Solve the following
Question: $\lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{x-a}$ Solution: Given $\lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{x-a}$ $\Rightarrow \lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{x-a}$ Now by adding and subtracting 2 to denominator for further simplification we get $=\lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{(x+2)-(a+2)}$ Now we have $\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}...
Read More →The space s described in time t by
Question: The space s described in time $t$ by a particle moving in a straight line is given by $S=t 5-40 t^{3}+30 t^{2}+80 t-250 .$ Find the minimum value of acceleration. Solution: Given : $s=t^{5}-40 t^{3}+30 t^{2}+80 t-250$ $\Rightarrow \frac{d s}{d t}=5 t^{4}-120 t^{2}+60 t+80$ Acceleration, $a=\frac{d^{2} s}{d t^{2}}=20 t^{3}-240 t+60$ $\Rightarrow \frac{d a}{d t}=60 t^{2}-240$ For maximum or minimum values of $a$, we must have $\frac{d a}{d t}=0$ $\Rightarrow 60 t^{2}-240=0$ $\Rightarrow ...
Read More →Prove the following
Question: $\lim _{x \rightarrow \frac{1}{2}} \frac{4 x^{2}-1}{2 x-1}$ Solution: Given $\lim _{x \rightarrow \frac{1}{2}} \frac{4 x^{2}-1}{2 x-1}$ The above equation can be written as $=\lim _{x \rightarrow \frac{1}{2}} \frac{(2 x)^{2}-1}{2 x-1}$ Using $a^{2}-b^{2}$ formula and expanding we get $=\lim _{x \rightarrow \frac{1}{2}} \frac{(2 x-1)(2 x+1)}{2 x-1}$ On simplifying and applying the limits we get $\Rightarrow \lim _{x \rightarrow \frac{1}{2}}(2 x+1)=2$ $\lim _{x \rightarrow \frac{1}{2}} \...
Read More →Prove the following
Question: 1. $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}$ Solution: 1. $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}$ Given $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}$ The above equation can be written as $=\lim _{x \rightarrow 3} \frac{(x-3)(x+3)}{x-3}$ On simplifying and applying limits we get $\lim _{x \rightarrow 3}(x+3)=6$ $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}=6$...
Read More →The total area of a page
Question: The total area of a page is $150 \mathrm{~cm}^{2}$. The combined width of the margin at the top and bottom is $3 \mathrm{~cm}$ and the side $2 \mathrm{~cm}$. What must be the dimensions of the page in order that the area of the printed matter may be maximum? Solution: Let $x$ and $y$ be the length and breadth of the rectangular page, respectively. Then, Area of the page $=150$ $\Rightarrow x y=150$ $\Rightarrow y=\frac{150}{x}$ ......(1) Area of the printed matter $=(x-3)(y-2)$ $\Right...
Read More →Find the equation of the hyperbola with vertices at (±6, 0) and foci at (±8, 0).
Question: Find the equation of the hyperbola with vertices at (6, 0) and foci at (8, 0). Solution: Given: Vertices at (6, 0) and foci at (8, 0) Need to find: The equation of the hyperbola Let, the equation of the parabola be: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ Vertices of the parabola is at $(\pm 6,0)$ That means $\mathrm{a}=6$ The foci are given at $(\pm 8,0)$ That means, ae = 8, where e is the eccentricity. $\Rightarrow 6 \mathrm{e}=8[$ As $\mathrm{a}=6]$ $\Rightarrow \mathrm{e}=\frac...
Read More →A straight line is drawn through a given point
Question: A straight line is drawn through a given point $P(1,4)$. Determine the least value of the sum of the intercepts on the coordinate axes. Solution: The equation of line passing through $(1,4)$ with slope $m$ is given by $y-4=m(x-1)$ ...(1) Substituting $y=0$, we get $0-4=m(x-1)$ $\Rightarrow \frac{-4}{m}=x-1$ $\Rightarrow x=\frac{m-4}{m}$ Substituting $x=0$, we get So, the intercepts on coordinate axes are $\frac{m-4}{m}$ and $-(m-4)$. Let $S$ be the sum of the intercepts. Then, $\mathrm...
Read More →Find the value
Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $5 y^{2}-9 x^{2}=36$ Solution: Given Equation: $5 y^{2}-9 x^{2}=36 \Rightarrow \frac{y^{2}}{36 / 5}-\frac{x^{2}}{4}=1$ Comparing with the equation of hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ we get, $a=\sqrt{\frac{36}{5}}=\frac{6}{\sqrt{5}}$ and $b=2$ (i) Length of Transverse axis $=2 a=\f...
