SEE_2081

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SEE 2081_RE1031_KOP

  1. In a survey of a group of 360 students, 100 students like basketball game, 60 like cricket game only and 100 do not like any of the two games.
    1. Using symbols B (basketball) and C (cricket), write the given information in terms of set notation.
    2. Present the above information in a Venn-diagram.
    3. Find the number of students who like both basketball and cricket games.
    4. If everyone who is not interested in any game liked cricket game in the second survey and others remained the same, what would be the number of students who like cricket game?
  2. Rajan borrowed a loan of Rs 10,000 from Ram for 2 years at the rate of 10% simple interest. Immediately, Rajan lent the same amount for the same time at 10% simple interest.
    1. According to the given context, which interest is more: simple interest or compound interest for 2 years? Write it.
    2. Find the profit Rajan got during the transaction of 2 years.
    3. How much more interest should Shyam need to pay to Rajan if Rajan lent the amount at semi-annual compound interest? Find it.
  3. The population of a village is 20,000. The population increases by 2% annually in the village.
    1. If the initial population is P and the annual rate of population growth is R% per annum, express the population after T years as PT.
    2. After how many years will the population of the village be 20,808? Find it.
    3. If the population increases at the rate of 3% per annum, by what number will the population of the village be increased in 2 years? Find it.
  4. An American dollar to the currency exchange rate is NRs 136.13 and selling rate was NRs 137.25 in a certain day.
    1. Which buying or selling rate is used when you exchange American dollar into Nepali rupees? Write it.
    2. How many Nepali rupees will he get by exchanging 1000 dollars? Find it.
    3. The American tourist spent NRs 1,01,817.50 and returned back with remaining Nepali rupees. How many American dollars can he/she exchange from remaining Nepali rupees, while returning back to own country? Find it.
  5. A group of students constructed a square based pyramid shaped tent having length of base side 24 meter and vertical height 5 meter.
    1. How many triangular surfaces are there in the square based pyramid? Write it.
    2. Find the slant height of the above square based tent.
    3. What is the total cost of cloths required to make triangular surfaces at the rate of Rs.125 per square metre? Find it.
  6. In the given figure, wooden cylinder and cone having equal base are shown.
    1. Write the formula to find the volume of a cone.
    2. Find the volume of the cone in the given objects.
    3. If the given wooden cylinder is drilled out in the conical shape, what will be the volume of remaining wood in the cylinder? Find it.
  7. The length of a wall is 10 m, width is 0.5 m and height is 2 m. Bricks of size 25 cm × 12 cm × 8 cm are used to build the wall. Also, 1/10 part of the wall is occupied by the clay joints.
    1. How many bricks are required to construct the wall? Find it.
    2. If 1000 bricks cost Rs 14,500, estimate the cost of bricks used in the wall at the rate of Rs 14,500 per 1000 bricks.
  8. Ramesh deposits money in a co-operative for 7 days by increasing the amount every day. He deposits Rs 10 on the first day, Rs 20 on the second day, Rs 40 on the third day and so on till the 7th day.
    1. What type of series is formed from the deposit amount according to the above context? Write it.
    2. How much amount will Ramesh deposit by the end of 7 days? Find it using formula.
    3. If Ramesh withdraws the amount deposited in the first day, how much amount will he receive at the end of the 7th days? Find it.
  9. The longer side of a rectangular field is 40 m more than the shorter side and its diagonal is 40 m more than its longer side.
    1. Write the relation among the length (l), breadth (b) and diagonal (d) of the field according to the above context.
    2. Find the length of the shorter side and longer side of the rectangular field.
    3. How many maximum numbers of plots of size 30 m × 20 m can be made from the rectangular field?
    1. Simplify:
      \(\frac{p+q}{pq} - \frac{q+r}{qr} - \frac{r+p}{rp}\)
    2. Solve:
      \( 3^{y} + 3^{-y} = 9{\frac{1}{9}} \)
  11. In the given figure, ∆ABC and ∆BCD are standing on same base BC and between same parallel lines AD and BC. From the point B, a perpendicular BP is drawn to the line AC.
    1. Write the name of triangle whose area is equal to area of ∆BAD in the given figure.
    2. If AC = 9 cm and BP = 6 cm, find the area of triangle BCD.
    3. In the given figure, PQRS is a trapezium, where PQ//SR. M and N are the mid points of the diagonals PR and QS respectively. Prove that: ∆MSR=∆NSR.
  12. In a triangle QR, ∡ QR 60°, QR 8 cm and Q 6 cm are given.)
    1. Construct a ∆ QR according to above measurements and also construct a rectangle RITA equal in area to the triangle.
    2. Why the areas of triangle and rectangle so formed are equal? Write reason.
  13. O is the centre of the given circle. Inscribed angles PAQ and PBQ are standing on the same arc PQ.
    1. Write the relationship between the circumference angles PAQ and PBQ.
    2. If the measures of central angle POQ is (12x + 4)° and the measures of inscribed angle PAQ is (3x + 20)°, find the value of x.
    3. Verify experimentally that central angle is double of the inscribed angle in the same arc. (Two circles having radii more than 3 cm are necessary.)
  14. In the given figure, height of the tower AB is 24.5 meter and height of a house CD is 4.5 meter. BC denotes the distance between tower and house.
    1. Define the angle of elevation.
    2. Find the value of AE.
    3. If ∠ADE = 30°, find the distance between the tower and the house.
    4. By how many degrees is the angle of elevation less or more when AE and ED are equal?Compare it.
  15. The marks obtained by the students in an exam of mathematics of 75 full marks are given in the following table.
    Obtained Marks 0–15 15–30 30–45 45–60 60–75
    Number of Students 2 5 4 6 3
    1. Illustrate the modal class from the above data.
    2. Find the median from the above table.
    3. Find the mean from the above table.
    4. Among all the participants in the exam, what percentage of students obtained marks below the modal class? Find it.
  16. A box contains 6 white and 10 black balls of same shape and size. Two balls are drawn at random one after another with replacement.
    1. If A and B are two independent events, write the multiplication law of probability.
    2. Show the probability of all the possible outcomes in a tree diagram.
    3. Find the probability of getting both balls of same color.
    4. By how much the the probability of getting both balls of different color is less or more than probability of getting both balls of white color? Find it.

