$QR$ at $X$ and $PR$ at $Z.$ $OZ,$ $OX,$ $OY$ are perpendicular to the sides $PR,$ $QR,$ $PQ.$ Here $PQR$ is an isosceles triangle with sides $PQ = PR$ and also from the figure, \[\Rightarrow...
$Δ ABC$ is an isosceles triangle such that $AB = AC.$ The vertical angle $BAC = 2θ$ Triangle is inscribed in the circle with center $O$ and radius $a.$ Draw $AM$ perpendicular to $BC.$ Since, $Δ...
As per the given question,
Let the radius and height of cone be r and h respectively \[Radius\text{ }of\text{ }sphere\text{ }=\text{ }R\] \[{{R}^{2}}~=\text{ }{{r}^{2}}~+\text{ }{{\left( h\text{ }-\text{ }R \right)}^{2}}\]...
As per the given question,
Let the length and breadth of rectangle $ABCD$ be $2x$ and $y$ respectively \[Radius\text{ }of\text{ }semicircle\text{ }=\text{ }r\text{ }\left( given \right)\] In triangle $OBA,$ where is the...
As per the given question,
Let the dimensions of the rectangle be $x$ and $y.$ Therefore, the perimeter of window \[=\text{ }x\text{ }+\text{ }y\text{ }+\text{ }x\text{ }+\text{ }x\text{ }+\text{ }y\text{ }=\text{ }12\]...
Let the radius of semicircle, length and breadth of rectangle be $r,$ $x$ and $y$ respectively \[AE\text{ }=\text{ }r\] \[AB\text{ }=\text{ }x=2r\text{ }\left( semicircle\text{ }is\text{...
Let the length, breath and height of tank be $I, b$ and $h$ respectively. Also, assume volume of tank as $V$ \[h\text{ }=\text{ }2\text{ }m\text{ }\left( given \right)\] \[V\text{ }=\text{ }8\text{...
Given length of rectangle sheet \[=\text{ }45\text{ }cm\] Breath of rectangle sheet \[=\text{ }24\text{ }cm\] Let the side length of each small square be $a.$ If by cutting a square from each corner...
Given side length of big square is $18 cm$ Let the side length of each small square be $a.$ If by cutting a square from each corner and folding up the flaps we will get a cuboidal box with Length,...
It is given that two sides of a triangle have lengths $a$ and $b$ and the angle between them is $θ.$ Let the area of triangle be $A$
As per the given question,
Let us say the sum of perimeter of square and circumference of circle be $L$ Given sum of the perimeters of a square and a circle. Assuming, \[side\text{ }of\text{ }square\text{ }=\text{ }a\text{...
Suppose the wire, which is to be made into a square and a triangle, is cut into two pieces of length $x$ and $y$ respectively. Then, \[x\text{ }+\text{ }y\text{ }=\text{ }20\text{ }\Rightarrow...
Suppose the given wire, which is to be made into a square and a circle, is cut into two pieces of length $x$ and $y$ $m$ respectively. Then, \[x\text{ }+\text{ }y\text{ }=\text{ }28\text{...
Solution: As per the given question,
Let $r$ $and$ $h$ be the radius and height of the cylinder, respectively. Then, Volume $(V)$ of the cylinder \[=\text{ }\pi {{r}^{2}}~h\] \[\to \text{ }100\text{ }=\text{ }\pi {{r}^{2}}~h\] \[\to...
Let the given two numbers be $x$ and $y$. Then, \[x\text{ }+\text{ }y\text{ }=\text{ }15\text{ }\ldots ..\text{ }\left( 1 \right)\] \[y\text{ }=\text{ }\left( 15\text{ }-\text{ }x \right)\] Now we...
As per the given question, \[1\text{ }+\text{ }3{{\left( {\scriptscriptstyle 1\!/\!{ }_2}\text{ }-\text{ }a \right)}^{2}}~\left( -1 \right)\text{ }=\text{ }0\] \[1\text{ }-\text{ }3{{\left(...
Let the two positive numbers be $a$ and $b$ Given \[a\text{ }+\text{ }b\text{ }=\text{ }64\text{ }\ldots \text{ }\left( 1 \right)\] We have, \[{{a}^{3}}~+\text{ }{{b}^{3}}\] is minima Assume,...
Which implies $S$ is minimum when \[a\text{ }=\text{ }15/2\text{ }and\text{ }b\text{ }=\text{ }15/2.\]
Let $f(x)=2 x^{3}-24 x+107$ $ \therefore f^{\prime}(x)=6 x^{2}-24=6\left(x^{2}-4\right) $ Now, for local maxima and local minima we have $f^{\prime}(x)=0$ $ \begin{array}{l} \Rightarrow...
(ii) Solution: (i) Given function is $f(x)=3 x^{4}-8 x^{3}+12 x^{2}-48 x+25$ on $[0,3]$ On differentiating we get $ \begin{array}{l} f^{\prime}(x)=12 x^{3}-24 x^{2}+24 x-48 \\...
(i) (ii) Given function is \[f\left( x \right)\text{ }=~{{\left( x\text{ }-\text{ }1 \right)}^{2}}~+\text{ }3\] On differentiation we get \[\Rightarrow \text{ }f'\left( x \right)~=\text{ }2\left(...
As per the given question,
Given \[y\text{ }=\text{ }a\text{ }log\text{ }x\text{ }+\text{ }b{{x}^{2}}~+\text{ }x\] On differentiating we get Given that extreme values exist at \[x\text{ }=\text{ }1,\text{ }2\] \[a\text{...
Given $f(x)=-(x-1)^{3}(x+1)^{2}$ $ \begin{array}{l} f^{\prime}(x)=-3(x-1)^{2}(x+1)^{2}-2(x-1)^{3}(x+1) \\ =-(x-1)^{2}(x+1)(3 x+3+2 x-2) \\ =-(x-1)^{2}(x+1)(5 x+1) \\ f^{\prime...
