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ALGEBRA ASSIGNMENT

A.P.
Q1. Attempt the following [each with 1 mark]
  1. Check whether the following numbers are Arithmetic Progression?  Justify. 3, 5, 7, 9, 11...
  2. Is the following list of numbers an Arithmetic Progression? Justify. -10, -13, -16, -19...
  3. Is the following list of numbers an Arithmetic Progression? Justify. 22, 26, 28, 31, ...
  4. Find the first five terms of the following sequence, whose 'nth' term terms is given : tn=4n-3
  5. Is the following list of numbers an Arithmetic Progression? Justify. 1, 3, 6, 10, ...
  6. Is the following list of numbers an Arithmetic Progression? Justify. 1, 4, 7, 10...
  7. For the sequence, find the next four terms. 1, 3, 7, 15, 31, ...
  8. For the sequence, find the next four terms. 12,16,118,154,…
  9. Find the first five terms of the following sequence, whose 'nth' term terms is given: tn=n3.
  10. Find the first four terms of the sequence whose nth term is 3n+1.
Q2. Attempt the following. [each with 2 marks]
  1. Find the twenty fifth term of the A.P. : 12,  16,  20,  24,  ...
  2. Find the first three terms of the sequence for which Sn is given: Sn=n2(n+1)
  3. For an A.P. if t4=24, and d= -10, then find its general term.
  4. For an A.P. if t4=12, and d= 1, then find its general term.

Q3. Solve the following (3 marks each)
  1. The 11th term and the 21st term of an A.P. are 16 and 29 respectively, find : the 1st term and the common difference.
  2. The taxi fare is Rs. 14 for the first kilometer and Rs. 2  for each additional kilometer. What will be fare for 10 kilometers?
  3. Find S10 if a=6, d=3.
  4. Vijay invests some amount in National saving certificate. For the first year he invests Rs. 500, for the second year he invests Rs. 700 and for the third year he invest Rs. 900 and so on. How much amount he has invested in 12 years?
  5. Find four consecutive terms in an A.P. whose sum is 12 and the sum of 3rd and 4th term is 14.
  6. The sum of first n terms of an A.P. is 3n+n2 then (i) find first term and sum of first two terms. (ii) find second, third and 15th term.
  7. Find the sum of all odd natural number from 1 to 150.
  8. How many three digit natural numbers are divisible by 4?
Q4. Attempt the following. (4 marks each)
  1. In a school, a plantation program was arranged on world environment day, on a ground of triangular shape. The trees are to be planted as shown in a figure. One plant in the first row, two in the second row, three in the third row and so on. If there are 25 rows then find the total number of plants to be planted.
  2. In winter, the temperature at a hill station from Monday to Friday is in A.P. The sum of the temperatures of Monday, Tuesday and Wednesday is zero and the sum of the temperatures of Thursday and Friday is 15. Find the temperature of each of the five days.
  3. Babubhai borrows Rs. 4000 and agrees to repay with a total interest of Rs. 500 in 10 instalments, each instalment being less than the preceding instalment by Rs. 10. What should be the first and the last instalment?
  4. If the 9th term of an A.P. is zero then prove that the 29th term is double the 19th term.
  5. How many two digit numbers leave the remainder 1 when divided by 5?
Q5. Attempt the following (5 marks each)
  1. Find three consecutive terms in an A.P. whose sum is – 3 and the product of their cubes is 512.

Quadratic Equation
Q1. Attempt the following [each with 1 mark]