SEE 2081_RE1031_MP

  1. A survey conducted among 160 people showed that the number of people who like only apple and only orange are 75 and 45 respectively. Among them, 23 people do not like any of these two fruits.
    1. If ‘A’ represents the set of people who like apple and ‘O’ represents the set of people who like orange, write the cardinality notation of the number of people who don’t like any of these fruits.
    2. Present the above information in a Venn diagram.
    3. Find the number of people who like apple.
    4. Compare the number of people who like apple and who like orange.
  2. Neeraj took a loan of Rs. 4,00,000 for 2 years at the rate of 10% annual compound interest. He paid Rs. 2,40,000 at the end of the first year.
    1. Write the relation among principal (P), annual compound interest rate (R), time (T) and compound interest (CI).
    2. Find the compound interest of the first year.
    3. How much total interest was paid by Neeraj in two years? Find it.
  3. Neelam bought a machine for Rs. 40,000. The price of the machine depreciates at the rate of 5% annually. The machine is sold for Rs. 36,100 after using for some years.
    1. By how much does the price of the machine depreciate in the first year? Find it.
    2. After how many years was the machine sold? Find it.
    3. Find the profit or loss percentage from selling the machine if she earns Rs. 4,900 from the rent of the machine.
  4. Ramesh had NRs. 2,07,345. When he went to the bank the exchange rate was as follows:
    Buying rate of $1 = Rs. 138.23
    Selling rate of $1 = Rs. 138.83
    1. Which exchange rate is used when Ramesh exchanges American dollar to Nepali rupees? Write it.
    2. Find the American dollar obtained from NRs. 2,07,345.
    3. By what percent Nepali currency is devaluated when the selling rate of 1 US dollar is Rs. 140.2183? Find it.
  5. The volume of a square based pyramid is 512 cubic cm and the length of the side of its base is 16 cm.
    1. How many plane surfaces are counted to find the total surface area of a square based pyramid? Write it.
    2. Find the vertical height of the pyramid.
    3. Find the total surface area of the pyramid.
  6. Jyoti bought a tank made up of a cylinder and a hemisphere from the local market. The total height of the tank is 3.5 meter and the radius of the base is 1.05 meter.
    1. How many curved surfaces are there in a combined solid made of a cylinder and a hemisphere? Write it.
    2. Find the volume of the tank.
    3. How much maximum liters of water is contained in the tank? Find it.
  7. The length, breadth and height of a rectangular room are 16 ft, 12 ft and 9 ft respectively. There are two square windows of dimension 4 ft × 4 ft and one door of dimension 6 ft × 2 ft.
    1. How much does it cost for carpeting the room at the rate of Rs.300 per sq. ft.? Find it.
    2. If the cost of coloring four walls and ceiling excluding doors and windows of the room is Rs.19,560, find the rate of coloring per square feet.
  8. Hari deposited Rs.1,000, Rs.2,000 and Rs.3,000 in bank on his son Aashish's first, second and third birthday respectively. In this way, he increases the deposit by Rs.1,000 on every birthday.
    1. Define mean in arithmetic series.
    2. How much total money is deposited up to 10th birthday? Find it.
  9. Kriti wants to fence her field having length twice of breadth. The area of the field is 800 square feet.
    1. Write down the standard form of quadratic equation.
    2. How much length of wire is required to fence the field once with wire? Find it.
    1. Simplify: \(\frac{x}{xy - y^{2}} + \frac{y}{xy - x^{2}}\)
    2. \(2^x + \frac{16}{2^x} = 10\)
  11. In the given figure, parallelograms PQRS and PQUT are standing on the same base PQ and between the same parallel lines PQ and TR.
    1. Write the relation between the areas of parallelograms PQRS and PQUT.
    2. Prove that area of ΔPQT is half of the area of parallelogram PQRS.
    3. Are the areas of ΔAPD and ΔBPQ equal in the given figure? Write with reason.
  12. In the given figure, O is the centre of the circle. The points M, N, P and L are on the circumference of the circle.
    1. Define inscribed angle.
    2. If the central angle ∠LOP = (9x + 2)° and the inscribed angle ∠LMP = (4x + 5)°, find the value of x.
    3. Verify experimentally that angles ∠LMP and ∠LNP are equal. (Two circles having at least 3 cm radii are necessary.)
    1. Construct a quadrilateral PQRS in which PQ = 5.4 cm, QR = 5.6 cm, RS = 5.4 cm, SP = 6.8 cm and ∡PQR = 75°. Then construct a triangle PSM equal in area to the quadrilateral PQRS.
    2. In the given adjoining paralle- logram ABCD, AE = BE. What percentage of area of a paralle logram ABCD is occupied by the triangle BEC? Find it.
  14. In the given situation, PQ represents the height of the house, RS represents the height of the tower and QS represents the distance between the house and the tower.
    1. Write the name of the angle of elevation of the top of the tower as observed from the roof of the house.
    2. Find the value of TR.
    3. Find the distance between the house and the tower.
    4. Is the angle of depression of 30° formed when the roof of the house is observed from a point 28 m below the top of the tower? Give reason.
  15. The following table represents the marks obtained by students in an internal examination of Mathematics with full marks 50. The median of the data is 29.

    Obtained Marks 0–10 10–20 20–30 30–40 40–50
    Number of Students 3 7 10 x 10
    1. Write the median class.
    2. Find the value of x.
    3. Find the mean mark of the given data.
    4. Find the ratio of students obtaining marks less than 20 and students obtaining marks 20 or more than 20.
  16. A bag contains 7 black balls and 4 red balls. Two balls are drawn randomly one after another without replacement.
    1. If B and R are two independent events, write the formula of P(B ∩ R).
    2. Show the probability of all possible outcomes in a tree diagram.
    3. Find the probability of getting both black balls.
    4. By how much is the probability of getting both red balls less than the probability of getting both black balls? Find it.

