Holidays Homework - X

Probability, Coordinate Geometry & Polynomials

1.       One card is drawn from a well shuffled pack of 52 playing cards. Find the probability that the card drawn is                                                                                                                        i) a non face card                                                                                                                           ii)         Neither a red card nor a queen

iii)        A king, a queen, a jack or an ace
iv)        A heart or a queen
v)         A king or a jack
vi)        Neither a king nor a queen
vii)       Either a king or a queen
viii)      A queen and a card of red order
ix)        A queen or a card of black order
x)         A black face card and a king
2.      Balls marked with 7 to 37 are put in a box. One ball is taken out randomly, find the probability that the ball taken out is                                                                                                          i)   A multiple of 2 and 3
ii)         A multiple of 2 or 3
iii)        A prime number
iv)        A multiple of 3 and odd number
v)         A perfect square 
3.      A number is chosen randomly from first 50 natural numbers, find the number chosen is divisible by 4 or 5
4.      A pair of dice is thrown simultaneously, find the probability of getting                                 i)          sum as 7                                                                                                                                  ii)         sum divisible by 3
iii)        Even on one die and prime on another
5.      The probability of selecting a green marble at random from a jar which contains only green, white and red marbles is ¼. The probability of selecting a white marble at random from the same jar is 1/3. If this jar contains 10 yellow marbles, what is the total number of marbles in the jar?
6.      Find the value of k for which the points (2,5), ( k,11/2) and (4,6) collinear
7.      Two opposite angular points of a square are (-1, 2) and (3,-2). Find the coordinates of other two angular points
8.      An equilateral triangle has two vertices at (3, 4) and (-2, 3). Find the coordinates of the third vertex
9.      Find the coordinates of the centre of a circle passing through the points (5,7), (6,6)               and (2,-2)
10.  If (10, 5), (8, 4) and (6, 6) are the mid points of the sides of a triangle, find the coordinates of its vertices. Hence find its area
11.  Find the ratio in which the point P (-6, a) divides the join of A (-3,-1) and B(-8,9). Also find the value of a
12.  The line segment joining the points (12, 0) and (-6, 15) is trisected at the points X (p, 5) and Y (0, q). Find the values of p and q
13.  Determine the ratio in which the line x – y – 2  = 0 divides the line joining (3,-1) and (8,9)
14.  Prove that ( a , b+c ),( b, c+a ) and ( c, a+b ) are collinear
15.    Find the circumcentre of a triangle whose vertices are (-2,-3), (-1, 0) and (7,-6). Also find its circumradius
16.  A (7,0), B (0,-24) are two vertices of a triangle whose third vertex is at origin. Calculate the length of the hypotenuse
17.    If A (-2,5), B(-1,3) and C(x,y) form an isosceles triangle, show that                                            6x – 16y + 19 = 0
18.  Find the condition that the point (x,y) should be equidistant from (2, 3) and (-1,2)
19.  If the distance between (11,3) and (3,y) is 10 units, find y
20.  Find a quadratic polynomial whose zeros are p + 2q and p – 2q
21.  If the sum of the zeros of the polynomial px² – 4x + 2p is same as their product, find the value of p
22.  Form a cubic polynomial whose zeros are 2, - 2 and 3
23.  If the product of two zeros of the polynomial 2x³ +6x² – 4x +9 is 3, find the third zero
24.  Obtain all other zeros of the polynomial x⁴ – 2x³ – 5x² + 8x + 4 if two of its zeros are                    1 ± √2
25.  Find the zeros of the polynomial x³ – 5x² – 16x + 80, if its two zeros are equal in magnitude but opposite in sign.