3-DIMENSION (XI)

              3- DIMENSIONS

                       DISTANCE 

1) Find the distance from the origin to each of the points

A) (2,2,3).                                   √17

B) (4,-1,2).                                  √21

C) (0, 4, -4).                                 4√2

D) (-4, -3, -2).                               √29

2) Find the distance between each of the following pairs of points:

A) (2,5,3) and (-3,2,1).               √38

B) (0,3,0) and (6,0,2).                    7

C) (-4, -2,0) and (3,3,5).              3√11

3) Show that the triangle with vertices (6,10,10), (1,0,-1),(6,-10,0) is a right angled triangle, and find its area.                      25√21sq,unit

4) Show that the triangle with vertices A(3,5-4), B(-1,1,2), C(-5,-5,-2) is isosceles.

5) Show that (4,2,4), (10,2-2) and (2,0,-4) are the vertices of an equilateral triangle.

6) Show that the points (1,-1,3), (2,-4,5) and (5,-13,11) are collinear.

7) Derive the equation of the locus of a point equidistant from the point (1,-2,3) and (-3,4,2).                 8x-12y+2z+15= 0

8) Derive the equation of the locus of a point twice as far from (-2,3,4) as from (3,-1,-2). 3x²+3y²+3z²-28x+14y+24z+27=0

9) Find the equation of the locus of a point whose distance from the Y-axis is equal to its distance from (2,1,-1).                   y²-2y-4x+2z+6=0

10) Find the equation of the locus of a point whose distance from xy-plane is equals to its distance from the point (-1,2,-3).                                           x²+y²+2x- 4y+ 6z+14= 0

11) A point moves so that the difference of the squares of its distances from the x-axis and the y-axis is constant. Find the equation of locus.                             y²- x² = a

12) Find the equation of the locus of a point whose distance from the z-axis is equal to its distance from the xy plane.                  x²+y²+z²= 0

  DIVISION or SECTION Formula

1) Find the coordinates of the points which divide the join of the points (2,-1,3) and (4,3,1) in the ratio 3:4 internally.  (20/7,5/7,15/7)

2) Find the Coordinates of the points which the line joining the points (2,-4,3), (-4,5,-6) in the ratio

A) 1:-4                                 (4,-7,6)

B) 2:1                                   (-2,2,-3)

3) Find the ratio in which the line joining the points (2,4,5),(3,5,-4) is divided by the yz plane.          -2:3

4) The three points A(0,0,0), B(2,-3,3), C (-2,3,-3) are collinear. Find in which ratio each point divides the segment joining the other two.            AB/BC= -1/2, BC/CA= -2/1, CA/AB=1/1.

5) Find the coordinates of the point which trisect AB given that (2,1,-3) and B(5,-8,3). (3,-2,1),(4,-5,1)

6) Find the coordinates of the point which is three-fifth of the way from (3,4,5) to (-2,-1,0).                 (0,1,2)

7) Show that the point (1,-1,2) is common to the lines which join (6,-7,0) to (16,-19 ,-4) and (0,3,-6) to (2,-5,10).          

8) Find the lengths of the medians of the triangle whose vertices are A(2,-3,1), B(-6,5,3), C(8,7,-7).          √91,√166,√217

9) Find the point of intersection of the medians of the triangle with vertices (-1,-3,-4), (4,-2,-7),(2,3,-8).              (5/3,-2/3,-19/3)

10) Find the ratio in which the join of (2,1,5) and (3,4,3) is divided by the plane 2x+2y-2z=1, also, find the coordinates of the point of division.           5:7,(29/12,9/4,25/6)

11) The midpoints of the sides of a triangle (1,5,0),+0,4,-2) and (2,3,4). find its vertices.           (1,2,3),(3,4,5)(-1,6,-7)

12) Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4) and C(-1,1,2). Find the Coordinates of the fourth verex D.                                                (1,-2,8)

13) What is the locus of a point for which

A) x= 0.                             yz- plane

B) y= 0.                             yz- plane

C) z= 0.                             xy- plane

D) x= a.                          The plane parallel to the yz- plane at a distance a unit from it.

E) y= b.                            The plane parallel to the xz plane at a distance b units from it.                  

F) z= c. The plane parallel to the xy plane at a distance c units from it.

14) What is the locus of a point for which.

A) x= 0, y= 0.                  The z-axis

B) y= 0, z=0.                   The x-axis

C) z= 0, x= 0.                  The y-axis

D) x= a, y= b.                  The line of intersection of the given planes x=a and y=b

E) y= b, z=c.                    The line of intersection of the given planes y=b and z=c

F) z= c, x= a.                 The line of intersection of the given planes z=c and x=a

15) Find the ratio in which the xy-plane divides the join of (-3,4,-8) and (5,-6,4). Also obtain the point of intersection of this line with the plane.                      2:1, (7/3,-8/3,0)

16) Given that P(3,2-4), Q(5,4,-6), R(9,8,-10) are collinear, find the ratio in which Q divides PR.      (38/16,57/16,17/16)

17) Using section formula, prove that the three points A(-2,3,5), B(1,2,3) and C(7,0,-1) are collinear.

18) Show that the point (1,2,3) is common to the lines which join A(4,8,12) to B(2,4,6) and (3,5,4) to D(5,8,5)

ANGLES, DIRECTION RATIOS, COSINES.....

1) The direction ratios of a line are 1, -2, -2. What are their direction cosines ?                     1/3,-2/3,-2/3

2) If k, l, m are angles which a line makes with the axes, prove that sin²k+ sin²l+ sin²m= 2.

3) Can a line have direction and angles 45°,60°, 120° ?         

4) Prove that 1, 1,1 cannot be direction cosines of a straight line.

5) Find the direction cosines and direction ratios of the line joining the points:

A) A(0,0 ,0), B(4,8,-8).

                              1/3,2/3,-2/3; 2, -2

B) A(1,3,5 ), B(-1,0-1).

                            -2/7, -3/7,-6/7; 2,3,6

C) A(5,6,-3), B(1,-6,3).

                            -2/7,-6/7,3/7; 2,6,-3

D) A(4,2,-6), B(-2,1,3).

    6/√118, 1/√118, -9/√118; 6,1, -9

6) By using direction ratios method, show that following set of points are collinear.

A) A(1,2,3), B(4,0,4) and C(-2,4,2)

B) (-2,4,7),(3,-6,-8), (1,-2,-2)

7) A line makes an angle of π/4 with each of the x-axis and the Y-Axes. Find the angle made by it with the z-axis.                                    π/2

8) If the line OP makes with x- axis an angle of measure 120° and with y-axis an angle of measure 60°. Find the angle made by the line with the Z-axis.                        45° or 135°

9) Find the angle between the vectors whose direction cosines are proportional to 2,3,-6 and 3,-4,5. 

                                cos⁻¹{(18√2)/35}

10) If k, l, m are the angles that line makes with the axes, then find cos m if 

A) cos k= 14/15, cos l= -1/3.  ±2/15

B) k= 60°, l,= 135°                      ±1/2

11) If the coordinates of A and B be (2,3,4 and (1,-2,1) respectively. prove that OA is perpendicular to OB, where O is the origin.

12) Show that the join of the points (1,2,3), (4,5,7) is parallel to the join the join the points (-4,3,-6) and (2,9,2).      

13) Find the angles between the lines whose direction ratios are

A) 5,-12,13; -3,4,5                                                                cos⁻¹(1/65)= 89°6'

B) 1,1,2; √3-1,-√3-1,4.                π/3

14) If P,Q,R are respectively (2,3,5), (-1,3,2) and (3,5,-2). find the direction cosines of the sides of the of the sides of the of the triangle PQR.                                -2/3,-1/3,2/3; 1/3√6, 2/3√6, -7/3√6,; 1/√2, 0, 1/√2

15) Prove that three points P, Q ,R, whose coordinates are respectively (3,2,-4),(5,4,-6) and (9,8,-10) are collinear and find the ratio in which in which Q divides PR.                1:2

16) Find the angle not greater than 90° between the lines joining the following pairs of points:

A) (8,2,0),(4,6,-7), and (-3,1,2),

                                   (-9,-2,4); 88°11'

B) (4,-2,3),(6,1,7) and (4,-2,3),(5,4,-2).                                         90°

C) (3,1,-2),(4,0,-4), and (4,-3,3),(6,-2,2).                                        π/3

17) Find the direction cosines of a line which is perpendicular to the lines with directions cosines proportional to 1, -2, -2; 0, 2, 1.

