Euler’s formula is true for
Question: Eulers formula is true for all three-dimensional shapes. Solution: False Eulers formula is true only for polyhedrons, i.e. F+V-E = 2 Where F = faces, V = vertices and E = edges...
Read More →Find the second order derivatives of each of the following functions:
Question: Find the second order derivatives of each of the following functions: $x^{3}+\tan x$ Solution: Basic idea: $\sqrt{S e c o n d}$ order derivative is nothing but derivative of derivative i.e. $\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)$ $\sqrt{T h e}$ idea of chain rule of differentiation: If $\mathrm{f}$ is any real-valued function which is the composition of two functions $u$ and $v$, i.e. $f=v(u(x))$. For the sake of simplicity just assume $t=u(x)$ Then $f=v(t) ...
Read More →Every cylinder has 2 opposite faces
Question: Every cylinder has 2 opposite faces as congruent circles, so it is also a prism. Solution: False The cylinder has a congruent cross-section which is a circle, so it could be called as a circular prism....
Read More →Pentagonal prism has 5 pentagons.
Question: Pentagonal prism has 5 pentagons. Solution: False Pentagonal prism has 2 pentagons, one on the top and other on the base....
Read More →Solve this
Question: If $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in AP, prove that (i) $\frac{(b+c)}{a}, \frac{(c+a)}{b}, \frac{(a+b)}{c}$ are in AP. (ii) $\frac{(b+c-a)}{a}, \frac{(c+a-b)}{b}, \frac{(a+b-c)}{c}$ are in AP. Solution: (i) $\frac{(b+c)}{a}, \frac{(c+a)}{b}, \frac{(a+b)}{c}$ are in A.P. To prove: $\frac{(b+c)}{a}, \frac{(c+a)}{b}, \frac{(a+b)}{c}$ are in A.P. Given: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P. Proof: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P. If each term o...
Read More →Regular octahedron has 8 congruent
Question: Regular octahedron has 8 congruent faces which are isosceles triangles. Solution: False A regular octahedron is obtained by joining two congruent square pyramids such that the vertices of the two square pyramids coincide. It has eight congruent equilateral triangular faces....
Read More →A polyhedron can have
Question: A polyhedron can have 4 faces. Solution: False A polyhedron can have atleast 4 faces....
Read More →The other name of cuboid
Question: The other name of cuboid is tetrahedron. Solution: False The other name of cuboid is rectangular prism....
Read More →A regular polyhedron is a solid
Question: A regular polyhedron is a solid made up of______faces. Solution: A regular polyhedron is a solid made up of congruent faces. [according to the definition of regular polyhedron]...
Read More →Total number of regular
Question: Total number of regular polyhedron is______ Solution: Total number of regular polyhedron is five, i.e. cube, octahedron, tetrahedron, dodecahedron and icosahedron....
Read More →If the solve the problem
Question: If $y=a\left\{x+\sqrt{x}^{2}+1\right\}^{n}+b\left\{x-\sqrt{x}^{2}+1\right\}^{-n}$, prove that $\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-n^{2}=0$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{d}{d x} x^{n}=n x^{n-1}$ (iii) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d}(\text { wou })}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}=\frac{\mathrm{dw}}{\mathrm{ds}} \cdot \frac{\mathrm{ds}}{\mathrm{dt...
Read More →A pentagonal prism has
Question: A pentagonal prism has ______ faces. Solution: A pentagonal prism has 7 faces....
Read More →If a, b, c are in AP, show that
Question: If a, b, c are in AP, show that (i) $(b+c-a),(c+a-b),(a+b-c)$ are in AP. (ii) $\left(b c-a^{2}\right),\left(c a-b^{2}\right),\left(a b-c^{2}\right)$ are in AP. Solution: (i) $(b+c-a),(c+a-b),(a+b-c)$ are in AP. To prove: $(b+c-a),(c+a-b),(a+b-c)$ are in AP. Given: a, b, c are in A.P. Proof: Let d be the common difference for the A.P. a,b,c Since a, b, c are in A.P $\Rightarrow b-a=c-b=$ common difference $\Rightarrow a-b=b-c=d$ $\Rightarrow 2(a-b)=2(b-c)=2 d \ldots$ (i) Considering ser...