  1. Is the given equation a quadratic equation: (y-2)(y+2)=0.
  2. Is the given equation a quadratic equation: y2-4=11y.
  3. Is the given equation a quadratic equation: 13=-5y2-y3
  4. Write the quadratic equation in standard form : x2=-7-10x.
  5. Determine the nature of the roots of the quadratic equation from its discriminant: y2-4y-1=0
  6. Write the quadratic equation in standard form: 7-4x-x2=0.
  7. Is the given equation a quadratic equation: 13=-5y2-y3.
  8. Determine the nature of the roots of the following equation from its discriminant: 2y2+5y-3=0.
  9. Write the equation in the standard form of ax2+bx+c=0, 8-3x-4x2=0.
Q2. Attempt the following. [each with 2 marks]
  1. Determine whether the given value of 'x' is a root of given quadratic equation. x2-2x+1=0, x=1
  2. If one root of the quadratic equation x2-7x+k=0 is 4, then find the value of k.
  3. If one root of the quadratic equation kx2-7x+12=0 is 3, then find the value of k.
  4. Solve the following quadratic equation by factorization method. x2-13x-30=0.
  5. Form the quadratic equation whose roots are 3 and 10.
  6. Solve the following quadratic equation by factorization method: 3x2+10=11x.
  7. Solve the following quadratic equation by factorization method: y2-3=0.
  8. If one root of the quadratic equation 3y2-ky+8=0 is 23 , then find the value of k.
  9. State whether k is the root of the given equation y2-(k-4)y-4k=0.
  10. Solve the following quadratic equation by factorization method: m2-84=0.
Q3. Solve the following (3 marks each)
  1. Solve the following quadratic equation by using formula: 5m2-2m=2.
  2. Solve the following quadratic equation by completing square method: p2-12p+32=0.
  3. Solve the following quadratic equation by using formula:  5m2-2m=2
  4. Solve the following quadratic equation by completing square method: Solve the following quadratic equation by completing square method: n2+3n-4=0.
  5. Solve the following quadratic equation by completing square method: m2-3m-1=0.
  6. Tinu is younger than Pinky by three years. The product of their ages is 180. Find their ages.
  7. Form the quadratic equation if its one of the root is 3-25
  8. Solve the following equation: x4-3x2+2=0.
  9. Form the quadratic equation whose one of the root is 1-35
  10. Solve the following quadratic equation by using formula 4x2+x-5=0.
Q4. Attempt the following. (4 marks each)
  1. If one root of the quadratic equation kx2-5x+2=0 is four times the other, find k.
  2. Solve the following equation: 2(y2-6y)2-8(y2-6y+3)-40=0.
  3. If the difference between the roots of the quadratic equation is 3 and difference between their cubes is 189, find the quadratic equation.
  4. Solve the following equation: (y2+5y)(y2+5y-2)-24=0.
  5. If the difference between the roots of the quadratic equation is 4 and difference between their cubes is 208, find the quadratic equation.
  6. Solve the following equation: 2y2+15y2=12.
  7. The length of one diagonal of a rhombus is less than the second diagonal by 4 cm.  The area of the rhombus is 30 sq. cm. Find the length of the diagonals.

Q5. Attempt the following (5 marks each)
  1. Solve the following equations: 12(x2+1x2)-56(x+1x)+89=0
  2. One diagonal of a rhombus is greater than other by 4cm. If the area of the rhombus is 96 cm2, find the side of the rhombus.
  3. Solve the following equation: 3(x2+1x2)-4(x-1x)-6=0
  4. A car covers a distance of 240 km with some speed. If its speed is increased by 20 km/hr, it will cover the same distance in 2 hours less. Find the speed of the car.
  5. The sum of the squares of five consecutive natural number is 1455. Find them.
  6. A businessman bought some items for Rs. 600, keeping 10 items for himself he sold the remaining items at a profit of Rs. 5 per item. From the amount received in this deal he could buy 15 more items. Find the original price of each item.
  7. Solve the following equation: 2(x2+1x2)-9(x+1x)+14=0
  8. Around a square pool there is a footpath of width 2m. If the area of the footpath is 54 times that of the pool. Find the area of the pool.
  9. A man travels by boat 36 km down a river and back in 8 hours. If the speed of his boat in still water is 12 km per hour, find the speed of the river current.
