Find the height of a cuboid of volume 100 cm
Question: Find the height of a cuboid of volume 100 cm3, whose length and breadth are 5 cm and 4 cm respectively. Solution: Let us suppose that the height of the cuboid is $h \mathrm{~cm}$. Given : Volume of the cuboid $=100 \mathrm{~cm}^{3}$ Length $=5 \mathrm{~cm}$ Breadth $=4 \mathrm{~cm}$ Now, volume of the cuboid $=$ length $\times$ breadth $\times$ height $\Rightarrow 100=5 \times 4 \times h$ $\Rightarrow 100=20 \times h$ $\therefore h=\frac{100}{20}=5 \mathrm{~cm}$...
Read More →Test the continuity of the function on f(x) at the origin:
Question: Test the continuity of the function onf(x) at the origin: $f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, x \neq 0 \\ 1, x=0\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{x}{|x|}, x \neq 0 \\ 1, x=0\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)$ $=\lim _{h \rightarrow 0} \frac{-h}{|-h|}=\lim _{h \rightarrow 0} \frac{-h}{h}=\lim _{h \rightarrow 0}-1=-1$ (RHL at $x...
Read More →Find the volume of a cube whose side is
Question: Find the volume of a cube whose side is (i) 4 cm (ii) 8 cm (iii) 1.5 dm (iv) 1.2 m (v) 25 mm Solution: (i) The side of the given cube is $4 \mathrm{~cm}$. $\therefore$ Volume of the cube $=(\text { side })^{3}=(4)^{3}=64 \mathrm{~cm}^{3}$ (ii) The side of the given cube is $8 \mathrm{~cm}$. $\therefore$ Volume of the cube $=(\text { side })^{3}=(8)^{3}=512 \mathrm{~cm}^{3}$ (iii) The side of the given cube $=1.5 \mathrm{dm}=1.5 \times 10 \mathrm{~cm}=15 \mathrm{~cm}$ $\therefore$ Volum...
Read More →Determine the AP whose fifth term is 19
Question: Determine the AP whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20 Solution: Let the first term of an AP be a and common difference d. Given, a5= 19and a13 a8 = 20 [given] $\therefore a_{5}=a+(5-1) d=19$ and $[a+(13-1) d]-[a+(8-1) d]=20 \quad\left[\because a_{n}=a+(n-1) d\right]$ $\Rightarrow \quad a+4 d=19 \quad \ldots$ (i) and $\quad a+12 d-a-7 d=20 \Rightarrow 5 d=20$ $\therefore$ $d=4$ On putting $d=4$ in Eq. (i), we get $a+4(4)=19$ $a+16=1...
Read More →Find the volume of a cuboid whose
Question: Find the volume of a cuboid whose (i) length = 12 cm, breadth = 8 cm, height = 6 cm (ii) length =1.2 m, breadth = 30 cm, height = 15 cm (iii) length = 15 cm, breadth = 2.5 dm, height = 8 cm. Solution: (i) In the given cuboid, we have: length $=12 \mathrm{~cm}$, breadth $=8 \mathrm{~cm}$ and height $=6 \mathrm{~cm}$ $\therefore$ Volume of the cuboid $=$ length $\times$ breadth $\times$ height $=12 \times 8 \times 6$ $=576 \mathrm{~cm}^{3}$ (ii) In the given cuboid, we have: length $=1.2...
Read More →The dimensions of a rectangular box are in the ratio of 2 : 3 : 4
Question: The dimensions of a rectangular box are in the ratio of 2 : 3 : 4 and the difference between the cost of covering it with sheet of paper at the rates of Rs 8 and Rs 9.50 per m2is Rs. 1248. Find the dimensions of the box. Solution: Suppose that the dimensions be $\mathrm{x}$ multiple of each other. The dimensions are in the ratio $2: 3: 4$. Hence, length $=2 \mathrm{x} \mathrm{m}$ Breadth $=3 \mathrm{x} \mathrm{m}$ Height $=4 \mathrm{x} \mathrm{m}$ So, total surface area of the rectangu...
Read More →Find a, b and c such that the
Question: Find a, b and c such that the following numbers are in AP, a, 7, b, 23 and c. Solution: Since a, 7, b, 23 and c are in AR $\therefore \quad 7-a=b-7=23-b=c-23=$ Common difference Taking second and third terms, we get $b-7=23-b$ $\Rightarrow \quad 2 b=30$ $\therefore \quad b=15$ Taking first and second terms, we get $7-a=b-7$ $\Rightarrow \quad 7-a=15-7 \quad[\because b=15]$ $\Rightarrow \quad 7-a=8$ $\therefore \quad a=-1$ Taking third and fourth terms, we get $23-b=c-23$ $\Rightarrow \...
Read More →The external dimensions of a closed wooden box are 48 cm, 36 cm, 30 cm.