Read More →If 4 km on a map is represented by
Question: If 4 km on a map is represented by 1 cm, then 16 km is represented by______cm. Solution: Given, 4 km on a map is represented by 1 cm, then 1 km on a map is represented by 1/4 cm. Hence, 16 km on a map is represented by x 16 = 4 cm...
Read More →In a three-dimensional shape,
Question: In a three-dimensional shape, diagonal is a line segment that joins two vertices that do not lie on the______face. Solution: In a three-dimensional shape, diagonal is a line segment that joins two vertices that do not lie on the same face....
Read More →The net of a rectangular prism
Question: The net of a rectangular prism has ______ rectangles. (Hint: Every square is a rectangle but every rectangle is not a square.) Solution: The net of a rectangular prism has six rectangles....
Read More →The given net can be
Question: The given net can be folded to make a ______. Solution: The given net can be folded to make a prism....
Read More →A pyramid on an n sided polygon
Question: A pyramid on an n sided polygon has ______ faces. Solution: A pyramid on an n sided polygon has n+1 faces....
Read More →In the figure,
Question: In the figure, the number of faces meeting at B is ________. Solution: The number of faces meeting at B is 4....
Read More →If the solve the problem
Question: If $\mathrm{y}=\mathrm{x}^{\mathrm{n}}\{\mathrm{a} \cos (\log \mathrm{x})+\mathrm{b} \sin (\log \mathrm{x})\}$, prove that $\mathrm{x}^{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+(1-2 \mathrm{n}) \frac{\mathrm{dy}}{\mathrm{dx}}+\left(1+\mathrm{n}^{2}\right) \mathrm{y}=0$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{\mathrm{d}}{\mathrm{dx}} \cos \mathrm{x}=\sin \mathrm{x}$ (iii) $\frac{d}{d x} \sin x=-\cos x$ (iv) $\frac{\mat...
Read More →Square prism is also called a _______.
Question: Square prism is also called a _______. Solution: Square prism is also called a cube. A cube is a platonic solid because all six of its faces are congruent squares....
Read More →According to the map,
Question: According to the map, the number of schools in the town is (a) 4 (b) 3 (c) 5 (d) 2 Solution: (c) 5...
Read More →The ratio of the number of general
Question: The ratio of the number of general stores and that of the ground is (a) 1 : 2 (b) 2 : 1 (c) 2 : 3 (d) 3 : 2 Solution: (d) 3: 2 By observing the given map, The number of general stores = 6 The number of ground = 4 Then, The ratio of the number of general stores and that of the ground is = 6/4 = 3/2 = 3: 2...
Read More →If the solve the problem
Question: If $y=A e^{-k t} \cos (p t+c)$, prove that $\frac{d^{2} y}{d t^{2}}+2 k \frac{d y}{d t}+n^{2} y=0$, where $n^{2}=p^{2}+k^{2}$ Solution: Formula: - (i) $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}_{1}$ and $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{y}_{2}$ (ii) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{e}^{\mathrm{ax}}=\mathrm{ae}^{\mathrm{ax}}$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \cos \mathrm{x}=\sin \mathrm{x}$ (iv) $\frac{d}{d x} \sin x=-\cos x$ (v) chain rule $\frac{\...
Read More →In a blueprint of a room,
Question: In a blueprint of a room, an architect has shown the height of the room as 33 cm. If the actual height of the room is 330 cm, then the scale used by her is (a) 1:11 (b) 1:10 (c) 1:100 (d) 1:3 Solution: (b) 1: 10 From the question it is given that, An architect has shown the height of the room as 33 cm The actual height of the room is 330 cm Then, the scale used by an architect is = Drawn size/actual size = 33/330 [divide both by 33] = 1/10 = 1: 10...
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