Comparison of parts of a
Question: Comparison of parts of a whole may be done by a (a) bar graph (b) pie chart (c) linear graph (c) line graph Solution: (b) There are various ways to represent and compare the data. One of them is pie chart. Pie chart is a pictorial representation of the data in which the whole is represented by a circle and the parts, by non-intersecting adjacent sectors. Hence, comparison of parts of a whole may be done by a pie chart....
Read More →A cuboidal tin’ box opened at the top has
Question: A cuboidal tin box opened at the top has dimensions 20 cm x 16 cm x 14 cm. What is the total area of metal sheet required to make 10 such boxes? Solution: Dimensions of cuboidal tin box are20 cm x 16cm x 14 cm, ...Area of metal sheet for 1 box = Surface area of cuboid = 2(lb + bh+hl) = 2(20 x 16+16 x 14+ 14 x 20) = 2(320 + 224 + 280) = 2(824) = 1648 cm2 ... Area of metal sheet required to make 10 such boxes = 10 x 1648= 16460cm2...
Read More →If a, b, c are in GP, prove that
Question: If a, b, c are in GP, prove that (i) $a\left(b^{2}+c^{2}\right)=c\left(a^{2}+b^{2}\right)$ (ii) $\frac{1}{\left(a^{2}-b^{2}\right)}+\frac{1}{b^{2}}=\frac{1}{\left(b^{2}-c^{2}\right)}$ (iii) $(a+2 b+2 c)(a-2 b+2 c)=a^{2}+4 c^{2}$ (iv) $a^{2} b^{2} c^{2}\left(\frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}\right)=a^{3}+b^{3}+c^{3}$ Solution: (i) $a\left(b^{2}+c^{2}\right)=c\left(a^{2}+b^{2}\right)$ To prove: $a\left(b^{2}+c^{2}\right)=c\left(a^{2}+b^{2}\right)$ Given: $a, b, c$ are in GP...
Read More →If the length of each edge
Question: If the length of each edge of a cube is tripled, what will be the change in its volume? Solution: Let the edge of a cube be a. If edge of the cube became tripled i.e. a = 3 x a = 3a ... Volume of the cube = a3 ... Volume of the cube with edge tripled = (3a)3= 27a3 Hence, volume is 27 times of the original volume....
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=x^{n-1} \log x$, then $x^{2} y_{2}+(3-2 n) x y_{1}$ is equal to A. $-(n-1)^{2} y$ B. $(n-1)^{2} y$ C. $-n^{2} y$ D. $n^{2} y$ Solution: Given: $y=x^{n-1} \log x$ $\frac{d y}{d x}=(n-1) x^{n-2} \log x+\frac{1}{x} x^{n-1}$ $=(n-1) x^{n-2} \log x+x^{n-2}$ $=x^{n-2}[(n-1) \log x+1]$ $x y_{1}=x^{n-1}[(n-1) \log x+1]$ $=(n-1) y+x^{n-1}$ $(3-2 n) x y_{1}=(3-2 n)\left[(n-1) y+x^{n-1}\right]$ $=\left(3 n-3-2 n^{2}+2 n\right) y+3 x^{n-1}-2 n ...
Read More →A cube of side 5 cm is cut into as many
Question: A cube of side 5 cm is cut into as many 1 cm cubes as possible. What is the ratio of the surface areas of the original cube to that of the sum of the surface areas of the smaller cubes? Solution: Surface area of a cube = 6 a2, where a is side of a cube.... Side of cube = 5cm ... Surface area of the cube = 6 x (5)2= 6 x 25 = 150cm2 Now, surface area of the cube with side 1 cm = 6 x (1)2= 6 cm2 ... Surface area of 5 cubes with side 1 cm = 5 x 6 = 30 cm2 Ratio of the surface area of the o...
Read More →The circumference of the front wheel
Question: The circumference of the front wheel of a cart is 3 m long and that of the back wheel is 4 m long. What is the distance travelled by the cart, when the front wheel makes five more revolutions than the rear wheel? Solution: Given, circumference of front wheel = 3 m Now, distance covered by front wheel of the cart in 1 revolution = Circumference of front wheel . ... Distance covered by front wheel in 5 revolutions = 3 x 5 = 15 m Hence, the distance covered by the cart is 15 m....
