Mathematics for Class 9
Contents
Fulfilment of Mathematics course for Secondary Education Examination preparing students
Derive the formula for the union of three sets.
In a survey of a group of people, it was found that 60% of the people liked apple, 70% liked orange and 400 people liked both of them. If 10% people liked none of them, then (i) draw a Venn – diagram to illustrate the above information. (ii) find the total number of people in the survey. (iii) find the number of people who liked apples onl
In a survey of a community, it was found that 65% liked folk songs, 50% liked modern songs, 30% liked both the songs and 450 liked neither of the songs, then (i) draw a Venn – diagram to illustrate the above information. (ii) find the total number of people participated in the survey. (iii) find the number of people who liked folk songs only.
Out of 300 students of a school, 58% liked Math, 32% liked English and Math, 20% liked Science and English, 22% liked Science and Math, 15% liked English only, 10% liked Science only and 5% liked all the three subjects. Using Venn – diagram, find the number of students who i) liked English, ii) liked Math only and iii) Liked none of the subjects.
In an advertisement, it is found that 70 applicants are expert in Statistics, 60 are expert in Computer, 50 in English, 30 in Statistics and Computer, 23 in Computer and English, 25 in English and Statistics and 20 are expert in all the three area. Using Venn – diagram, find i) the number of applicants expert in Computer only, ii) the total number of applicants.
In an examination, 45% students passed in written test, 40% students passed in speaking test and 50% of them passed in listening test. If 10% passed both in written and speaking tests, 20% in both speaking and listening, 15% passed in both written and listening tests and 7% of them failed in all three area, then using Venn – diagram, find the number of students passed (i) in all area (ii) only in one area (iii) only in two area (iv) at least in one area.
Derivation of Compound Amount and Interest
Find the compound interest of $5500 for two years at the rate of 4% p.a. (with and without using compound formula)
Find the compound amount of $ 5500 for 3 years at the rate of 4% p.a.
A sum of $ 5270 is deposited in a bank at the rate of 8% p.a. If the bank provides half yearly compound interest, find the compound amount and interest after 2 years.
The difference between compound and simple interest of a certain sum of money for two years at the rate of 7% is $34.30. Find the sum.
A bank pays 10% p.a. interest compounded half yearly and a tax of 5% is imposed on the interest incurred. Find the interest on the first and second year if a sum of $ 1234 is deposited for two years.
A bank pays interests 8% for the first year, 10% for the second year and 12% for the third year. If the interests are compounded annually, find the compound amount and compound interest of the sum $ 25000 deposited for three years.
A bank compounds interest yearly at a certain rate . If the compound amount of $ 5000 at the end of first year is $ 5600, find a) rate of annual compound interest, b) compound amount at the end of third year.
If the yearly compound interest of a sum in 2 years at the rate of 12% p.a. interest is $316.80 more than the simple interest, find the sum.
A sum amounts to $ 8830 in 2 years and $ 9713 in 3 years at a certain rate of annual compound interest. Find the a) rate of annual compound interest, b) amount of the principal.
Divide a sum of $ 63000 in two parts so that one part at the end of two years and another part at the end of three years produce equal amount, compound interests being calculated at the rate of 10% p.a.
The population of a city in 2011 was 50000. If the annual population growth rate was 1.2%, what would be the population in 2013? Also calculate the increment of the population of the city.
Three years ago, the price of an electric heater was $ 5000. If the rate of increase in the price was 4% per annum. Find the increase in the price.
The price of a plot of land is $ 150 per sq. ft. If the rate of increase in price is 7% p.a., find its price before 3 years.
The number of students in a college at present is 729. Two years ago, it was 625. Find the annual rate of increment of enrollment.
The number of bacteria present in a sample of urine at 5 a.m. was 5×10⁵. The number increased by 4%, 6% and 8% in successive three hours. Calculate the number present in the urine sample taken at 8 a.m.