                                     2/3, -1/3, 2/3

18) Find the direction ratios of a perpendicular to the two lines determined by the pairs of points (2,3,-4),(-3,3,-2), and (-1,4,2),(3,5,1).                                                -2,3,-5

19) For what value of x will the line through (4,1,2),(5,x,0) be parallel to the line through (2,1,1) and (3,3,-1). 

                                                 x= 3

20) For what value of x the lines in the above(19) Problem be perpendicular?                            -3/2

21) show that the points (4,7,8), (2,3,4),(-1,-2,1) and (1,2,5) are the vertices of a Parallelogram.

22) Show that the points (5,-1,1),(7,-4,7),(1,-6,10) and (-1,-3,4) are the vertices of a rhombus.

23) Find the foot of the perpendicular drawn from the point A(1,0,3) to the join of the points B (4,7,1) and C(3,5,3) .     5/3,7/3,17/3

24) A(1,0,4) and B(0,-11,3), C(2,-3,1) are three points and D is the foot of the perpendicular from A on BC. Find the co-ordinate of D. 

                                22/9,-11/9,5/9

25) Calculate the cosine of the angle A of the triangle with the vertices A(1,-1,2),B(6,11, 2), C(1, 2,6).                                         36/65

26) if A,B,C,D are the points (6,-6,0),(1,-7,6),(3,-4,4),(2,-9,2) respectively, prove that AB is perpendicular to CD.

27) Find the angle between any two diagonals of a cube.         Cosk=1/3

MISCELLANEOUS QUESTIONS

1) If a line makes angles 90°,135°, 45° with the positive x,y and z Axis respectively. find its direction cosines.                          0,-1/√2,1/√2

2) If a line has the direction ratios -18,12,-4, then what are the direction cosines?                                                                  -9/11,6/11,-2/11

3) Find the distance between the points (7,4,-5) and (1,6,-2) show that these two points are collinear with the point (-5,8,1).                7 units

4) Find the ratio in which the join of (-3,5,6) and (4,6,-5) is divided by the yz-plane.                                      3:4

5) Show that the line joining the points A(7,4,2) and B(3,-2,5) is parallel to the line joining the points to C(2,-3,5) and D(-6,-15,11).        

6) Find the angle between the lines whose direction ratios are 2,3,6 and 1,2,2.                          Cosk= 20/21

7) If A(6,-6,0), B(-1,-7,6), C(3,-4,4) and D(2,-9,2) be 4 points in space, show that AB perpendicular to CD.

8) Show that the point (0,7,10),(-1,6,6) and (4,9,6) form an isosceles right-angles triangle.

   (Hints: show that angle between two sides = 90° by using the formula Cosk = l'.l"+ m'.m" + n'.n". Also, show that two angles are equal.)

SECTION FORMULA A- Z

           SECTION FORMULA

EXERCISE -A

1) Calculate the coordinates of the point P which divides the line segment joining

a) A(1,3) and B(5,9) in the ratio 1: 2.   (7/3, 5)

b) A(-4,6) and B(3, -5) in the ratio 3: 2.                 (1/5, -3/5)

c) A(0, -3) and B(4, -1) in the ratio 1: 2.                 (4/3, -7/3)

d) A(1, -3) and B(-5,9) in the ratio 2: 15.                 (5/17, -27/17)

e) (8, 9) and B(-7,4) in the ratio 

     i) internally 2:3.                  (2,7)

     ii) externally 4:3              (-52, -11)

f)  (-4,4) and B(1,7) in the ratio externally 2:1          (6,10) or (-9,1)

g) (3,4) and B(-6,2) in the ratio externally 3:2          (-24,-2) or (21,8)

h) (1,-2) and B(4,7) in the ratio internally 2:1          (2,1) or (3,4)

i) Calculate the coordinates of the point P which divides the line joining A(-1,3) and B(5,9) in the ratio 1: 2.               (1, 5)

j) Find the coordinates of the point which divides the the line segment PQ joining the points P(6,4) and Q(7,-5), in the ratio 3:2. (33/5, -7/5).

k) Find the coordinates of the point which divides the line segment joining the points (9,5) and (-7, -3), in the ratio 5:3. (-1,0)

l) Write down the coordinates of the point which divides the segment joining the points (1, -2) and (6,8) in the ratio 2:3. (3,2)

EXERCISE - B

1)a) In what ratio is the line joining (2, -3) and (5,6) divided by x-axis.                 1:2

b) In what ratio does the join of (4, 3) and (2, -6) divided by x-axis . Also find the coordinates of the point of intersection.                           1:2, (10/3,0)  

c) The line segment joining A(2,3) and B(6,-5) is intersected by x-axis at point K. Write down the ordinate of K. Hence, find the ratio in which K divides AB.          0, 3:5   

2) a) In what ratio is the line joining (2, -4) and (-3,6) divided by y-axis.                2:3

b) In what ratio does the join of (-4, 7) and (3, 0) divided by y-axis . Also find the coordinates of the point of intersection.                   4:3, (0,3)  

c) The line segment joining M(5,7) and N(-3,2) is intersected by y-axis at point L. Write down the abscissa of L. Hence, find the ratio in which L divides MN. Also, find the coordinates of L.      0, 5:3, (0, 31/8)

d) In what ratio is the line joining the points (4,5) and (1,2) divided by the y-axis. Find also the coordinates of point of division.            4:1, (0,1)

e) Find the ratio in which the point (-2,2) divides the line segment joining the points (-4,6) & (1/2, -3).                                                 4:5

f) find the ratio in a which the point (-5,-20) divides line segment joining the point (4,7) and (1,-2). 3:2 ext

g) Find the ratio in which the point (-1,0) divides the line segment joining the points (-7,-3) and (9,5). 3:5

3) The line joining P(-4,5) and Q(3,2) intersects the y-axis at R. PM and QN are perpendiculars from P and Q on the x-axis. Find

i) the ratio PR: RQ.                      4:3

ii) the coordinates of R.       (0,23/7)

4) Find the ratio in which the line joining A(6,5) and B (4,-3) is divided by the line y= 2.               

5)a) The line joining the points A(-3, -10) and B(-2, 6) is the divided by the point P such that PB/AB = 1/5. Find the coordinates of P.                 (-11/5, 14/5)

b) P is the point on the line joining A(4,3) and B(-2,6) such that 5AP= 2BP. Find the coordinates of P.     (16/7, 27/7)

c) Given, two fixed points A(0, 10) and B(-30, 0). Calculate the coordinates of a point P which lies in AB such that:

 i) 2AP= 3PB.                         (-18,4)

ii) 3AP= AB.                     (-10, 20/3)

iii) 7PB = AB.              (-180/7, 10/7)

d) Given, two fixed points P(-3, 4) and Q(5, -2). Calculate the coordinates of a point A and B in PQ such that:

 i) 5AP= 3PQ                  (9/5,2/5)

ii) 5PB= 2PQ.                   (7/3, 0)

e) Find a point on the line through A(5,-4) and B(-3,2) that is twice as far from A as from B.         (-1/3,0), (-11,8)

6) if the point (9,2) divide the line segment joining the points P(6,8) and Q (x,y) in the ratio 3:7, find the coordinates of Q. (16,-12)

7) if the point (6,3) divides the segment of the line from (4,5) to Q(x,y) in the ratio 2:5, find the coordinates (x,y) of Q. What are the coordinates of the midpoint of PQ. (11,-2), (15/2,3/2)

8) Find the coordinates of the point which divides the join of (2,3) and (5,3) internally in the ratio 1:2 .                                                       (3,1)

9) in what ratio is the line joining the points (4,5) and (1,2) divided by the Y-axis. Find also the coordinates of the point of division.                         4:1 ext. (0,1)

