FREQUENCY DISTRIBUTION

FREQUENCY DISTRIBUTION
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1) The age of children in my colony. Draw the frequency distribution.
7   8    5    4    9    8    5    7    6    8
9   6    7    9    8    7    9    9    6    5
8   9    4    5    5    8    9    6.

2) Following data gives the number of children in 40 families:
1, 2, 6, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 4, 4, 3, 2, 2, 0, 0, 1, 2, 2, 4, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 4, 1, 1, 6, 2, 2.
Represent it in the form of a frequency distribution.

3) The marks obtained by 40 students of a class in an examination are given below. Present the data in the form of a frequency distribution using equal class-size, one such Class being 10-15(15 not included).
3, 20, 14, 1, 21, 14, 4, 24, 16, 14, 18, 12, 6, 13, 5, 25, 9, 3, 7, 19, 20, 4, 10, 13, 7, 18, 2, 5, 8, 10, 16, 8, 17, 18, 8, 24, 21, 6, 23, 15.

4) Construct a frequency table for the following ages (in years) of 30 students using equal
class- interval, one of them being 9-12, where 12 is not included.
18, 12, 8, 7, 11, 16, 21, 9, 6, 14, 16, 17, 23, 19, 15, 21, 24, 9, 13, 18, 15, 7, 18, 24, 23, 17, 9, 21, 11, 16.

5) Construct a frequency table with equal Class- interval from the following data on the monthly wages(in rupees) of 28 labourers working in a factory, taking one of the class interval as 210-230 (230 included):
220, 268, 258, 242, 210, 268, 272, 242, 311, 290, 300, 320, 319, 305, 304, 319, 307, 293, 255, 279, 210, 240, 280, 317, 307, 215, 257, 237

6) The weekly wages(in rupees) of 30 workers in a factory are given below:
630, 636, 690, 610, 635, 637, 639, 646, 698, 690, 620, 660, 633, 634, 656, 646, 605, 609, 613, 640, 686, 635, 637, 679, 640, 668, 690, 607, 640, 690.
Represent the data in the form of a frequency distribution with class size 10.

7) Given marks obtained by some  students in an examination:
3, 25, 48, 23, 18, 14, 10, 47, 38, 45, 11, 20, 40, 39, 36, 37, 34, 35, 5, 6, 17, 21,39, 43,28, 43, 25, 12, 3, 8, 17, 21,, 48, 34, 16, 20, 19, 32, 19,  21, 28, 32, 21, 20, 23.
Arrange the data in ascending order and present it was as a grouped data in:
A) Discontinues Interval form, taking class-intervals 1-10, 11-20, etc
B) Continuous Interval form, taking class- interval 1-10, 10-20 ..

8) The electric bills (in rupees) of houses in a locality are given:
147, 166, 135, 179, 176, 116, 115, 155, 154, 156, 121, 115, 156, 176, 140, 141, 189, 190, 177, 209, 212, 143, 203, 210, 189, 177, 213, 119, 198, 145, 134, 133, 197, 185.
Construct a frequency distribution table with class size 10.

9)  Convert the following distribution from discontinuous to continuous form:
     Marks:         Frequency
      1-10                 7
     11-20                5
     21-30                9
     31-40               11
     41-50                6
     51-60               12
     61-70                4

10) Given the ages(in years) of 360 patients in a hospital.
Age(in years)  Number of patients
10-20                     90
20-30                     50
30-40                     60
40-50                     80
50-60                     50
60-70                     30
Construct the cumulative frequency table for the above data
also Construct
i) Mte than form
ii) Less than form

11) Construct the cumulative frequency table from the frequency table given below:
Class interval        Frequency
0-6                                7
6-12                             11
12-18                             8
18-24                            14
24-30                            12

12) Construct the  frequency table from the cumulative frequency table given below:
Class interval        Frequency
0- 8                               8
8-16                             21
16-24                            26
24-32                            33
32-40                            42

13) Construct a frequency table from the following data:
Age (in years)   No. of students
Less than 10                6
Less than 20               14
Less than 30                30
Less than 40                52
Less than 50                65
Less than 60                70

14) Convert the following distribution to exclusive form:
Class interval        Frequency
30-34                            7
35-39                           10
40-44                            15
45-49                             6
50-54                             2
55-59                             10
Use this table to find:
A) The true class limits of the fourth class interval.
B) The class-boundaries of the fifth Class interval
C) The class mark of the third class interval
D) The class size of the sixth class interval.

SURDS

                    SURDS

                   *********

A)

1) a) Express 5 as a biquadratic surd.

2)Express as pure surd. 

a) 5 ³√3    

b) 3 ³√5

c) ⁵√(128a¹¹b⁶)

3) Express  as complete surd

a) 2x² y ⁵√z⁴  

b) p³ q² ⁴√r³ 

4) Express as mixed surd.  

a) ³√1080 

b) ³√192

c) ⁴√1280

5) Arrange in ascending as  well as descending order of magnitude 

a) √5 , ²√11, 2 ⁶√2

b) ⁴√3, ⁵√4, ¹⁰√12

c) 2√2, 2 ³√2 , 2 ⁶√5

6) Simplify

a) ³√24 - ³√192 + ³√81

b) (√5 +√3)²(4 - √15)

c) 3√48 - 4√75 + 5√192

d) ³√56 - ³√875 + ³√189

e) √(x³y) + √(xy³) + √(xyz²)

f) (1+√2 - √3)(2 + √2 + √6)

g)(√50+√32-√18)/(√75 -3√3 +√12)

7) Express 4 as a cubic surd

8) Express x²y as fifth order surd

9) Find the square of √(x+y)-√(x-y)

10) Find the cube of 2√3 - 3√2

B) Rationalise the denominator

1) 3/√2                                  3√2/2

2) 3/(2+√3).                       3(2- √3)

3) 2/(√5 - √3)                       (√5+√3)

4) 3/(4√5 - 2).             3(2√5 +1)/38

5) 5/(4√5 - 5√3).             (4√5+5√3)

6) (3√2+ 2√3)/(3√2 - 2√3).  (5+2√6)

7) √3/(√2+ √3 - √5).   (2+√6+√10)/4

8) 1/(1 + √2 - √3).         (√2+2+√6)/4

9) 12/(3 +√5 + 2√2).   (1+√5+√2-√10)

