Sample Paper – 2008

**Class – XSubject – Mathematics**

- The vertices of a triangle are A(3,4), B(7,2) and C(-2, -5).Find the length of the median through the vertex B.
- In what ration does the point (1/2, 6) divide the line segment joining the points (3,5) and (-7,9).
- The co-ordinates of one end of the diameter of a circle are (3,5) and the co-ordinates of its center are (7,4). Find out the co-ordinates of the other end of the diameter.
- A is a point on the y-axis who’s ordinate is 5 and B is the point (3, 1). Calculate the length of AB
- The mid point of the line segment joining (2a, 4) and (2, 3b) is (1, 2a +1). Find the values of a and b.
- Prove that the points (3, 0), (6, 4) and (-1, 3) are vertices of a right-angled triangle. Also, prove that these are the vertices of an isosceles triangle.
- In what ratio is the line segment joining the points (-2, -3) and (3, 7) divided by they-axis? Also, find the coordinates of the point of division.
- A (5, -1), B (-3, -2) and C (-1, 8) are the vertices of triangle ABC, find the length of median through A and the coordinates of the centroid.
- Show that the points A (1, 2), B(S, 4), C(3, 8) and D(— 1,6) are the vertices of a square.
- Find the co-ordinates of the point equidistant from three given points A (5, 1), B (- 3, -7) and C (7, -1).
- Find the value of
*p*for which the point (-1, 3), (2, p) and (5, -1) are collinear. - If (-2, -1); (a, 0); (4, b) and (1, 2) are the vertices of a parallelogram, find the values of a and b.
- Prove that the points (-4, -1); (-2, -4); (4, 0) and (2, 3) are vertices of a rectangle.
- The vertices of a triangle are (—1, 3); (1, —1) and (5, 1). Find the lengths of medians through vertices (—1, 3) and (5, 1).
- By distance formula, show that the points (1, -1) (5, 2) and (9, 5) are collinear.

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- Show that the points A (2, -2), B (14, 10), C (11, 13) and D (-1, 1) are the vertices of a rectangle.
- Determine the ratio in which the points (6, a) divides the join of A (-3, -1) and B (-8, 9). Also find the value of “a”.
- Find the point on the x-axis which is equidistant from the points (-2, 5) and (2, -3).find the coordinates of a point which divides internally the line segment joining the points (-3,-4) and (-8, 7) in the ratio of 7:5.
- Find the value of k for which the points (2, 5), (k, 11/2) and (4, 6) are collinear.
- Find the lengths of the median of the triangle whose vertices are (1,-1), (0, 4) and (-5,3) .
- Find the co-ordinates of the points of trisection of the line segment joining the points (3, -3) and (6, 9).
- The center of a circle of radius 13 units is the point (3, 6). P (7, 9) is a point inside the circle. APB is a chord of the circle such that AP = PB. Calculate the length of AB.
- The co-ordinates of the mid-point of the line joining the points (3p, 4) and (-2, 2q) are (5, p). Find the values of p and q.
- If ‘a’ is the length of one of the sides of an equilateral triangle ABC base BC lies on x-axis and vertex B is at the origin, find the coordinates of the vertices of the triangle ABC.
- The coordinates of the mid-point of the line joining the points (2p+1, 4) and (5, q – 1) are (2p, q). Find the value of p and q.
- Find the value of K for which the points with coordinates (3, 2), (4, K) and (5, 3) are collinear.
- A is a point on the y-axis whose ordinate is 5 and B is the point (-3, 1). Calculate the length of AB.
- The distance between A (1, 3) and B(x, 7) is 5. Find the possible values of x
- Show that the points (3, 3), (9, 0) and (12, 21) are the vertices of a right angled triangle.
- The center of a circle is (1, -2) and one end of a diameter is (-3, 2), find the co-ordinates of the other end.
- Find the area of the quadrilateral whose vertices , taken in order are (-6,-2), (-3,-5) ,(4,-2) and (2,4)
- Find area of the square whose points are A(5,6) B( 1,5) C(2,1) D( 6,2).
- Determine the ratio in which the points P (m, 6) divides the join of A (-4, 3) and B (2, 8). Also find the value of m.
- Find the value of p for which the points (-1, 3), (2, p) and (5,-1) are collinear.
- Calculate the ratio in which the line joining A (6, 5) and B (4,-3) is divided by the line y=2. Also find the point of division.
- If the two opposite sides of the square are (5, 4) and (-1, 6). Find the other two.
- If the points (x,y) is equidistant from the points (2,1) and (1,-2). Show that x+3y=0.
- Show that the points (1, 1), (-1,-1) and ( form an isosceles triangle.
- Find the coordinates of third side of the triangle if its two vertices are (-1, 4) and (5, 2) and mid point of one side is (0, 3).
- Find area of triangle formed by (5, 6), (8, 9) and (2, 1).
- Show that the points (0, -1), (-2, 3), (6, 7) and (8, 3) are the vertices of a rectangle.
- The points A (0, 3), B (-2, a) and C (-1, 4) are the vertices of a right angled triangle at A, find the value of a.
- Find the value of k for which the points with coordinates (7, 5), (k, 11/2) and (2, 6) are collinear.

- Find the ratio in which the point (-3,p) divides the line segment joining the points (-5, -4) and (-2,3). Also find the value of p.(3)
- Determine the ration in which the line 3x + y – 9 = 0 divides the segment joining the pt (1,3) and (2,7).(3)
- Find the value of x if the distance between the points (x, 1) and (3,2) is 5. (3)
- If the mid points of sides of the triangle are (2,6), (6.4) and (4,2) find its vertices.(5)
- The coordinates of the mid-point of the line joining the points
*(3p, 4) and (-2, 2q) are (5, p)*. Find the values of p and q. - Find the ratio in which the line-segment joining the points (6, 4) and (1, -7) is divided by x-axis.
- Find the value of m for which the points with coordinates (3, 5), (m, 6) and are collinear.
- Find the value of k for which the points with coordinates (3, 2), (4, k) and (5, 3) are collinear.
- If the point P(x, y) is equidistant from the points A (5, 1) and B (-1, 5), prove that 3x - 2y = 0(5)
- The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point P lies on the line 2x - y + k = 0, find the value of k.(5)
- The vertices of a triangle are A(3,4), B(7,2) and C(-2, -5).Find the length of the median through the vertex B.
- The co-ordinates of one end of the diameter of a circle are (3,5) and the co-ordinates of its center are (7,4). Find out the co-ordinates of the other end of the diameter
- Prove that the points (3, 0), (6, 4) and (-1, 3) are vertices of a right-angled triangle. Also, prove that these are the vertices of an isosceles triangle.
- If (-2, -1); (a, 0); (4, b) and (1, 2) are the vertices of a parallelogram, find the values of a and b.
- If the distance
*of P (x, y) from A (6,3) and B (-3,6)*are equal prove that*3x = y.* - If the pt
*A(-2, -1) B(1, 0) C(x, 3) and D(1, y)*lie on the ends of parallelogram find the value of X and y. - Find the coordinates of point’s which trisect the line segment joining (1 , 2 ) and (-3 , 4 ).
- The area of a triangle is 5.Two of it’s vertices are (2,1) and (3, -2). The third vertex lies on y = x + 3.find the third vertex.(5)