  10. Solve the following equation: 9(x2+1x2)-3(x-1x)-20=0
Linear equation in two unknown variables
Q1. Attempt the following [each with 1 mark]
  1. Find the value of the following determinant: |-3 8 6 0 |
  2. What is the equation of Y – axis? Hence, find the point of intersection of Y – axis and the line y = 3x + 2.
  3. Write the equation of X – axis and Y – axis ?
  4. Write the co – ordinates of the point of intersection of X – axis and Y – axis?
  5. If the value of the determinant |m 2 -5 7 | is 31, find m.
  6. If Dx= -15, & D=-5, are the values of the determinants for certain equations in x and y, find x.
  7. If (a , 3 ) is the point lying on the graph of the equation 5x + 2y = - 4, then find a.
  8. Examine whether the point (2 , 5) lies on the graph of the equation 3x – y = 1.
  9. If Dy= -5, & D=-5, are the values of the determinants for certain simultaneous equations in x and y, find y.
Q2. Attempt the following. [each with 2 marks]
  1. If x = 5 and  y = 3 is the solution of 3x+ky=3, find k.
  2. If Dx= -18, & D=3, are the values of the determinants for certain simultaneous equations in x and y, find x.
  3. Express the following information in mathematical form using two variables. The perimeter of a rectangle is 36 cm.
  4. If Dx= -1, & D=-8, are the values of the determinants for certain simultaneous equations in x and y, find x.
  5. Find the value of the following determinant |15 20 7 4 |.
  6. Examine whether the point (- 2 , 5) lies on the graph of the equation 3x – y = 1.
  7. If the point (3 , 2) lies on the graph of the equation 5x+ay=19, then find a.
  8. What is the equation of X – axis? Hence, find the point of intersection of the graph of the equation x+y=3 with the X – axis.
  9. Find the value of k for which the given simultaneous equations have infinitely many solutions: 4y=kx-10, 3x=2y+5.
  10. Solve the following quadratic equation by using formula: y=5x-102;4x+5=-y.
Q3. Solve the following (3 marks each)
  1. Without actually solving the simultaneous equations given below, decide whether simultaneous equations have unique solution, no solution or infinitely many solutions. x-2y3=3;2x-4y=92.
  2. Find the value of Dx for the following simultaneous equation: 5x=10-2y;y=3x-11.
  3. Without actually solving the simultaneous equations given below, decide whether simultaneous equations have unique solution, no solution or infinitely many solutions. 3y=2-x;3x=6-9y.
  4. Find  the value of following determinant: |1.2 0.03 0.57 -0.23 |
  5. Without actually solving the simultaneous equations given below, decide whether simultaneous equations have unique solution, no solution or infinitely many solutions. 3x-7y=15;6x=4y+10.
  6. If 12x+13y=29; and 13x+12y=21, Find x+y.
  7. Without actually solving the simultaneous equations given below, decide whether simultaneous equations have unique solution, no solution or infinitely many solutions. 3x+5y=16; 4x-y=6.
  8. Without actually solving the simultaneous equations given below, decide whether simultaneous equations have unique solution, no solution or infinitely many solutions. 8y=x-10;2x=3y+7.
  9. Point (3, - 1) and (6, 1) lie on the line represented by the equation px+qy=9, find the values of p and q.
  10. Solve the following simultaneous equations using Cramer's rule: 3x-y=7;x+4y=11.
Q4. Attempt the following. (4 marks each)
  1. Without drawing the graphs, shows that the following equations are of concurrent lines: y=5x-3;    y=4-2x;    2x+3y=8.
  2. Solve the following simultaneous equations: ax+by=5 and bx+ay=3, where a and b are constant.
  3. If (3, 1) is the point of intersection of lines ax+by=7 and bx+ay=5 , find the values of a and b.
  4. Solve the following simultaneous equations using Cramer's rule 3x+2y+11=0, 7x-4y=9.
  5. Find the value of k for which are given simultaneous equations have infinitely many solutions: kx-y+3-k=0 ; 4x-ky+k=0 .
