Q. 82. Find the sum of all the natural numbers less than 100 which are divisible by 6.
Q. 83. The sum of n terms of an A.P.; be 3n2 – n and its common difference is 6,find its first and 27th term.
Q. 84. The sum of nth terms of an A.P. is 3n2 + 5n, then find its nth term.
Q. 85. If the pth term of an A.P.is q and the qth term is p, prove that its nth term is (p + q – n ) and (p+q)th term is zero.
Q. 86. If m times the mth term of an A.P. is equal to n times the nth term, show that the (m+n)th terms of an A.P is zero.
Q. 87. The 4th term of an A.P. is three times the first and 7th term exceeds twice the third term by 1.find the A.P.
Q. 88. For what value of n ,the nth terms of the arithmetic progressions 63,65,67……..and 3,10,17,…………are equal?
Q. 89. 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.
Q. 90. 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.
Q. 91. 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.
Q. 92. The mid point of the line segment joining (2a, 4) and (2, 3b) is (1, 2a +1). Find the values of a and 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.
Q. 93. 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.
Q. 94. Show that the points A (1, 2), B(S, 4), C(3, 8) and D(— 1,6) are the vertices of a square.
Q. 95. Find the co-ordinates of the point equidistant from three given points A (5, 1), B (- 3, -7) and C (7, -1).
Q. 96. Find the value of p for which the point (-1, 3), (2, p) and (5, -1) are collinear.
Q. 97. If (-2, -1); (a, 0); (4, b) and (1, 2) are the vertices of a parallelogram, find the values of a and b.
Q. 98. Prove that the points (-4, -1); (-2, -4); (4, 0) and (2, 3) are vertices of a rectangle.
Q. 99. 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).
Q. 100. Show that the points A (2, -2), B (14, 10), C (11, 13) and D (-1, 1) are the vertices of a rectangle.
Q. 101. 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”.