How many years it takes to increase the population of a city from 100000 to 102010 if the annual rate of increment is 1%.
Suren bought a television set for $ 350. After 3 years, he sold it at 12% p.a. depreciation. Find the depreciated price and depreciated amount.
Two years ago, a plot of land of 10 acers was purchased for $20000. Due to recession, a plot of one acre land can now be sold for $1800. Find the rate of depreciation of the price.
The price of the share of a manufacturing company is decreasing at the rate of 10% p.a. If all the shares of the company are sold out for $729. (a) Find the price of shares before 3 years. (b) If the shares were purchased 3 years ago at $1/share , find the number of shares purchased.
Based on given exchange rate table, find the difference between buying and selling price of a) ₹ 700 b) $ 300 c) € 555 d) ₩ 678
Today’s exchange rate between US dollar and Nepali rupees is $1 = रू 127.95. How many dollars can be exchanged for रू 50,000.
Today’s exchange rate between US dollar and Nepali rupees is $1 = रू 127.95. How many Nepali rupees can be exchanged for $ 400.
Based on given exchange rate table, change 1 euro into Korean won.
If $300 = € 280 and रू 7100 = € 50 , then find how many dollars are equal to रू 21500?
According to exchange rate declared by Nepal Rastra Bank on 2 May 2024, the buying and selling rate for Hong Kong dollar 1 are Rs. 17.02 and Rs. 17.10 respectively. How many Hong Kong dollar can be exchanged for Rs. 75,600?
According to exchange rate declared by Nepal Rastra Bank on 2 May 2024, the buying and selling rate for Hong Kong dollar 1 are Rs. 17.02 and Rs. 17.10 respectively. How many rupees can be exchanged for HK$ 2710?
The fair of air ticket from Kathmandu to Japan is Rs. 117800. But the ticket can be booked from Naruhito for 148900 Japanese Yen. Which option is cheaper and by what percentage? (¥ 10 = Rs. 8.50)
A merchant ordered 100 mobile sets from Malaysia for MR 72000. Find the cost price of mobile sets in Nepalese rupees if the custom duty is 20% and VAT is 13%. (Exchange rate: MR 1 = Rs. 28)
How many Nepalese rupees are equivalent to 5000 pounds if £ 100 = € 94, € 1 = ¥ 168, ¥ 159 = $1 and $ 1 = रू 134 ?
Rs. 500000 are exchanged into dollars at the rate of $ 1 = Rs. 140. Two days later rupees deflated by 5% and exchanged dollars are again exchanged into rupees. Find the loss or profit on these exchanges.
Introduction
Area of base Area of lateral surfaces
Find the slant height and length of the edge of square based pyramid given in the figure.
Find the edge and height of the square based pyramid given in the figure.
Find the length of the edge of square based pyramid given in the figure.
Find the total surface area and volume of the square based pyramid given in the figure.
Find the total surface area and volume of the square based pyramid given in the figure.
Find the total surface area and volume of the square based pyramid given in the figure.
Find the total surface area and volume of the square based pyramid given in the figure.
A square based pyramid is of height 50m. If area of its base is 3600 m2, find the total surface area and volume of the pyramid.
An aquarium is in the shape of a square based pyramid. If height and area of the base of the aquarium are 40 cm. and 3600 cm2 respectively, find how much water is needed to completely fill the aquarium.
Find the volume of a cone if its base area is 50 cm2 and height is 30cm.
If the base area and curved surface area of a cone are 100cm² and 250cm² respectively, find its total surface area.
Find slant height and curved surface area of the given cone.
Find slant height and total surface area of the given cone.
If the volume of a conical shaped tent is 2.916𝜋 m³ and height 2.7m, find its slant height and area of the base.
If the height of a right cone is 6 times its radius and volume of the cone is equal to 54𝜋 cm³. Find its total surface area.
Three solid cones of equal heights and radii 6cm, 8cm and 10cm are molten together to form a single cone of same height. Find the diameter of the base circle of the cone so formed.
