Volume of a cylinder with
Question: Volume of a cylinder with radiusT? and height r is_______. Solution: h2r Given, radius of cylinder = h and height of cylinder = r. Now, volume of a cylinder = x (Radius)2x Height = x h2x r = h2r...
Read More →Total surface area of a cylinder
Question: Total surface area of a cylinder of radius h and height r is_______ . Solution: 2h(r+ h) Given, radius of cylinder = h and height of cylinder = r ...Total surface area of a cylinder = Curved surface area + Area of top surface + Area of base = 2 x x Radius x Height + (Radius)2+ (Radius)2 = 2hr+ h2+ h2 = 2rh + 2h2 = 2h(r + h)...
Read More →Curved surface area of a cylinder
Question: Curved surface area of a cylinder of radius h and height r is_______. Solution: 2hr (or) 2rh We know that, the curved surface area of a cylinder of radius h and height r = 2 x Radius x Height . = 2 x h x r=2hr = 2rh...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $\mathrm{y}=\log _{\mathrm{e}}\left(\frac{\mathrm{x}}{\mathrm{a}+\mathrm{bx}}\right)^{2}$, then $\mathrm{x}^{3} \mathrm{y}_{2}=$ A. $\left(x y_{1}-y\right)^{2}$ B. $(x+y)^{2}$ c. $\left(\frac{\mathrm{y}-\mathrm{xy}_{1}}{\mathrm{y}_{1}}\right)^{2}$ D. none of these Solution: Given: $y=\left(\log _{e}\left(\frac{x}{a+b x}\right)\right)^{2}$ $=2 \log _{e}\left(\frac{x}{a+b x}\right)$ $\frac{d y}{d x}=2\left(\frac{1}{\frac{x}{a+b x}}\right...
Read More →Opposite faces of a cuboid are
Question: Opposite faces of a cuboid are _________ in area. Solution: Equal Explanation: A cuboid is made up of 6 rectangular faces, but the opposite sides have equal length and breadth. Hence, the opposite areas are equal....
Read More →All six faces of a cuboid are
Question: All six faces of a cuboid are __________ in shape and of ______ area. Solution: Rectangular shape, different Explanation: It is known that, a cuboid is made up of 6 rectangular face which different lengths and breadths. Hence, it has different area....
Read More →Three numbers are in AP, and their sum is 21.
Question: Three numbers are in AP, and their sum is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three numbers in GP. Find the numbers. Solution: To find: Three numbers Given: Three numbers are in A.P. Their sum is 21 Formula used: When $a, b, c$ are in GP, $b^{2}=a c$ Let the numbers be $a-d, a, a+d$ According to first condition $a+d+a+a-d=21$ $\Rightarrow 3 a=21$ $\Rightarrow a=7$ Hence numbers are $7-d, 7,7+d$ When second number is reduced by 1 and third...
Read More →A trapezium with 3 equal sides and
Question: A trapezium with 3 equal sides and one side double the equal side can be divided into __________ equilateral triangles of _______ area. Solution: 3, equal areas Explanation: By using SSS congruency rule of triangle, we can show that a trapezium can be divided into three equilateral triangle with equal areas....
Read More →The perimeter of a rectangle becomes
Question: The perimeter of a rectangle becomes __________ times its original perimeter, if its length and breadth are doubled. Solution: Two times Explanation: We know that the perimeter of a rectangle is 2(l+b) When the length and breadth of the perimeter are doubled, we will get P = 2(2l +2b) Now take 2 outside, P = 2 [2(l+b)]...
Read More →If the diagonal d of a quadrilateral
Question: If the diagonal d of a quadrilateral is doubled and the heights h1 and h2 falling on d are halved, then the area of quadrilateral is __________. Solution: (h1+h2) d Explanation: Assume that ABCD be a quadrilateral, h1and h2are the heights on the diagonal BD = d, then, the area of a quadrilateral be = (1/2)(h1+h2) BD Since the diagonal is doubled and the heights are halved, we will get = (1/2) [ (h1/2) +(h2/2) ] 2d = (h1+h2) d...
