Maths Problems for Class 8 with a Flavour of Physics

By Jitender Singh 17-11-2022

Calculations

The charge on an electron is $q=1.6\times10^{-19}$ Coulomb. The mass of the electron is $m=9.1\times10^{-31}$ kg. Find charge to mass ratio ($q/m$) for an electron.
The electrostatic force between two particles of charges $q_1$ and $q_2$ separated by a distance $r$ is given by \begin{align} F_e=\frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}. \end{align} Take $\epsilon_0=9\times10^{-12}$, $q_1=1.6\times10^{-19}$, $q_2=1.6\times10^{-19}$ and $r=10^{-15}$ to find electrostatic repulsion force between two protons in a nucleus. The force you get is in Newton.
The gravitational force between two particles of masses $m_1$ and $m_2$ separated by a distance $r$ is given by \begin{align} F_g= \frac{G m_1 m_2}{r^2}. \end{align} Take $G=6.67\times10^{-11}$, $m_1=1.67\times10^{-27}$, $m_2=1.67\times10^{-27}$ and $r=10^{-15}$ to find gravitational attraction force between two protons in a nucleus. The force you get is in Newton.
Find $F_e/F_g$ for two protons. The electrostatic force is much much larger than the gravitational force.
The average kinetic energy of an atom at temperature $T$ Kelvin is given by $E=\frac{3}{2}kT$, where $k=1.38\times10^{-23}$. Calculate the average kinetic energy of an atom at room temperature (T = 300 Kelvin). The energy you get is in Joule. The kinetic energy due to temperature is also called thermal energy.
The energy is converted from Joule to electron-volt (a useful unit in modern physics) by dividing energy in Joule by charge of an electron. Convert the average kinetic energy of an atom at room remperature into electron-volt.
The temperature at the core of the sun is $1.5\times10^{7}$ Kelvin. Find average kinetic energy of a proton at this temperature. The electrostatic potential energy between two charges $q_1$ and $q_2$ separated by a distance $r$ is given by \begin{align} U=\frac{q_1 q_2}{4\pi\epsilon_0 r}. \end{align} Find the distance between two protons at which their electrostatic potential energy is equal to their average kinetic energy at the core of the sun.
The energy of a photon of wavelength $\lambda$ is given by \begin{align} E=\frac{hc}{\lambda}, \end{align} where $h=6.64\times10^{-34}$ and $c=3\times10^{8}$. The energy you get is in Joule. Find energy of a photon of visible light of wavelength $\lambda=5\times10^{-7}$ metre. Convert this energy into electron-volt.
Find the energy of a Ultra Vilot (UV) light photon of wavelength $\lambda=10^{-7}$ metre. Convert this energy in electron-volt. UV photons of this energy can eject electrons from the surface of metals. This phenomenon is called photo-electric effect.
The solar energy falling on one square metre area of the earth in one second is 1400 Joule. Assume that all photons are of wavelenegth $5\times10^{-7}$. How many photons are falling in this area in one second.
Avogadro number is the number of carbon atoms in 12 gram of carbon. The carbon atom has 6 protons, 6 neutrons, and 6 electrons. The mass of the electron is negligible in comparison to that of proton. The mass of the neutron is approximately equal to the mass of the proton. Can you find the approximate value of Avogadro number. It is very large.

Volume and Surface Area

A field is 80 m long and 50 m broad. In one corner of the field, a pit which is 10 m long, 7.5 m broad and 8 m deep has been dug out. The soil taken out of it is evenly spread over the remaining part of the field. Find the rise in the level of the field.
Find the volume of wood used to make an open box whose external dimensions are 36 cm by 45 cm by 16.5 cm, the thickness of the wood being 1.5 cm thick throughout.
Find the length of the longest pole that can be put inside the above box.
How many soap of size 7 cm by 5.5 cm by 5 cm can be put into this box. Think!
Find the breadth and the height of a cuboid with square base, being given the volume 63 $\mathrm{in}^3$ and surface area 102 $\mathrm{in}^2$.
A rectangular piece of paper 22 cm by 6 cm is folded without overlapping to make a cylinder of height 6 cm. Find the curved surface area and volume of the cylinder.
A rectangular vessel 22 cm by 16 cm by 14 cm is full of water. If the total water is poured into an empty cylindrical vessel of radius 8 cm, find the height of water in the cylindrical vessel.
A well of inner diameter 14 m is dug to a depth of 12 m. The soil taken out of it has been evenly spread all around it to a width of 7 m to form an embankment. Find the height of the embankment so formed.
A road roller takes 750 complete revolutions to move once over to level a road. Find the area of the road if the diameter of the road roller is 84 cm and its length is 1m.
The length of a metallic tube is 1 m, its thickness is 1 cm and its inner diameter is 12 cm. Find the mass of the tube if the density of the metal is 7.7 grams per cubic centimeter.