10) (√3 - √2)/(√3 + √2).           5-2√6

11) {√(x+a) -√(x-a)}/{√(x+a)+√(x-a)}.                     {x- √(x² - a²)}/a

C) Simplify 

1) 1/(√6 -√5) - 3/(√7 - √2) - 4/(√6 + √2).                                2√2+ √5+ √7

2)√{(√12 -√8)(√3 +√2)}/√{5 + √24}.                     (5√2 - 4√3)

3) (3+√5)/(3-√5)  - (3 -√5)/(3+√5).   3√5

4) (√6 -2)/(√75 - √32 - √48 + √18).  √2/7

5) √5/(√3 +√2) - 3√3/(√2 + √5) + 2√2/(√5 +√3).                          0

6) {√2(2+√3)}/{√3(√3 +1)   - {√2(2 - √3)}/{√3(√3 - 1)}.                 -1)√6

7) {√2(√3+1)(2-√3)}/{(√2 -1)(3√3-5)(2+√2)}.              (2+√3)

8) √(ax)/{√a +√x - √(a+x)} - √(ax)/{√a +√x + √(a+x)}.      √(a+x)

D)

1) If x= √{(√5 +1)/(√5 -1)}, Show that x² - x -1=0

2) If x= √3 / 2 find the value of {√(1+x)+√(1-x)}/{√(1+x)- √(1-x)}

3) If x= {³√(a+b)+³√(a -b)}/{³√(a+b) - ³√(a -b)}, Show that bx³ - 3ax² +3bx - a = 0

4) If x= √2 +√3, show that x⁻² = 5 - 2√6

5) If x= (√3 +1)/(√3 -1) and xy =1, find the value of 

a) (x²+xy+y²)/(x² - xy +y²)

b) x⁴ + x²y² +y⁴

6) If a= √3/ 2, show that √(1+a) +√(1- a) = 2a

7) If x= 2√2/3 find the value of {√(1+x)+√(1-x)}/{√(1+x) - √(1-x)}

8) If x= √{(n-1)/(n+1)}, Show that {x/(x-1)}² + {x-(x+1)}²= n(n -1)

9) If x = 4+ 2√3, find the value of (√x - 2/√x)

10) If x = 4 + √15, Show that √x + 1/√x = √10

11) If x= √3/2 then find the value of (1+x)/{1+√(1+x)} +(1-x)/{1-√(1-x)}

12) If x= {√(a+2b) +√(a-2b)}/{√(a+2b) - √(a - 2b)} , probe that, bx² - ax + b= 0

13) If x={q- √(p² - 4q)}/{q+√(p² -4q)} , show that, (q² - p² +4q)(x²+1) - 2(q² + p² - 4q) = 0

14) If 2x = √3 + 1/√3, prove that √(x² -1)/{x - √(x² -1)} = 1

15) If x= 2+ √3 Find the value of x³- 6x² + 9x +8

16) If x = 3 +√2 Find the value of x⁴ - 5x³ + 2x² + x + 8

17) If x={√(a+2)-√(a-2)}/{√(a+2)+ √(a -2)} and x⁴ - 5x³ + 8x² - 5x +1=0, then show that a= 2 or 3

18) Prove √[-√3 + √{3 + 8√(7+4√3)}] = 2

19) If √(y -z) +√(z -x) + √(x -y) = 0, prove that, x= y = z

20) If 2x = √a + 1/√a, show that √(x² -1)/{x - √(x² -1)}= (a -1)/2

21) If x = 1+ √2 + √3, find the values of 2x⁴ - 8x³ - 5x² + 26x - 28

E) Find the value of

1) √[2 +√{2 + √(2 + ....... up to ∞)}]

2) √[³√{b √(a ³√b ..... up to ∞)}]

3) √[6 + √{6 +√(6+ .....up to ∞)}]

E) Find the square root of .

1) 28 - 6√3          2) 8+ 3√7

3) 5 + 2√6           4) (1/2) (2+ √3)

5) 18 + 6√5         6) 28+ 5√12

7) 17 - 12√2        8) 8 - √15

9) x+y+z+2√(xy + zx)

10) x+ √(x² - y²)

11) a + b + √(2ab + b²)

12) 1+ x² + √(1+ x² + x⁴)

13) (a+b)² - 4(a - b)√(ab)

14) 6+ 4√3            15) √45 + √40

16) √48 - √45

17) 8+ 2√2 - 2√5 - 2√10

18) 11+ 6√2 + 4√3 + 2√6

19) (1/2) (3x+1) - √(2x²- x -6)

20) 9+ 6√3

21) 5 + √6 - √10 - √15

22) √{(x-y)(y-z)} + √{(y-z)(z-x)} + √{(z-x)(x-y)}

23) 2√{(a -b)(b-c)} - 2√{(b-c)(c-a)} - 2√{(c -a)(a-b)}

CARTESIAN CO-ORDINATES XI

DISTANCE FORMULA

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                  EXERCISE-1

                 ---------------------

1) Find the distance between the points:

a) A(2,3) , B(-6,3).                            8

b) C(-1,-1), D(8,11).                       15

c) P(-8, -3), Q(-2, -5).                  2√10

d) R(a+b, a- b), S(a-b), a+b).    2b√2

e) (ap², 2ap) and (aq², 2aq). 

f) (a + p sinx, b + p cosx) and 

(a + q sinx, b+ q cosx).              |p- q|

g) (am², 2am), (a/m², -2a/m).     a(m + 1/m)²

h) Prove that the distance between the points (a cos t, a sin t) and (cos p, a sin p) is |2a sin(k - p)/2|.

i) Find the length of the sides of the triangle whose vertices are A(3,4), B(2, -1) and C(4, -6).    √26,√29,√101

j) A line is of length 10, and one end is at the point (-3,2). If the ordinate of the other end be 10, Prove that the abscissa will be 3 or -9.    

k) Which one among the points (2,3), (-3,1) and (0,4) is nearest to the origin ?                                 (-3,1)

l) Find the distance between the points A(x₁,y₁) and (x₂, y₂), when 

  i) AB is parallel to the x-axis.

  ii) AB is parallel to the y-axis.  

m) A is a point on the x-axis with abscissa -8 and B is a point on the y-axis with the ordinate 15. Find the distance AB.                          17 units