  6. Solve the following simultaneous equations using Cramer's rule: 4x+3y-4=0;6x=8-5y.
  7. Solve the following simultaneous equations using graphical method: x+y=8;x-y=2.
  8. Solve the following equation using graphical method: 4x=y-5, y=2x+1.
  9. Solve the following equation using graphical method: x+2y=5:   y=-2x-2
  10. Solve the following simultaneous equations: 27x-2+31y+3=85;31x-2+27y+3 =89.
Q5. Attempt the following (5 marks each)
  1. Solve the following simultaneous equations using graphical method: 3x+4y+5=0;y=x+4.
  2. Solve the following simultaneous equations: 13x+15y=115 ; 12x+13y=112.
  3. Draw the graphs representing the equations 2x=y+2 & 4x+3y=24 on the same graph paper. Find the area of the triangle formed by these lines and X – axis.
  4. Solve 2x+6y=13;3x+4y=12
  5. Solve   13x+15y=115; 12x+13y=112.
  6. Students of a school were made to stand in rows for drill. If 3 students less were standing in each row, 10 more rows were required and if 5 students more were standing in each row then the number of rows was reduced by 10. Find he number of students participating in the drill.
  7. A bus covers a certain distance with uniform speed. If the speed of the bus would have been increased by 15 km/hr, it would have taken two hours less to cover the same distance and if the speed of the bus would have been decreased by 5 km / hr, it would have taken one hour more to cover the same distance. Find the distance covered.
  8. Some part of a journey of 555 km was completed by a car with speed 60 km/hr then the speed is increased by 15 km/hr and the journey is completed. If it takes 8 hours to reach, find the time taken and distance covered by 60 km/h speed.
  9. Durga's mother gave some 10 rupee and some 5 rupee notes to her, which amounts to Rs. 190. Durga said, "If the number of 10 rupee notes and 5 rupee notes would have been interchanged, I would have Rs. 185 in my hand. So how many notes of rupee 10 and rupee 5 were given to Durga?
  10. Sharad bought a table and a fan together for Rs. 5000. After sometime he sold the table at the gain of 25% and the fan at a gain of 20%. Thus he gained 23% on the whole. Find the cost of the fan.
  11. The weight of a bucket is 15kg, when it is filled with water 35 of its capacity while it weighs 19 kg, if it filled with 45 of its capacity. Find the weight of bucket, it is completely filled with water.
  12. A person deposits Rs. x in savings bank account at the rate of 5% per annum and Rs. y in fixed deposit at 10% per annum. At the end of one year he gets Rs. 400 as total interest. If he deposits Rs. y in saving bank account and Rs. x in fixed deposit he would get Rs. 350 as total interest. Find the total amount he deposited.
  13. A man travels by boat 36 km down a river and back in 8 hours. If the speed of his boat in still water is 12 km per hour. Find the speed of the river current.
  14. AB is a segment. The point P is on the perpendicular bisector of segment AB such that the length of AP exceeds length of AB by 7 cm. if the perimeter of ∆ ABP is 38 cm. Find the sides of ∆ ABP.
Probability
Probability

Q1. Attempt the following [each with 1 mark]
    1. In the following experiment write the sample space S, number of sample points n(S), events P, Q, n(P), and n(Q). A die is thrown: P is the event of getting an odd number. Q is the event of getting an even number.
    2. In the following experiment write the sample space S, number of sample points n(S), events P, Q, R using set and n(P), n(Q) and n(R). There are 3 red, 3white and 3 green balls in a bag. One ball is drawn at random from a bag. P is the event that the ball is red. Q is the event that the ball is not green. R is the event that ball is red or white.
    3. In each of the following experiments, write the sample space S, number of sample point n (S), events A, B and n(A), n(B). A coin is tossed three times. A is the event that head appears once, B is the event that head appears at the most twice.
    4. One card is drawn from a well – shuffled deck of 52 cards. Find the probability of getting king of red colour.
    5. In the following experiment write the sample space S, number of sample points n(S), events P, Q, R using set and n(P) , n(Q) and n(R). A die is thrown: P is the event of getting an odd number. Q is the event of getting an even number. R is the event of getting a prime number.
    6. The probability that at least one of the event A and B occurs is 0.6. If A and B occur simultaneously with probably 0.2, evaluate P(A) + P(B).
    7. If two coins are tossed then find the probability of the events. At least one tail turns up.
    8. In the following experiment, write the sample space S, number of sample point n(S), event A, B, n(A), n(B). Two coins are tossed, A is the event of getting at most one head, B is the event of getting both heads.
