eSaral - JEE | NEET | Class 8-10 Exam Prep App

Aasheesh can paint his doll in 20 minutes and his sister Chinki can do so in 25 minutes.

Question: Aasheesh can paint his doll in 20 minutes and his sister Chinki can do so in 25 minutes. They paint the doll together for five minutes. At this juncture they have a quarrel and Chinki withdraws from painting. In how many minutes will Aasheesh finish the painting of the remaining doll? Solution: Aasheesh can paint a doll in 20 minutes, and Chinki can do the same in 25 minutes. $\therefore$ Work done by Aasheesh in 1 minute $=\frac{1}{20}$ $\therefore$ Work done by Chinki in 1 minute $=\...

From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out.

Question: From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. Solution: We have, the height of the cone $=$ the height of the cylinder $=h=2.8 \mathrm{~cm}$ and the radius of the base, $r=\frac{4.2}{2}=2.1 \mathrm{~cm}$ The slant height of the cone, $l=\sqrt{r^{2}+h^{2}}$ $=\sqrt{2.1^{2}+2.8^{2}}$ $=\sqrt{4.41+7.84}$ $=\sqrt{12.25}$ $=3.5 \mathrm{~cm}$ Now, the total...

Construct a rhombus whose side is of length

Question: Construct a rhombus whose side is of length 3.4 cm and one of its angles is 45. Solution: We know that, in rhombus all sides are equal. To construct a rhombus whose side is of length $3.4 \mathrm{~cm}$ and one of its angle is $45^{\circ}$, use the following steps 1. Draw a line segment AS of length $3.4 \mathrm{~cm}$. 2. Now, generate an angle $45^{\circ}$ at both ends $A$ and $B$ of line segment $A B$ and plot the parallel lines $A X$ and BY. 3. Cut $A D$ and $S C$ of length $3.4 \mat...

A can do a piece of work in 40 days and B in 45 days.

Question: Acan do a piece of work in 40 days andBin 45 days. They work together for 10 days and thenBgoes away. In how many days willAfinish the remaining work? Solution: It is given that A can finish the work in 40 days and B can finish the same work in 45 days. $\therefore$ Work done by A in 1 day $=\frac{1}{40}$ Work done by B in 1 day $=\frac{1}{45}$ $\therefore$ Work done by $(\mathrm{A}+\mathrm{B})$ in 1 day $=\frac{1}{40}+\frac{1}{45}$ $=\frac{9+8}{360}=\frac{17}{360}$ $\therefore$ Work d...

Find the matrix X satisfying the equation

Question: Find the matrixXsatisfying the equation $\left[\begin{array}{ll}2 1 \\ 5 3\end{array}\right] X\left[\begin{array}{ll}5 3 \\ 3 2\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}2 1\end{array}\right.$ $5 \quad 3], B=\left[\begin{array}{ll}5 3\end{array}\right]$ $\left.\begin{array}{ll}3 2\end{array}\right]$ and $I=\left[\begin{array}{ll}1 0\end{array}\right.$ $\left.\begin{array}{ll}0 1\end{array}\right]$ $\Rightarrow|\mathrm...

Construct a rectangle whose adjacent

Question: Construct a rectangle whose adjacent sides are of lengths 5 cm and 3.5 cm. Solution: We know that, each angle of a rectangle is right angle (i.e., 90) and its opposite sides are equal and parallel.To construct a rectangle whose adjacent sides are of lengths 5 cm and 3.5 cm, use the 1 following steps Draw a line segment BC of length 5 cm. Now, generate an angle of 90 at points B and C of the line segment BC and plot the parallel lines BX and CY at these points. Cut AB and CD of length 3...

A and B can do a piece of work in 20 days and B in 15 days.