2) Find the distance of the following point  from the origin:          

a) (6, -6).                                  6√2

b) (-7, -24).                                     25   

c) (a+b, a - b).                √{2(a²+b²)}

d) (ap+bq), aq - bp).     √{(a²+b²)(p²+ q²)

e) Show that the distance of the point (a cosx + b sinx, a sinx - b cosx) from origin is √(a²+b²) units

3) a) If the distance between the points (-3,3) and (4, y) be 5√2 units, find the value of y.                  4 or 2

b) If the distance between the points (3,5) and (x, 8) be 5 units, find the value of x.                   7 or -1

c) A point with ordinate 3 lies on the line joining two points (4,1) and (-2,7). Find its abscissa.                  2

d) If the distance between the points P(3,4) and Q(5, k) be √13 units, find the coordinates of Q.       (5,7) or (5, 1)

e) Find the radius of the circle that has its centre at (0, -4) and passes through (√13,2).                            7

4)a) If a point P(x,y) is equidistance from the point A(6, -1) and B(2,3). Find the relation between x, y.   x- y= 3

b) If the points (a,b) and (b, a) are equidistant from the point (x,y), prove that x= y.

c) Find the coordinates of the point which lies on y- axis and is equidistant from the point (2,3) and (-1,2).                                          (0,4)

d) Find a point on the x-axis which is equidistant from the point A(7,6) and (-3, 4).                                 (3,0) 

e) For what value of k will the point (cos 2k, sin 2k) be equidistant from the two axes ?                 nπ/2 ± π/8, where n is zero or any integer.

f) Find a point on the y-axis which is equidistant from A(-4,3), B(5,2).    (0,-2).

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

                  EXERCISE--2

                  -------------------

1) Show that the points are the vertices of an isosceles triangle.

a) (1,4), (4,1), (8,8).

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

c) (3,3), (-3, -5), (-5,-3).

2) Show that the vertices are right angled triangle.

a) (3,4), (-1,7) , (-3, -4).

b) (1,3), (6,5), (5, -7).

c) (-1,3), (0,5), (3,1).

3) Show that the points are the vertices of an isosceles right angled triangle.         

a) A(7,10), B(-2,5) and C(3, -4).

b) (3,1), (9,7), (-3,7).

c) (3,3), (-2, -2), (8, -2).

4) Show that the points are the equilateral triangle

a) A(1,1), B(-1, -1) and C(-√3, √3).

b) (3,2), (1,0), (2- √3, 1+ √3).

c) If (1,1), (-1, -1), (a, -a) are the vertices then the value of a.     ±√3

d) In an equilateral triangle ABC, if the coordinates of the vertices B and C are (2a, 6a) and (2a+ √3 a, 5a) respectively, find the coordinates of the vertex A.     (2a, 4a) or (2a+ √3 a, 7a)

e) If the origin is situated inside an equilateral triangle and the coordinates of two vertices of the triangle be (3,2) and (-3,2), then find the coordinates of the third vertex.      (0, 2- 3√3)

f) Vertices of an equilateral triangle are (a,b), (a+ r cos t, b + r sin t), (a + r cos k, b + r sin k), find the value of | t - k|.                                           π/3

5) Show that the points are the angular parts of rectangle.

a) A(2,-2), B(8,4), C(5,7) and D(-1,1).

6) Show that are the vertices of a square.

a) A(3,2), B(0, 5), C(-3,2) and D(0,-1).

b) (-2,-7), (2,-4), (-1,0), (-5,-3).

c) (2,1), (0,0), (-1,2), (1,3).

7) Show that are the vertices of a parallelogram. 

a) A(1,-2), B(3, 6), C(5,10) and D(3, 2) 

b) (-2,-1), (1,0), (4,3) and (1,2).

c) (-1,2), (6,-3), (4,-10), (-3,-5).

d) (2,4), (3,8), (5,1), (4,-3).

8) Show that the points are the vertices of a rhombus.

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

b) (0,5), (-2,-2),(5,0),(7,7)

c) (0,0), (0,10), (8,16), (8,6).

d) (2,5), (6,8), (9,12), (5,9).

          EXERCISE --3

         ----------------------

** Find the area of:

1) triangle:

a) A(-3, -5), B(5,2) and C(-9, -3).    29

b) (4,4),(3,-16),(3,-2)

c) (a,0), (0,b), (x,y).                1/2(bx+ ay- ab)

d) (a, 1/a), (b, 1/b), (c, 1/c).    

e) (a, b+c), (a, b - c), (-a, c).       |2ac|

f) (0,0), (cos t, sin t), (cos 2t, sin 2t).                                      1/2 |sin t|

g) (a, bc), (b, ca), (c, ab)

h) Sides are y+2x= 3, 4y+ x=5, 5y+3x= 1.                                      3.5

i) (a cos t, b sin t), (a cos k, b sin k) and (a cos g, b sin g).  

j) A(3,4), B(5,2), C(x,y); if the area of the triangle ABC is 3 square units, show that x+ y -10 = 0.

k) The coordinates of the points A, B, C are (6,3), (-3,5), (4,-2) respectively and that of the point P(x,y). Show that the ratio of the areas of ∆ PBC and ∆ABC is |x+ y -2|/7.

l) A and B are two points (3,4) and (5,-2). Find point P such that PA= PB and ∆PAB= 10.                    (7,2)

2) Find the area of quadrilateral whose vertices are 

a) A(-4, 5), B(0,7), C(5, -5) and D(-4, -2).                                                60.5

b) (-2,-3), (6,-5), (18,9), (0,12).    206

c) (1,1), (3,4), (5,-2), (4,-7).        41/2

3) If the area of the quadrilateral whose angular points A,B, C, D taken in order are (1,2), (-5,6), (7,-4), (-2,k) be zero, find k.                       3

4) The coordinates of quadrilateral ABCD are (-3,4),(-1,-2),(5,6) and (x, -4) respectively. If ∆ABD = 2∆ACD, find the value of x.                    14.2

5) Show that the points are collinear:

a) A(-5,1), B(5, 4) and C(10, 7) are collinear.

b) A(3, -2), B(5, 2) and C(8,8) are collinear.

c) (1,2), B(2,-1), (3,-4).

d) (a, b+c), (b, c+a), (c, a+b).

6) If the three points (1,2), (2,4),(t,6) are collinear, find t.               3

7) Find the value of k for which the points A(-2,3), B(1,2) and C(k, 0)  are collinear.            7

8) Find the area of the triangle with Vertices A(3,1), B(2k, 3k), C(k, 2k). Show that the three distinct points A, B, C are collinear when k= - 2.