दिइएको संयुक्त ठोस वस्तुको (क) आधार सतहको (ख) वक्रसतहको र (ग) पूरा सतहको क्षेत्रफल निकाल्नु होस। Find the area of (a) base (b) curved surfaces and (c) total surface of the given combined solid.
Find the total surface area of the given combined solid.
दिइएको संयुक्त ठोस वस्तुको पूरा सतहको क्षेत्रफल पत्तालगाउनुहोस। Find the total surface area of the given combined solid.
The volume of the solid shown in the figure is 408 cm³. Find its total height.
A square based water tank is 10ft. high and side of the base is 15 ft. Find the capacity of the tank in liters. (1 ft3 = 28.317 liters) If cost of electricity for filling the tank is Rs. 0.40 per 1000 liters, find total cost of filling the tank.
A parking area has the shape as in the figure below. A flooring tile of size 0.5ft × 0.5ft costs Rs. 55. It requires 5 workers in order to pave the area with tiles in 4 days and wage of a worker per day is Rs. 1500. Find (a) the area of parking land (b) number of tiles needed to floor the area and (c) total cost of flooring.
A combined solid is made by joining a pyramid having slant height 1.22m over a cuboid of size 0.4m×0.4m×3m as shown in figure. If the surfaces were covered gold plates at the rate of Rs. 200000/m², find the total cost of gold plating.
The cost of painting at the rate of Rs 2 per cm² over the whole surface of a solid cone having diameter of the base 20 cm is Rs 1760. Find the volume of the solid.
Derivation of formula
Derivation of formula
Derivation of formula for arithmetic means
Find the formula for the arithmetic mean between 𝑎 and 𝑏.
The first term of an arithmetic sequence is 1 and its common difference is also 1. If sum of its terms is 55, find the number of terms of the sequence.
A person retires from the job completing a service of 30 years. Initially, the monthly salary was Rs. 8010 and Rs. 600 was yearly increment in monthly salary. Find the total of the salaries in 30 years.
The sum of three numbers in arithmetic sequence is 27. If the product of the first and last number is 56, find the numbers.
The sum of three numbers in arithmetic sequence is 27. If the product of the these numbers is 504, find the numbers.
Derivation of term formula of a geometric sequence.
Derivation of sum formula of a gemetric sequence.
Derivation of formula for geometric means
Find the geometric mean between 𝑎 and 𝑏.
Find the geometric mean between 2 and 128.
Find five geometric means between 2 and 128.
Fourteen geometric means have been inserted between 3 and 3⁹. Find the 9th geometric mean.
If the numbers 𝑥−1, 𝑥+2, 𝑥+8 are in geometric sequence, find the value of 𝑥.
Find the 7th term and sum to up to 7 terms of the geometric series 1+3+ 9+27+…..
The fifth and eighth terms of a geometric sequence are 162 and 4374. Find its tenth term. correction needed
Is 3125 a term of the sequence {5, 25, 125, .... }?
Find the sum of the series: 3+ 6 + 12 + 24 +…..+ 384.
In a geometric series, the first term is 3, ratio is 2 and sum of its 𝑛 terms is 765. Find 𝑛.
How many terms of the G.S. 26 + 65 + 325/2 + ... will have a sum 67431/16?
The sum of 5 terms of a geometric series with common ratio 1.5 is 26375. Find the difference between first and last terms.
Solve the quadratic equation by factorization method: 2x²+3x-2=0.
Solve the quadratic equation 10x²+19x-15=0 by factorization.
Solve the quadratic equation 6x²-7x+2=0 by factorization.
Solve the quadratic equation ax²+bx+c=0 by completing perfect square.
Solve the quadratic equation x²+3x+2=0 by completing perfect square.
Solve the equation x²-3x+2=0 by using quadratic formula.
Solve the equation √3x²-6x+2√3=0 by using quadratic formula.
