Curved Surface Area (C. S. A) = rl $\Rightarrow \pi r l=60 \pi \mathrm{cm}^{2}$ ⟹ r * 8 = 60 ⟹ r = 7.5 cm Therefore the radius of the base of the cone is 7.5 cm....
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Question: $\left(x^{2}+x y\right) d y=\left(x^{2}+y^{2}\right) d x$ Solution: The given differential equation i.e., $\left(x^{2}+x y\right) d y=\left(x^{2}+y^{2}\right) d x$ can be written as: $\frac{d y}{d x}=\frac{x^{2}+y^{2}}{x^{2}+x y}$ ...(1) Let $F(x, y)=\frac{x^{2}+y^{2}}{x^{2}+x y}$ Now, $F(\lambda x, \lambda y)=\frac{(\lambda x)^{2}+(\lambda y)^{2}}{(\lambda x)^{2}+(\lambda x)(\lambda y)}=\frac{x^{2}+y^{2}}{x^{2}+x y}=\lambda^{0} \cdot F(x, y)$ This shows that equation (1) is a homogene...
Read More →If 3 sin x + 5 cos x = 5,
Question: If 3 sinx+ 5 cosx= 5, then write the value of 5 sinx 3 cosx. Solution: $3 \sin x+5 \cos x=5$ (Given) Squaring both the sides: $9 \sin ^{2} x+25 \cos ^{2} x+30 \sin x \cos x=25$ $30 \sin x \cos x=25-9 \sin ^{2} x-25 \cos ^{2} x$ (1) We have to find the value of $5 \sin \theta-3 \cos \theta$. $(5 \sin x-3 \cos \mathrm{x})^{2}=25 \sin ^{2} \mathrm{x}+9 \cos ^{2} x-30 \sin x \cos x$ $(5 \sin x-3 \cos x)^{2}=25 \sin ^{2} x+9 \cos ^{2} x-\left(25-9 \sin ^{2} x-25 \cos ^{2} x\right) \quad[$ F...
Read More →In the given figure, if PB || CF and DP || EF, then ADDE=
Question: In the given figure, if PB || CF and DP || EF, thenADDE= (a)34(b)13(c)14(d)23 Solution: Given: PB||CF and DP||EF. AB = 2 cm and AC = 8 cm. To find: AD: DE According toBASIC PROPORTIONALITY THEOREM, if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. In ∆ACF, PB || CF. $\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{\mathrm{AP}}{\mathrm{PF}}$ $\frac{\mathrm{AP}}{\mathrm{PF}}=\frac{2}{8-2}$ $\frac{\mathrm{AP}}{\m...
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