1
In a survey conducted among 320 participants of a conference, it was found that 60 liked to sing only song and 100 liked to dance only. Also the ratio of the number of people who didn't like any of these two categories and who like both was 3:1.
(a)If S and D be the sets of people who like to sing and dance respectively, write the cardinality notation for the set of people who like to sing only.
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(b)Present the given information in a Venn-diagram.
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(c)Find the number of people who like to sing.
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(d)By what percentage of the participants did not like any of these two singing and dancing? Find it.
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2
A person plans to deposit Rs. 5,00,000 in a bank for 2 years. The bank offers to provide the interest at the rate of 5% per year with yearly compound interest and 4.5% per year with half yearly compound interest to the person.
(a)Write the formula to calculate half yearly compound interest.
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(b)If the person planned to deposit the money at the half yearly compound interest rate in the bank, how much money should the bank provide as the interest at the end of 2 years? Find it.
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(c)Which proposal of the bank does the person accept to deposit the money to gain more interest? Give logic with calculation.
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3
The present population of a rural municipality is 15,606. The annual population growth rate of the municipality is 2%.
(a)If the present population is denoted by \(P_0\), population growth rate is R% and population after T years is \(P_T\), write the relationship among \(P_0\), T, R and \(P_T\).
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(b)What was the population of the rural municipality before 2 years? Find it.
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(c)If the population growth rate was 3% in the previous year, what would be the present population of the municipality? Calculate it.
2
4
Mohan exchanged NRs. 80,640 into American dollar at the rate of US $1 equal to NRs. 144. After a few days, Nepali currency was devaluated by 0.5% in comparison to American dollar.
(a)How many American dollars did Mohan exchange? Find it.
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(b)How many Nepali rupees did Mohan receive by exchanging American dollar after devaluation? Find it.
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(c)How much percent profit or loss did Mohan make in that transaction? Find it.
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5
The given figure is a square based solid pyramid, where BC = 18 cm and OP = 12 cm.
Square-based pyramid PABCD with apex P, base side BC = 18 cm, height OP = 12 cm
(a)If the length of base side of a pyramid is 'a', vertical height 'h' and volume is V, then express V in terms of 'a' and 'h'.
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(b)Find the volume of the pyramid.
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(c)Find the area of triangular surfaces of the pyramid.
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6
A toy is made with the combination of cone and hemi-sphere having same base. The total height of the toy is 31 cm and radius of the base is 7 cm.
(a)How many surfaces should be there in the toy? Write it.
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(b)Find the surface area of the conical part of the toy.
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(c)By how much the surface area of conical part is more or less than the surface area of the hemi-spherical part? Compare with calculation.
2
7
The given figure ABCD is a quadrilateral shaped land in which BF ⊥ AC, DE ⊥ AC, AC = 40 m, DE = 10 m, and BF = 15 m.
(a)Find the area of the land.
2
(b)Does Rs. 34,000 sufficient to level the land at the rate of Rs. 700 per 100 square meters? Calculate it.
1
8
The first term of an arithmetic series is 24 and sum of its first five terms is 100.
(a)Write the formula to find the sum of the first 'n' terms of an arithmetic series whose first term is a and common difference 'd'.
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(b)Find the common difference of the series.
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(c)How many terms should be taken from the first to make the sum of the series 156? Find using formula.
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9
The numerical product of the present ages of a mother and daughter is 1,500. Ten years ago the mother was twice as old as daughter.
(a)Write the roots of quadratic equation \(ax^2 + bx + c = 0;\ a \neq 0\).
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(b)Make a quadratic equation according to the given condition.
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(c)Find the present ages of mother and daughter.
2
10
(a)Simplify: \(\dfrac{1}{m-n} - \dfrac{m-n}{m^2-n^2}\)
2
(b)Solve: \(16^{x} - 5 \times 4^{x+1} + 64 = 0\)
3
11
Parallelograms PQRS and TQRM and \(\triangle TQR\) are on the same base QR and between the same parallel lines QR and TS.
(a)Write the relation between the area of parallelograms PQRS and TQRM.
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(b)Prove that the area of parallelogram PQRS is double the area of \(\triangle TQR\).
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(c)Construct a quadrilateral ABCD in which AB = 5.4 cm, BC = 5.6 cm, CD = 5.4 cm, AD = 6.8 cm and ∠ABC = 75°. Also, construct a triangle equal in area to the quadrilateral ABCD.
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12
In a circle with centre O, the centre angle AOB and circumference angles ADB and ACB are on the same arc AB.
(a)Write the relationship between the inscribed angles ADB and ACB.
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(b)If the measure of central angle is \((5x)^\circ\) and the measure of inscribed angle is \((2x+10)^\circ\), find the value of x.
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(c)Verify experimentally that the relationship between the inscribed angles ADB and ACB. (At least two circles with radii not less than 3 cm are necessary).
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13
In the given figure, AD ∥ BE and AB ∥ DC.
(a)Prove that: DC = DE.
2
(b)Prove that the quadrilateral ADEB is an isosceles trapezium.
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14
A man observes the top of a tower 12 meter high walking 12 meter away from the foot of the tower and finds the angle of elevation to be \(x^\circ\).
(a)Define the angle of elevation.
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(b)Make a diagram according to the given context.
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(c)Find the value of x.
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(d)How many meter should the observer be moved farther from that place to get the angle of elevation of the top of the tower 30°? Find it.
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15
In the following given data, the age of some people are given, where the first quartile \((Q_1) = 25\).
| Ages (in years) | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|---|
| No. of people | 9 | 11 | p | 20 | 30 | 16 |
(a)What is the class of first quartile? Write it.
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(b)Find the value of p.
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(c)Find the mean of above data.
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(d)What is the percentage of the number of people whose ages are less than 20 years? Find it.
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16
A box contains 5 white and 3 black balls of the same size and shape.
(a)Define a mutually exclusive event.
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(b)Two balls are drawn randomly one after another from the box without replacement. Show the probability of all the possible outcomes in a tree diagram.
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(c)Find the probability of getting both the white balls.
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(d)What is the difference between the probabilities of both balls being white if two balls are drawn one after the another with replacement and without replacement? Find it.
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