Solve the equation 1⁄ (x-2) + 1 ⁄ (x+1)=1⁄2 by using quadratic formula.
If 10 is subtracted from the square of a number, the difference is 10, find the number.
If the sum of a number and its square is 42, find the number.
If the product of two consecutive odd numbers is 143, find them.
If the sum of a number and its reciprocal is 26/5, find it.
The difference of the ages of two sisters is 6 years and the product of their ages is 315. Find their ages.
The difference of the ages of husband and wife is 5 years and the sum of the squares of the ages is 2125. Find their ages.
The breadth of a rectangular plot is 9 feet less than its length. If the area of the plot is 2 kattha, find the dimension of the plot. ( 1 kattha = 3645 square feet )
Write in simplest form : (x²-2x) ⁄ (x²-4) and (a²-b²) ⁄ (a-b)².
Simplify (i) 1⁄ (x-y)+1⁄(y-x) (ii) x⁄ (x-y)+y⁄(y-x)
Simplify (i) 1⁄ (x-y)+1⁄(x-y) (ii) (x³+y³) ⁄ (x²-xy+y²)+(x³-y³) ⁄ (x²+xy+y²)
Simplify (i) 1 ⁄ (a+2b)+1⁄(a-2b)+2b(4b²-a²).
Simplify x⁄(x-y)(z-x)+y1⁄(y-x)(y-z)+z⁄(z-y)(z-x).
Simplify (a²-(x-y)²)⁄ [(a+y)²-x²]+(x²-(a-y)²)⁄ [(a+x)²-y²]+(y²-(a-x)²)⁄ [(x+y)²-a²]
Simplify: 4x³ ⁄ (x⁴-b⁴) -8x⁷ ⁄ (x⁸-b⁸)
Simplify 3⁄(x²-4x+3)+2⁄(2-x²-x)-1⁄(x²-x-6)
Solve the equation and check the result: 2^x=64
Solve the equation and check the result: 3^(x-4) = 1⁄243.
Solve the equation 8^[3x⁄(x-3)]=16.
Solve the equation: 3^(𝑥+1)+3^(𝑥+2)+3^(𝑥) + 2=353.
Solve the equation: 8^(x-1)-2^(3x-2)+8=0
Solve the equation: 2^(x) + 1⁄2^(x) = 8 1⁄8.
Solve the equation: 7^(x) + 7^(2-x)=2402⁄7.
Solve the equation: If x = 5^(1⁄3)+5^(-1⁄3), prove that 5x³-15x=26.
Construct a parallelogram having adjacent sides measuring 4cm, 4.5cm and one angle measuring 60°. Construct another parallelogram equal in area but having one angle of 45°.
Construct a parallelogram having adjacent sides measuring 4cm, 4.5cm and one angle measuring 60°. Construct a triangle equal in area but having one angle of 45°.
Construct a parallelogram having adjacent sides measuring 4cm, 4.5cm and one angle measuring 60°. Construct a triangle equal in area having one side 5.5cm.
Construct a triangle having sides 8cm, 5.5cm, 5.7cm and construct a parallelogram of equal area with an angle 60°.
Construct a quadrilateral having sides 5.6cm, 4.2cm, 2cm and 5cm and one diagonal 5.8cm. Also construct a triangle having area equal to that of the quadrilateral.
Parallelograms standing on the same base and lying between the same parallel lines are equal in area.
Parallelograms standing on the same base and lying between the same parallel lines are equal in area.
A rectangle and a parallelogram standing on same base and lying between same parallel lines are equal in area.
The area of a triangle is half of the area of a parallelogram if both stand on same base and lie between the same parallel lines.
Two triangles having same base and lying between same parallel lines are equal in area.
Half the area of any parallelogram is equal to the area of the parallelogram obtained by joining its mid points taken in order.
In the adjoining figure, 𝐴𝐵𝐶𝐷𝐸 is a pentagon. Prove that ∆EDA=∆ABC.