Question: The external dimensions of a closed wooden box are 48 cm, 36 cm, 30 cm. The box is made of 1.5 cm thick wood. How many bricks of size 6 cm 3 cm 0.75 cm can be put in this box? Solution: The outer dimensions of the closed wooden box are $48 \mathrm{~cm} \times 36 \mathrm{~cm} \times 30 \mathrm{~cm} .$ Also, the box is made of a $1.5 \mathrm{~cm}$ thick wood, so the inner dimensions of the box will be $(2 \times 1.5=3) \mathrm{cm}$ less. i. e., the inner dimensions of the box are $45 \ma...
Read More →The dimensions of a cinema hall are 100 m, 50 m and 18 m.
Question: The dimensions of a cinema hall are 100 m, 50 m and 18 m. How many persons can sit in the hall, if each person requires 150 m3of air? Solution: The dimensions of a cinema hall are $100 \mathrm{~m} \times 50 \mathrm{~m} \times 18 \mathrm{~m}$. i.e., volume of air in the cinema hall $=100 \times 50 \times 18=90000 \mathrm{~m}^{3}$ It is given that each person requires $150 \mathrm{~m}^{3}$ of air. $\therefore$ The number of persons that can sit in the cinema hall $=\frac{\text { volume o...
Read More →Write the following sets in roster from:
Question: Write the following sets in roster from: C = {x : x is a two-digit number such that the sum of its digits is 9}. Solution: The elements of this set are 18, 27, 36, 45, 54, 63, 72, 81 and 90 So, C = {18, 27, 36, 45, 54, 63, 72, 81, 90}...
Read More →A metal cube of edge 12 cm is melted and formed into three smaller cubes.
Question: A metal cube of edge 12 cm is melted and formed into three smaller cubes. If the edges of the two smaller cubes are 6 cm and 8 cm, find the edge of the third smaller cube. Solution: Let the edge of the third cube be $\mathrm{x} \mathrm{cm}$. Three small cubes are formed by melting the cube of edge $12 \mathrm{~cm}$. Edges of two small cubes are $6 \mathrm{~cm}$ and $8 \mathrm{~cm}$. Now, volume of a cube $=(\text { side })^{3}$ Volume of the big cube $=$ sum of the volumes of the three...
Read More →Write the following sets in roster from:
Question: Write the following sets in roster from: B = {x : x is an integer and 4 x 6}. Solution: Integers = -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, The elements of this set are -3, -2, -1, 0, 1, 2, 3, 4 and 5 only. So, B = {-3, -2, -1, 0, 1, 2, 3, 4, 5}...
Read More →The length of a hall is 18 m and the width 12 m.
Question: The length of a hall is 18 m and the width 12 m. The sum of the areas of the floor and the flat roof is equal to the sum of the areas of the four walls. Find the height of the wall. Solution: Length of the hall $=18 \mathrm{~m}$ Its width $=12 \mathrm{~m}$ Suppose that the height of the wall is $\mathrm{h} \mathrm{m}$. Also, sum of the areas of the floor and the flat roof $=$ sum of the areas of the four walls $\Rightarrow 2 \times($ length $\times$ breadth $)=2 \times($ length $+$ bre...
Read More →Write the first three terms of the
Question: Write the first three terms of the AP's, when a and $d$ are as given below (i) $a=\frac{1}{2}, d=\frac{-1}{6}$ (ii) $a=-5, d=-3$ (iii) $a=\sqrt{2}, d=\frac{1}{\sqrt{2}}$ Solution: (i) Given that, first term $(a)=\frac{1}{2}$ and common difference $(d)=-\frac{1}{6}$ $\because$ nth term of an AP, $T_{n}=a+(n-1) d$ $\therefore$ Second term of an AP, $T_{2}=a+d=\frac{1}{2}-\frac{1}{6}=\frac{2}{6}=\frac{1}{3}$ and third term of an AP, $T_{3}=a+2 d=\frac{1}{2}-\frac{2}{6}=\frac{1}{2}-\frac{1...
Read More →Write the following sets in roster from:
Question: Write the following sets in roster from: A = {x : x is a natural number, 30 x 36}. Solution: Natural numbers = 1, 2, , 30, 31, 32, 33, 34, 35, 36, The elements of this set are 30, 31, 32, 33, 34 and 35 only So, A = {30, 31, 32, 33, 34, 35}...
Read More →The cost of preparing the walls of a room 12 m
Question: The cost of preparing the walls of a room 12 m long at the rate of Rs 1.35 per square metre is Rs 340.20 and the cost of matting the floor at 85 paise per square metre is Rs 91.80. Find the height of the room. Solution: The cost of preparing 4 walls of a room whose length is $12 \mathrm{~m}$ is $\mathrm{Rs} 340.20$ at a rate of $\mathrm{Rs} 1.35 / \mathrm{m}^{2}$. $\therefore$ Area of the four walls of the room $=\frac{\text { total cost }}{\text { rate }}=\frac{\mathrm{Rs} 340.20}{\ma...
Read More →Let A be the set of all even whole numbers less than 10.