SEE 2081_RE1031_BP

  1. In the survey conducted on 120 households of a village, 70 households are doing homestay business, 50 households are doing agriculture business and 30 households do other work. The sets of household doing homestay and agriculture business are denoted by H and A respectively.
    1. Write the cardinality notation of the number of households who do not do any of the homestay and agriculture business.
    2. Present the above information in a Venn diagram.
    3. Find the number of households doing only one business.
    4. If 10 households who do other work start homestay business, find the ratio of homestay and agriculture business households.
  2. Ramesh borrowed a loan of Rs. 2,00,000 from the bank at an annual compound interest rate of 7% per annum. After some time, he repaid Rs. 2,28,980 including the principal and interest to the bank.
    1. How much interest has Ramesh paid?
    2. For how many years has Ramesh used the loan?
    3. If the interest rate is reduced by 1%, how much amount would be paid by Ramesh?
  3. The present population of village A is 4500 and village B is 5000. The annual population growth rate of village A is 2%.
    1. What does P denote in the population after T years \(P_T = P\left(1 + \frac{R}{100}\right)^T\)?
    2. If 200 people are added by migration in village A after 1 year, what will be the population after 1 year?
    3. If the population of village B decreases by the same growth rate of village A, what will be the population of village B after 2 years?
  4. When Rita came to Nepal from America, she brought 2500 American dollars. The exchange rate was as follows:
    Buying rate of $1 Selling rate of $1
    Rs.127.35 Rs.127.95
    1. Which exchange rate is used to exchange Nepali currency by Rita?
    2. What is the profit or loss for Rita if she exchanges the dollars into Nepali currency and then exchanges back into dollars after two days?Find it.
    3. On the same day, if $1 = Rs.127.35 and Rs.160 = INR Rs.100, how many dollars can be exchanged with Indian Rs.1,20,000?
  5. The diagram represents a square based pyramid. Each side of the base is 6 meters and the vertical height is 4 meters.
    1. Write the formula for finding the area of all triangular surfaces of the pyramid.
    2. Find the volume of the pyramid.
    3. Find the total surface area of the pyramid.
  6. Given solid object is made up of a cone and a hemisphere. The total height of the object is 17 cm and the diameter of the base is 10 cm.
    1. Write the relation between height and radius of the hemisphere.
    2. Find the total surface area of the solid object.
    3. Will Rs.150 be sufficient to color the surface at the rate of 40 paisa per square cm?
  7. A rectangular room has length 12 ft, breadth 8 ft and height 9 ft. There are two windows of size 2.5 ft × 3 ft and two doors of size 6 ft × 2 ft.
    1. Find the total area of four walls, floor and ceiling.
    2. How much less or more is the cost of coloring the four walls excluding the doors and windows at the rate of Rs.175 per square feet than Rs.50,000? Find it.
  8. A shop sells shoes and slippers (in pairs) during winter season according to the following table.
    Day 1st Day 2nd Day 3rd Day 4th Day 5th Day
    No. of Shoes 2 4 8 16 ..........
    No. of Slippers 3 6 9 12 ..........
    1. Write the relationship between arithmetic mean and geometric mean.
    2. How many slippers are sold up to 8th days? Find it.
    3. Compare the total number of shoes and slippers sold up to 8th days.
  9. The area and perimeter of a rectangular field are 150 sq. m and 50 m respectively.
    1. Find the length and breadth of the field by forming a quadratic equation.
    2. How much equal part should be subtracted from the length and breadth to get the area 84 sq. m?
    1. Write \( \frac{1}{x - 6} \) in positive index of x.
    2. Simplify: \( \frac{a^2}{a - 2b} + \frac{4b^2}{2b - a} \)
    3. Solve: \( 4^{x-2} = 0.25 \)
  11. In the given figure, AE // BD, ED // AC and BE // CD.
    1. Write the relationship between the area of parallelogram and triangle standing on the same base and between the same parallel lines.
    2. Prove that ΔABE = ΔBCD.
    3. Compare the area of triangle ABE and trapezium ACDE.
  12. In the given figure, PQRS is a cyclic quadrilateral. The side PQ is produced to point T.
    1. Write the relation between ∠PSQ and ∠PRQ.
    2. Verify experimentally that ∠SPR and ∠SQR are equal. (Two circles with radii at least 3 cm are necessary.)
    3. Prove that ∠RQT = ∠PSR.
  13. In quadrilateral PQRS, PQ = 5 cm, QR = 4.5 cm, RS = SP = 6 cm and QS = 6.5 cm.
    1. Construct quadrilateral PQRS and then construct a triangle equal to the quadrilateral in area.
    2. Why are the areas of the quadrilateral and triangle equal? Give reason.
  14. In the given figure, the height of tower CD is 15√3 m and the height of house AB is 10√3 m. The angle from the roof of the house to the top of the tower is 30°.
    1. What type of angle is formed when the top of the tower is observed from the roof of the house?
    2. Find the height of part DE of the tower.
    3. Calculate the distance between the house and the tower.
    4. Find the angle of depression of the top of the house from the basement.
  15. The marks obtained by 50 students of a class in an exam of mathematics with full mark 60 are given below:
    Marks 10–20 20–30 30–40 40–50 50–60
    Number of Students 7 13 15 10 5
    1. Write the class of mode.
    2. Find the value of first quartile (Q₁).
    3. Calculate the average mark.
    4. Find the average mark of students who obtained 40 or more marks.
  16. In a box, 5 red and 3 white balls of same size and shape are kept. Two balls are drawn one after another without replacement.
    1. Write the addition law of probability of mutually exclusive events.
    2. Two balls are drawn one after another without replacement from the box. Show the probability of all the possible outcomes in a tree diagram.
    3. Find the probability of getting both white balls from the tree diagram.
    4. Find the difference between the probability of both balls being white if two balls are drawn one after another with replacement and without replacement.