In the given figure, 𝐴𝐵𝐶𝐷 is a quadrilateral. 𝐶𝐸 is drawn parallel to diagonal 𝐷𝐵. 𝐴𝐵 is extended to meet 𝐶𝐸 at 𝐸. Prove that area of quad. 𝐴𝐵𝐶𝐷 is equal to area of ∆𝐴𝐷𝐸.
In the given figure, 𝐴𝐵𝐶𝐷 is a parallelogram. 𝐷𝐶 is extended to 𝐸 and 𝐴𝐶, 𝐴𝐸, 𝐵𝐸 are joined. 𝐴𝐸 and 𝐵𝐶 intersect at 𝑂. Prove: area of ∆𝐴𝑂𝐶 = area of ∆𝐵𝑂𝐸.
A point is taken on each of two adjacent sides a parallelogram. Prove that the area of triangles formed by joining each point to the end points of the opposite side are equal.
The central angle of a circle is double of the inscribed angle standing on the same base.
The inscribed angles of a circle standing on the same arc are equal.
The opposite angles of a cyclic quadrilateral are supplementary.
The central angle of a circle is double of the inscribed angle standing on the same base.
The inscribed angles of a circle standing on the same arc are equal.
The opposite angles of a cyclic quadrilateral are supplementary.
Find the mean temperature of the 7 days recorded as follows: 21ºC, 26ºC, 31ºC, 43ºC, 36ºC, 33ºC, 34ºC
Find the mean height of students in a sport group: Height in cm: 160 170 180 190 200 210 Number of students: 3 10 25 40 17 5
Find the mean of marks obtained by students in a class: Marks obtained: 0 – 05 05 – 10 10 – 15 15 – 20 20 – 25 Number of students: 6 12 24 36 12
Find the mean of daily expenses of 500 households: Expense in Rs.: 500 – 1000 1000 – 1500 1500 – 2000 2000 – 2500 No. of households: 109 221 120 50
From the following intermediate result, find the value of unknown: 1. ∑𝑥 = 25, 𝑛 = 90, 𝑥 ̅ = ? 2.∑𝑥 = 432 , 𝑥 ̅ = 12 , 𝑛 = ? 3. 𝑥 ̅ = 36.5, 𝑛 = 60, ∑𝑥 = ?
The figures give the weight of 40 patients recorded in a day at a hospital. Compute the mean weight (kg) of the data. Also prepare a discrete frequency distribution table and then compute the mean of weight from the distribution table. Compare the results obtained. 53, 45, 72, 40, 65, 45, 53, 49, 56, 40, 75, 49, 63, 75, 65, 72, 72, 94, 48, 40, 82, 49, 45, 83, 72, 90, 48, 90, 75, 90, 82, 90, 45, 90, 72, 56, 60, 60, 63, 65
By taking 15 as the width of the class interval, prepare a continuous frequency table for the given data and compute the mean of the data. 53, 45, 72, 40, 65, 45, 53, 49, 56, 40, 75, 49, 63, 75, 65, 72, 72, 94, 48, 40, 82, 49, 45, 83, 72, 90, 48, 90, 75, 90, 82, 90, 45, 90, 72, 56, 60, 60, 63, 65
If the mean of the data is Rs. 1361, find the value of the missing frequency. Expense in Rs. : 500 – 1000 1000 – 1500 1500 – 2000 2000 – 2500 Number: 109 221 𝑘 50
Find the median of 7 day's temperatures recorded as follows: 21ºC, 26ºC, 31ºC, 43ºC, 36ºC, 33ºC, 34ºC.