Read More →Ratio of area of a circle to the area
Question: Ratio of area of a circle to the area of a square whose side equals radius of circle, is 1 : TC. Solution: False Given, side of a square equals radius of a circle. Then, area of the square = r2 and area of the circle =r2 where r is a radius of the circle. Now, the ratio of area of the circle to area of the square = r2:r2 = : 1....
Read More →The surface area of a cube formed
Question: The surface area of a cube formed by cutting a cuboid of dimensions 2 x 1 x 1 in 2 equal parts, is 2 sq units. Solution: False The dimensions of the given cuboid are 2 x 1 x 1. It is sliced into two equal parts, which are cubes. Then, the dimensions of the cube, so formed are 1 x 1 x 1. ...The surface area of the cube so formed = 6 (Side)2= 6 x (1)2= 6sq units Hence, the surface area of the sliced cube is 6 sq units....
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $\frac{d}{d x}\left\{x^{n}-a_{1} x^{n-1}+a_{2} x^{n-2}+\ldots+(-1)^{n} a_{n}\right\}$ $e^{x}=x^{n} e^{x}$ Then the value of $a_{r}, 0r \leq n$, is equal to A. $\frac{\mathrm{n} !}{\mathrm{r} !}$ B. $\frac{(n-r) !}{r !}$ C. $\frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) !}$ D. none of these Solution: Given: $\frac{d}{d x}\left\{x^{n}-a_{1} x^{n-1}+a_{2} x^{n-2}+\cdots+(-1)^{n} a_{n}\right\} e^{x}=x^{n} e^{x}$ $\frac{d}{d x}\left\{a_{0}(-1...
Read More →A cube of side 3 cm painted on all its faces,
Question: A cube of side 3 cm painted on all its faces, when sliced into 1 cu cm cubes, will have exactly 1 cube with none of its faces painted. Solution: True Given, a cube of side 3 cm is painted on all its faces. Now, it is sliced into 1 cu cm cubes. Then, there will be 8 corner cubes that have 3 sides painted, 6 centre cubes with only one side painted and only 1 cube in the middle that has no side painted....
Read More →The area of a trapezium becomes 4 times,
Question: The area of a trapezium becomes 4 times, if its height gets doubled. Solution: False We know that, Area of a trapezium = 1/2 (a + b) x h where, a and b are the lengths of parallel sides and h is the altitude (height). Now, if the height gets doubled, then Area of trapezium = 1/2(a + b) x 2h = 2(1/2(a + b) x h) Hence, the area is doubled. So, the statement is false....
Read More →If (a – b), (b – c), (c – a) are in GP then prove that
Question: If $(a-b),(b-c),(c-a)$ are in GP then prove that $(a+b+c)^{2}=3(a b+b c+$ ca). Solution: To prove: $(a+b+c)^{2}=3(a b+b c+c a)$. Given: $(a-b),(b-c),(c-a)$ are in GP Formula used: When $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}, \mathrm{b}^{2}=\mathrm{ac}$ As, $(a-b),(b-c),(c-a)$ are in GP $\Rightarrow(b-c)^{2}=(a-b)(c-a)$ $\Rightarrow b^{2}-2 c b+c^{2}=a c-a^{2}-b c+a b$ $\Rightarrow a^{2}+b^{2}+c^{2}-b c-a c-a b=0$ Adding 3(ab + bc + ac) both side $\Rightarrow a^{2}+b^{...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y^{1 / n}+y^{-1 / n}=2 x$, then $\left(x^{2}-1\right) y_{2}+x y_{1}=$ A. $-n^{2} y$ B. $n^{2} y$ C. 0 D. none of these Solution: Given: $\mathrm{y}^{1 / \mathrm{n}}+\mathrm{y}^{-1 / \mathrm{n}}=2 \mathrm{x}$ $\frac{1}{n} y^{\frac{1}{n}-1} \frac{d y}{d x}+\frac{-1}{n} y^{\frac{-1}{n}-1} \frac{d y}{d x}=2$ $\frac{1}{n} \frac{d y}{d x}\left\{y^{\frac{1}{n}-1}-y^{\frac{-1}{n}-1}\right\}=2$...
Read More →The surface area of a cuboid formed
Question: The surface area of a cuboid formed by joining face-to-face 3 cubes of side x is 3 times the surface area of a cube of side x. Solution: False Three cubes having side x are joined face-to-face, then the cuboid so formed has the same height and breadth as the cubes but its length will be thrice that of the cubes. Hence, the length, breadth and height of the cuboid so formed are 3x, x and x, respectively. Then, its surface area = 2 (lb + bh + hl) = 2(3xxx+xxx+xx 3x)=2(3x2+x2+ 3x2) = 2 x ...