    9. A and B are two events on a sample space S such that P(A) = 0.8, P(B) = 0.6, P(AUB) = 0.6, find P(AnB).
    10. A box contains 3 red, 3 white and 3 green balls, A ball is selected at random. Find the probability that ball picked up is a red ball.

Q2. Attempt the following [each with 2 mark]
  1. If two coins are tossed then find the probability of the events: at least one tail turns up.
  2. One card is drawn from a well – shuffled pack of 52 cards. Find the probability of getting the jack of hearts.
  3. Two digit number are formed from the digits 0, 1, 2, 3, 4 where digits are not repeated. A is the event that the number formed is even. Write S, A , n(S) and n(A).
  4. Two coins are tossed. Find the probability of the events head appears on both the coins.
  5. In the following experiment write the sample space S, number of sample points n(S), events P, Q using set and n(P), n(Q). Form two digit number using the digits, 0, 1, 2, 3, 4, 5 without repeating the digits. P is the event that the number so formed is even. Q is the event that the number so formed is divisible by 3.
  6. In the following experiments write the sample space S, number of sample points n(S), events P, Q using set and n(P), n(Q). A coin is tossed and a die is thrown simultaneously: P is the event of getting head and a odd number. Q is the event of getting either H or T and an even number.
  7. In the following experiment write the sample space A, number of sample point n(S), event A, B, C and n(A), n(B), n(C). A die is thrown. A is the event that prime number comes up, b is the event that the number is divisible by three comes up, C is the  event that the perfect square comes up.
  8. A coin is tossed three times then find the probability of getting head on middle coin.
  9. P(A) = 3/4, P(B') = 1/3, and P(A n B) = 1/2, then find P(A U B).
  10. Sachin buys fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish. What is the probability that the fish taken out is a male fish?
Q3. Solve the following (3 marks each)
  1. There are three boys and two girls. A committee of two is to be formed, find the probability of events that the committee contains at least one girl.
  2. In the following experiment write the sample space S, number of sample points n(S), events P, Q, R using set and n(P), n(Q) and n(R). There are 3 men and 2 women. A 'Gramswachaatta Abhiyan' committee of two is to be formed. P is the event that the committee should contain at least one woman. Q is the event that the committee should contain one man and one woman. R is the event that there is not woman in the committee.
  3. One lottery ticket is drawn at random from a bag containing 20 tickets numbered from 1 to 20. Find the probability that the number on the ticket drawn is either even or square of an integer.
  4. Two fair dice are thrown, find the probability that sum of the points on their uppermost faces is a perfect square or divisible by 4.
  5. In a survey conducted among 400 students of X standard in Pune district, 187 offered to join Science faculty after X std. and 125 students offered to join Commerce faculty after X std. If a student is selected at random from this group. Find the probability that the student prefers Science or Commerce faculty.

Q4. Solve the following (4 marks each)
  1. A card is drawn at random from a well shuffled pack of cards. Find the probability that the card drawn is : a diamond card or a king.
  2. In the following experiment, write the sample space A, number of sample point n(S), events A, B, C and n(A), n(B), n(C). Also find complementary events, mutually exclusive events: Two dies are thrown, A is the event that the sum of the numbers on their upper face is at least nine, B is the event that the sum of the number on their upper face is divisible by 8, C is the event that the same number on the upper faces of both dice.
  3. One lottery ticket is drawn at random from a bag containing 20 tickets numbered from 1 to 20. Find the probability that the number on the ticket drawn is divisible by 3  or 5.
  4. What is the probability that a leap year has 53 Sundays?
  5. Three horses A, B and C are in a race, A is twice as like to win as B and B is twice as like to win as C, What are their probabilities of winning?
  6. Two dice are thrown find the probability of getting:
a. The sum of the numbers on their upper faces is divisible by 9.
b. The sum of the numbers on their upper faces is at most 3.
c. The number on the upper face of the first die is less than the number on the upper face of the second die.

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