Find the median height of students in a sport group: Height in cm: 160 170 180 190 200 210 Number of students: 3 10 25 40 17 4
Find the median of marks obtained by students in a class: Marks obtained: 0 – 05 05 – 10 10 – 15 15 – 20 20 – 25 Number of students: 6 12 24 36 12
If the median of the data is 145.45, find the missing frequency 𝑘. Expense in Rs. : 50 – 100 100 – 150 150 – 200 200 – 250 Number : 10 22 𝑘 11
Find the median of marks obtained by students in a class: Marks obtained: < 60 < 70 < 80 < 90 < 100 Number of students: 2 5 11 16 20
Prepare a discrete frequency table and compute median for the data given below. 53, 45, 72, 40, 65, 45, 53, 49, 56, 40, 75, 49, 63, 75, 65, 72, 72, 94, 48, 40, 82, 49, 45, 83, 72, 90, 48, 90, 75, 90, 82, 90, 45, 90, 72, 56, 60, 60, 63, 65
Prepare a continuous frequency table for the raw data given below by taking width of class interval 15. Also compute median from the table. 53, 45, 72, 40, 65, 45, 53, 49, 56, 40, 75, 49, 63, 75, 65, 72, 72, 94, 48, 40, 82, 49, 45, 83, 72, 90, 48, 90, 75, 90, 82, 90, 45, 90, 72, 56, 60, 60, 63, 65
Find the mode of the following individual data: 53, 45, 72, 40, 56, 45, 53, 49, 56, 40, 75, 56, 63
Find the mode of the following distribution: Variable (𝑥) : 30 80 70 60 40 90 50 Number (𝑓) : 7 6 17 22 15 3 29
Find the modal marks from the distribution given below: Marks obtained: 45 – 50 50 – 55 55 – 60 60 – 65 65 – 70 Number of students: 6 12 24 36 12
Find the lower and upper quartile of the following distribution: 5, 11, 23, 16, 20, 44, 34, 17, 55, 60, 29, 64, 73, 6, 75
Find the lower and upper quartile of the following distribution: Height in cm: 160 175 190 225 245 300 Number : 3 10 25 40 17 4
If the lower quartile of distribution = 11.25, find the missing frequency. Time (Hrs.) : 0 – 05 05 – 10 10 – 15 15 – 20 20 – 25 25 – 30 Frequency : 6 12 24 ? 12 6
If 𝐴 and 𝐵 are exclusive events, then 𝑃 (𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵).
If 𝐴 and 𝐵 are any two events, then 𝑃 (𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) - 𝑃 (𝐴 ∩ 𝐵)
A container contains 4 red, 6 black and 5 yellow identical balls. If a ball is drawn randomly, find the probability of getting red ball.
A container contains 4 red, 6 black and 5 yellow identical balls. If a ball is drawn randomly, find the probability of getting a red or yellow ball.
In a picnic group of 30 people, 22 liked tea, 17 liked coffee and 13 liked both tea and coffee. If one person is selected at random, find the probability of getting the person as tea or coffee lover.
An alphabet is drawn randomly from the word “COCONUT”. Find the probability of getting a vowel.
A dice and a coin are rolled together. Find the probability of getting 3 on the dice and Head on the coin.
A container contains 4 red and 5 yellow identical balls. If 2 balls are drawn at random one after another without replacement, present the outcomes in a tree diagram and write the probabilities.
The angle of elevation of a tower situated on the other side of 100ft wide road is 30°. Find the height of the building.
The angle of elevation of a 100ft tall building situated on the other side of a river is 30°. Find the width of the river.
A tree is broken due to wind into two parts and the upper part makes an angle of 60° with the ground. If the length of the broken upper part is 15m, find the total length of the tree.
A man finds that the angle of elevation of a 100m tower is 60° . If the distance between man and tower is 56.7m, find the height of the man.
From the top of a tower 100ft high, the angle of depression at the top of a tree is 30°. If the distance between the tower and tree is 138ft, find the height of the tree.
A pole is fixed at the centre of a circular pond of radius 30m. The length of the pole above the water surface is 30m. What will be the angle of elevation of the top of the pole observed from the edge of the pond?
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