9) If (a,0), (0,b) and (1,1) are collinear show that 1/a + 1/b = 1.

10) Find the condition of collinearity of the points (a,b), (A' , b') and (a- A' , b- b').

11) The three points P, Q, R are collinear. If the coordinates of the points P and Q are (3,4), (7,7) respectively and PR= 10 units, find the coordinates of the point R.      (11, 10) or (-5,-2)

12) If the points (a,b),(A' , b') and (a- a', b - b') are collinear, show that ab'= a'b.

              EXERCISE-4

               ----------------

1) Find the coordinates of the midpoint of the line joining points

a) (-2,-5), (3,-1).                      (1/2,-3)

b) (0,0),(8,-5).                        (4,-5/2)

c) (-4,3),(6,-7).                           (1,-2)

d) Find the midpoints of the sides of a triangle whose vertices are A(1,-1), B(4,-1), C(4,3).      (5/2,-1), (4,1),(5/2,1)

e) Find the centre if the end points of a diameter are A(-5,7) and B(3,-11).                                    (-1,-2)

f) If M is the midpoint of AB, find the coordinates of:

   i) A if the coordinates of M and B        are M(2,8) and B(-4,19).      (8,-3)

 ii) B if the coordinates of A and M are A(-1,2), M(-2,4).                 (-3,6)

g) The vertices of ∆ ABC are A(-1,3), B(1,1) and C(5,1). Find the length of the median to

   i) AB.                                      √26

  ii) AC.                                         √2

  iii) BC.                                      2√5

h) A circle has its centre at the origin and a radius of √12. State whether each of the following points is on, outside or inside the circle:

  i) (1, -√7).                            Outside

 ii) (3,5).                                 Outside

 iii) (2, 2√2).                                  On

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

j) 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)

2) Find the coordinates of the point which divides the join of:

a) A(-5, 11) and B(4, -7) in the ratio 2 :7.                                           (-3,7)

b) (5,-2), (9,6) in the ratio of 3:1.       (8,4)

c) (-4,4),(1,7) in 2:1 externally.  (6,10)

d) (3,4),(-6,2) in 3:2 internally.   (21,8)

e) (a+b, a-b) and (a-b, a+b) internally and externally in the ratio a: b.            

3) a) In what ratio is the line segment joining the points A(-4,2) and B(8, 3) divided by the y-axis ? Also find the point of intersection.

      1:2, (0, 7/3)

b) Find the ratio in which the x-axis cuts the join of the points A(4,5) and B(-10, -2). Also, find the point of intersection.                    5:2, (-6,0)

c) In what ratio does the point (1,-7/2) divides the join of (-2,-4) and (2,-10/3) ?.                        3:1

d) In what ratio is the line joining the points:

  i) (2,-3) and (5,6) divided by the x-axis.                                              1:2

 ii) (3,-6) and (-6,8) divided by the y-axis ?                                             1:2

e) Find the ratio in which the axes divide the line joining the points (2,5) and (1,9).     5:9 and 2:1 externally

f)  In what ratio is the line joining A(-1,1) and B(5,7) divided by the line x+y = 4.       1:2

g) Find the coordinates of the points of trisection of the line joining the points (2,3) and (6,5).   (10/3,11/3),(14/3,13/3)

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

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

j) The line joining the points (3,2) and (6,8) is divided into four equal parts, find the coordinates of the points of section.        (15/4,7/2), (9/2,5),(21/4,13/2)

k) Determine the coordinates of the vertices of a triangle if the middle points of its sides have the coordinates (2,-3),(4,2) and (-5,-2).   (-7,-7),(11,1),(-3,3)

l) If the point (9,2) divides 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)

m) If the point (6,3) divides the segment of the line P(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)

4) Find the centroid of the triangle:

a) (-4, 6),(2,-2),(2,5).                  (0,3)

5) Find the coordinates of the in-centre of the triangle whose vertices are:

a) (-36,7),(20,7),(0,-8).            (-1,0)

b) (-1,5),(3,2),(-1,-1).           (1/2,3/4)

c) (0,0),(2,0),(0,3).            

d) (0,0),(3,0),(0,4).                      (1,1)

6) Find the ortho-centre of the triangle with vertices

a) (2,3),(4,6),(-1,1).                 (14,-9)

b) (-3,1),(2,4),(5,1).                   (2,4)

c) The ortho-centre of a triangle whose vertices are (-3,1) and (2,4) is (2,4). Find the third vertex of the triangle.                                  (5,-1)

7) Find the circum-centre of the triangle whose vertices are:

a) (-3,0),(3,2),(5,-2).         (6/7,-11/7)

b) (-2,3),(1,2),(4,-1).                 (-3,-5)

c) The coordinates of the circum-centre of a triangle ABC are (8,3). If the coordinates of the vertices A, B ,C are (x, -9),(y, -2( and (-5,3) respectively, then find the values of x and y.          x= 3,13 and y= -4,20

d) Find the circum-centre of the triangle formed by three points (-3,1),(1,3) and (3,0). Find the circum-radius of the triangle.     (-1/16, 1/8); √(2405)/16 units

8) Find the area of ∆ ABC, the midpoints of whose sides AB, BC and CA are D(3, -1), E(5, 3) and F(1, -3) respectively.                              8

9) If the points A(-2, -1), B(1,0), C(x,3) and D(1, y) are the vertices of a parallelogram, find the values of x and y. 4, 2

10) If the coordinates of the points A, B and S are respectively (at², 2at), (a/t², -2a/t) and (a,0), Prove 1/SA + 1/SB = 1/a.

                EXERCISE--5

                  --------------

1) 

   

DIRECTION SENSE TEST (CA)

 DIRECTION SENSE TEST - 1

Here, the Questions consists of a sort of direction puzzle. A successive follow-up of directions is formulated and the candidate is required to ascertain the final direction or the distance between two points. The test is meant to judge the candidate's ability to trace and follow correctly and sense the direction correctly.