Question: Let A be the set of all even whole numbers less than 10. (i) Write A in the roster from. (ii) Fill in the blanks with the approximate symbol or ϵ : (a) 0 . A (b) 10 . A (c) 3 . A (d) 6 . A Solution: (i) Whole numbers are 0, 1, 2, 3, Even whole numbers less than 10 are 0, 2, 4, 6, 8, 9 So, A = {0, 2, 4, 6, 8} (ii) (a) Here, A = {0, 2, 4, 6, 8} Hence, 0 A (b) Here, A = {0, 2, 4, 6, 8} As 10 is not in a set A Hence, 10 A (c) Here, A = {0, 2, 4, 6, 8} As 3 is not in a set A Hence, 3 A (d) ...
Read More →Three cubes whose edges measure 3 cm, 4 cm, and 5 cm respectively are melted to form a new cube.
Question: Three cubes whose edges measure 3 cm, 4 cm, and 5 cm respectively are melted to form a new cube. Find the surface area of the new cube formed. Solution: Three cubes of edges $3 \mathrm{~cm}, 4 \mathrm{~cm}$ and $5 \mathrm{~cm}$ are melted and molded to form a new cube. i. e., volume of the new cube = sum of the volumes of the three cubes $=(3)^{3}+(4)^{3}+(5)^{3}$ $=27+64+125$ $=216 \mathrm{~cm}^{3}$ We know that volume of a cube $=(\text { side })^{3}$ $\Rightarrow 216=(\text { side }...
Read More →Two cubes, each of volume 512 cm
Question: Two cubes, each of volume 512 cm3are joined end to end. Find the surface area of the resulting cuboid. Solution: Two cubes each of volume $512 \mathrm{~cm}^{3}$ are joined end to end. Now, volume of a cube $=(\text { side })^{3}$ $\Rightarrow 512=(\text { side })^{3}$ $\Rightarrow$ Side of the cube $=\sqrt[3]{512}=8 \mathrm{~cm}$ If the cubes area joined side by side, then the length of the resulting cuboid is $2 \times 8 \mathrm{~cm}=16 \mathrm{~cm}$. Breadth $=8 \mathrm{~cm}$ Height ...
Read More →A field is 150 m long and 100 m wide.
Question: A field is 150 m long and 100 m wide. A plot (outside the field) 50 m long and 30 m wide is dug to a depth of 8 m and the earth taken out from the plot is spread evenly in the field. By how much is the level of field raised? Solution: The dimensions of the plot dug outside the field are $50 \mathrm{~m} \times 30 \mathrm{~m} \times 8 \mathrm{~m}$. Hence, volume of the earth dug - out from the plot $=50 \times 30 \times 8=12000 \mathrm{~m}^{3}$ Suppose that the level of the earth rises b...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. The collection of all interesting dramas written by Shakespeare Solution: As the collection of all interesting dramas written by Shakespeare is not well - defined because it depends on person interest. , this is not a set....
Read More →Solve this
Question: If $A=\left[\begin{array}{ll}2 4 \\ 4 3\end{array}\right], X=\left[\begin{array}{l}n \\ 1\end{array}\right], B=\left[\begin{array}{c}8 \\ 11\end{array}\right]$ and $A X=B$, then find $n$. Solution: Here, $\left[\begin{array}{ll}2 4 \\ 4 3\end{array}\right]\left[\begin{array}{l}n \\ 1\end{array}\right]=\left[\begin{array}{c}8 \\ 11\end{array}\right]$ $\Rightarrow\left[\begin{array}{l}2 n+4 \\ 4 n+3\end{array}\right]=\left[\begin{array}{c}8 \\ 11\end{array}\right]$ $\Rightarrow 2 n+4=8$ ...
Read More →The dimensions of a room are 12.5 m by 9 m by 7 m.
Question: The dimensions of a room are 12.5 m by 9 m by 7 m. There are 2 doors and 4 windows in the room; each door measures 2.5 m by 1.2 m and each window 1.5 m by 1 m. Find the cost of painting the walls at Rs 3.50 per square metre. Solution: The dimensions of the room are $12.5 \mathrm{~m} \times 9 \mathrm{~m} \times 7 \mathrm{~m}$. Hence, the surface area of walls $=2 \times($ length $\times$ height $+$ breadth $\times$ height $)$ $=2 \times(12.5 \times 7+9 \times 7)$ $=301 \mathrm{~m}^{2}$ ...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. The collection of all persons of Kolkata whose assessed annual incomes exceed (say) Rs 20 lakh in the 4 financial years 2016-17. Solution: As the collection of all persons of Kolkata whose assessed annual incomes exceed (say) Rs 20 lakh in the 4 financial years 2016-17 is known and well defined. , this is a set...
Read More →Solve this
Question: If $\left[\begin{array}{lll}1 0 0 \\ 0 0 1 \\ 0 1 0\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}2 \\ -1 \\ 3\end{array}\right]$, find $x, y, z$ Solution: Here, $\left[\begin{array}{lll}1 0 0 \\ 0 0 1 \\ 0 1 0\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}2 \\ -1 \\ 3\end{array}\right]$ $\Rightarrow\left[\begin{array}{l}x \\ z \\ y\end{array}\right]=\left[\begin{array}{c}2 \\ -1 \\ 3\end{array}\r...
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