SEE 2081_RE1031_GP

  1. In a survey of 450 people, 200 people liked tea and 250 people liked coffee. But 50 people did not like any of these two drinks.
    1. If T and C denote the set of people who like tea and coffee respectively, write the cardinality of n(T ∪ C).
    2. Present the above information in a Venn diagram.
    3. Find the number of people who liked tea only.
    4. Compare the number of people who like both tea and coffee with the number of people who like coffee only.
  2. A farmer deposited Rs.50,000 in a co-operative for 2 years to get the annual compound interest at the rate of 8% per annum.
    1. How many times the interest is calculated in the quarterly compound interest in one year? Write it.
    2. How much annual compound interest will the farmer receive at the end of 2 years? Find it.
    3. By how much the semi-annual compound interest is more than the annual compound interest of the same sum at the same rate and for the same period of time? Find it.
  3. An electric bus is purchased for Rs.45,00,000. Using the bus for 2 years Rs.12,00,000 is earned. The value of the bus depreciates at the rate of 10% per annum.
    1. If the initial price of bus is V₀, annual rate of depreciation is R and price of the bus after T years is Vₜ, express Vₜ in terms of V₀, R and T.
    2. How much the price of the bus depreciated in first year? Find it.
    3. If the bus will be sold after 2 years, what will be the percentage of profit or loss? Find it.
  4. Nabin went to bank to exchange American dollars to visit abroad. In that day the buying rate of 1 dollar was Rs.138.23 and selling rate was Rs.138.83.
    1. By how much the selling rate is more than the buying rate? Find it.
    2. How much Nepali rupees can be exchanged with American dollar 500? Find it.
    3. After some days the selling rate of dollar 1 becomes Rs.139.80 then by what percent the Nepali currency was devaluated? Find it.
  5. The vertical height of a square based pyramid is 12 cm and its base side is 10 cm.
    1. How many triangular surfaces are there in a square based pyramid? Write it.
    2. Find the volume of the pyramid.
    3. Find the total surface area of the pyramid.
  6. In the given figure, a combined solid object is formed with the combination of cylinder and cone having same radius. In the solid object, the length of cylindrical part is 28 cm and the slant height of conical part is 17 cm. The volume of the cylindrical part is 5632 cubic cm.
    1. What type is the shape of the base of solid object? Write it.
    2. Compare the height of conical part and the length of cylindrical part.
    3. Is the volume of cylindrical part five times the volume of conical part? Justify with calculation.
  7. The length, breadth and height of a rectangular room are 12 m, 8 m and 3 m respectively. There are two square windows with edges 2 m and a door of size 1.5 m × 1 m in the room.
    1. Find the area of the floor of the room.
    2. How much does it cost to coloring the four walls and ceiling of the room excluding the area occupied by the windows and door at the rate of Rs.15 per square meter? Calculate it.
  8. There are 7 arithmetic means between 3 and 27.
    1. Write the formula to calculate arithmetic mean between a and b.
    2. What is the 5th mean of the given sequence? Find it.
    3. Which one is greater by how much in arithmetic mean and geometric mean between 3 and 27? Compare it.
  9. The perimeter and area of a rectangular ground are 44 meter and 120 square meters respectively.
    1. Write the formula to solve the quadratic equation ax² + bx + c = 0, a ≠ 0.
    2. Find the length and breadth of the ground.
    3. If the ground is made a square by reducing the length side, by what percent the area will be increased or decreased? Find it.
    1. Simplify: a/(a−b) + b/(b−a)
    2. Solve: 2ˣ + 1/(2ˣ) = 2½
  11. In the given figure, EC // AB, DA // CB and DF ⟂ BC.
    1. Write the relation between areas of triangle and parallelogram standing on the same base and between same parallel lines.
    2. If BC = 6 cm and DF = 8 cm find the area of ΔABE.
    3. In the given figure, if area of ΔAOB and area of ΔCOD are equal, then prove that AD ∥ BC.
  12. In the given diagram, O is the centre of the circle and PQRS is a cyclic quadrilateral.
    1. Write the relationship between angle ∠QRS and reflex ∠QOS.
    2. If ∠QPS = 46°, find the value of ∠QOS.
    3. Verify experimentally that: ∡QPS + ∡QRS = 180°. (Two circles having radii more than 3 cm are necessary.)
  13. In quadrilateral PQRS, PQ = 5 cm, QR = 4.5 cm, RS = SP = 6 cm and QS = 6.5 cm.
    1. Construct quadrilateral PQRS according to the above measurements and then construct a triangle which is equal to the quadrilateral in area.
    2. Why the area of the quadrilateral and triangle are equal? Give reason.
  14. In the given figure, from the top of a tower AB 30 meter high, the angle of depression of the roof of a house is 30°. The distance between tower and house is 10√3 meter.
    1. Write the definition of the angle of elevation.
    2. What is the angle of elevation of the top of the tower from the roof of the house? Write it.
    3. Find the height of the house.
    4. What is the value of ∠CAH when AE = EC? Give reason.
  15. The marks obtained by 15 students in an examination with full mark 50 are given in table.
    Obtained Marks 0–10 10–20 20–30 30–40 40–50
    Number of Students 5 3 4 2 1
    1. Write the formula to calculate the first quartile of continuous series.
    2. Find the first quartile of the given data.
    3. Calculate the mean of the given data.
  16. A bag contains 7 black and 4 red balls of same shape and size. Two balls are drawn randomly one after another without replacement.
    1. If B and R be two independent events then write the formula of P(B ∩ R).
    2. Show the probability of all possible outcomes in a tree diagram.
    3. Find the probability of getting both black balls.
    4. By how much the probability of getting both red balls is more or less than the probability of getting both black balls? Find it.

SEE 2081_RE1031_LP

  1. In a survey of 450 people, 200 people liked tea and 250 people liked coffee. But 50 people did not like any of these two drinks.
    1. If T and C denote the set of people who like tea and coffee respectively, write the cardinality of n(T ∪ C).
    2. Present the above information in a Venn diagram.
    3. Find the number of people who liked tea only.
    4. Compare the number of people who like both tea and coffee with the number of people who like coffee only.
  2. A farmer deposited Rs.50,000 in a co-operative for 2 years to get the annual compound interest at the rate of 8% per annum.
    1. How many times the interest is calculated in the quarterly compound interest in one year? Write it.
    2. How much annual compound interest will the farmer receive at the end of 2 years? Find it.
    3. By how much the semi-annual compound interest is more than the annual compound interest of the same sum at the same rate and for the same period of time? Find it.
  3. An electric bus is purchased for Rs.45,00,000. Using the bus for 2 years, Rs.12,00,000 is earned. The value of the bus depreciates at the rate of 10% per annum.
    1. If the initial price of bus is V₀, annual rate of depreciation is R and price of the bus after T years is Vₜ then express Vₜ in terms of V₀, R and T.
    2. How much the price of the bus depreciated in first year? Find it.
    3. If the bus will be sold after 2 years, what will be the percentage of profit or loss? Find it.
  4. Nabin went to bank to exchange American dollars to visit abroad. In that day the buying rate of 1 dollar was Rs.138.23 and selling rate was Rs.138.83.
    1. By how much the selling rate is more than the buying rate? Find it.
    2. How much Nepali rupees can be exchanged with American dollar 500? Find it.
    3. After some days the selling rate of dollar 1 becomes Rs.139.80 then by what percent the Nepali currency was devaluated? Find it.
  5. The vertical height of a square based pyramid is 12 cm and its base side is 10 cm.
    1. How many triangular surfaces are there in a square based pyramid? Write it.
    2. Find the volume of the pyramid.
    3. Find the total surface area of the pyramid.
  6. A solid object is made up of a cone and a cylinder is given in the figure.
    1. How many curved surfaces are there in the given solid object? Write it.
    2. Find the height of cone.
    3. Compare the volume of cone and cylinder.
  7. The length, breadth and height of a rectangular classroom are 18 ft, 14 ft and 10 ft respectively. In the classroom, there are two windows with size 6 ft × 4 ft and two doors with size 6 ft × 3 ft.
    1. How much does it cost to paint four walls and ceiling of the classroom excluding doors and windows at the rate of Rs.40 per square feet? Find it.
    2. If a painter paints 202 square feet in a day, how many days will two painters take to paint the classroom? Find it.
  8. There are 7 arithmetic means between 3 and 27.
    1. Write the formula to calculate arithmetic mean between a and b.
    2. What is the 5th mean of the given sequence? Find it.
    3. Which one is greater by how much in arithmetic mean and geometric mean between 3 and 27? Compare it.
  9. The perimeter and area of a rectangular ground are 44 meter and 120 square meters respectively.
    1. Write the formula to solve the quadratic equation ax² + bx + c = 0, a ≠ 0.
    2. Find the length and breadth of the ground.
    3. If the ground is made a square by reducing the length side, by what percent the area will be increased or decreased? Find it.
    1. Simplify: a/(a − b) + b/(b − a)
    2. Solve: 2ˣ + 1/(2ˣ) = 5½
  11. In the adjoining figure, ΔPQR, parallelograms PQRS and PQTU are standing on the same base PQ and between the same parallel lines PQ and UR.
    1. Write the relation between the area of parallelograms PQRS and PQTU.
    2. Prove that the area of ΔPQR is half of the area of parallelogram PQTU.
  12. In the given figure, O is the centre of the circle and PQRS is a cyclic quadrilateral.
    1. Write the relationship between angle ∠QRS and reflex ∠QOS.
    2. If ∠QPS = 46°, find the value of ∠QOS.
    3. (Experimentally verify that the opposite angles ABC and ADC of cyclic quadrilateral ABCD are supplementary. (Two circles with at least 3 cm radii are necessary.)
  13. Construct a triangle CAT having sides AT = 4.4 cm, AC = 5.5 cm and ∠CAT = 60°. Construct another triangle BAT whose area is equal to the area of the given triangle, where AB = 6.2 cm.
    1. Why the area of ΔCAT and ΔBAT are equal? Give a reason.
    2. In the parallelogram ROSE, if P and Q are any points of sides ES and ER respectively, prove that ΔROP = ΔSOQ.
  14. In the given figure, height of the electric pole (PQ) is 18 meter and height of a man (RS) is 1.5 meter. SQ represents the distance between electric pole and man, where ∠PRT = 30°.
    1. Define the angle of elevation.
    2. Find the value of PT.
    3. Find the distance between the electric pole and the man.
    4. By how many degrees will the angle of elevation be less or more when PT and TR are equal? Find it.
  15. The marks obtained by 20 students in an examination with full marks 50 are given in the following table.
    Marks obtained 0–10 10–20 20–30 30–40 40–50
    No. of students 2 3 4 7 4
    1. Write the modal class of the given data.
    2. Find the median from the given data.
    3. Calculate the average mark from the given data.
    4. How many maximum number of students could be there who obtained the marks less than the average mark? Find it.
  16. Two cards are drawn randomly one after another without replacement from a well shuffled deck of 52 cards.
    1. If P(A∩B) = P(A) × P(B), what type of events are A and B? Write it.
    2. Show the probability of all possible outcomes of getting and not getting king cards in a tree diagram.
    3. Find the probability of getting both king cards.
    4. Is the probability of getting both ace of diamond possible? Give reason.

