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Answers to Your Questions - Page 12
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Question 120. There are two tangents to a cirle from same point. the distance between the point and centre is equal to the diameter of the circle. Then prove that the triange formed when a chord is made from the point of contact of tangent is equilateral by Himanshu Zharbade. Answered on Jan 13, 2007 at 7:26 p.m.
Answer 120. Download Now
Question 121. Are there geometric figures which are always similar? by Ritika. Answered on Jan 16, 2007 at 8:20 p.m.
Answer 121. Yes. Circles, equilateral triangles and all other regular polygons are always similar.
Question 122. Find the sum of all two digit numbers which when divided by 9, yield 1 as remainder, by Nishajith. Answered on Jan 20, 2008 at 8:09 pm I.S.T
Answer 122. The Numbers of this type are 10, 19, 28, 37,...,91.
                      First term = a = 10
                      common difference = d = 19 - 10  = 9
                      last term = Tn = 91
                      a + (n - 1)d = 91
                      10 + (n - 1)9 = 91
                      (n - 1)9 = 81
                       n - 1 = 9
                       n = 9 + 1 =  10 
                       Sum = S10 = n/2 (a + l)
                                       = 10/2 ( 10 + 91)
                                       = 5 (101)
                                       = 505
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Question 123. Write a linear equation which passes through x = 2 and y = 3. How many such lines are possible.by Raman. Answered on Jan 21, 2007 at 10:21 pm I.S.T.
Answer 123. x + y = 5. Infinitely many lines are possible.  
Question 124. How many tangents can be drawn to a circle from a point in its interior? by Shahnaz. Answered on Jan 25, 2008 at 10:43 pm I.S.T.
Answer 124. None. Tangents can be drawn from an exterior point or a point on the circle.
Question 125. The sum of ages of A and B  is 49 years. A said to B, " I am twice as old as you were when I was as old as you are. Find their present ages by Ravi. Answered on Jan 31, 2007 at 9:14 pm I.S.T.
Answer 125. Let age of A = x years
                     Let age of B = y years
                     Difference of their ages = (x - y) years
                     according to first condition
                     x = 2 [ y - (x - y) ]
                     x = 2 ( y - x + y )
                     x = 2 ( 2y - x )
                     x =  4y - 2x
                     3x = 4y  ..............(i)
                     according to second condition
                     x + y = 49
                     x = 49 - y ............(ii)
                     From i and ii
                     3 (49 - y) = 4y
                     147 - 3y = 4y
                     7y = 147
                      y = 21
                      Substituting in ii
                      x = 28
                      Age of A = 28 years
                      Age of A = 21 years
Question 126. Suppose you drop a dice in a rectangular region with sides 6 m and 4 m. A circle is of diameter 1 m is drawn inside it. What is probability that it will land inside the circle. by Neerja. Answered on Feb 7, 2008 at 10:38 p.m.
Answer 126. Area of rectangular region = 6 × 4 
                                                               = 24 square metre
                     Area of circle = pie/4  square metre
                     Required probability = (pie/4)/24
                                                     = pie/96
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Question 127. prove that the area of triangle whose coordinates are (t,t-2),(t+23,t+2)and (t+3,t) is independent of t by Puneet Singh. Answered on Feb 10, 2007 at 11:11 pm
Answer 127. Area of triangle = 1/2 | t(t + 2 - t) + (t + 23)(t - t + 2) + (t + 3)(t - 2 - t - 2)|
                                                = 1/2 | 2t + 2t + 46 - 4t - 12|
                                                = 1/2 |34|
                                                = 17 square units
                                               So we see area of triangle does not contain t and hence is independent of t
Question 128. determine the ratio in which the line 2x+y-4=0 divides the line segment joining the points A(2,-2) and B(3,7) by Sukhmani Lamba. Answered on Feb 12, 2007 at 9:12 p.m.
Answer 128. Let the coordinates of point of intersection be (x,y) and let it divide the line segment AB in ratio
                               k:1.                       
                               So x = (2 + 3k)/(k+1)  and y = (- 2 + 7k)/(k+1)
                               Putting these values in the equation of given line we get
                              2(2 + 3k)/(k+1)  +  (- 2 + 7k)/(k+1)  -  4 = 0
                              (4 + 6k)/(k+1)  +  (- 2 + 7k)/(k+1)  -  4 = 0
                              (4 + 6k - 2 + 7k)/(k+1)   =  4
                              (2 + 13k)/(k+1)   =  4
                               2 + 13k = 4k + 4
                              9k = 2
                              k = 2/9
                              Required ratio = 2:9
Question 129. (x-2) and (x-1/2) are the factorsof the polynomials qx^2+5x+r prove that q=r, by Arushi Gautam. Answered on Feb 14, 2007 at 9:37 pm I.S.T. 
Answer 129. Let f(x) = qx^2+5x+r 
                                Since x - 2 is a factor therefore f(2) = 0 by factor theorem
                                q(2^2) + 5(2) + r = 0
                                4q + r = - 10   ........(i)
                                Since x - 1/2 is a factor therefore f(1/2) = 0 by factor theorem
                                q(1/2)^2 + 5(1/2) + r = 0
                                q/4 + r = - 5/2  .....(ii)
                                     (i) - (ii)
                                 4q - q/4 = - 10 + 5/2
            (×4)             16q - q  = - 40 + 10
                                 15 q = - 30
                                  q = - 2
                                  Substituting in (i)
                                  4(- 2) + r = - 10
                                   r = - 10 + 8
                                   r = - 2
                                   Therefore p = q = -2
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