Read More →The strength of a beam varies as the product of its breadth and square of its depth.
Question: The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a. Solution: Let the breadth, height and strength of the beam be $b, h$ and $S$, respectively. $a^{2}=\frac{h^{2}+b^{2}}{4}$ $\Rightarrow 4 a^{2}-b^{2}=h^{2}$ ....(1) Here, Strength of beam, $S=K b h^{2}$ $\Rightarrow S=k b\left(4 R^{2}-b^{2}\right)$ [Where $K$ is some constant] $\Rightarrow S=k\left(b 4 a^{2}-b^{3...
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Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $3 y^{2}-x^{2}=108$ Solution: Given Equation: $3 y^{2}-x^{2}=108 \Rightarrow \frac{y^{2}}{36}-\frac{x^{2}}{108}=1$ Comparing with the equation of hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ we get, $a=6$ and $b=\sqrt{108}=6 \sqrt{3}$ (i) Length of Transverse axis $=2 a=12$ units. Length of Co...
Read More →Find the value
Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $\frac{\mathrm{y}^{2}}{9}-\frac{\mathrm{x}^{2}}{27}=1$ Solution: Given Equation: $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$ Comparing with the equation of hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ we get, $a=3$ and $b=\sqrt{27}=3 \sqrt{3}$ (i) Length of Transverse axis $=2 \mathrm{a}=6$ units. Le...
Read More →A given quantity of metal is to be cast into a half cylinder
Question: A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is$:(\pi+2)$ Solution: Volume, $V=\frac{1}{2} \pi l\left(\frac{D}{2}\right)^{2}$ $\Rightarrow V=\frac{\pi D^{2} l}{8}$ $\Rightarrow l=\frac{8 V}{\pi D^{2}}$ ....(1) Total surface area $=\frac{\pi D^{2}}{4}+l D+\frac{\pi D l}{2}$ $\Rightar...
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Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $\frac{y^{2}}{16}-\frac{x^{2}}{49}=1$ Solution: Given Equation: $\frac{y^{2}}{16}-\frac{x^{2}}{49}=1$ Comparing with the equation of hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ we get, a = 4 and b = 7 (i) Length of Transverse axis $=2 \mathrm{a}=8$ units. Length of Conjugate axis $=2 b=14$ un...
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Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $24 x^{2}-25 y^{2}=600$ Solution: Given Equation: $24 x^{2}-25 y^{2}=600 \Rightarrow$ $\frac{x^{2}}{25}-\frac{y^{2}}{24}=1$ Comparing with the equation of hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ we get, $a=5$ and $b=\sqrt{24}=2 \sqrt{6}$ (i) Length of Transverse axis $=2 a=10$ units. Leng...
Read More →The sum of the surface areas of a sphere and a cube is given.
Question: The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube. Solution: Letrbe the radius of the sphere,xbe the side of the cube andSbe the sum of the surface area of both. Then, $S=4 \pi r^{2}+6 x^{2}$ $\Rightarrow x=\left(\frac{S-4 \pi r^{2}}{6}\right)^{\frac{1}{2}}$ .....(1) Sum of volumes, $V=\frac{4}{3} \pi r^{3}+x^{3}$ $\Rightarrow V=\frac{4 \pi r^{3}}{3}+\left[\frac{\le...
Read More →The sum of the surface areas of a sphere and a cube is given.
Question: The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube. Solution: Letrbe the radius of the sphere,xbe the side of the cube andSbe the sum of the surface area of both. Then, $S=4 \pi r^{2}+6 x^{2}$ $\Rightarrow x=\left(\frac{S-4 \pi r^{2}}{6}\right)^{\frac{1}{2}}$ .....(1) Sum of volumes, $V=\frac{4}{3} \pi r^{3}+x^{3}$ $\Rightarrow V=\frac{4 \pi r^{3}}{3}+\left[\frac{\le...
Read More →Find the value
Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $25 x^{2}-9 y^{2}=225$ Solution: Given Equation: $25 x^{2}-9 y^{2}=225 \Rightarrow$ $\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$ Comparing with the equation of hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ we get $a=3$ and $b=5$ (i) Length of Transverse axis = 2a = 6 units Length of Conjugate axis = 2b...