* A man is facing west. He turns 45° in the clockwise direction and then another 180° in the same direction and then 270° in the anti-clockwise direction. Which direction is he facing now ?
A) south.                B) northwest
C) west.                  D) Southwest

2) If you are facing northeast and move 10m forward, turn left and move 7.5m, then you are
A) north of your initial position
B) south of your position
C) east of your initial position
D) 12m from your initial position
E) both C and D

3) One day, Ravi left home and cycled 10 km southwards, turned right and cycled 5 km and turned right and cycled 10km and turned left and cycled 10km. How many kilometres will he have to cycle to reach his home straight ?
A) 10km b) 15 km. C) 20. D) 25

4) A child is looking for his father. He went 90 metres in the east before turning to his right. He went 20m before turning to his right again to look for his father at his uncle's place 30m from this point. His father was not there. From there, he went 100m to his north before meeting his father in a street. How far did the son meet his father from the starting point?
A) 80m.B)100m.C) 140m D) 260m

5) Raju faces towards north. Turning to his right, he walks 25m. He then turns to his left walks 30m. Next, he moves 25m to his right. He then turns to his right again and walks 55m. Finally, he turns to the right and moves 40m. In which direction is he from starting point?
A) South-west.     B) South
C) Northwest.      D) southeast

6) Deepa moved a distance of 75m towards the north. She then turned to the left and walking for about 25m, turned left again and walked 80m. Finally, she turned to the right at an angle of 45°. In which direction was she moving finally?
A) Northeast.      B) Northwest
C) southwest.     D) southeast

7) Amisha walks 10km towards North. From there she walks 6km towards south. Then, she walks 3km towards East. How far and in which direction is she with reference to his starting point?
A) 5km west.       B) 7km west
C) 7km east. D) 5km northeast

8) Divesh left for his office in his car. He drove 15km towards north and then 10km towards west. He then turned to the south and covered 5km. Further, he turned to the east and moved 8km. Finally, he turned right and drove 10km. How far and in which direction is he from his starting point?
A) 2km west.      B) 5km east
C) 3km north. D) 6km south. E) n

1) A man is facing south. He turns 135° in the anti-clockwise direction and then 180° in the clockwise direction. Which direction is he facing now?
A) northeast.       B) northwest
C) southeast.       D) southwest

2) A man is facing northwest. He turns 90° in the clockwise direction and then 135° in the anti-clockwise direction.. which direction is he facing now?
A) east. B) west. C) nort. D) south

3) A man is facing towards west and turns through 45° clockwise, again 180° clockwise and then turns through 270° anti-clockwise. In which direction is he facing now?
A) west.                 B) Northwest
C) south.               D) southwest

4) I am facing east. I turned 100° in the clockwise direction and then 145° in the anti-clockwise direction. Which direction am I facing now ?
A) East.                   B) North-east
C) North.                 D) southwest

5) A river flows west to east and on the way turns left and goes in a semicircle round a hillock, and then turns left at right angles. In which direction is the river finally flowing?
A) west.B) east.C) north. D) south

6) You go north, turn right, then right again and then go to the left. In which direction are you now ?
A) north. B) south.C) east.D) west

7) I am standing at the centre of a Circular field. I go down south to the edge of the field and then turning left I walk along the boundary of the field equal to three-eighths of its length. Then I turn west and go right across to the opposite point on the boundary. In which direction am I from the starting point?
A) northwest.            B) north
C) southwest.           D) west

8) Deepak starts walking straight towards East. After walking 75m, he turns to the left and walks 25m straight. Again he turns to the left, walks a distance of 40m straight, again he turns to the left and walks a distance of 25m. How far is he from the starting point?
A) 25m              b) 50m
c)115m             d) 140m    e) none)

9) Kunal walks 10kilograms towards north. From there, he walks 6 kilometres towards south. Then, he walks 3 kilometres towards East. How far and in which direction is he with reference to his starting point?

A) 5 kilometres west
B) 5 kilometres Northeast
C) 7 kilometres east
D) 7 kilometres west

10) Rohan walks a distance of 3km towards north, then turns to his left and walks for 2km. He again turns left and walks for 3km. At this point he turns to his left and walks for 3km. How many kilometres is he from the starting point?
A) 1km. B) 2km. C) 3km. D) 5km

11) Manisha walked 40m towards north, took a left turn and walked 20m. She again took a left turn and walked 40m. How far and in which direction is she from the starting point?
A) 20m east.       B) 29m north
C) 20m south.    D) 100m south
D) none of these

12) Sita walks 14m towards west, then turns to her right and walks 14m and then turns to her left and walks 10m. Again turning to her left she walks 14m. What is the shortest distance (in meters) between her starting point and the present position?
A) 10. B) 24. C) 28.  D) 38

13) A man leaves for his office from his home. He walks towards East. After moving a distance of 20m, he turns south and walks 10m. Then he walks 35m towards the west and further 5m towards the north. He then turns towards East and walks 15m. What is the straight distance (in metres) between his initial and final positions?
A) 0.            B) 5.           C) 10.
D) cannot determined.  E) none

14) Namrata walks towards southeast a distance of 7km, then she moves towards west and travels a distance of 14m. From here, she moves towards northwest a distance of 7m and finally she moves a distance of 4m towards East and stood at this point. How far is the starting point from where she stood?
A) 3m. B) 4m. C) 10m. D) 11m

15) Gopi starts from his home towards west. After walking a distance of 30m, he turned towards right and walked 20m. He then turned left and moving a distance of 10m, turned to his left again and walked 40m. He now turns to the left and walks 5m. Finally he turns to his left. In which direction is he walking now.
A) north.     B) South.   C) east
D) South-west    e) west

16) A rat runs 20' towards East and turns to right, runs 10' and turns to right, runs 9' and again turns to left, runs 5' and then turns to left, runs 12' and finally turns to left and runs 6'. Now, which direction is the rat facing?
A) east b) west c) north d) south

17) Ajit walked 30m towards East, took a right turn and walked 40m. Then he took a left turn and walked 30m. In which direction is he now from the starting point?
A) northeast b) east c) southeast
D) south.  E) none

18) Maya starts at point T, walks straight to point U which is 4ft away. She turns left at 90° and walks to W Which is 4ft away, turns 90° right and goes 3ft to P, turn 90° right and walks 1ft to Q, turns left at 90° and goes to V, which is 1ft away and once again turns 90° right and goes to R, 3ft away. What is the distance between T and R ?
A) 4ft b) 5ft. C) 7ft.  D) 8ft.