Question: AandBcan do a piece of work in 20 days andBin 15 days. They work together for 2 days and thenAgoes away. In how many days willBfinish the remaining work? Solution: It is given that A can finish the work in 20 days and B can finish the same work in 15 days. $\therefore$ Work done by A in 1 day $=\frac{1}{20}$ Work done by B in 1 day $=\frac{1}{15}$ $\therefore$ Work done by $(\mathrm{A}+\mathrm{B})$ in 1 day $=\frac{1}{20}+\frac{1}{15}$ $=\frac{3+4}{60}=\frac{7}{60}$ $\therefore$ Work d...

A and B can finish a work in 20 days.

Question: $A$ and $B$ can finish a work in 20 days. A alone can do $\frac{1}{5}$ th of the work in 12 days. In how many days can $B$ alone do it? Solution: It is given that $\mathrm{A}$ and $\mathrm{B}$ can finish the work in 20 days. $\therefore$ Work done by $(\mathrm{A}+\mathrm{B})$ in 1 day $=\frac{1}{20}$ Now, A alone can do $\frac{1}{5}$ th of the work in 12 days. $\therefore$ Time taken by A alone to complete the work $=(5 \times 12)=60$ days $\Rightarrow$ Work done by A in 1 day $=\frac{...

Construct a square of side 3 cm.

Question: Construct a square of side 3 cm. Thinking Process Firstly, draw a line segment of given length. In both ends of segment draw an angle of 90 from these line segment draw a line upto the given length. Further draw an angle of 90 from these line and join the other line, to get the required construction. Solution: We know that, each angle of a square is right angle (i.e., 90).To construct a square of side 3 cm, use the following steps. Draw a line segment AS of length 3 cm. Now, generate a...

From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and base radius 6 cm is hollowed out.

Question: From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and base radius 6 cm is hollowed out. Find the volume of the remaining solid. Also, find the total surface area of the remaining solid. Solution: Volume of the solid left $=$ Volume of cylinder $-$ Volume of cone $=\pi r^{2} h-\frac{1}{3} \pi r^{2} h=\frac{2}{3} \times \frac{22}{7} \times 8 \times 6 \times 6=603.428 \mathrm{~cm}^{3}$ The slant length of the cone, $l=\sqrt{r^{2}+h^{2}}=\sqrt{36+6...

$A$ and $B$ can polish the floors of a building in 10 days.

Question: $A$ and $B$ can polish the floors of a building in 10 days. A alone can do $\frac{1}{4}$ th of it in 12 days. In how many days can $B$ alone polish the floor? Solution: It is given that $\mathrm{A}$ and $\mathrm{B}$ can polish the floors of the building in 10 days. $\therefore$ Work done by $(\mathrm{A}+\mathrm{B})$ in 1 day $=\frac{1}{10}$ Now, A alone can do $\frac{1}{4}$ th of the work in 12 days. $\therefore$ Time taken by $\mathrm{A}$ alone to do the complete work $=(4 \times 12)=...

A solid is in the form of a right circular cone mounted on a hemisphere.

Question: A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 2.1 cm and the height of the cone is 4 cm. The solid is placed in a cylindrical tub full of water in such a way that the whole solid is submerged in water. If the radius of the cylinder is 5 cm and its height is 9.8 cm, find the volume of the water left in the tub. Solution: The object is shown in the figure below. Radius of hemisphere = 2.1 cm Volume of hemisphere $=\frac{2}{3} \pi...

Construct a ΔABC in which BC = 5 cm,

Question: Construct a ΔABC in which BC = 5 cm, B = 60 and Solution: Given, in $\triangle A B C, B C=5 \mathrm{~cm}, \angle B=60^{\circ}$ and $A C+A B=7.5 \mathrm{~cm}$. To construct the triangle ABC use the following steps. 1. Draw the base $B C=5 \mathrm{~cm}$. 2. At the point $B$ make an $\angle X B C=60^{\circ}$. 3. Cut a line segment $B D$ equal to $A B+A C=7.5 \mathrm{~cm}$ from the ray $B X$. 4. Join DC. 5. Make an $\angle D C Y=\angle B D C$. 6. Let $\mathrm{CY}$ intersect $\mathrm{BX}$ a...