Read More →The surface area of a cylinder
Question: The surface area of a cylinder which exactly fits in a cube of side b is __________. Solution: b2 Explanation: When the cylinder exactly fits in the cube of side b, the height equals to the edges of the cube and the radius equal to half the edges of a cube. It means that, h = b, and r = b/2 Then the CSA of a cylinder be = 2rh = 2 (b/2)(b) = b2...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=\frac{a x+b}{x^{2}+c}$, then $\left(2 x y_{1}+y\right) y_{3}=$ A. $3\left(x y_{2}+y_{1}\right) y_{2}$ B. $3\left(x y_{2}+y_{2}\right) y_{2}$ C. $3\left(x y_{2}+y_{1}\right) y_{1}$ D. none of these Solution: Given: $y=\frac{a x+b}{x^{2}+c}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{a}\left(\mathrm{x}^{2}+\mathrm{c}\right)-2 \mathrm{x}(\mathrm{ax}+\mathrm{b})}{\left(\mathrm{x}^{2}+\mathrm{c}\right)^{2}}$ $=\frac{-a x^{2}-2 b x+a ...
Read More →The volume of a cylinder
Question: The volume of a cylinder which exactly fits in a cube of side a is __________. Solution: a3/4 Explanation: When the cylinder exactly fits in the cube of side a, the height equals to the edges of the cube and the radius equal to half the edges of a cube. It means that, h = a,and r = a/2 Then the volume of a cylinder be = r2h = (a/2)2(a) = a3/4...
Read More →The curved surface area of a cylinder
Question: The curved surface area of a cylinder is reduced by ____________ per cent if the height is half of the original height. Solution: 50% Explanation: The CSA of cylinder with radius r and height h is 2rh When the height is halved, then new CSA is 2r (h/2) = rh Hence, the percentage reduction in CSA = [(2rh rh) (100)]/ 2rh = 50%...
Read More →The volume of a cylinder becomes
Question: The volume of a cylinder becomes __________ the original volume if its radius becomes half of the original radius. Solution: times Explanation: Volume of cylinder = r2h (when radius is r and height is h) When the radius is halved, then it becomes V = (r/2)2h V = (r2h)...
Read More →Three numbers are in AP, and their sum is 15. If 1, 4, 19
Question: Three numbers are in AP, and their sum is 15. If 1, 4, 19 be added to them respectively, then they are in GP. Find the numbers. Solution: To find: The numbers Given: Three numbers are in A.P. Their sum is 15 Formula used: When $a, b, c$ are in GP, $b^{2}=a c$ Let the numbers be a - d, a, a + d According to first condition a + d + a +a d = 15 $\Rightarrow 3 a=15$ $\Rightarrow a=5$ Hence numbers are 5 - d, 5, 5 + d When 1, 4, 19 be added to them respectively then the numbers become 5 d +...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=\frac{2}{\sqrt{a^{2}-b^{2}}} \tan ^{-1}\left(\frac{a-b}{a+b} \tan \frac{x}{2}\right), ab0$, then A. $\mathrm{y}_{1}=\frac{-1}{\mathrm{a}+\mathrm{b} \cos \mathrm{x}}$ B. $y_{2}=\frac{b \sin x}{(a+b \cos x)^{2}}$ C. $\mathrm{y}_{1}=\frac{1}{\mathrm{a}-\mathrm{b} \cos \mathrm{x}}$ D. $\mathrm{y}_{2}=\frac{-\mathrm{b} \sin \mathrm{x}}{(\mathrm{a}-\mathrm{b} \cos \mathrm{x})^{2}}$ Solution: Given: $y=\frac{2}{\sqrt{\left(a^{2}-b^{2}\righ...
Read More →If a cube fits exactly in a cylinder
Question: If a cube fits exactly in a cylinder with height h, then the volume of the cube is __________ and surface area of the cube is __________. Solution: volume is h3and surface area is 6h2 Explanation: Each side of a cube = h Thus, volume of cube = h3 Surface area of a cube = 6 (h2)...
Read More →The surface area of a cuboid formed
Question: The surface area of a cuboid formed by joining two cubes of side a face-to-face, is_______. Solution: 10a2 We have, two cubes of side a. These two cubes are joined face-to-face, then the resultant solid figure is a cuboid which has same breadth and height as the joined cubes has length twice of the length of a cube, i.e. l = 2a,b= aandh = a Thus, the total surface area of the cuboid = 2 (lb + bh + hi) = 2 (2a x a + a x a + a x 2a) = 2 (2 a2+ a2+ 2 a2)=2 x 5a2= 10a2...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=e^{\tan x}$, then $\left(\cos ^{2} x\right) y_{2}=$ A. $(1-\sin 2 x) y_{1}$ B. $-(1+\sin 2 x) y_{1}$ C. $(1+\sin 2 x) y_{1}$ D. none of these Solution: Given: $y=e^{\tan x}$ $\frac{d y}{d x}=e^{\tan x}(\sec x)^{2}$ $\frac{d^{2} y}{d x^{2}}=e^{\tan x}(\sec x)^{2}(\sec x)^{2}+e^{\tan x} \times 2 \sec x \times \tan x \times \sec x$ $=e^{\tan x}(\sec x)^{2}\left[(\sec x)^{2}+2 \tan x\right]$ $\left(\cos ^{2} x\right) y_{2}=e^{\tan x}\le...