Linear Equations

Twenty-four is divided into two parts such that 7 times the first part added to 5 times the second part makes 146. Find each part.
The sum of the digits of a two-digit number is 12. If the new number formed by reversing the digits is greater than the original number by 54, find the original number.
The denominator of a rational number is greater than its numerator by 3. If 3 is subtracted from the numerator and 2 is added to its denominator, the new number becomes 1/5. Find the original number.
The length of a rectangle exceeds its breadth by 7 cm. If the length is decreased by 4 cm and the breadth is increased by 3 cm, the area of the new rectangle is the same as the area of the original rectangle. Find the length and the breadth of the original rectangle.
An altitude of a triangle is five-thirds the length of its corresponding base. If the altitude be increased by 4 cm and the base decreased by 2 cm, the area of the triangle remains the same. Find the base and the altitude of the triangle.
Two angles of a triangle are in the ratio 4:5. If the sum of these angles is equal to the third angle, find the angles of the triangle.
A boat goes downstream from one port to another in 9 hours. It covers the same distance upstream in 10 hours. If the speed of the stream be 1 km/hr, find the speed of the boat in still water and the distance between the ports.
The distance between two stations is 300 km. Two trains start simultaneously from these stations and move towards each other. The speed of one of them is 7 km/hr more than that of the other. If the distance between them after two hours of their start is 34 km, find the speed of each train.
The difference between the ages of two cousins is 10 years. Fifteen years ago, if the elder one was twice as old as the younger one, find their present ages.
Half of a flock of sheep are grazing in the field and three-fourths of the remaining are playing nearby. The rest 9 are drinking water from the river. Find the number of sheep in the flock.

Rational Numberes

Express following rational numbers in standard form: (a) $\frac{-14}{49}$ (b) $\frac{24}{-64}$ (c) $\frac{-36}{-63}$
Arrange the numbers $\frac{-3}{5}$, $\frac{7}{-10}$ and $\frac{-5}{8}$ in ascending order.
Arrange the numbers $-2$, $\frac{-13}{6}$, $\frac{8}{-3}$ and $\frac{1}{3}$ in descending order.
Represent $\frac{13}{5}$ and $\frac{-13}{5}$ on the number line.
The sum of two rational numbers is $-5$. If one of them is $\frac{-13}{6}$, find the other.
What number should be added to $\frac{-7}{8}$ to get $\frac{4}{9}$.
What number should be subtracted from $\frac{-5}{7}$ to get $-1$.
Find the value of $\frac{-16}{7}\times\left(\frac{-8}{9}+\frac{-7}{6}\right)$.
The product of two numbers is $\frac{-28}{27}$. If one of the number is $\frac{-4}{9}$, find the other.
By what number should $\frac{-33}{8}$ be divided to get $\frac{-11}{2}$?
A basket contains three types of fruits weighing $19\frac{1}{3}$ kg in all. If $8\frac{1}{9}$ kg of these be apples, $3\frac{1}{6}$ kg be oranges and the rest pears, what is the weight of the pears in the basket?
A car is moving at an average speed of $60\frac{2}{5}$ km/hr. How much distance will it cover in $6\frac{1}{4}$ hours?
The area of a room is $65\frac{1}{4}$ square metre. If its breadth is $5\frac{7}{16}$ metres, what is its length?
After reading $\frac{7}{9}$ of a book, 40 pages are left. How many pages are there in the book?
If $\frac{3}{5}$ of a number exceeds its $\frac{2}{7}$ by 44, find the number.