A, B and C can reap a field in

Question: $A, B$ and $C$ can reap a field in $15 \frac{3}{4}$ days; $B, C$ and $D$ in 14 days; $C, D$ and $A$ in 18 days; $D, A$ and $B$ in 21 days. In what time can $A, B, C$ and $D$ together reap it? Solution: Time taken by $(\mathrm{A}+\mathrm{B}+\mathrm{C})$ to do the work $=15 \frac{3}{4}$ days $=\frac{63}{4}$ days Time taken by $(\mathrm{B}+\mathrm{C}+\mathrm{D})$ to do the work $=14$ days Time taken by $(\mathrm{C}+\mathrm{D}+\mathrm{A})$ to do the work $=18$ days Time taken by $(\mathrm{...

A wooden article was made by scooping out a hemisphere from each end of a cylinder, as shown in the figure.

Question: A wooden article was made by scooping out a hemisphere from each end of a cylinder, as shown in the figure. If the height of the cylinder is 20 cm and its base is of diameter 7 cm, find the total surface area of the article when it is ready. Solution: The heighth of cylinder = 20 cm and diameter of its base = 7 cm. $\Rightarrow$ The radius $r$ of its base $=3.5 \mathrm{~cm}$. $\Rightarrow$ Curved surface area of cylinder $=2 \pi r h=2 \times \frac{22}{7} \times 3.5 \times 20=440 \mathr...

Construct a triangle whose sides are 3.6 cm ,

Question: Construct a triangle whose sides are 3.6 cm , 3.0 cm and 4. 8 cm. Bisect the smallest angle and .measure each part. Thinking Process Angle opposite to smallest side is smallest. Solution: To construct a triangle $A B C$ in which $A B=3.6 \mathrm{~cm}, A C=3.0 \mathrm{~cm}$ and $B C=4.8 \mathrm{~cm}$, use the following steps. 1. Draw a line segment $B C$ of length $4.8 \mathrm{~cm}$. 2. From B, point A is at a distance of $3.6 \mathrm{~cm}$. So, having B as centre, draw an arc of radius...

A and B can do a piece of work in 12 days; B and C in 15 days; C and A in 20 days.

Question: AandBcan do a piece of work in 12 days;BandCin 15 days;CandAin 20 days. How much time willAalone take to finish the work? Solution: Time taken by $(\mathrm{A}+\mathrm{B})$ to do the work $=12$ days Time taken by $(\mathrm{B}+\mathrm{C})$ to do the work $=15$ days Time taken by $(\mathrm{A}+\mathrm{C})$ to do the work $=20$ days Now, Work done by $(\mathrm{A}+\mathrm{B})=\frac{1}{12}$ Work done by $(\mathrm{B}+\mathrm{C})=\frac{1}{15}$ Work done by $(\mathrm{A}+\mathrm{C})=\frac{1}{20}$...

A and B can do a piece of work in 18 days;

Question: AandBcan do a piece of work in 18 days;BandCin 24 days andAandCin 36 days. In what time can they do it, all working together? Solution: Time taken by $(\mathrm{A}+\mathrm{B})$ to do the work $=18$ days Time taken by $(\mathrm{B}+\mathrm{C})$ to do the work $=24$ days Time taken by $(\mathrm{A}+\mathrm{C})$ to do the work $=36$ days Now, Work done by $(\mathrm{A}+\mathrm{B})=\frac{1}{18}$ Work done by $(\mathrm{B}+\mathrm{C})=\frac{1}{24}$ Work done by $(\mathrm{A}+\mathrm{C})=\frac{1}{...

Draw an angle of 80° with the help of protractor.