EXERCISE - C

1)a) In what ratio does the point P(3,3) divide the join of A(1, 4) and B(7, 1) ?           1:2

b) In what ratio does the point (1,a) divide the join of (-1, 4) and B(4, -1) ? Also find the value of a.         2:3, 2

c) In what ratio does the point (a,6) divide the join of A(-4, 3) and B(2, 8) ? Also find the value of a?     3:2, -2/5

d) A point (-4,1) divides the line joining A(2,-2) & B in the ratio 3:5. Find the point B.                   (-14,6)  

e) In what ratio does the point (2,-5) divide the line joining the points (-3,5) and (4, -9).                        5:2

  

EXERCISE - D             

1) a) Find the coordinates of the points of trisection of the line joining the points (-3,0) and (6,6).       (0,2),(3,4)

b) Show that the line segment joining the point (-5,8) and (10, -4) is trisected by coordinates axes.        

c) Show that A(3, -2) is a point of trisection of the line segment joining the points (2,1) and (5, -8). Also, find the coordinates of other point of trisection.          (4, -5)  

d) The line joining the points (3,-1) and (-6,5) is trisected. Find the coordinates of the Points of trisection.                         (0,1),(-3,3)

e) The line joining the points (2,3) and (6,5) is trisected. Find the coordinates of the Points of trisection.       (10/3,11/3),(14/3,13/3)

f) Find the coordinates of the point trisection of the line segment joining the points P(-2,3) and Q(3, -1) that is nearest to P.      (-1/3, 5/3)

g) A line segment directed from (-3,2) to (1,-4) is trebled. Find the coordinates of the terminal point. (9, -16)

8) The line segment joining the point (2, -2) and (4,6) is extended each way a distance equal to half of its own length; Find the co-ordinate of its terminal points. (5,10),(1, -6)

2) Calculate the coordinates of points which divide the join of (8,6) and (2,3) into four equal parts.       (13/2,21/4),(5,9/2),(7/2,15/4)

3)  A(2,5), B(-1,2) , C(5,8) are the coordinates of the vertices of the triangle ABC. Points P and Q lie on AB and AC respectively, such that AP: PB = AQ: QC = 1: 2. Find

i) coordinates of P and Q.     (1,4)

ii) Show that PQ= BC/3.

 

EXERCISE - E

1) Find the mid point of the line segment joining the points:

a) (-6, 7) and (3, 5).               (-3/2,6)

b) (5, 3) and (-1,7).                      (2,2)

c) (3, 1) and (3,-3).                     (3,-1)

d) (8, -7) and (-4, 3).                   (2,-2)

2)a) Points A and B have coordinates (3,5) and (x, y) respectively. The mid point of AB is (2, 3). Find the values of x & y.     1,1

b) The midpoint of the line A(2,p) and B(q,4) is (3,5). Calculate the numerical values of p and q.        6,4

c) If R (8,17) be the midpoint of the line segment joining the points P (5, -3) and Q(x,y), find the coordinates of Q.                     (21,37)

d) Midpoint of a line is (-4,-2) and one end of the line is (-6,4). Find the coordinates of the other end. (-2,-8)

3) A(5, 3), B(-1, 1) and C(7,-3) are the vertices of triangle ABC. If L is the midpoint of AB and M is the midpoint of AC, Show that LM= 1/2 BC.

4) A line meets x-axis at P and y-axis at Q. If the coordinates of the midpoint of PQ are (-2,3); find the co-ordinates of P and Q. Also, find the length of PQ.    (-4,0),(0,6) 2√13

5) Given M is the midpoint of AB, find the coordinates of :

a) A; if M(1,7) B(-5,10).             (7,4)

b) B; if A(3,-1) M(-1,3).              (-5,7)

6) (-5,2),(3, -6) and (7,4) are the vertices of a triangle. Find the lengths of all its medians.   10, √109 , √85

7)a) One end of the diameter of a circle is (-2,5). Find the coordinates of the other end of it, if the to centre of the circle is (2,-1).                 (6,-7)

b) Midpoint of a line is (-4, -2) and one end of the line is (-6, 4). Find the coordinates of the other end.   (-2,-8)

c) The centre of a circle is (3,4) and one end of the diameter is (6,8), find the other end.       (0,0)

EXERCISE - F

1) A(2,5), B(1,0), C(-4,3) and D(-3,8) are the vertices of quadrilateral ABCD. Find the co-ordinates of the mid points of AC and BD.    

Give a special name to the quadrilateral.        (-1,4) parallelogram

2) P(4,2) and Q(-1,5) are the vertices of parallelogram PQRS and (-3,2) are the coordinates of the point of intersection of its diagonals, Find the co-ordinate of R and S.                           (-10,2),(-5,-1)

3)a) A(-1,0), B(1,3) and D(3,5)are the vertices of a parallelogram ABCD. Find the coordinates of vertex C.   (5,8)

b) The vertices of a parallelogram are (0,0),(a,0) and (b,c). Find the coordinates of the fourth vertex.     (b - a, c)

4) The points (2, -1),(-1,4) and (-2,2) are midpoints of the sides of a triangle. Find its vertices.    (1,-3),(3,1),(-5,7)

5) Points A(-5, x), B(y,7) and C(1,-3) are collinear (i e. lie on the same straight line) such that AB = BC. Calculate the values of x and y.         17, -2

6) Points P(a, -4), Q(-2, b) and R(0, 2) are collinear, such that PR= 2QR. Calculate the vertices of a and b.   -4, -1

7) Co-ordinates of A and B are (-3, a) and (1, a+4). The midpoint of AB is (-1,1). Find the value of a.      -1

8) The vertices of a parallelogram are (0,0), (a,0) and (b,c). Find the coordinates of the fourth vertex. (b-a,c)

EXERCISE - G

1)a)  Calculate the coordinates of the centroid of the triangle ABC, if A=( 7, -2), B(0, 1) and C= (-1,4).     (2,1)

b) If (2,-8),(14,-3) and (-10,8) are the vertices of a triangle, find its centroid.           (2, -1)

3) The Co-ordinates of the centroid of a triangle PQR are (2,-5) . If Q= (-6,5) and R=(11,8); calculate the coordinates of vertex P.   (1,-28)

4)a) A(5, x), B(-4,3) and C(y, -2) are the vertices of the triangle ABC whose centroid is the origin. Calculate the values of x and y.     -1

b) Find the third vertex of triangle if two of its vertices are at (-1,4) and (5,2) and the centroid at (0,-3).      (-4, -15)

c) Vertices of a triangle are (2,1), (5,2) and (3,4). Find the coordinates of the centroid and the circumcentre.       (10/3,7/3),(13/4, 9/5)

d) Show that the centroid of the triangle formed by the points (a, b- c), (b, c -a), (c, a- b) lies on x-axis.   

5) Vertices of a triangle are (2,1), (5,2) and (3,4). Find the co-ordinates of the centroid and circumcentre.   (10/3,7/3); (13/4,9/5)

EXERCISE - H

1)a)  Find the vertices of the triangle of coordinates of the midpoints of whose sides are (0,1/2),(1/2,1/2) and (1/2,0).           (1,0),(0,0),(0,1)

b) Find the vertices of the triangle of coordinates of the midpoints of whose sides are (3,4),(-3, -1) and (9, -4).                        (3,-9),(15,1),(-9,7)

c) Determine the co-ordinates of the middle points of the sides of the triangle whose vertices have co-ordinates (3,2),(-1, -2) and (-5, -4) (1,0),(-3,-3), (-1,-1)

2) Find the third vertex of a triangle if two of its vertices are at (4,-6) and (2,-2) and the median meet at (8/3,-1).                         (2,5)

EXERCISE - I

1) Show that the point (-2, -1) is equidistance from the point (4,6)  and (3,7).

2) Show that the points (0,2),(4,1) and (16,-2) lie in a straight line.

3) show that the points (-4,0), (6,3) and (36, 12) lie in a straight line.

4) Find the coordinates of the point equidistant from (2,6), (-2, 2) and (-5,-1).           (-2,3)

5) Prove that midpoint of the line segment joining the points  (2,1) and (6,5) lies on the line joining the points (-4, -5) and (9,8).            

