- Find the distance between P(b+c, c+a) and Q(c+a, a+b).
- For what value of x, the distance between the points (x,2) and (3,4) be 8 units.
- Find the point on x-axis which is equidistant from the points (-2,5) and (2,-3).
- Find the point on y-axis which is equidistant from the points (-5,-2) and (3,2).
- 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.
- For what value of k, the point P(0,2) is equidistant from the points (3,k) and (k,5).
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Prove that the points A(1,1), B(-2,7) and C(3,-3) are collinear.
- 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.
- Find the coordinates of the points which divide the line segment joining the points (-2,0) and (0,8) in four equal parts.
- 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.
- 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.
- 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).
- 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.
- Find the ratio in which line segment joining the points (1,3) and (2,7) is divided by the line 3x+y-9=0.
- 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.
- Find the value of p for which the points (-1,3),(2,p),(5,-1) are collinear.
- Three consecutive vertices of a parallelogram are (-2,-1), (1,0) and (4,3). Find the coordinates of the fourth vertex.
- 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.
- Prove that the points (p,0), (0,q) and (1,1) are collinear if, 1 + 1 = 1
p q - 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.
- 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.
- 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.
Friday, February 27, 2009
Coordinate Geometry
Real Numbers
- 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.
- Show that one and only one out of n,n+2 and n+4 is divisible by 3, where n is any positive integer.
- Show that 4n can never end with digit 0.
- Find the HCF and LCM of 10224 and 1608 using prime factorization method and verify the answer using Euclid’s Lemma.
- Find the HCF of 144, 180, 192 using Euclid Algorithm.
- Without actual division, state whether 19/3125 is terminating or non-terminating.
- Prove that √3 is irrational number.
- Prove that √3-√5 is an irrational number.
- Prove that 4-2√5 is an irrational number.
- By Euclid’s division algorithm show that square of any positive integer is of the form 3n or 3n+1.
- By Euclid’s division algorithm show that cube of any positive integer is of the form 9n , 9n+1 or 9n+8.
- Given HCF(306,657)=9, find the lcm of (306,657).
Arithmetic Progression
- Write the 3rd , 5th and 8th term of the sequence whose nth term is given by tn = (4n+1)/2
- Show that a-b, a , a+b are three consecutive terms of an A.P.
- If 2, 3k-1, 8 are in A.P. then what is the value of k?
- Write first four terms of the A.p. whose first term is -1 and common difference is -2.
- Find the nth term of the AP 3,5,7,9,11……. Also find its 16th term.
- Which term of the A.P. 5,9,13,17,….. is 81?
- Is 51 a term of the A.P. 5,8,11,14…….. ?
- The sixth term of an A.P. is -10 and its 10th term is -26. Determine the 15th term.
- If the 8th term of an V is 31 and its 15th term is 16 more than the 11th term, find the A.P.
- The 8th term of an A.P. is zero. Prove that its 38th term is triple its 18th term.
- Which term of the A.P. 24, 21, 18, 15…….. is the first negative term?
- Find three terms in A.P. whose sum is 21 and product is 231?
- Divide 24 in three parts such that they are in A.P. and their product is 440.
- 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.
- In an A.P. prove that tm+n +tm-n = 2tm
- Find the 20th term from the end of the A.P. 3, 8, 13,……,253.
- How many 2-digit numbers are divisible by 3?
- How many 3-digit numbers are divisible by 7?
- How many numbers between 121 and 446 are divisible by 7?
- Which term of the A.P. 3, 15, 27….. will be 132 more than its 54th term?
- 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.
- Find the sum 3+11+19+……+803.
- The angles of a triangle are in A.P. The greatest angle is twice the least. Find all the angles.
- 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.
- 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.
- If Sn = 5n2 – 3, find the A.P. and also find its 20th term.
- Find four numbers in A.P. whose sum is 20 and sum of whose squares is 120.
- 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.
- 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.
- 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.
- 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
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
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