Solution: (i) $ \begin{array}{l} \text { Given } f(x)=(x-1)(x-2)^{2} \\ f(x)=(x-2)^{2}+2(x-1)(x-2) \\ =(x-2)(x-2+2 x-2) \\ =(x-2)(3 x-4) \\ f^{\prime}(x)=(3 x-4)+3(x-2) \end{array} $ For maxima and...
Given $\mathrm{f}(\mathrm{x})=\mathrm{x} \mathrm{e}^{x}$ $ \begin{array}{l} f(x)=e^{x}+x e^{i}=e^{x}(x+1) \\ f^{\prime}(x)=e^{2}(x+1)+e^{t} \\ =e^{2}(x+2) \end{array} $ For maxima and minima, $...
(i) Given $f(x)=(x-1)(x+2)^{2}$ $ \begin{array}{l} \therefore \mathrm{f}(\mathrm{x})=(\mathrm{x}+2)^{2}+2(\mathrm{x}-1)(\mathrm{x}+2) \\ =(\mathrm{x}+2)(\mathrm{x}+2+2 \mathrm{x}-2) \\...
(i) Given $f(x)=x^{4}-62 x^{2+} 120 x+9$ $ \begin{array}{l} \therefore \mathrm{f}(\mathrm{x})=4 \mathrm{x}^{\mathrm{a}}-124 x+120=4\left(\mathrm{x}^{a}-31 \mathrm{x}+30\right) \\...
Given \[f\text{ }\left( x \right)\text{ }=\text{ }sin\text{ }2x\] Differentiate w.r.t x, we get \[f'\left( x \right)\text{ }=\text{ }2\text{ }cos\text{ }2x,\text{ }0\text{ }<\text{ }x\text{...
Given, $f(x)=x^{3}-6 x^{2}+9 x+15$ Differentiate with respect to $x$, we get, $f^{\prime}(x)=3 x^{2}-12 x+9=3\left(x^{2}-4 x+3\right)$ $=3(x-3)(x-1)$ For all maxima and minima, $ \begin{array}{l}...
As per the given question, Therefore \[x\text{ }=\text{ }0,\] now for the values close to \[x\text{ }=\text{ }0,\] and to the left of \[0,\text{ }f'\left( x \right)\text{ }>\text{ }0\] Also for...
Differentiate with respect to $x$, we get, $ \begin{array}{l} f(x)=(x+2)^{2}+2(x-1)(x+2) \\ =(x+2)(x+2+2 x-2) \\ =(x+2)(3 x) \end{array} $ For all maxima and minima, $ \begin{array}{l} f(x)=0 \\...
Given, $f(x)=x^{2}(x-1)^{2}$ Differentiate with respect to $x$, we get, $ \begin{array}{l} {\left x=3 x^{2}(x-1)^{2}+2 x^{2}(x-1)\right.} \\ =[x-1]\left(3 x^{2}(x-1)+2 x^{2}\right) \\ =[x-1]\left(3...
Given, $f(x)=x^{2}-3 x$ Differentiate with respect to $x$ then we get, $ f(x)=3 x^{2}-3 $ $\mathrm{Now}, \mathrm{f}(x)=0$ $ 3 x^{2}=3 \Rightarrow x=\pm 1 $ Again differentiate $f(x)=3 x^{2}-3$ $...
Given $f(x)=(x-5)^{4}$ Differentiate with respect to $x$ $ f(x)=4(x-5)^{2} $ For local maxima and minima $ \begin{array}{l} f(x)=0 \\ =4(x-5)^{2}=0 \\ =x-5=0 \\ x=5 \end{array} $ $f(x)$ changes from...
Given $f(x)=|\sin 4 x+3|$ on $R$ We know that $-1 \leq \sin 4 x \leq 1$ $ \begin{array}{l} \Rightarrow 2 \leq \sin 4 x+3 \leq 4 \\ \Rightarrow 2 \leq|\sin 4 x+3| \leq 4 \end{array} $ Hence, the...
Given $f(x)=\sin 2 x+5$ on $R$ We know that $-1 \leq \sin 2 x \leq 1$ $ \begin{array}{l} \Rightarrow-1+5 \leq \sin 2 x+5 \leq 1+5 \\ \Rightarrow 4 \leq \sin 2 x+5 \leq 6 \end{array} $ Hence, the...
Given $f(x)=|x+2|$ on $R$ $\Rightarrow f(x) \geq 0$ for all $x \in R$ So, the minimum value of $f(x)$ is 0, which attains at $x=-2$ Hence, $f(x)=|x+2|$ does not have the maximum value.
Given $f(x)=-(x-1)^{2}+2$ It can be observed that $(x-1)^{2} \geq 0$ for every $x \in R$ Therefore, $f(x)=-(x-1)^{2}+2 \leq 2$ for every $x \in R$ The maximum value of $f$ is attained when $(x-1)=0$...
Given $f(x)=4 x^{2}-4 x+4$ on $R$ $=4 x^{2}-4 x+1+3$ By grouping the above equation we get, $ =(2 x-1)^{2}+3 $ Since, $(2 x-1)^{2} \geq 0$ $ \begin{array}{l} =(2 x-1)^{2}+3 \geq 3 \\ =f(x) \geq f(1...
Answer: Given, 23n – 7n – 1 23n – 7n – 1 = 8n – 7n – 1 Using binomial theorem, 8n = 7n + 1 8n = (1 + 7) n 8n = nC0 + nC1 (7)1 + nC2 (7)2 + nC3 (7)3 + nC4 (7)2 + nC5 (7)1 + … + nCn (7) n 8n = 1 + 7n...