6) Show that the points A(8,12), B(-2,7) and C(2,9) lie on a straight line. Also find the ratio in which the line segment AB is divided at C.                                           3:2

MISCELLANEOUS - 1

1) Three points A, B and P have  coordinates (a,b), (c,d) and [a+k(c-a), (b + k(c - b)] respectively. Show that for all values k, P lies on AB and also determine the ratio in which P divides AB.              k: 1-k

2) Find the vertices of the triangle of coordinates of the midpoints of whose sides are (0, 1/2), (1/2, 1/2) and (1/2,0).                (1,0),(0,0),(0,1)

3) The centre of a circle is (3,4) and one end of the diameter is (6,8), find the other end.            (0,0)

4) A point (-4,1) divides the line joining A(2, -2) and B in the ratio 3:5. Find the point B.              (-14,6)

5) In what ratio does the point (2,-5) divide the line joining the points (-3,5) and (4,-9)?                 5:2

6) The line joining the points (3,-1) and (-6,5) is trisected. Find the coordinates of the point of trisection.                  (0,1) and (-3,3)

7) If (3,4), (-3,-1) and (9,-4) are the middle points of the sides of a triangle, what are the coordinates of its vertices?                                                               (3,-9),(15,1) and (-9,7)

8) Find the point on the line through A(5,-4) and B(-3,2) that is twice as far from A as from B.                                     (-1/3,0) and (-11,8)

9) If (2,-8), (14,-3) and (-10,8) are the vertices of a triangle, find its centroid.                                    (2, -1)

10) Find the third vertex of a triangle if two of its vertices are at (-1,4) and (5,2) and the centroid at (0, -3).                                       (-4,-15)

11) Find the third vertex of a triangle if two of its vertices are at (4,-6) and (2, -2) and the median meet at (8/3, -1).                        (2,5)

12) Show that the centroid of the triangle formed by the points (a, b-c), (b, c-a), (c, a-b) lies on x axis.

13) Find the coordinates of the centre of the circle inscribed in the triangle whose vertices are (4,-2), (-2,4) and (5,5).                   (5/2,5/2)

14) If A(5, -1), B(-1,7), C(1, 2) are the vertices of a triangle, find the length of the internal bisectors of angle A .                                            (14√2)/√3

PROBABILITY For XII

              PROBABILITY

               **************

1) A bag contains 4 white 2 black balls. Two balls are drawn at random, find the probability that they are white.                            2/5

2) An urn contains 9 balls, 2 of which are white, 3 blue and 4 black. 3 balls are drawn at random from the URN. What is the chance that 

i) the three balls will be different colours?                                     2/7

ii) 2 balls will be of the same colour and the third a different colour ?                                                        55/84

iii) three balls will be of the same colour ?                                  5/84

3) There are 10 persons who are to be seated around a circular table. Find the probability that two particular persons will always sit together.                                       2/9

4) One card is drawn from a full pack of 52 cards. Find the probability that the card drawn is

A) either a spade or a diamond. 1/2

B) either spade or a king.   1/2, 4/13

5) There are 50 tokens marked 1,2,3....50. One token is selected at random. Find the probability that it is a multiple of 4 or 6.               8/25

6) A pair of fair dice is thrown. Find the probability of getting a sum 7, When it is known that the digit in the first dice is greater than that of second.                                     1/5

7) The probability that a boy will not pass MBA examination is 3/5 and that a girl will not pass is 4/5 . Calculate the probability that at least one of them passes the examination.                           13/25

8) An article manufactured by a company consists of two parts A and B. in the process of manufacture of part A, 9 out of 100 are likly to be defective. similarly 5 out of 100 are likely to be defective in the manufacture of part B. Calculate the probability that the assembled article will not be defective.                                0.8645.

9) 8 Men in a company of 25 are graduates. If 3 men are picked out of the 25 at random, what is the probability that.

A) they are all a graduates.  14/575 

B) at least one graduate.       81/115

10) A certain player, say X is known to win with probability 0.3, if the track is fast and 0.4. If the track is slow. For Monday, there is 0.7 probability of a fast track and 0.3 probability of a slow track. What is the probability that player X will win on Monday.                                 0.33

11) There are 4 hotels in a town. If 3 men check into hotels in a day. what is the probability that each check into a different hotel ?      3/8

12) A sample of 3 items is selected at random from a box containing 12 items of which 3 are defective. Find the possible number of defective combination of the said 3 selected items along with probability of a defective combination.            0.618

13) A die is is located in such a way that each odd number is twice as likely to occur as each even number. Find

A) the probability that the number rolled is a perfect square.         1/3

B) the probability that the number rolled is a perfect square provide it is greater than 3.                         1/9

14) Given below are the weekly wages (in a rupees) of six workers in a factory: 62, 90, 78, 85, 79 and 68.

If two of these workers are selected at random to serve as representatives, what is the probability that at least one will have a wage lower then the average ?                                                  3/5

15) 100 students randomly selected from a group students are cross-classified by age and educational qualification as a below:

Qualification   Age.                    Total

          25-above  26-28  over 28

Graduate 24         19          11         54

Post Gr.   11         16          19         46

 Total       35         35           30      100

A student is selected from this group. Find the probability that:

A) his age is between 26- 28 years. 

                                                    0.35

B) He is a graduate.                 0.54

C) his age is between 26-28 years and he is a graduate.               0.19

D) his age is between 26-28 years assuming that he is a graduate and.

                                                    0.35

E) he is a graduate assuming that his age is between 26-28 years.

                                                    0.54

16) A bag contains 4 red and 3 blue balls. Two drawings of 2 balls are made. Find the probability of drawing first 2 red balls and the second 2 blue balls.

A) If the balls are returned to the bag after the first draw;            2/49

B) if the balls are not returned after the first draw.                             3/35

17) An URN contains 7 red and 4 blue balls. if two balls are drawn at random with replacement, find the probability of getting

A) 1 red and 1 blue ball.        56/121

B) two blue balls.                   16/121

18) One bag contains 4 white and 2 black balls. Another bag contains 3 white and 5 black balls. If one ball is drawn from each bag. Find the probability that one is white and one is black.                           13/24

19) A purse contains two silver and four gold coins. A second purse contains 4 silver and three gold coins. if a coin is taken out at random from one of the two purses, what is the probability that it is a silver coin ?                            19/42

20) Two urns contains respectively 10 white, 6 red and 9 black balls and 3 white, 7 red and 15 black balls. One ball is drawn from each uen. Find the probability that:

A) both balls are red,            42/625

B) both balls are same colour. 

                                               207/625

21) A bag contains 5 red and 4 black balls. A ball is drawn at random from the bag and put into another bag which contains 3 red and 7 black balls. A ball is drawn randomly from the second bag. What is the probability that it is red ?                                                 32/99

22) A can solve 80% of the problems given in statistics and B can solve 60% . What is the probability that at least one of them will solve a problem selected at random?                                       0.92

23) A problem in business statistics is given to four students A,B,C,D. their chances of solving it are 1/2, 1/3, 1/4, 1/5 respectively. What is the probability that the problem will be solved ?                                   4/5

24) The probability that three man hit a a target are respectively 1/6, 1/4 and 1/3. Each man shoots once at the target. What is the probability that exactly one of them hits the target ?                                      31/72

25) there are four letters and 4 corresponding envelopes. letters are placed one in each envelope. Find the probability that no letter goes to the corresponding addressed envelope.                  3/8

26) A man has 2 French, 3 German and 4 Spanish friends. He invites one or more of them in his birthday party. Find the probability that there will be 

A) at least one German friend. 

                                                448/511

B) at least one friend from each country.                                  315/511

27) An URN contains 2 mangoes, 3 apples and 4 oranges. Any number of fruits are selected from it. Find the probability that in the selection there will be 

A) at least one mango.           40/59

B) at least one fruit of each type. 

                                                   24/59.

28) 6 different books are arranged in a shelf. Find the probability that two particular books will always be together.                                       1/3

29) How many losses of a coin are needed so that the probability of getting at least one head is 87.5% ? 

                                                        3

30) A coin is tossed 3 times in succession. Find the probability of obtaining at least two heads.      1/2

31) Two dice are rolled. Find the probability that 

A) the total of the number on the die is 9 or greater;.                     5/18

B) the number in the first dice is greater than that in the second die. 