19) A villager went to meet his uncle in another village situated 5km away in the northeast direction of his own village. From there he came to meet his
father-in-law living in a village situated 4km in the south of his uncle's village. How far away and in what direction is he now?
A) 3km in the north
B) 3km in the east
C) 4km in the east
D) 4km in the west

20) A person starts from a point A and travels 3km eastwards to B and then turns left and travels thrice that distance to reach C. He again turns left and travels five times the distance he covered between A and B and reaches his destination D. The shortest distance between the starting point and the destination is:
A)12km.B) 15km C)16km d)18km

21) A girl leaves from her home. She first walks 30m in northwest direction and then 30m in southwest direction. Next, she walks 30m in southeast direction. Finally, she turns towards her house. In which direction is she moving ?
A) northeast.     B) northwest
C) southeast. C) southwest e) n

22) Shyam walks 10m towards the south. Turning to the left, he walks 20m and then moves to his right. After moving a distance of 20m, he turns to the right and walks 20m. Finally, he turns to the right and walks a distance of 10m. How far and in which direction is he from the starting point ?
A) 10m north.    B) 20m south
C) 20m north d) 10m south. E) n

23) Kashish goes 30m north, then turns right and walks 40m, then again turns right and walks 20m, then again turns right and walks 40m. How many metres is he from his original position?
A) 0. B) 10. C) 20. D) 40. E) none

24) I am facing South. I turn right and walk 20m. Then I turn right again and walk 10m. Then I turn left and walk 10m and then turning right walk 20m. Then I turn right again and walk 60m. In which direction am I from the starting point?
A) north.                B) northwest
C) east.                  D) northeast

25) A man walks 30m towards south. Then, turning to his right, he walks 30m. Then, turning to his left, he walks 20m. Again, he turns to his left and walks 30m. How far is he from his initial position?
A)20m B) 30m.C)60m D)80m e) n

26) Rohit walked 25m towards south. Then he turned to his left and walked 20m. He then turned to his left and walked 25m. He again turned to his right and walked 15m. At which distance is he from the starting point and in which direction?
A) 35 metres East
B) 35 metres North
C) 40m East. D) 60m East. E) n

27) Starting from a point P, Sachin walked 20m towards south. He turned left and walked 30m. He then turned left and walked 20m. He again turned left and walked 40m and reached a point Q. How far and in which direction is the point Q from the point P ?

A) 20m west.    B) 10m east
C) 10m west. D) 10m north e) n

28) Ramakant walks northwards, After a while, he turns to his right and a little further to his left. Finally, after walking a distance of one kilometre, , he turns to his left again. In which direction is he moving now ?
A) north b) south c) east d) west

29) A man walks 1km towards East and then he turns to South and walks 5km. Again he turns to East and walks 2km, after his turns to North and walks 9km. Now, how far is he from his starting point?
A) 3km  b) 4km. C) 5km. D) 7km

30) Raj travelled from a point X straight to Y at a distance of 80m. He turned right and walked 50m, then again turned right and walked 70m. Finally, he turned right and walked 50m. How far is he from the starting point?
A) 10m.B) 20m.C)50m d)70m e)n

31) Laxman went 15km to the west from my house, then turned left and walked 20kms. He then turned east and moved 25kms and finally left covered 20kms. How far was he from his home
A) 5kms.                       b)10kms
C)40kms                      d)80kms

32) From his house, Lokesh went 15kms to the North. Then he turned West and covered 10kms. Then, he turned South and covered 5kms. Finally, turning to east, he covered 10kms. In which direction is he from his home?
A) east. B) west c) north d) south

33) Going 50m to the south of her house, Radhika turns left and walks another 20m. Then, turning to the north, she walks 30m and then starts walking to her house. In which direction is she moving now?
A) North-west.         B) north
C) southeast.           D) east

34) A walks 10m in front and 10m to the right. Then every time turning to his left, he walks 5, 15 and 15m respectively. How far is he now from his starting point?
A) 5m           b) 10m       c) 15m
d) 20m         e) 23m

35) Amar walks 20m north. Then he turns right and walks 30m. Then he turns right and walks 35m. Then he turns left and walks 15m. Then he again turns left and walks 15m. In which direction and how many metres away is he from his original position?
A) 15m west.            B) 30m east
C) 30m west.            D) 45m east

36) A child is looking for his father. He went 90m in the east before turning to his right. He went 20m before turning to his right again to look for his father at his uncle's place 30m from this point. His father was not there. From here he went 100m to the north before meeting his father in a street. How far did the son meet his father from the starting point?
A) 80m.B)100m.C)140m d) 260m

37) The door of Aditya's house faces the east. From the back side of his house, he walks straight 50m, then turns to the right and walks 50m again. Finally, he turns towards left and stops after 25m. Now, Aditya is in which direction from the starting point?
A) southeast.      B) Northeast
C) southwest.     D) North-west

DIRECTION SENSE TEST - 2

1) Two buses start from the opposite points of a main road, 150kms apart. The first bus runs for 25kms and takes a right turn and then runs for 15kms. It then turns left and runs for another 25kms and takes the direction back to reach the main road. In the meantime, due to a minor breakdown, the other bus has run only 35kms along the main road. What would be the distance between the two buses at this point?

A) 65kms b) 75. C) 80.  D) 85kms

2) X and Y start moving towards each other from two places 200m apart. After walking 60m, Y turns left and goes 20m, then he turns right and goes 40m. He then turns right again and comes back to the road on which he had started walking. If X and Y walk with the same speed, what is the distance between them now?

A) 20m. B) 30m. C) 40m. D) 50m

3) If A is to the south of B and C is to the east of B, in what direction is A with respect to C ?

A) Northeast.      B) North-west

C) southeast.      D) southwest

4) A is 40m southwest of B. C is southeast of B. Then, C is in which direction of A ?

A) east                     b) west 

c) Northeast           d) south

5) There are four towns P, Q, R,T. Q is to the southwest of P, R is to the east of Q and southeast of P, and T is to the north of R in line with QP. In which direction of P is T located ?

A) southeast.        B) north

C) Northeast.        D) east

6) Of the five villages P, Q,R and T situated close to each other, P is to the west of Q, R is to the south of P, T is to the north of Q and S is to the east of T, then, R is in which direction with respect to S?

A) North-west.    B) southeast

C) southwest.   

D) data inadequate.    E) none

7) P,Q,R,S,T,U,V,W are sitting around a round table in the same order, for group discussion at equal distances. Their position are clockwise. If V sits in the north, then what will be the position of S ?

A) east.                   B) southeast

C) south.                D) southwest

9) Ravi wants to go to the University. He starts from his home which is in the east and comes to a crossing. The road to the left ends in a theatre, straight ahead is the hospital. In which direction is the University?

