Wednesday, March 11, 2009

Polynomials

Polynomials




  1. Find the zeroes of the polynomial f(x) = x2 + 7x +12 and verify the relation between its zeroes and coefficients.
  2. Find the quadratic polynomial whose zeroes are 2/3 and -1/4.
  3. Find the quadratic polynomial whose zeroes are 2 and -6.
  4. Find the quadratic polynomial, the sum and product of whose zeroes are √2 and -12 resp.
  5. Find the zeroes of the quadratic polynomial (x2 – 5) and verify the relation between its zeroes and its coefficients.
  6. Verify that 2, -3 and 1/2 are the zeroes of the cubic polynomial f(x) = 2x3 + x2 - 13x + 6 and then verify the relation between its zeroes and coefficients.
  7. Find a cubic polynomial whose zeroes are 3,5 and -2.
  8. 4x3 - 8x2 + 8x + 1 when divided by g(x) gives (2x-1) as quotient and (x+3) as remainder. Find g(x).
  9. Find all the zeroes of the polynomial f(x)=2x4 - 3x3 - 5x2 + 9x – 3, it being given that two of its zeroes are √3 and -√3.
  10. Divide 12-17x-5x2 by 3-5x and verify the division algorithm.
  11. It being given that 1 is one of the zeroes of the polynomial 7x - x3 -6. Find its other zeroes.
  12. If the polynomial x4 - 6x3 + 16x2 - 25x + 10 is divided by another polynomial x2 - 2x + k, the remainder comes out to be x + a. Find the values of k and a.
  13. If the zeroes of the polynomial x3 - 3x2 + x +1 are a-b, a and a=b, find the values of a and b.
  14. Give examples of polynomials p(x), g(x), q(x) and r(x), where p(x) is divided by g(x) and q(x) and r(x) are quotient and remainder resp. which satisfy the division algorithm and
    a. deg p(x) = deg r(x)      b. deg q(x) = deg r(x)            c. deg r(x) =0                 d. r(x) = 0
  15. If x + a is a factor of 2x3 - 3x2 + x +10, find a.
  16. If one zero of the polynomial (a2 + 9)x2 + 13x +6a is reciprocal of the other, find the value of a.
  17. If the product of the zeroes of the polynomial ax2 - 6x - 6 is 4, find the value of a.



Friday, February 27, 2009

Coordinate Geometry

Coordinate Geometry




  1. Find the distance between P(b+c, c+a) and Q(c+a, a+b).
  2. For what value of x, the distance between the points (x,2) and (3,4) be 8 units.
  3. Find the point on x-axis which is equidistant from the points (-2,5) and (2,-3).
  4. Find the point on y-axis which is equidistant from the points (-5,-2) and (3,2).
  5. If the distances of the point P(x,y) from the points A(5,1) and B(-1,5) are equal, show that 3x=2y.
  6. For what value of k, the point P(0,2) is equidistant from the points (3,k) and (k,5).
  7. Prove that the points A(a,a), B(-a,-a) and C(-√3a,√3a) are the vertices of a an equilateral triangle. Calculate the area of this triangle.
  8. Prove that A(-3,0), B(1,-3) and C(4,1) are the vertices of an isosceles right angled triangle. Find the area of this triangle.
  9. Prove that the points A(1,-3), B(13,9), C(10,12) and D(-2,0) taken in order are the angular points of a rectangle.
  10. Prove that the points A(1,2), B(5,4), C(3,8) and D(-1,6) taken in order are the angular points of a square.
  11. Show that P(2,-1), Q(3,4), R(-2,3) and S(-3,-2) are four angular points of a rhombus but not a square. Also find its area.
  12. Find the coordinates of the circumcentre of a triangle whose vertices are A(4,6), B(0,4), C(6,2). Also find its circumradius.
  13. Prove that the points A(1,1), B(-2,7) and C(3,-3) are collinear.
  14. Find the coordinates of the point which divides the line segment joining the points A(4, -3) and B(9,7) in the ratio 3:2.
  15. Find the coordinates of the points which divide the line segment joining the points (-2,0) and (0,8) in four equal parts.
  16. Find the ratio in which the point P(m,6) divides the join of A(-4,3) and B(2,8). Also find the value of m.
  17. If A and B are the points (-2,-2) and (2,-4) respectively, find the coordinates of a point P on the line segment AB such that AP=(3/7)AB.
  18. Find the coordinates of point A, where AB is a diameter of a circle whose centre is (2,-3) and B is the point (1,4).
  19. Find the ratio in which line segment joining the points (1,-5) and (-4,5) is divided by x-axis. Also find the point of division.
  20. Find the ratio in which line segment joining the points (1,3) and (2,7) is divided by the line 3x+y-9=0.
  21. If the points A(6,1), B(8,2), C(9,4) and D(x,3) are vertices of a parallelogram taken in order. Find the value of x.
  22. Find the value of p for which the points (-1,3),(2,p),(5,-1) are collinear.
  23. Three consecutive vertices of a parallelogram are (-2,-1), (1,0) and (4,3). Find the coordinates of the fourth vertex.
  24. Two vertices of a triangle are (1,2) and (3,5). If the centroid of the triangle is at origin, find the coordinates of the third vertex.