                                                     5/12

32) In a game of Bridge what is the probability that a hand will contains all 4 Kings.                         114/4165

33) An URN contains 8 white 3 red balls. if two balls are drawn at random, find the probability that

A) both are white.                   28/55

B) both are red.                          3/55

C) one of each colour.             24/55

34) Find the probability of finding at least one king out of 10 cards drawn from a 52 cards.       349/595

35) From a box containing 3 blue and 5 red balls. 3 balls are drawn at random. Find the chance of getting 

A) at least one red ball.           55/56

B) at most One red ball.              2/7

36) two letters are drawn from the word HOME. Find the probability that

A) both are vowels.                     1/6

B) at least one is vowel.              5/6

C) One of the letter is chosen M.

                                                       1/2

37) There are 3 Geologist, 4 Engineers, 2 Statistician and 1 Doctor. A committee of 4 among them is to be formed. Find the probability that the committee

A) consists of one of each kind

B) has at least one geologist.

C) has the doctor as a member and 3 others.                    4/35, 5/6, 2/5

 

38) A 4 digit number is formed by the digit 1, 2, 3, 4 with no repetition. Find the probability that the numbers is 

A) odd.                                          1/2

B) divisible by 4.                          1/4

39) A 5 digited number is formed by the digits 0, 1, 2, 3, 4 with no repetition. Find the probability that the number is 

A) odd.                                         3/8

B) divisible by 4.                        5/16

40) four cards are drawn from a pack of 52 cards. What is the probability that they are from four different suits?             2197/20825

41) A and B stand in a line at random with 10 other people. Find the chance that there will be 3 people between them.             4/33

42) The first 12 letters of the alphabet are written at random. Find the probability that there are exactly 4 letters between C and D. 

                                                   7/66

43) The letters of the word EDUCATION are arranged at random. Find the probability that there will be exactly 4 letters between A and E.                         1/9

44) The letters of the word SUNDAY are arranged at random. Find the chance that the arrangement will

A) begin with S.                         1/16

B) begin with S but not end with Y

C) Vowels will occupy odd places. 

                                            2/15, 1/5

45) The letters of the word DIRECTOR are arranged at random. Find the probability that the vowels will be will be always together. 3/28

46) 6 coins are tossed. Find the probability of getting

A) exactly 2 heads.                15/64

B) at least two heads.             57/64

47) From a group of 10 players, six players are selected at random. Find the probability that Mr X and Mr Y will 

A) never be in the selection.     2/15

B) always be in the selection.     1/3

48) Some fruits are selected at random from a basket containing 3 Apples, 4 mangoes and 5 oranges. Find the probability that the selection will contents.

A) at least one Apple.           90/119

B) at least one fruit of each type. 

                                                 60/119

49) A man has got 2 German, 3 Spanish and 4 French friends. He invites at random some of them. Find the probability that he will invite.

A) at least one German.      384/511

B) at least one friend from each country.                                     45/73

50) 8 counters marked 1,,2,.....,8 four counters, are are selected at random. Find the chance of getting at least one odd and one even counter.                                    34/35

51) A sub-committee of 6 members is to be formed from 7 men and 4 ladies. Calculate the probability that sub committee will consist of 

A) exact two ladies.                   5/11

B) at least two ladies.              53/66

52) Find the chance of throwing at least 8 in a single cast with two dice.                                             5/12

53) In a packet of 10 watches, three are known to be defective. If two watches are selected at random from this packet, what is the probability that at least one of them is defective.                              8/15

54) 40% of the student in a class are girls. If 60% and 70% of the boys and girls respectively of the class pass is certain test. What is the probability that a student selected from this class will have passed the test ?                   16/25

55) In a group of 14 males and 6 females, 8 and 3 of the males and females respectively are aged about 40 years. What is the probability that a person selected at random from this group is aged above 40, given that selected person is a female?                    1/2

56) In a family of four children, what is the probability that

A) all of them will have different birthdays.           364.363.362/365³

B) two of them will have the same birthday?                 6.364.363/365³

57) A bag contains 5 white 4 black balls. One ball is drawn from the bag and replaced and then a second draw of a ball is made. What is the chance that the two balls drawn are of different colours?                                   40/81

58) A box contains 8 red and 5 white balls. Two successive draws of 3 balls are made. Find the probability that the first drawing will give 3 white balls and the second 3 red balls. if the balls are drawn

A) with replacement.      140/20449

B) without replacement.        7/429

59) Boxes 1 and 2 contain respectively 4 white, 3 red and 3 blue balls; and 5 white, 4 red and 3 blue balls. if one ball is drawn at random from each box, What is the probability that both balls are of the same colour.                         41/120

60) Two boxes contain respectively 4 white and 3 red balls; and 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. Find the probability that the ball is white.                                    61/140

61) An URN contains 5 white and 3 black balls, and a second urn contains 4 white and 5 black balls. One of the urn is chosen at random and 2 balls are drawn from it. Find the probability that one is white and other is black.                     275/504

62) Each of two identical bags contain 5 white and 5 red red balls. One ball is transferred at random from the second Bag to the first and then one ball is drawn from the first bag. Find the probability that the ball drawn is red.                 1/2

63) The odds against a certain event are 5:2 and the odds in favour of another event, independent of the former are 6:5. Find the chance that at least one of the event will happen.                                    52/77

64) It is 8:5 against a person who is now 30 years old to live 40 years more and 3:5 in favour of a person who is now 40 years old to live 40 years more. Find the probability that at least one of these persons will be alive 40 years hence.        59/91

65) A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in stating the same fact ?                        35%

66) A person is known to hit 4 out of 5 shots. whereas another person is known to hit 3 out of 4 shots. Find the probability of hitting a target if they both try.             19/20

67) A can solve solve 80% of the problems in this book and B can solve 70%. A problem of this book is selected at random. What is the chance that the problem will be solved if they both try.             47/50

68) The probability that a student passes in Statistics test is 2/3 and the probability that he passes both Statistics and Mathematics test is 14/45. The probability that he passes at least one test is 4/5. What is the probability that he passes the Mathematics test?    4/9

69) Presuming the daily demand to be independent, you are to find the probability that over a two-day period the number of request at some service station will be

A) 9                                               0.48

B) 10 if the past record indicates that daily demand has either been 4 with probability 0.4 or 5 with probability 0.6.                             0.36

70) Mr. X is called per an interview for 3 separate post. At the first interview there are 5 candidates, at the 2nd 4 candidates and at the 3rd 3 candidates. if the selection of each of them is equally likely, find the probability that Mr. X will be selected for the least one post.  3/5

71) A six-faced die is so biased that it is twice as likely to show an even number as an odd number when thrown. it is thrown twice. What is the probability that the sum of two numbers thrown is even.           5/9

72) In a group of equal number of men and women, 10% men and 45% women are unemployed. What is the probability that a person selected at random is employed ? 

                                                  29/40

73) A problem of statistics is given to A, B and C whose chances of solving it are 1/3,1/4 and 1/5 respectively. Find the probability that the problem will be solved by at least one of them.                 3/5

74) Three shots are fired at a moving target. The probability of hitting the target at the first, second and the third shot are 1/2, 3/10, and 1/10. Calculate the probability that them target gets hit.          137/200

75) The probability of A, B, C solving a problem are 1/3, 2/7 and 3/8 respectively. if all of them try to solve the problem simultaneously, find the probability that exactly one of them will solve it.               25/56

76) In a given race the odds in favour of four horses A, B, C, D are 1:3, 1:4, 1:5, 1:6 respectively. Assuming that a dead heat is impossible, find the chance that one of them wins the race.  57/140

77) A, B, and C in that order, toss a coin. The first one to throw a head wins. What are their respective chances of winning? Assuming that the game may continue indefinitely.

                                        4/7, 2/7,1/7

78) Markmen A and B compete by taking turns to shoot at a target. Odds in favour of A hitting the target (in a single try) are 3:2 andthe odds in favour of B hitting the target (in a single try) are 4:3.

Calculate the probability of A winning the competition if he gets the first chance to shoot.     21/29

PROBLEM ON QUADRATIC EQUATIONS

1) The product of two consecutive integers is 56. Find the integers.

2) The sum of the squares of two consecutive natural numbers is 41. Find the numbers.

3) Find the natural numbers which differ by 5 and the sum of whose squares is 97.

4) The sum of a number and its reciprocal is 4.25. find the number.

5) Two natural numbers differ by 3. Find the numbers, if sum of their reciprocals is 7/10.

6) Divide 15 into two parts such that the sum of their reciprocals is 3/10.