SEE 2081_RE1031_KAP

  1. Out of 100 students of class ten of a school, 60 liked English and 50 like mathematics. But 20 did not like any of these two subjects. The sets of students who liked English and Mathematics are denoted by 'E' and 'M' respectively.
    1. Write the set of the students who did not like any of these two subjects in the cardinality notation.
    2. Present the given information in a Venn diagram assuming x for the number of students who liked both subjects.
    3. Find the number of students who liked exactly one subject.
    4. If 30 students did not like any of these two subjects, what will be the effect in the number of students who liked both subjects? Find it.
  2. Aashal deposited Rs.1,00,000 in a bank. The bank provides 8% per annum interest compounded semi annually.
    1. Write the formula to calculate annually compound interest.
    2. How much interest does Aashal receive in 2 years? Find it.
    3. If the bank provides yearly compound interest for same rate and same period of time, how much would be profit or loss for him? Find it.
  3. The population of a village is 10,000. Annual population growth rate is 4%. At the end of first year, 100 people migrated from that village to other places.
    1. Find the population of the village after one year.
    2. If nobody were migrated in second year, what would be the population of the village after 2 years? Find it.
    3. If nobody was migrated in first year, what would be the difference in population growth in 2 years? Find it.
  4. According to currency exchange rate of Nepal Rastra Bank, 1 American dollar equals to NRs.138.83 in a day. Nepali currency was devaluated by 2% in the comparison of dollar after some days.
    1. What is called currency exchange? Write it.
    2. How many Nepali rupees can be exchanged with American dollar ($) 1500 before devaluation? Find it.
    3. After devaluation, how many American dollar can be exchanged with NRs.7,08,033? Find it.
  5. In the square based pyramid given in the figure, AH = 26 cm and AD = 24 cm.
    1. Write the relation of HD and EF.
    2. Find the value of EF.
    3. Find the total surface area of the pyramid.
  6. The given figure is of a combined solid object made with the combination of cylinder and hemisphere. The total height of the solid is 17 cm and circumference of the base is 44 cm.
    1. Write the formula to calculate volume of solid object.
    2. What is the volume of hemispherical part? Find it.
    3. Compare between volume of cylinder and hemisphere.
  7. A rectangular tank of 5 m × 1 m × 4 m is filled with water at the rate of 50 paisa per litre.
    1. The water containing in full tank is enough for 20 families distributed equally for one month. How much cost of water should one family have to pay in one year? Find it.
    2. If length, breadth and height each of the tank is increased by 1 m, by how many times the capacity of tank is increased? Find it.
  8. The number of words learned by a child daily in the double than previous day is shown in the following table.
    Day 1st 2nd 3rd 4th ........ ........
    Number of words 3 6 12 24 ........ ........
    1. In which sequence is the child learning words? Write it.
    2. How many words does the child learn upto 8 days? Find it using formula.
    3. In how many days will the child learn 6141 words? Find it.
  9. A two digit number is four times the sum of digits and three times the product of digits.
    1. If one’s place digit is y and ten’s place digit is x, write the two digit number in algebraic form.
    2. Make a quadratic equation in terms of x according to given conditions.
    3. Find the number.
  10. Simplify:
    1. 1/(b−1) − 1/(b+1)
    2. Solve: 7x + 7−x = 7 1/7
  11. In the figure, triangles APQ and BPQ are standing on the same base PQ and between the same parallel lines AB and PQ.
    1. Write the relation between the area of triangle APQ and triangle BPQ.
    2. If the perpendicular distance between AB and PQ is 8 cm and AB = 10 cm, find the area of ΔAPB.
  12. In a quadrilateral PQRS, PQ = 5.1 cm, QR = 7 cm, RS = 4.6 cm, SP = 5.4 cm and QS = 6.6 cm are given.
    1. Construct the quadrilateral PQRS according to above measurement and then construct a triangle which is equal to the quadrilateral in area.
    2. Why the areas of triangle and quadrilateral so constructed are equal? Give reason.
  13. In the figure, O is the center of the circle and ABDC is a cyclic quadrilateral.
    1. What is the relation between inscribed angles standing on same arc of a circle? Write it.
    2. If inscribed angle ∠BAC = 35°, find the value of ∠BOC.
    3. If arc BDC and arc ACD are equal, prove that AB // CD.
    4. Verify experimentally that ∠BAC and ∠BDC are supplementary.(Two circles having at least 3 cm radii are necessary.)
  14. The height of Himali is 1.22 m. She observed the top of the school building standing 36 m far from the base of school building found the angle of 30°.
    1. What is the name of the angle found when Himali observed at the top of school building according to the given context? Write it.
    2. Sketch the figure from the above context.
    3. Find the height of the school building.
    4. How many meters should Himali have to walk nearer or farther from that place to make the angle of the top of the building to be 60°? Give reason.
  15. The age of 300 students of a school are given in the table.
    Age in years 0–4 4–8 8–12 12–16 16–20
    No. of students 50 65 75 60 50
    1. What does c.f denote in the first quartile (Q1) = L + (N/4 − c.f)/f × i ? Write it.
    2. Calculate the mean of the given data.
    3. Find the median of the given data.
    4. Find the percentage of number of students who obtained more marks than median class.
  16. From a well shuffled pack of 52 cards two cards are drawn randomly one after another without replacement.
    1. Write the multiplicative law of probability.
    2. Show the probability of all possible outcomes of getting and not getting face cards in a tree diagram.
    3. Calculate the probability of getting both face cards.
    4. Compare between the probability of getting both face cards and the probability of not getting both face cards.