Exponents

Evaluate $\left(\frac{-3}{4}\right)^{-4}$.
Evaluate $\left(\frac{8}{5}\right)^{-2}\times \left(\frac{8}{5}\right)^2$.
Evaluate $\left(\frac{-2}{7}\right)^{-4}\times\left(\frac{-5}{7}\right)^2$.
Evaluate $\left\{\left(\frac{-3}{2}\right)^2\right\}^{-3}$.
By what number should $\left(\frac{1}{2}\right)^{-1}$ be multiplied so that the product is $\left(\frac{-5}{4}\right)^{-1}$?
By what number should $\left(\frac{-3}{2}\right)^{-3}$ be divided so that the quotient is $\left(\frac{9}{4}\right)^{-2}$?
If $5^{2x+1} \div 25=125$, find the value of $x$.

Square, Cube, Square Root, Cube Root

Find the least square number (perfect square) which is exactly divisible by each of the numbers 8, 12, 15 and 20.
The area of a square field is 60025 $m^2$. A man cycles along its boundary at 18 km/h. In how much time will he return to the starting point.
Evaluate $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{0.9}$ (each upto two decimal places).
Find the length of each side of a square whose area is equal to the area of a rectangle of length 1.6 m and breadth 3.4 m.
Find $\frac{\sqrt{80}}{\sqrt{405}}$.
Evaluate cube root of (a) 1728 (b) $\frac{729}{1000}$ (c) $\frac{-27}{125}$.

Playing with numbers

The sum of digits of a two digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. Find the original number.
Find all possible values of x for which the 4-digit number 754x is not divisible by 3.
Find two numbers whose product is a 1-digit number and the sum is a 2-digit number.

Algebraic Expressions

The perimeter of a triangle is $6x^2-4x+9$ and two of its sides are $x^2-2x+1$ and $3x^2-5x+3$. Find the third side of the triangle.
Find (a) $(4x^2+5) (4x^2+5)$ (b) $(4x^2-5) (4x^2-5)$ (c) $(4x^2+5) (4x^2-5)$
Find quotient and remainder when we divide ($8x^4+10x^3-5x^2-4x+1$) by ($2x^2+x-1$).
Find the value of expression ($36x^2+25y^2-60xy$), when $x=2/3$ and $y=1/5$.
If $(x-\frac{1}{x}=5)$, find the value of (a) $x^2+\frac{1}{x^2}$ (b) $x^4+\frac{1}{x^4}$.
If $x-y=7$ and $xy=9$, find the value of $(x^2+y^2)$.

Factorization

Factorize the following expressions:

$12x^2 y^3 -21x^3 y^2$
$1+x+xy+x^2y$
$ab(x^2+y^2)+xy(a^2+b^2)$
$x^3-3x^2+x-3$
$25(x+y)^2-36(x-2y)^2$
$4x^2-y^2+6y-9$
$1-6x+9x^2$
$(x+y)^2-4xy$
$x^2+x-56$
$3x^2-4x-4$
$28-31x-5x^2$
$6x^2-5x-6$
$2x^2-x-6$
$x^2+x-30$
$7x^2-2x-5$
$4x^2-16$
$3x^2-4x-4$
$-2x^2+3x+9$
$-x^2+7x+18$

Percentage

Write in ascending order $16\frac{2}{3}$ percent , $\frac{2}{15}$ and 0.12.
A number is increased by 20 percent and then decreased by 20 percent . Find the net increase or decrease in per cent?
Out of her total monthly salary, Sarita spends 30 percent on house rent and 60 percent of the rest on household expenditure. If she saves Rs 10500, what is her total monthly salary?
The price of sugar goes up by 20 percent . By how much per cent must a housewife reduce her consumption of sugar so that the expenditure on sugar remains the same?
The value of machine depreciates every year by 10 percent . If the present value of the machine be Rs 99000, what was its value last year?
A's income is 60 percent more than that of B. By what percent is B's income less than A's?
The price of petrol goes up by 10 percent . By how much per cent must a motorist reduce the consumption of petrol so that the expenditure on it remains unchanged?
An alloy contains 40 percent copper, 30 percent nickel and rest zinc. Find the mass of zinc in 1 kg of the alloy.
Gunpowder contains 75 percent nitre and 10 percent sulphur. Find the amount of gunpowder which carries 9 kg nitre. What amount of gunpowder would contains 2.5 kg sulphur?
Find the percentage of pure gold in 22-carat gold, if 24-carat gold is 100 percent pure.

Three dimensional figures

Write the number of vertices (V), faces (F) and edges (E) in (a) cuboid (b) cube (c) prism (d) square pyramid (e) triangular pyramid - tetrahedron.
Show that $F-E+V=2$ for above three dimensional figures.

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