Question: Draw an angle of 80 with the help of protractor. Then, construct angles of (1) 40 (2) 160 and (3) 120. Solution: First, draw an angle of 80 say QOA = 180 with the help of protractor. Now, use the the following steps to construct angles of (1)402() 160 (3) 120 Taking O as centre and any radius draw an arc which intersect OA at E and OO at F. Taking E and F as centres and radius more than EF draw arcs which intersect each other at P. Join OP Thus, POA = 40 [ 40 = x 80] Now, taking F as c...

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends.

Question: A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area. Solution: We have, the total height of the capsule $=14 \mathrm{~mm}$ and the radius of the capsule, $r=\frac{5}{2} \mathrm{~mm}$ Also, the height of the cylinder, $h=14-\left(2 \times \frac{5}{2}\right)=14-5=9 \mathrm{~mm}$ Now, the surface area of the capsule = CSA of the cylinder $+2...

A, B and C working together can do a piece of work in 8 hours.

Question: A,BandCworking together can do a piece of work in 8 hours.Aalone can do it in 20 hours andBalone can do it in 24 hours. In how many hours willCalone do the same work? Solution: Time taken by A to do the work $=20$ hours Time taken by $\mathrm{B}$ to do the work $=24$ hours Time taken by $(\mathrm{A}+\mathrm{B}+\mathrm{C})$ to do the work $=8$ hours Now, Work done by $\mathrm{A}=\frac{1}{20}$ Work done by $\mathrm{B}=\frac{1}{24}$ Work done by $(\mathrm{A}+\mathrm{B}+\mathrm{C})=\frac{1...

Draw a line segment AB of 4 cm in length.

Question: Draw a line segment AB of 4 cm in length. Draw a line perpendicular to AB through A and B, respectively. Are these lines parallel? Solution: To draw a line perpendicular to AB through A and B, respectively. Use the following steps of construction. Draw a line segment AB = 4 cm. Taking 4 as centre and radius more than AB (i.e., 2 cm) draw an arc say it intersect AB at E. Taking E as centre and with same radius as above draw an arc which intersect previous arc at F. Again, taking F as ce...

Sita can finish typing a 100 page document in 9 hours.

Question: Sita can finish typing a 100 page document in 9 hours. Mita in 6 hours and Rita in 12 hours. How long will they take to type a 100 page document if they work together? Solution: Time taken by Sita to do the work $=9$ hours Time taken by Mita to do the work $=6$ hours Time taken by Rita to do the work $=12$ hours Now, Work done by Sita $=\frac{1}{9}$ Work done by Mita $=\frac{1}{6}$ Work done by Rita $=\frac{1}{12}$ $\therefore$ Work done by them together $=\frac{1}{9}+\frac{1}{6}+\frac...

Draw an angle of 110° with the help

Question: Draw an angle of 110 with the help of a protractor and bisect it. Measure , each angle. Solution: Draw BXA = 110 with the help of a protractor. Now, we use the following steps for required construction Taking X as centre and any radius daw an arc to Intersect the rays XA and XB, say at E and D, respectively. Taking D and E as centres and with the radius more than DE, draw arcs to intersecteach other, say at F. Draw the ray XF.Thus, ray XF is the required bisector of the angle B X A. On...

Mohan takes 9 hours to mow a large lawn.

Question: Mohan takes 9 hours to mow a large lawn. He and Sohan together can mow it in 4 hours. How long will Sohan take to mow the lawn if he works alone? Solution: Time taken by Mohan to do the work $=9$ hours Time taken by Mohan and Sohan to do the work $=4$ hours $\therefore$ Work done by Mohan $=\frac{1}{9}$ Work done by Mohan and Sohan $=\frac{1}{4}$ $\therefore$ Work done by Sohan $=\frac{1}{4}-\frac{1}{9}$ $=\frac{9-4}{36}=\frac{5}{36}$ Thus, Sohan can do the work in $\frac{36}{5}$ hours...

All Study Material