A) north.                 B) south

C) east.                   D) west

10) Of the six members of a panel sitting in a row. A is to the left of D, but on the right of E. C is on the right of X, but is on the left of B who is to the left of F. Which two members are sitting right in the middle?

A) A and C.            B) C and B

C) D and B.            D) D and C

11) A, B, C, D are playing cards. A and B are partners. D faces towards north. If A faces towards west, then who faces towards South?

A) B.                            B) C.

 C) D.         D) Data inadequate

12) P, Q, R and S are playing a game of carrom. P, R and S, Q are partners. S is to the right of R who is facing west. Then, Q is facing

A) north b)south c) east d) west

13) The town of Paranda is located on Green Lake. The town of Akram is west of Paranda. Tokhada is east of Akram but west of Paranda. Kakran is east of Bopri but west of Tokhada and Akram. If they are all in the same district, Which town is the farthest west ?

A) Paranda.           B) Kakran

C) Akram.              D) Bopri

14) Five boys are standing in a row facing east. Deepak is to the left of Sameer, Tushar and Shailendra, Sameer, Tushar and Shailendra are to the left of Sushil. Shailendra is between Sameer and Tushar. If Tushar is fourth from the left, how far is Sameer from the right?

A) first.  B) second. C) third

D) fourth.    E) fifth

15) Two ladies and two men are playing cards and are seated at North, East, South and West of a table. No lady is facing East. Persons sitting opposite to each other are not of the same sex. One men is facing south. Which direction are the ladies facing?

A) East and West. 

B) south and east

C) north and east

D) north and west. E) none

16) The post office is to the east of the school while my house is to the south of the school. The market is to the north of the post office. If the distance of the market from the post office is equal to the distance of my house from the school, in which direction is the market with respect to my school?

A) north.                B) east

C) Northeast.      D) southwest

*** A, B, C, D ,E, F, G, H and I are nine houses. C is 2km east of B. A is 1km north of B and H is 2km south of A. G is 1km west of H while D is 3km east of G and F is 2km north of G. I is situated just in middle of B and C while E is just in middle of H and D.

17) Distance between E and G is

a) 1km. b) 1.5km. c) 2km. d)5km

18) Distance between E and I is:

i) 1km. ii) 2km.  iii) 3km iv) 4km

19) Distance between A and F is:

i) 1km. ii) 1.41km iii) 2km iii) 3km

** On a playing ground, Dinesh, Kunal, Nitin, Atul and Prashant are standing as described below facing the north.

i) Kunal is 40m to the right of Atul

ii) Dinesh is 60m to the south of Kunal.

iii) Nitin is 25m to the west of Atul

iv) Prashant is 90m to the north of Dinesh.

20) Who is to the Northeast of the person who is to the left of Kunal

i) Dinesh.   ii) Nitin.   iii) Atul

iv) Either Nitin or Dinesh. v) n

21) If a boy walks from Nitin, meets Atul followed by Kunal, Dinesh and then Prashant, how many metres has he walked if he has travelled the straight distance all through ?

i) 155m.           ii) 185m.  iii) 215m

iv) 245m.         v) none

*** Seven villages A, B, C, D, E, F and G are situated as follows:

- E is 2km to the west of B

- F is 2km to the north of A

- C is 1km to the west of A

- D is 2km to the south of G

- G is 2km to the east of C

- D is exactly in the middle of B and E.

22) A is in the middle of

i) E and C.             ii) E and G

iii) F and G.          iv) G and C

23) Which two villages are the farthest from one another?

i) D and C.           ii) F and E

iii) F and B.         iv) G and E

24) How far is E from F(in km) as the crow flies?

i) 4.  ii) √20.    iii) 5.    iv) √26

25) Lokesh's school bus is facing North when it reaches his school. After starting from Lokesh's house, it turns right twice and then left before reaching the school. What direction was the bus facing when it left the bus stop in front of Lokesh's house?

A) north.  B) south.    C) east

D) west.      E) none

26) I start from my home and go 2km straight. Then, I turn towards my right and go 1km. I turn again towards my right and go 1km again. If I am North-west from my house, then in which direction did I go in the beginning?

A) north. B) south.     C) east

D) west.     E) southeast

27) After walking 6km, I turned right and covered a distance of 2km, then turned left and covered a distance of 10km. In the end, I was moving towards the north. From which direction did I start my journey?

A) north b) south c)east d) west

28) A postman was returning to the post office which was in front of him to the north. When the post office was 100m away from him, he turned to the left and moved 50m to deliver the last letter at Shantivilla . He then moved in the same direction for 40m, turned to his right and moved 100m. How many metres was he away from the post office?

A) 0. B) 90. C) 150. D) 100. E) n

29) A boy rode his bicycle northwards, then turned left and rode one km and again turned left and rode 2km. He found himself exactly one km west of his starting point. How far did he ride northwards initially?

A) 1km b) 2km c) 3km d) 5km

30) If 'Southeast' is called 'East', 

' North-west is called ' West', ' Southwest' is called ' south' and so on, what will ' north' be called

A) east.            B) Northeast 

c) North-west d) south.   E) none

31) If southeast becomes north, Northeast becomes west and so on, what will west become?

A) Northeast.    B) North-west

C) southeast. D) southwest 

e) south

32) A direction pole was situated on the crossing. Due to an accident the pole turned in such a manner that the pointer Which was showing east, started showing south. One traveller went to the wrong direction thinking it to be west. In what direction actually he was traveling?

33) A watch reads 4.30. If the minute hand points East, in what direction will the hour hand point?

A) North.               B) North-west

C) southeast. D) northeast. E) N

34) It is 3 o'clock in a watch. If the minute hand points towards the Northeast, then the hour hand will point towards the

A) south.                B) southwest

C) North-west.      D) Southeast

35) A clock is so placed that at 12noon its minute hand points towards Northeast. In which direction does its hour hand point at 1.30 p.m ?

A) north b) south c) east d) west

36) If the above clock is turned through an angle of 135° in an anti-clockwise direction, in which direction will its minute hand point at 8.45 p.m ?

A) north b) south c) east d) west

***

 i) Six flats on a floor in two rows facing north and south are allotted to P,Q,R,S,T U

ii) Q gets a north facing flat and is not next to S.

iii) S and U get diagonally opposite flats.

iv) R, next to U, gets a south facing flat and T gets a north facing flat.