Read More →Find the values of k for which
Question: Find the values of k for which k + 12, k 6 and 3 are in GP. Solution: To find: Value of k Given: k + 12, k 6 and 3 are in GP Formula used: (i) when $a, b, c$ are in GP $b^{2}=a c$ As, $k+12, k-6$ and 3 are in GP $\Rightarrow(k-6)^{2}=(k+12)(3)$ $\Rightarrow \mathrm{k}^{2}-12 \mathrm{k}+36=3 \mathrm{k}+36$ $\Rightarrow \mathrm{k}^{2}-15 \mathrm{k}=0$ $\Rightarrow \mathrm{k}(\mathrm{k}-15)=0$ $\Rightarrow \mathrm{k}=0$, Or $\mathrm{k}=15$ Ans) We have two values of k as 0 or 15...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=\left(\sin ^{-1} x\right)^{2}$, then $\left(1-x^{2}\right) y_{2}$ is equal to A. $x y_{1}+2$ B. $x y_{1}-2$ C. $-x y_{1}+2$ D. none of these Solution: Given: $y=\left(\sin ^{-1} x\right)^{2}$ $\frac{d y}{d x}=2 \sin ^{-1} x \frac{1}{\sqrt{1-x^{2}}}$ $\frac{d^{2} y}{d x^{2}}=2\left\{\left(\frac{1}{\sqrt{1-x^{2}}}\right)^{2}+\sin ^{-1} x \frac{\frac{2 x}{2 \sqrt{1-x^{2}}}}{\left(\sqrt{1-x^{2}}\right)^{2}}\right\}$ $=2\left\{\frac{1}{1...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=\sin \left(m \sin ^{-1} x\right)$, then $\left(1-x^{2}\right) y_{2}-x y_{1}$ is equal to A. $m^{2} y$ B. my C. $-m^{2} y$ D. none of these Solution: Given: $y=\sin \left(m \sin ^{-1} x\right)$ $\frac{d y}{d x}=m \cos \left(m \sin ^{-1} x\right) \frac{1}{\sqrt{\left(1-x^{2}\right)}}$ $x \frac{d y}{d x}=\cos \left(m \sin ^{-1} x\right) \frac{m x}{\sqrt{\left(1-x^{2}\right)}}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ $=m\lef...
Read More →If a, b, c are GP, then show that
Question: If a, b, c are GP, then show that $\frac{1}{\log _{a} m} \frac{1}{\log _{b} m}, \frac{1}{\log _{c} m}$ are in AP Solution: To prove: $\frac{1}{\log _{a} m}, \frac{1}{\log _{b} m^{\prime}}, \frac{1}{\log _{c} m}$ are in AP. Given: a, b, c are in GP Formula used: (i) $\frac{1}{\log _{a} m}=\log _{m} a=\frac{\log a}{\log m}$ As, a, b, c are in GP $\Rightarrow \frac{b}{a}=\frac{c}{b}$ Taking log both side $\log \frac{b}{a}=\log \frac{c}{b}$ $\Rightarrow \log b-\log a=\log c-\log b$ $\Right...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $x=f(t)$ and $y=g(t)$, then $\frac{d^{2} y}{d x^{2}}$ is equal to A. $\frac{f^{\prime} g^{\prime \prime}-g^{\prime} f^{\prime \prime}}{\left(f^{\prime}\right)^{3}}$ B. $\frac{f^{\prime} g^{\prime \prime}-g^{\prime} f^{\prime \prime}}{\left(f^{\prime}\right)^{2}}$ C. $\frac{g^{\prime \prime}}{f^{\prime \prime}}$ D. $\frac{f^{\prime \prime} g^{\prime}-g^{\prime \prime} f^{\prime}}{\left(g^{\prime}\right)^{3}}$ Solution: Given: $x=f(t)$ a...
Read More →