Read More →If a, b, c are in GP, prove that
Question: If a, b, c are in GP, prove that $\frac{a^{2}+a b+b^{2}}{a b+b c+c a}=\frac{b+a}{c+b}$ Solution: To prove: $\frac{a^{2}+a b+b^{2}}{a b+b c+c a}=\frac{b+a}{c+b}$ Given: a, b, c are in GP Formula used: When $a, b, c$ are in GP, $b^{2}=a c$ a, b, c are in GP $\Rightarrow \mathrm{b}^{2}=\mathrm{ac} \ldots$ (i) $\Rightarrow \mathrm{b}=\sqrt{\mathrm{ac}}$ (ii) Taking LHS = $\frac{a^{2}+a b+b^{2}}{a b+b c+c a}$ Substituting the value $b^{2}=a c$ from eqn. (i) $\mathrm{LHS}=\frac{\mathrm{a}^{2...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $x=f(t) \cos t-f^{\prime}(t) \sin t$ and $y=f(t) \sin t+f^{\prime}(t) \cos t, t h e n\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}=$ A. $f(t)-f^{\prime \prime}(t)$ B. $\left\{f(t)-f^{\prime \prime}(t)\right\}^{2}$ C. $\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$ D. none of these Solution: Given: $x=f(t) \cos t-f^{\prime}(t) \sin t$ $y=f(t) \sin t+f^{\prime}(t) \cos t$ $\frac{d x}{d t}=f^{\prime}(t) \cos t-f(t) \sin...
Read More →The areas of any two faces
Question: The areas of any two faces of a cuboid are equefl. Solution: False A cuboid has rectangular faces with different lengths and breadths. Only opposite faces of cuboid have the same length and breadth. Therefore, areas of only opposite faces of a cuboid are equal....
Read More →The areas of any two faces
Question: The areas of any two faces of a cube are equal. Solution: True Since, all the faces of a cube are squares of same side length, therefore the areas of any two faces of a cube are equal....
Read More →__________ surface area of room = area
Question: __________ surface area of room = area of 4 walls. Solution: Lateral Explanation: We know that, the rooms are in the cuboid shape. The walls are considered as the lateral faces of the cuboid shaped room....
Read More →__________ of a solid is the measurement
Question: __________ of a solid is the measurement of the space occupied by it. Solution: The space occupied by any solids or three dimensional shaped are always measured in terms of volume....
Read More →In the above question,
Question: In the above question, curved surface area of A is ________ curved surface area of B. Solution: Same Explanation: For cylinder A, h= 10 cm and r = 10/ Thus, CSA of cylinder A = 2rh = 200 For cylinder B, h= 12 cm and r = 5/ Thus, CSA of cylinder B = 2rh = 200...
Read More →The sum of three numbers in GP is 56.
Question: The sum of three numbers in GP is 56. If 1, 7, 21 be subtracted from them respectively, we obtain the numbers in AP. Find the numbers Solution: To find: Three numbers Given: Three numbers are in G.P. Their sum is 56 Formula used: When $a, b, c$ are in $G P, b^{2}=a c$ Let the three numbers in GP be $a, a r, a r^{2}$ According to condition :- $a+a r+a r^{2}=56$ $a\left(1+r+r^{2}\right)=56 \ldots$ (i) 1, 7, 21 be subtracted from them respectively, we obtain the numbers as :- $a-1, a r-7,...
Read More →Two cylinders A and B are formed by folding
Question: Two cylinders A and B are formed by folding a rectangular sheet of dimensions 20 cm 10 cm along its length and also along its breadth respectively. Then volume of A is ________ of volume of B. Solution: Twice Explanation: Rectangular sheet dimension is 20 cm 10 cm When a cylinder is folded along its length, which is 20 cm, then the resultant cylinder is with height 10 cm. Again, if a cylinder is folded along its breadth, which is 10 cm, then the resultant cylinder is with height 20 cm ...
Read More →Area of a rhombus = 1/2
Question: Area of a rhombus = 1/2 product of_______. Solution: diagonals We know that, the area of a rhombus = Half of the product of its diagonals =1/2 [Product of diagonals]...
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