7) A can do a piece of work in x days and B can do the same work in x+16 days. If both working together can do it in 15 days; find x

8) The square of a number added to one-fifth of it, is equal to 26. Find the number.

9) The sum of the squares of two consecutive positive even Numbers is 52. Find the numbers.

10) Find two consecutive positive odd numbers, the sum of whose squares is 74.

11) Two numbers are in the ratio 3:5. Find the numbers; if the difference between their squares is 144.

12) Three positive numbers are in the ratio 1/2:1/3:1/4. Find the numbers; if the sum of their squares is 244.

13) Divide 20 into two parts such that three times the square of one part exceeds the other part by 10.

14) The sum of two numbers is 32 and their product is 175. Find the numbers.

15) Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60.

16) The ages of two sisters are 11 years and 14 years. In how many years time will the product of their ages be 304 ?

17) A stone is thrown vertically downwards and the formula

d= 16t²+ 4t gives the distance, d metres, that it falls in t seconds. How long does it take to fall 420 metres?

18) One pipe can fill a cistern in 3 hours less than the other. The two pipes together can fill the cistern in 6 hours 40 minutes. Find the time that each pipe will take to fill the cistern.

20) Out of the three consecutive positive integers, the middle number is p. If three times the square of the largest is greater than the sum of the squares of the other two numbers by 67; Calculate the value of p.

21) The sides of a right angled triangle containing the right angle are 4x cm and (2x - 1)cm. If the area of the triangle is 30cm²; find the length of its sides.

22) The hypotenuse of a right angled triangle is 26 cm and the sum of other two sides is 34 cm. Find the lengths of its sides.

23) The hypotenuse of a right angled triangle exceeds one side by 1 cm and other side by 18 cm; find the length of the sides of the triangle.

24) The length of of a rectangle plot exceeds its breadth by 12 m and the area is 1260m². Find its dimensions.

25) The perimeter of a rectangle plot is 104m and its area is 640 m². Find its dimensions.

26) A footpath of uniform width runs round the inside of a rectangular field 32m long and 24m wide. If the path occupies 208m², find the width of the footpath.

27) Two squares have sides x cm and (x+4) cm. The sum of their areas is 656cm². Find the side.

28) The length of a rectangle board exceeds its breadth by 8cm. If the length were decreased by 4cm and the breadth doubled, the area of the board would be increased by 256cm². Find the length of the board.

29) An area is paved with square tiles of a certain size and the number required is 600. If the tiles had been 1cm smaller each way, 864 would have been needed. Find the size of the larger tiles.

30) A farmer has 70m of fencing, with which he encloses three sides of a rectangular sheep pen; the fourth side being a wall. If the area of the pen is 600m², find the length of its shorter side.

31) A square lawn is bound on three sides by a path 4m wide. If the area of the path is 7/8 that of the lawn, find the dimensions of the lawn.

32) The area of a big room is 300 m². If the length were decreased by 5m and the breadth increased by 5m; the area would be unaltered. Find the length of the room.

33) The speed of an ordinary train is x km/hr and that of an express train is (x+25)km/hr

i) Find the time taken by each train to cover 300 km.

ii) If the ordinary train takes 2hrs more than the express train: Calculate speed of the train.

34) If the speed of a car is increased by 10km/hr, it takes 18 minutes less to cover a distance of 36km. Find the speed of the car.

35) If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200km. Find the speed of the aeroplane.

36) A train covers a distance of 300 km/hr. Another train covers the same distance at a speed of (x-5) km/hr.

I) Find the time which each train takes to cover the distance between the stations.

ii) If the second train takes 3 hours more than the first train, find the speed of each train.

37) A girl goes to her friend's house, which is at a distance of 12 km. She covers half of the distance at a speed of X km/hr. and the remaining at a speed of (x+2) km/hr. If she takes 2hrs 30 minutes to cover the whole distances; find x

38) A car made a run of 390 km in 'x' hours. If the speed had been 4 km per hour more, it would have taken 2 hours less for the journey. Find x

39) Rs 250 is divided equally among a certain number of children; if there were 25 children more, each would have received 50 paise less. Find the number of children.

40) A trader bought a number of articles for Rs1200. Ten were damaged and he sold each of the rest at Rs2 more than what he paid for it, thus getting a profit of Rs69 on the whole transaction.

41) Mr. Mehra sends his servant to the market to buy oranges worth Rs15. The servant having eaten three oranges on the way, Mr. Mehra pays 25 paise per orange more than the market price. Find number of oranges.

42) A man bought an article for Rs x and sold it for Rs 16. If his loss was x% find the cost price of the article.

43) A trader bought an article for Rs and sold it for Rs52, thereby making profit of (x-10)% on his outlay. Calculate the cost price, for trader.

44) By selling chair for Rs 75, Mohan gained as much % as its cost. Find the cost of chair.

45) A goods train leaves a station at 6 p.m., followed by an express train which leaves at 8pm. and travel 39 km per hour faster than the goods train. The express train arrives at a station, 180 km away, 36 minutes before the goods train. Find their speed.

46) The product of the digits of a two digit number is 24. If its units digit exceeds twice its ten's digit by 2; find the number.

MEDIAN For Class IX

1) Find the median of the following values of a variate:

A) 5, 10,3, 7, 2, 9, 6, 2, 11.               6

B) 30, 5, 21, 42, 13, 10, 27, 33, 17, 8.                                                   19

C) 37, 31, 42, 43, 46, 25, 39, 43, 32.                                                 39

D) 35, 15, 37, 87, 65, 56, 17, 3, 52, 39.                                                   38

E) 33, 73, 89, 108, 94, 140,94, 85, 100, 120.                                    94

F) 25, 34, 31, 23, 22, 26, 35, 29, 20, 32.                                            27.5

G) 2, 10, 9, 9, 5, 2, 3, 7, 11.          7

H) 31,38, 27,28, 36, 25, 35,40.    33

I) 15, 6, 16, 8, 22, 21, 9, 18,25.    16

J) 37, 49, 15, 87, 65, 25, 17, 3, 52, 39.                                                38

K) 92, 35, 67, 85, 72, 81, 56, 51, 42, 69.                                        68

L) 83, 37, 70, 29, 45, 63, 41, 70, 30, 54.                                    49.5

M) 9, 10, 8, 2, 4, 4, 3, 9, 1, 5, 6, 2, 4. Also find mean.                 4, 34/7.

MEAN For Class- VIII/IX

                       MEAN

1) Find the mean of first 7 even natural numbers.                       8

2) The number of children in 25 families of a locality are recorded as follows: 3,1,4,0, 2,2, 1,1, 2, 3, 3, 2, 2, 2, 5, 0, 1, 4, 1, 2, 1, 2, 3, 0, 1, 4. Find the mean number of children per family.                                    2.

3) The marks obtained by by 10 students in a test are 56, 54, 71, 60, 62, 72, 59, 64, 70, 52. Find

A) the mean marks of the student

B) The mean marks of the student if 5 extra marks are given to each student.        62, 67

4) A student obtained 60, 75 and 85 marks, respectively in three monthly examination in Physics and 95 marks in the final examination. The three monthly examination are of equal weightage whereas the final examination is waghted twice as much as a monthly examination. Find his mean marks for Physics. 82

5) Find the mean of

A) first five natural natural numbers.

B) first five positive odd integers

C) first five multiples of 3 multiples of 3.