Read More →Find the value
Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $3 x^{2}-2 y^{2}=6$ Solution: Given Equation: $3 x^{2}-2 y^{2}=6 \Rightarrow \frac{x^{2}}{2}-\frac{y^{2}}{3}=1$ Comparing with the equation of hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ we get, $a=\sqrt{2}$ and $b=\sqrt{3}$ (i) Length of Transverse axis $=2 a=2 \sqrt{2}$ units. Length of Con...
Read More →A box of constant volume c is to be twice as long as it is wide.
Question: A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions? Solution: Let $l, b$ and $h$ be the length, breadth and height of the box, respectively. Volume of the box $=c$ Given : $l=2 b$ $\cdots(1)$ $\Rightarrow c=l b h$ $\Rightarrow c=2 b^{2} h$ $\Rightarrow h=\frac{c}{2 b^{2}}$ .......(2) Let cost of the material required for bottom be $...
Read More →Find the value
Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $x^{2}-y^{2}=1$ Solution: Given Equation: $x^{2}-y^{2}=1$ Comparing with the equation of hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ we get $a=1$ and $b=1$ (i) Length of Transverse axis $=2 \mathrm{a}=2$ units. Length of Conjugate axis $=2 b=2$ units. (ii) Coordinates of the vertices $=(\pm a...
Read More →If the distance between the points (a, 0, 1)
Question: If the distance between the points (a, 0, 1) and (0, 1, 2) is 27, then the value of a is (A) 5 (B) 5 (C) 5 (D) none of these Solution: (B) 5 Explanation: Let $\mathrm{P}$ be the point whose coordinate is $(\mathrm{a}, 0,1)$ and $\mathrm{Q}$ represents the point $(0$, $1,2)$. Given, $P Q=V 27$ From distance formula we have $P Q=\sqrt{(a-0)^{2}+(0-1)^{2}+(1-2)^{2}}=\sqrt{a^{2}+2}$ $\Rightarrow \sqrt{27}=\sqrt{a^{2}+2}$ Squaring on both sides $a^{2}+2=27$ $\Rightarrow a^{2}=25$ $\Rightarr...
Read More →Distance of the point (3, 4, 5)
Question: Distance of the point (3, 4, 5) from the origin (0, 0, 0) is (A) 50 (B) 3 (C) 4 (D) 5 Solution: (A) 50 Explanation: Let $P$ be the point whose coordinate is $(3,4,5)$ and $Q$ represents the origin. From distance formula we can write as $P Q=\sqrt{(3-0)^{2}+(4-0)^{2}+(5-0)^{2}}$ $=\sqrt{9+16+25}$ $=\sqrt{50}$ $\therefore$ Distance of the point $(3,4,5)$ from the origin $(0,0,0)$ is $\sqrt{50}$ units. Hence, option (A) is the only correct choice....
Read More →What is the length of foot of perpendicular
Question: What is the length of foot of perpendicular drawn from the point P (3, 4, 5) on y-axis (A) 41 (B) 34 (C) 5 (D) none of these Solution: (B) 34 Explanation: As we know that y-axis lies on x y plane and y z. So, its distance from x y and y z plane is 0. By basic definition of three-dimension coordinate we can say that x-coordinate and zcoordinate are 0. As, perpendicular is drawn from point P to y-axis, so distance of point of intersection of this line from x z plane remains the same. y-c...
Read More →Find the value
Question: Find the (i) lengths of the axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity and (v) length of the rectum of each of the following the hyperbola : $\frac{x^{2}}{25}-\frac{y^{2}}{4}=1$ Solution: Given Equation: $\frac{x^{2}}{25}-\frac{y^{2}}{4}=1$ Comparing with the equation of hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ we get, $a=5$ and $b=2$ (i) Length of Transverse axis $=2 \mathrm{a}=10$ units. Length of Conjugate axis $=2 b=4$ unit...
Read More →An open tank is to be constructed
Question: An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width. Solution: Let $l, h, V$ and $S$ be the length, height, volume and surface area of the tank to be constructed. Since volume, $V$ is constant, $l^{2} h=V$ $\Rightarrow h=\frac{V}{l^{2}}$ .....(1) Surface area, $S=l^{2}+4 l h$ $\Rightarrow S=l^{2}+\frac{4 V}{l}$ $\left[\begin{array}{ll}...
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