SEE 2081_RE1031_SP

  1. In a survey conducted among 120 students studying in class Ten of a secondary school, it was found that 60 students liked cricket game, 55 students liked basketball game and 20 students did not like any of these games.
    1. If C and B denote the sets of students who liked cricket and basketball game respectively, write the cardinality of n(B ∪ C).
    2. Present the above information in a Venn-diagram.
    3. Find the number of students who liked cricket game only.
    4. Compare the number of students who liked cricket game only and who liked basketball game only.
  2. Aatmik wants to deposit Rs.4,00,000 in a bank for 2 years. The bank offers 10% per annum compound interest to Aatmik with three alternatives (annual compound interest, semi-annual compound interest and quarterly compound interest).
    1. Which option among the above three alternatives Aatmik has to use to get more interest? Write it.
    2. How much compound interest does he receive at the end of 2 years compounded semi annually? Find it.
  3. A photocopy machine is purchased for Rs.80,000. After using it for 2 years, only Rs.30,000 is earned. The price of machine depreciates annually at the rate of 20% and the machine is sold after 2 years.
    1. The initial price of a machine is V₀, annual rate of compound depreciation is R and the price of machine after T years is Vₜ, express Vₜ in terms of V₀, R and T.
    2. Find the total profit or loss amount on selling the machine.
    3. If he had sold the machine after using it one year more, by how much the selling price is less or more than the purchased price? Compare it.
  4. A businessman exchanged Australian dollars with NRs.1,29,090 at the exchange rate of Australian dollar 1 NRs.86.06. After some days, Nepali currency was revaluated by 2% in comparison to Australian dollar and on that day he exchanged the Australian dollars into Nepali currency again.
    1. How many Australian dollars did the businessman exchange? Find it.
    2. How many Nepali rupees did the businessman receive when he exchanged Australian dollar after revaluation in Nepali currency? Find it.
    3. What profit or loss percent did the businessman make in that transaction? Find it.
  5. The vertical height of the square based pyramid is 24 cm and the length of one side of base is 20 cm.
    1. Write the formula to find the volume of the pyramid.
    2. Find the total surface area of the pyramid.
  6. In the figure, a metallic solid made of hemisphere and cone is given, where the height of cone is 24 cm and diameter of base is 14 cm.
    1. Write the formula to find the slant height of cone when vertical height and radius of base are given.
    2. Find the volume of the solid object.
    3. If the solid object is melted and turned into a cylindrical object of radius 7 cm, what is the height of cylinder? Calculate it.
  7. The volume and height of a square based room are 75 cubic meter and 3 meter respectively. The area occupied by a door and two windows in the room is 6 square meter.
    1. What is the total cost of plastering the four walls without door and windows at the rate of Rs.200 per square meter? Find it.
    2. If the rate of plastering per square meter is increased by one-forth, then what will be the increment in the total cost of plastering the walls? Find it.
  8. Hira collected following sum of money in the first 5 days of month Baishakh.
    Baishakh-1 Baishakh-2 Baishakh-3 Baishakh-4 Baishakh-5
    Rs.10 Rs.20 Rs.40 Rs.80 Rs.160
    1. What is the mean value of the amount collected on 2nd Baishakh and 4th Baishakh? Write it.
    2. How much money will be collected by 10th day? Find using formula.
    3. Up to how many days of Baishakh can Rs.1,63,830 be collected? Find it.
  9. The length of rectangular field is twice of its breadth and its area is 200 square meter.
    1. Write the standard form of quadratic equation.
    2. Find the length and breadth of the rectangular field.
    3. How many maximum numbers of pieces having size 5 m × 4 m can be made in the field? Also present diagram.
    1. Simplify: 1/(x−y) − 1/(x+y)
    2. If x² = 32/3 + 3−2/3 − 2, prove that: 3x³ + 9x = 8
  11. In the given figure, parallelogram EBCF and square ABCD are on the same base BC and between the same parallel lines AF and BC.
    1. Write the relation between the areas of parallelograms standing on the same base and between same parallel lines.
    2. Prove that: Area of parallelogram EBCF = Area of square ABCD.
    3. In the given figure PQRS is a parallelogram and M is the mid-point of TR. Prove that: ΔTQM = 1/2 (ΔPQT + ΔSRT)
    1. Construct a triangle ABC having BC = 6.4 cm, AB = 5.6 cm and AC = 6 cm. Also construct a triangle DAB having one side 7 cm equal in area to ΔABC.
    2. Why the area of ΔABC and ΔDAB are equal? Give reason.
  13. Central angle AOB and inscribed angles ADB and ACB are standing on the same arc AB in a circle with center O.
    1. Write the relation between the inscribed angles standing on the same arc.
    2. Experimentally verify that, the central angle AOB is double of the inscribed angle ACB.
    3. The measure of central angle is (5x)° and the measure of inscribed angle is (2x + 10)° standing on the same arc in a circle, find the value of x.
  14. A tree x meter high is broken by the wind, at the height 6 meter from the ground so that its top touches the ground and makes an angle 30° with the ground.
    1. What is called the angle of elevation? Write it.
    2. Express the length broken part of the tree in terms of x.
    3. What was the height of the tree before broken? Find it.
    4. What height should the tree be broken so that its top makes an angle of 45° with ground? Find it.
  15. The first quartile of the given data is 35.
    Obtained Marks 0-20 20-40 40-60 60-80 80-100
    Number of students 2 x 8 5 1
    1. Illustrate the class where the first quartile lies.
    2. Find the value of x.
    3. Find the mode from the given data.
    4. Find the ratio of students who are above and below the first quartile class.
  16. A married couple has given birth of two children in the interval of five years.
    1. Define independent events.
    2. Show all the possible outcomes in a tree diagram.
    3. Find the probability of having both daughters.
    4. By how much the probability of getting both children son is less or more than the maximum probability? Calculate it.