E) all factors of 10.        3,5,6,9,4.5

6) find the mean of the numbers 96, 98, 100, 102, 104.                      100

7) find the mean of the numbers 994, 996, 998, 1000, 1002.        998

8) find the mean of x, x+2, x+4, x+6, and x+8.                                       x+4

9) If the mean of 6, 8, 5, 7, x and 4 is 7, find the value of x.                    12

10) Following are the weight(in kg) of 10 newly born babies in a hospital on a particular day: 3.4, 3.6, 4.5, 3.9, 4.1, 3.8, 4.5, 4.4, 3.6. Find the mean.                           5.6

11) The traffic police recorded the speed(in km/hr) of 10 motorists as 47, 53, 49, 60, 39, 42, 55, 57, 52, 48. Later an error in the recording instruments was found. Find the correct average speed of the motorists if the instrument recorded the speed 5 km/hr less in each each case.                      45.2

12) If the mean of 6, 4, 7, p and 10 is 8, find the value of p.                13

13) If the mean of 5 numbers is 7, find the mean of another 5 numbers numbers which are obtained by adding 2 to each case of the 5 numbers.                                         9

14) Find the sum of the deviations of the variate values 2, 4, 6, 8, 14 from their mean.                            0

15) The mean of 8 numbers is 15. If each number is multiplied by 2, what will be the new mean.        30

16) The mean is 21 numbers is 15. If each number is multiplied by 2, what will be the new mean.         30

17) There are 45 students In a  class, of which 15 are girls. The average weight of 15 girls is 45 kg and that of 30 boys is 52kg. Find the mean weight in kg of the entire class.                                       49.67

18) A school has 4 sections in Class X having 40, 35, 45 and 42 students. The mean marks obtained in a Chemistry Test are 50, 60, 55, and 45 respectively, for the 4 sections. Determine the overall average of marks per students. 52.2

19) mean of 25 observations was found to be 78.4. but later on it was discovered that 96 was misread as 69. Find the correct mean.     79.48

20) Mean of 40 observations was 160. It was detected on rechecking that the value 165 was wrongly occupied as 125 for computation of a mean. Find the correct mean. 16.1

21) The mean weight per student in a group of seven students is 55 kg. The individual weight(in kg) of 6 of them are 52, 58, 55, 53, 56, and 54. Find the weight of the 7th student.                                    57 kg

22) The mean weight of 150 students in a certain class is 60 kg. The mean weight of boys in the class of 70 kg and that of the girls is 55kg. Find the number of boys and the number of girls in the class.                                    50, 100

23) In a class, the average score of girls in an examination in 73 and that of a boy is 71. The average score for the whole class 71.8. Find the percentage of girls and the boys in the boys in the class.    40, 60%

24) A train travels between two stations A and B, while going from A to B, its average speed is 100 km/hr, and when coming back from B to A, its average speed is 150 km/hr. Find the average speed during the whole journey.            120

25) The mean height of 10 students is 153 centimetre. it is discovered later on that while calculating the mean the reading 151cm was wrongly read as as 141cm. Find the correctly mean.                         154

26) The mean of five number is 27. If one number is excluded, their mean is it 25. Find the excluded number.                                       35

27) There are 50 students In a class of which 40 are and the rest girls. The average weight of the the class is 44 kg and the average weight of the girls is 40 kg. Find the average weight of the the boys.                45

28) There are 120 boys in a class in which twenty of them are girls and the rest boys. If the average marks in Maths of the boys is 65% and that of the girl is 80%. Find the average mark of the class.     56.83

29) The average score of boys in an examination of a school is 71 and that of girls is 73. The average of the school in that examination in 71.8. Find the ratio of the number of boys to the number of girls appeared in the examination.      3:2

30) The mean of marks secured by 25 students of of section A of class 47, that of 35 students of section B is 51 and that of 30 students of Section C is 53  students of Section C is 53 . Find the mean of the marks of students of three sections of class sections of class X.   50.55

31) The average monthly wage of a group of 10 persons is Rs.1500. One member of the group whose monthly wage is Rs1350, left the group and is replaced by a new member whose monthly wage is Rs.1200. Find the new monthly wage.                                       1485

32) The mean of monthly salary of 10 members of a group is Rs1445. One more member whose monthly salary is Rs1500 has joined the group. Find the mean of monthly salary of 11 members of the group. 1450

33) The mean of marks secured by hundred students was found to be 40. later on it was discovered that score of 53 was misread 83. Find the correct mean.                     39.7

34) The aggregate monthly expenditure of a family was Rs 4050 during the first three months. Rs 4260 during the next four months and Rs4326 during the last five months of a year. If the total saving during the year be Rs 6120, find the average average monthly income.                      4678

35) the average of 11 results is 50. If the average of the first six results is 49 and that of last is 52, find the sixth sixth result.                       56

36) A batsman in his 12th innings makes a score 63 runs and thereby increases his average score by 2. What is his average after the 12th innings.                                       41.

37) The average age of 8 persons in a committee is increased by 2 years, when 2 men aged 35 years and 45 years are substituted by two women. Find the average age of these women.                              48

38) A Car owner buys petrol at Rs 17, Rs19 and Rs 20 per litre for 3 consecutive years. compute the average cost pee litre, If he spends Rs 6460 per year.                    18.58

39) The arithmatic mean of 3, 7, 5, x, 8, -3, is 4. Find x.                      4

SIMPLE INTEREST(C)

1) Find the simple interest on ₹850 for 7 years at 9% p.a.

A) ₹500  B) ₹535.50 C) ₹453.05 D) ₹553.35

2) Find how much would be the simple intrest on ₹1000 for 20 years at 5% p.a

A)₹ 1000 B) ₹ 1500 C)₹1450 D) ₹1550

3) At what rate percent p.a will ₹ 450 amount to ₹ 810 in 10 years?

A) 2%  B) 2.50%  C) 3%  D) 4% 

4) At what rate p.a will simple interest on ₹8650 amount to ₹5190 in 8 yrs.

A) 7% B) 7.5%  C) 8% D) none

5) What principle amount to ₹459 in 5 years at 7% p.a simple interest ?

A) ₹350  B) ₹335.50 C) ₹340 D) none

6) Find What sum of money will amount to ₹1401.80 in 7 years 9% p.a simple interest.

A) ₹860 B) ₹853 C) 845 D) 850

7) In what time will ₹1250 amount to ₹2150 at 9% p.s simple interest ?

A) 6 years B) 5 years C) 7 years D) 5.5 years 

8) In what time will ₹1450 amount to ₹ 2320 at 8% p.a. simple interest ?

A) 6.5 years  B) 5 years C) 7.5 years D) 5.5 years

9) A sum of ₹720 amounts to ₹ 1044 in 5 years. What is the rate of interest percent per annum ? What Sum will amount to 2363.50 in 7 years at the same rate ?

A) 8%, ₹1450 

B) 9% , ₹1450

C)  10%, 1450 

D) 9%, 1500

10) At what rate will the interest on ₹900 in 4 years be the same as the interest on ₹720 for 6 years at 5%.

A)  6%  B) 5%  C) 7.5% D)  5.5% 

11) The simple interest on ₹300 for 4 years together with that on ₹500 for 3 years is ₹162, the rate being same in both the cases. Find the rate percent of interest.

A) 6.5% B) 5% C) 6% D) 5.5 

12) A sum of money amounts to ₹3576 in 14/3 years at 10.5% simple interest. When will it double itself at the same rate ?

A) 9 years B) 9.5 years C) 200/21 D) 5.5 years 

13) A person finds that a fall of interest from 4% to 3.5% p.a. diminishes his yearly income by ₹60. What is his capital ?

A) ₹22000 B) ₹23000 C) ₹23500 D) ₹ 24000 

14) In what time will ₹ 4000 at 9% interest p.a. produce the same Income as 6000 in 5 years at 8% simple intrest.

A) 6.5 years B) 5 years C) 20/3 years D) 6.33 years

15) The principal and interest for 5 years are together ₹396 and the interest is 9/25 of the principal. Find the principal and the rate of interest.

A) ₹ 225 , 7% B) ₹225,64/9% C) ₹250, 7.5%  D) none

16) At what rate percent per annum simple interest on a certain sum of money for 20 years be equal to 4/9th of its amount for that period ?

A) 6% B) 5% C) 7.5% D) 4%

17) A pressure cooker is available for Rs 250 cash or Rs 100 cash down payment followed by ₹165 after 6 months. Find the rate of interest charged under the installment plan.

A) 16% B) 15% C) 20%  D) 14%

18) A lens ₹500 to B and a certain sum to C at the same time 8%p.a. simple interest. If in 4 years he altogether receives ₹210 as interest from the two, find the sum of money lent to C.

A)₹ 156.25 B) ₹165 C) ₹145 D) ₹ 156 

19) A man deposited ₹5000 on 20th April in a bank paying interest 4%p.a. He  withdraw ₹3000 on 15th May and deposited ₹4000 on 6th June. How much interest was due to him on 30th June following ?

A) ₹34  B) ₹34.20 C) ₹24.30 D) ₹34.30

20) What sum will amount to ₹5200 in 6 years at the same rate of simple interest at which ₹1706 amount to ₹2412 in 20 year ?