  25. Prove that the points (p,0), (0,q) and (1,1) are collinear if, 1 + 1 = 1
    p q

  26. The coordinates of mid points of the sides of a triangle are (1,1), (2,-3) and (3,4) Find the coordinates of the vertices of triangle.
  27. If the points (-1,3), (1,-1) and (5,1) are the vertices of a triangle, find the length of median through the third vertex.
  28. The line segment joining the points (3,-4) and (1,2) is trisected at the points P(p,-2) and Q(5/3,q). Find the values of p and q.

Real Numbers

Real Numbers



  1. Show that every positive even integer is of the form 4q, 4q+2 and every positive odd integer is of the form 4q+1, 4q+3.
  2. Show that one and only one out of n,n+2 and n+4 is divisible by 3, where n is any positive integer.
  3. Show that 4n can never end with digit 0.
  4. Find the HCF and LCM of 10224 and 1608 using prime factorization method and verify the answer using Euclid’s Lemma.
  5. Find the HCF of 144, 180, 192 using Euclid Algorithm.
  6. Without actual division, state whether 19/3125 is terminating or non-terminating.
  7. Prove that √3 is irrational number.
  8. Prove that √3-√5 is an irrational number.
  9. Prove that 4-2√5 is an irrational number.
  10. By Euclid’s division algorithm show that square of any positive integer is of the form 3n or 3n+1.
  11. By Euclid’s division algorithm show that cube of any positive integer is of the form 9n , 9n+1 or 9n+8.
  12. Given HCF(306,657)=9, find the lcm of (306,657).

Arithmetic Progression

Arithmetic Progressions





  1. Write the 3rd , 5th and 8th term of the sequence whose nth term is given by tn = (4n+1)/2
  2. Show that a-b, a , a+b are three consecutive terms of an A.P.
  3. If 2, 3k-1, 8 are in A.P. then what is the value of k?
  4. Write first four terms of the A.p. whose first term is -1 and common difference is -2.
  5. Find the nth term of the AP 3,5,7,9,11……. Also find its 16th term.
  6. Which term of the A.P. 5,9,13,17,….. is 81?
  7. Is 51 a term of the A.P. 5,8,11,14…….. ?
  8. The sixth term of an A.P. is -10 and its 10th term is -26. Determine the 15th term.
  9. If the 8th term of an V is 31 and its 15th term is 16 more than the 11th term, find the A.P.
  10. The 8th term of an A.P. is zero. Prove that its 38th term is triple its 18th term.
  11. Which term of the A.P. 24, 21, 18, 15…….. is the first negative term?
  12. Find three terms in A.P. whose sum is 21 and product is 231?
  13. Divide 24 in three parts such that they are in A.P. and their product is 440.
  14. If 10 times the 10th term of an A.P. is equal to the 15 times the 15th term, show that its 25th term is zero.
  15. In an A.P. prove that tm+n +tm-n = 2tm
  16. Find the 20th term from the end of the A.P. 3, 8, 13,……,253.
  17. How many 2-digit numbers are divisible by 3?
  18. How many 3-digit numbers are divisible by 7?
  19. How many numbers between 121 and 446 are divisible by 7?
  20. Which term of the A.P. 3, 15, 27….. will be 132 more than its 54th term?
  21. The fourth term of an A.P. is equal to 3 times its first term and its seventh term exceeds twice the third term by 1. Find the common difference and first term of the A.P.
  22. Find the sum 3+11+19+……+803.
  23. The angles of a triangle are in A.P. The greatest angle is twice the least. Find all the angles.
  24. If the sum of first 7 terms of an A.P. is 49 and that of 17 terms is 289, find the sum of first n terms.
  25. Find the common difference of an A.P. whose first term is 5 and sum of first four terms is half of the sum of next four terms.
  26. If Sn = 5n2 – 3, find the A.P. and also find its 20th term.
  27. Find four numbers in A.P. whose sum is 20 and sum of whose squares is 120.
  28. If p<th term of an A.P. is q and qth term is p then show that its (p+q)th term is zero.
  29. If m times the mth term of an A.P. is equal to n times the nth term and m≠n, show that its (m+n)th term is zero.
  30. If pth, qth, rth terms of an A.P. are a,b,c respectively, then show that a(q-r)+b(r-p)+c(p-q)=0.
  31. If mth term of an A.P. is 1/n and its nth term is 1/m, show that its (mn)th term is 1.




Thursday, February 12, 2009

Class X

1. State Euclid division lemma.
2. State Fundamental Theorem of Arithmetic.
3. Find the HCF of 105 and 245 by Euclid division algorithm.
4. Express 296 as a product of its primes
5. Find the HCF and LCM of 75 and 160 by Fundamental theorem of Arithmetic and verify LCM x HCF = product of two numbers
6. If HCF of 30 and 45 is 15. Find the LCM.
7. Prove 5 + 2√3 is irrational
8. Check whether 17/210 is terminating or non-terminating.
9. Find the zeros and verify the relation between zeros and
coefficients of (i) x2 + 11x + 30 (ii) x2 - 9