SEE 2081_RE2023_OPT Math

Group 'A'[10*1=10]

  1. What type of the function is represented by \( f(x) = x^{2} + 5x + 6 \)? Write it.

  2. Write the statement of Remainder Theorem.

  3. In which condition a function \( f(x) \) is continuous at the point \( x = a \)? Write it.

  4. What is the determinant value of a singular matrix? Write it.

  5. Write the condition of parallelism of two lines having slopes \( m_{1} \) and \( m_{2} \).

  6. In which condition a circle is formed when a plane surface intersects a cone? Write it.

  7. Express \( \cos 2\theta \) in terms of \( \tan \theta \).

  8. If \( \cos A = 1 \), write the acute angle value of \( A \).

  9. If \( \beta \) is the angle between two vectors \( \vec{p} \) and \( \vec{q} \), then write the formula to find the value of \( \cos \beta \).

  10. The inversion point of a point \( P \) is \( P' \) in a circle having centre \( O \) and radius \( r \). If the distance of the point \( P \) and \( P' \) from the centre \( O \) are \( OP \) and \( OP' \), write the relationship among \( OP \), \( OP' \), and \( r \).

Group 'B'[8*2=16]
  1. If \( f(x) = \dfrac{x - 3}{5} \) and \( g(x) = 5x + 3 \), find the value of \( (f \circ g)(2) \).
  2. Find the vertex of the parabola formed from the equation \( y = x^{2} - 5x + 6 \).
  3. Find the inverse matrix of the matrix \[ A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix} \]
  4. The line passing through the points \( (-2, -5) \) and \( (1, a) \) is perpendicular to the line having equation \( 2x - y + 5 = 0 \). Find the value of \( a \).
  5. Prove that: \[ \sqrt{1 - \sin A} = \sin\left(\frac{A}{2}\right) - \cos\left(\frac{A}{2}\right), \quad \text{where } 0 \le A \le \pi. \]
  6. Solve: \[ 2\cos^{2}\theta - 2\sin^{2}\theta = 1 \]
  7. Prove that \( \vec{a} = 3\vec{i} + 2\vec{j} \) and \( \vec{b} = 4\vec{i} - 6\vec{j} \) are perpendicular to each other.
  8. The interquartile range of a continuous data is \( 20 \) and the first quartile \( (Q_{1}) = 10 \). Find the coefficient of quartile deviation.
Group 'C'[11*3=33]
  1. If \( f(x) = 3x - 5 \), \( g(x) = \dfrac{2x + 7}{3} \) and \( (f \circ g^{-1})(x) = f(x) \), find the value of \( x \).
  2. If the third term and sixth term of a geometric series are \( 12 \) and \( 96 \) respectively, what is the arithmetic mean between the first term and the sum of the first four terms of the series? Find it.
  3. If \( f(x) = 3x - 5 \) is a real valued function.
  4. Find \( f(3.9) \), \( f(3.99) \), \( f(4.01) \) and \( f(4.001) \).
  5. Find \( \lim_{x \to 4^{+}} f(x) \) and \( \lim_{x \to 4^{-}} f(x) \).
  6. Is the function \( f(x) \) continuous at the point \( x = 4 \)? Give reason.
  7. Solve using Cramer's rule: \[ \begin{aligned} 4x + 3y &= -18 \\ 2x - 5y &= 4 \end{aligned} \]
  8. Find the angle between a pair of lines represented by the equation \( 2x^{2} + 7xy + 3y^{2} = 0 \).
  9. Prove that: \[ \cos 20^\circ \cdot \cos 40^\circ \cdot \cos 60^\circ \cdot \cos 80^\circ = \frac{1}{16}. \]
  10. If \( A + B + C = \pi \), then prove that: \[ \sin 2A + \sin 2B + \sin 2C = 4 \sin A \cdot \sin B \cdot \sin C. \]
  11. From the top of a tower, the angles of depression of the roof and basement of a building \( 20 \) meter high are \( 30^\circ \) and \( 45^\circ \) respectively. Find the height of the tower.
  12. A triangle \( PQR \) having the vertices \( P(1,2) \), \( Q(4,1) \) and \( R(2,5) \) is transformed by a \( 2 \times 2 \) matrix so that the coordinates of the image are \( P'(5,2) \), \( Q'(6,1) \) and \( R'(12,5) \). Find the \( 2 \times 2 \) matrix.
  13. Find the mean deviation from mean of the given data.
    14. Marks obtained 5–15 15–25 25–35 35–45 45–55
    Frequency 3 5 4 5 3
  14. Find the standard deviation from the given data.
    15. Class Interval 2–4 4–6 6–8 8–10
    Frequency 3 4 2 1
  15. Find the maximum value of the objective function \( P = 4x + 6y \) under the constraints: \[ x + 2y \le 8,\quad 3x + 2y \le 12,\quad x \ge 0,\; y \ge 0. \]
  16. One end of a diameter of the circle having equation \( x^{2} + y^{2} - 4x - 6y - 12 = 0 \) is \( (5,4) \). Find the coordinates of the other end of the diameter.
  17. In triangle \( XYZ \), the midpoints of the sides \( XY \) and \( YZ \) are \( A \) and \( B \) respectively. Prove by vector method that \( AB \parallel XZ \).
  18. A triangle \( ABC \) with vertices \( A(1,2) \), \( B(4,-1) \) and \( C(2,5) \) is reflected successively on the lines \( x = 5 \) and \( y = -2 \). Find vertices of the images so obtained. Plot the triangle and images on the same graph paper.