A) 2000 B) 3000 C) 4000 D) none

21) A man borrowed ₹ 20000 at 12% p.a simple interest from a bank. After 1.5 years he paid ₹12000 to the bank. Find how much he will have to pay after 2 years more to clear the loan.

A) ₹14300 B) ₹14384 C) ₹15384 D) 15000

22) A man borrowed ₹ 800 at the beginning of the year at a certain rate of interest. After 7 months he again borrowed ₹240 at double the previous rate of interest, If at the end of the year he has to pay ₹50 as total interest on the two sums. Find the rate of interest.

A) 6% B) 5.5% C) 5% D) none 

23) A man undertakes to pay back the loan of ₹4000 in monthly installments of ₹500 plus interest at 6% on outstanding balances. Find the average rate of interest earned by the lender.

A) 3% B) 27/8% C) 3.33% D) 3.66%

24) A man borrows two sums of money differing by ₹100 at the same time, one at 5% per annum and the other at 6.5% p a. both at simple interest. At the end of 5 years he pay back both the loans with interest. If he pays the same amount in respect of each loan. Find the sums borrowed by him .

A) 2100, 2200 B) 1900, 2000 C) 2000, 2100 D) 2200, 2300

25) Mr. Moi deposited a total sum of ₹15000 in two different banks. One of the them pays 8% interest and other pays 9% intrest. At the end of 1 year he received ₹1301.50 as intrest.  What sumd respectively were deposited in these two Banks.

A) 4809, 10200 B) 4850, 10150 C)  5000, 10000 D) 4500, 10500

26) Mr. Zen deposited a sum of ₹10000 in a bank. After 2 years, he withdraw ₹ 4000 and at the end of 5 years he received an amount of ₹7520. Find the rate of simple interest .

A) 6%  B) 5% C) 4% D) 5.5% 

27) What is the simple interest on ₹600 at 6% per annum for 5 years ?

A)₹ 200 B) ₹180 C) ₹260 D) ₹240

27) Moi deposited ₹7200 in a finance company, which pays 15% simple interest p.a. Find the interest and the amount he is expected to get after 4 years and 6 months.

A) 4800,12000 B) 8400, 15600 C) 4860, 12060 D)  3860 , 11060

28) Zen deposited ₹520 in a bank. The bank pay simple interest 8% p.a. Find the interest and amount to be received by Zen after 2 years.

A) 83.20, 603.20 B) 80.20, 600.20 C) 73.20, 599.20  D) none

29) Ramsukhlal, the farmer borrowed ₹2400 at 12% simple interest per annum. At the end of 2.5 years, he cleared his account by paying ₹1200 and a cow. What is the cost of his cow ?

A) 1820 B) 1720 C) 1920 D) 2000

30) What is the interest on ₹1200 at 6% simple interest per annum for 146 days?

A) ₹27.56 B) ₹28.00 C) ₹26.67 D) ₹28.80

31) Find the simple interest and amount of ₹ 365 from 1st January 2005 to 25th October 2005 at 5% per anuman.

A) 14.00 , 379.00 B)  15.85, 380.85 C) 14.85, 379.85  D) none

32) What principle amount to ₹17200 in 6 years 12% simple interest p.a ?

A) ₹12000 B) ₹10000 C) ₹11000 D) ₹10500 

33) A sum fetched a total simple interest of ₹4016.25 at the rate of 9% in simple interest o.a in 5 years ? The sum is:

A) 8900 B) ₹9000 C) ₹9925 D) ₹8925

34) Ram invested ₹3000 at simple intrest for 3 years and received ₹ 4170. What was the rate of interest applied?

A) 11% B) 11.5% C) 12% D) 13% 

35) If the simple interest on a certain sum of money for 2 years is one fifth of the sum,----p.a. is the rate of interest 

A) 9% B) 10% C) 11%  D) none

36) Mrs. Chi  invested ₹3500 for 4 years and ₹4500 for 5 years at the same rate of interest, If she received a total simple interest of ₹4380, what was the rate of interest?

A)  10% B) 12.5% C) 12% D) 11.56% 

37) In what time will ₹2000 amount to ₹ 3960 at 14% S. I p.s ?

A) 7 years B) 8 years C) 6 years D) 5.5 years 

38) At what time will Rs1200 amount to ₹1344 at 6% p a ?

A) 3 years B) 2 years C) 2.5 years D) n

39) The difference between the simple interest and compound interest on ₹60 for 1 year at 10% p.a, reckoned half yearly is:

A) ₹ 2.00 B) ₹1.00  C) ₹1.50 D) ₹2.50

40) At what rate will a sum of money double itself with simple interest in 16 years?

A) 6.67% B) 6.25% C) 12.5% D) none

41) A sum of money amount to₹3576 in 4.67 years at 19.5% simple interest. When will it double itself at the same rate ?

A) 200/21 B) 29/31 C) 29/13 D? none

42) A sum of ₹5700 is lent out in two parts in such a way that intrest on one part @8% for 5 years is equal to that on the other part at 1/2% for 15 years. Find the sum lent at 8% (simple interest being reckoned)

A) 600 B) 900 C) 1200 D) 1700

43) A sum of money doubles itself in 8 years. In how many years will it triple itself ?

A)12 years B) 16 years C) 24 years D) n

44) A sum of money triples itself in 8 years. In how many years will it be 5 times ? simple interest being reckoned.

A) 12 years B) 16 years C) 24 years D) 13.33 years 

45) In What time will ₹8000 amount to ₹40000 at 4% p.a(simple interest being Reckoned)

A) 100 years B) 50 years C) 110 years D) 160 years

46) Mr. Zen takes a loan of ₹ 525 at 4% p a simple interest for Mr. Moi. He pays Mr  Moi ₹250 at the end of 1st year. How much should he pay at the end of 2nd year in order to clear his dues?

A) 307.84 B) 286.44  C) 300.84 D) 310.54

47) A person bought a motorcycle under the following scheme : Down payment of ₹15000 and the rest amount at 8% per annum for 2 years. In this way he paid ₹28920 in total. Find the actual price of the motorbike (assume simple interest)

A)₹ 26000 B)₹27000 C) ₹27200 D) 26500

48) Moizen borrows ₹ 7000 at simple interest from the village moneylender. At the end of 3 years, he again borrows ₹3000 and closes his account after paying ₹4615 as interest after 8 years from the time he made the first borrowing. Find the rate of interest?

A) 3.5% B) 4.5% C) 5.5% D) 6.5% 

49) Some amount was lent at 6% per annum simple interest. After 1 year ₹6800 is repaid and the rest of the amount is repaid at 5% per annum in the following year. If the second year's interest is 11/20 of the first year's interest, find what amount of money was lent out.

A) ₹17000 B) 16800 C) 16500 D) 20000

50) At what rate percent p.a simple interest on a certain a money for 20 years be equals to 4/9th of its amount for that period ?

A) 5% B) 4% C) 6% D) 7%

51) The difference between compound and simple interest on a certain sum of money is 40 for first two years and ₹122 for the first three years. Find sum, If the rate is same in both the cases.

A) 8000 B)12000 C) 15000  D) 16000

52) The compound interest on a certain sum for 2 years is 71.40 and the simple interest is 70. What is the rate of interest ?

A) 3%  B) 4% ac)  5%  d) 6%

53) If the difference between compound interest and simple interest on a certain sum of money for 3 years at 10% per annum is ₹248, find the sum 

A) 6000 B) 7000 C) 8000 D) 9000

54) Zen gives a loan of ₹680 to Moi and recovered it at the rate of ₹40 per month for 20 months. What is the rate of simple interest charged?

A)  20%pa B) 25%la C) 24%pa D) 19%pa

55) ₹ 6400 amounts ₹7840 in 2 years at simple interest.  How much will a sum of ₹84 invested at the same rate simple interest amount to in 4 years ?

A)110.59 B) 121.80 C) 111.80 D) 123.89

57) Find the ratio in which a sum has to divided into 3 parts so that the amounts accrued on the three parts at simple interest after 2,3 and 4 years respectively are the same. It is given that the rate of interest is 5% p.a in each case.

A) 130:115:109

B) 276:264:253

C) 110:114:119 D) none