SEE 2081_RE2021A_OPT Math

Group 'A'[10*1=10]
  1. Write the definition of cubic function.
  2. If the first term of an arithmetic series is ‘a’, common difference ‘d’ and the number of terms ‘n’, then write the formula to find the sum of the first n terms of the series.
  3. Write the name of the set of numbers which is continuous.
  4. In which condition does the inverse matrix of the given matrix does not exist? Write it.
  5. If the slopes of two lines are m₁ and m₂ respectively, then in which condition these two lines are perpendicular to each other? Write it.
  6. Which geometrical figure is formed if a plane surface intersects a cone parallel to its base? Write it.
  7. Express sin3A in terms of sinA.
  8. Express sinA − sinB in the product form.
  9. If j is a unit vector along Y-axis, what is the value of (j)²? Write it.
  10. The radius of the inversion circle is ‘r’ and the centre is ‘O’. If the image of a point A is A′, what is equal to OA × OA′? Write it.
Group 'B'[8*2=16]
  1. If f(x) = x³ + mx² − x + 7 is divided by x − 3, the remainder is 4. Find the value of m using remainder theorem.
  2. Present the inequality x − y ≥ 2 in the graph.
  3. If D = 10, Dₓ = |4 2; 2 6| and Dᵧ = |4 1; −2 7| then find the values of x and y.
  4. If two lines having equations 3x − y + 7 = 0 and kx + 2y + 8 = 0 are parallel, find the value of k.
  5. Prove that: (sin50° − sin30°) / (cos30° − cos50°) = cot40°
  6. If 3tan²θ − 9 = 0 then find the value of θ. (0° ≤ θ ≤ 90°)
  7. If a · b = 48, |a| = 6√2 and |b| = 8 then find the angle between a and b.
  8. If the value of first quartile (Q₁) of any data is 43 and the quartile deviation (Q.D.) is 6.5, find the coefficient of quartile deviation.
Group 'C'[11*3=33]
  1. If two functions are f(x) = (2x − 5)/3 and g(x) = x + 4, find (fog)⁻¹(3).
  2. Solve quadratic equation x² + 2x − 3 = 0 by graphical method.
  3. If the function f(x) = {3x − 1 for x ≥ 2, x + 3 for x < 2} is defined, is the function f(x) continuous at x = 2? Give reason.
  4. Solve by matrix method: 4x − 3y = 11 and 3x + 7y + 1 = 0.
  5. Find the equation of a circle having centre (3, 5) and passing through the centre of the circle x² + y² + 4x − 6y − 36 = 0.
  6. Prove that: (cos²A − sin²B) / (sinA cosA + sinB cosB) = cot(A + B)
  7. If A + B + C = 180° then prove that: cosA + cosB + cosC = 1 + 4 sin(A/2) sin(B/2) sin(C/2)
  8. An observer observes the roof of the house 20 m high to the top and bottom of the television tower and found the angle of elevation and angle of depression to be 45° and 30° respectively. Find the height of the tower.
  9. The centre and scale factor of the enlargement E are (−3, −4) and 2 respectively. R represents the reflection in the line y = 0.
  10. In which point the combined transformation EoR transforms a point P(x, y)? Find it.
  11. Transform the ΔABC with vertices A (2, 0), B (3, 1) and C (1, 1) using the combined transformation EoR.
  12. Present the ΔABC and the image in the same graph paper.

SEE 2081_RE2021B_OPT Math

Group 'A'[10*1=10]
  1. Write the definition of quadratic function.
  2. Write the formula to find the sum of a geometric series having n terms, where common ratio (r) is more than 1.
  3. Write lim x→a f(x) in sentence.
  4. If the inverse matrix of a square matrix A of order 2 × 2 is A⁻¹, what is equal to A·A⁻¹? Write it.
  5. The slopes of two straight lines are m₁ and m₂ respectively. Write the formula to find the angle between them.
  6. In which condition a plane surface intersects a cone to form a parabola? Write it.
  7. Express cos2A in terms of cosA.
  8. Write sinC − sinD in terms of product of sine and cosine.
  9. If a⃗ and b⃗ are perpendicular, then write the value of a⃗ · b⃗.
  10. If inversion point of a point P is P′ in a circle with centre O and radius r, what is equal to OP × OP′? Write it.
Group 'B'[8*2=16]
  1. If one factor of polynomial f(x) = x³ − 5x² + (k + 1)x + 8 is (x − 2), find the value of k.
  2. Draw the graph of the inequality 2x + y ≤ 3.
  3. Find the values of Dx and Dy using Crammer's rule from the equations 2x − y = 5 and x − 2y = 1.
  4. If two lines having equations cx + dy + e = 0 and fx + gy + h = 0 are parallel to each other then prove that cg − df = 0.
  5. Prove that: (sinθ + sin(θ/2)) / (cosθ + cos(θ/2)) = tan(θ/2).
  6. Solve: √3 tanA − 1 = 0, [0° ≤ A ≤ 180°].
  7. If |a⃗| = 4, |b⃗| = 6 and a⃗ · b⃗ = 12 then find angle between a⃗ and b⃗.
  8. In a continuous series, third quartile is two times of the first quartile. If the sum of the first quartile and third quartile is 90, find the quartile deviation.
Group 'C'[11*3=33]
  1. If g(x) = 4x − 17, f(x) = \frac{2x + 8}{5} and gog(x) = f⁻¹(x), find the value of x.
  2. Solve graphically: x² + x − 2 = 0.
  3. If the function f(x) = {2x + 4 for x < 3, 4x − 2 for x ≥ 3} is defined, is the function f(x) continuous at x = 3? Give reason.
  4. Solve by matrix method: x − y = 2 and 4x − 3y = 1.
  5. Find the equation of the circle whose centre is (2, 3) and passes through the centre of the circle x² + y² − 10x + 4y + 13 = 0.
  6. Prove that: \cot\left(\frac{A}{2}+\frac{\pi}{4}\right)-\tan\left(\frac{A}{2}-\frac{\pi}{4}\right)=\frac{2\cos A}{1+\sin A}.
  7. If A + B + C = π then prove that: cosA + cosB − cosC = 4cos(A/2)cos(B/2)sin(C/2) − 1.
  8. The angles of depression and elevation of the pinnacle of a temple 10 m high with pinnacle are found to be 60° and 30° respectively from the top and bottom of a tower. Find the height of the tower.
  9. If the matrix [[a, 2], [b, 2]] transforms a unit square to the parallelogram [[0, 4, c, 2], [0, 1, 3, d]] then find the values of a, b, c and d.
  10. Find the mean deviation from median of the given data.
    11. Marks obtained 0–10 10–20 20–30 30–40 40–50
    No. of students 3 8 5 6 4
  11. Find the standard deviation from the given data.
    12. Age (in years) 0–4 4–8 8–12 12–16 16–20 20–24
    No. of students 7 8 10 12 9 6
Group 'D'[4*4=16]
  1. The sum of three numbers in an arithmetic series is 18. If the geometric mean between the first and third numbers is 4√2 then find the numbers.
  2. Find the separate equations of a pair of lines represented by the equation x² − 2xy cosecθ + y² = 0. Also find the angle between them.
  3. PQ is the diameter of semi-circle with centre O and M is a point on the circumference of the semi-circle. Prove by vector method that ∠PMQ = 90°.
  4. A triangle PQR with vertices P (4, 3), Q ( 2, 0) and R (5, 2) is translated by T = and the image so obtained is rotated through 90° in negative direction about origin. Find the co-ordinates of the vertices of the obtained images. Plot the given triangle and the images in the same graph paper.

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