37) which of the following combinations get south facing flats ?

i) QTS.  ii) UPT.  iii) URP

iv) data inadequate.   v) none

38) Whose flat is between Q and S 

i) T.              ii) U               iii) R 

iv) P             v) data inadequate

39) If the flats of T and P are interchanged, whose flat will be next to that of U ?

i) P.  ii) Q.  iii) R.      iv) T.   v) none

40) The flats of which of the other pairs than SU, are diagonally opposite to each other?

i) QP.  ii) QR.  iii) PT iv) TS v) None 

41) To arrive at the answers to the above questions, which of the following statements can be dispensed with?

A) none B) (i) only C) (ii) only D) (iii) only E) none of these.

42) A man is performing yoga with his head down and legs up. His face is towards the west, in which direction will his left hand be ?

A) north b) south c) east d) west

43) One morning after sunrise, Gopal was standing facing a pole. The shadow of the pole fell exactly to his right. Which direction was he facing?

44) One morning after sunrise, Reeta and Kavita were talking to each other face to face at Tilak Square. If Kavita's shadow was exactly to the right of Reeta, which direction Kavita was facing?

A) north        b) south.   C) east

D) data inadequate.   E) none

45) One morning after sunrise, Vikram and Shailesh were standing in a lawn with their backs towards each other. Vikram's shadow fell exactly towards left hand side.  Which direction was Shailesh facing?

A) east b) west c) north d) south

46) One evening before sunset two friends Sumit and Mohit were talking to each other face to face. If Mohit's shadow was exactly to his right side, which direction was Sumit facing ?

A) north.   B) south.   C) west

D) data inadequate.  E) none

47) Anuj started walking positioning his back towards the sun. After sometime, he turned left, then turned right and then towards the left again. In which direction is he going now ?

A) North or South

B) east or west

C) north or west

D) south or west

DETERMINANT (A- Z)

EXERCISE - A

A) Prove by property :

1) 13       33         55 

     11       31         53      = 0

     16       36         58

2)   7         3           4

      5          2          3  =0

      4          3          1

3) 19        25       26

     20        25       27      = 0

     21        25       28

4) 91       92        93

     94       95        96       = 0

     97       98        99

5) 265     240      219

     240     225      198      = 0

     219     198      181 

) 43      1       6

   35     7       4 = 0

   17     3       2

6) a - b   b - c      c - a

     b - c   c - a      a - b      = 0

     c - a   a - b      b - c 

7)   x+1       x+2      x+a

       x+2       x+3      x+b        = 0

       x+3       x+4      x+c
Where a, b, c are in A. P

8)     0         x - y      x - z
       y - x        0         y - z       = 0
       z - x      z - y        0

9)     1           x           y+z

        1           y           z+x       = 0

        1           z           x+y

10)     1          bc       a(b+c)
           1          ca       b(c+a)    = 0
           1          ab       c(a+b) 

11)   x+a       x+b        x+c
         y+a        y+b       y+c      = 0
         z+a        z+b       z+c

12)  a+2b     a+4b     a+6b
        a+3b     a+5b     a+7b     = 0
        a+5b     a+7b     a+9b

13)    0         (a - b)³    (a - c)³

      (b -a)³         0        (b - c)³    = 0

      (c - a)³    (c - b)³        0

14)   z           0               y

        0          - z               x        = 0

      - x          - y               0

15)  b² - ab      b- c         bc - ca

       ab - a²      a - b         b² - ab     = 0

       bc - ac      c- a         ab - a²

16)   1           bc         a(b+c)

         1           ca         b(c+a)    = 0

         1           ab         c(a+b)

17)   1           bc        bc(b+c)

         1           ca        ca(c+a)   = 0

         1           ab        ab(a+b)

18)  b²c²       bc         b+c

        c²a²       ca         c+a      = 0

        a²b²       ab         a+b 

19) a) If x+y+z= 0 then show that

           x         y          z

           x²       y²         z²           = 0

         y+z    z+x       x+y

b)       1         1          1

           x         y           z =0

           x³       y³          z³

20) 1       a         a²- bc

       1       b         b²- ca = 0

       1       c         c²- ab 

21) cosA       sinA        sin(A+D)

       cosB       sinB        sin(B+D)   = 0

       cosC       sinC        sin(C+D) 

22) y+x        y-x      y-x 

       y-x         y+x      y-x  =0

       y-x         y-x      y+x

EXERCISE - B  

1)  13        3          23
      30        7          53      =1
      39        9          70

2)   1²        2²         3²
       2²        3²         4²     = - 8
       3²        4²         5²

3)   1!         2!         3!
       2!         3!         4!     = 24

       3!         4!         5!

4)     2          3         - 4

        1          3         - 1      = 51

        3          1           4 

5) 1     374    1893

    1     372    1892   = 1

    1     371    1891 

6)  3      21          4

    15     292      14

    16    193       17    =    241

    38     398      38

EXERCISE - C 

1)   1           1          1
       1        1+x        1      = xy
       1          1        1+y 

2)    3x+y         2x          x

       4x+3y       3x         3x       = x³

       5x+6y       4x         6x 

3)    - x²        xy         xz

         xy       - y²        yz     = (2xyz)²

         xz        yz       - z² 

4)      a+b            a            a

       5a+4b        4a          2a        = a³

     10a+8b        8a          3a 

5) a+d      a+d+k   a+d+c

      c           c+b          c       = abc 

      d           d+k        d+c 

6)    b+c        a            a

        b          c+a          b          = 4abc

        c            c          a+b 

7) x²+y²+1  x²+2y²+3  x²+3y²+4

         y²+2        2y²+6       3y²+8  = x²y²

         y²+1        2y²+3       3y²+4 

8)      a²             bc        c²+ac

      a²+ab          b²           ac   = 4a²b²c²

         ab          b²+bc       c² 

9)    b+ c      a  - c       a -  b

       b - c       c + a       b - a      = 8abc

       c - b       c -  a       a + b

10) 1+a     1        1

          1     1+b     1 = ab + bc + ca + abc

          1       1     1+c

11) -bc      bc+b²     bc+c²

    ca+a²     -ca        ca+a² =(bc + ca + ab)³

    ab+a²    ab+b²     -ab

12) (a²+b²)/c          c             c

           a             (b²+c²)/a      a       =  4abc

           b                   b       (c²+a²)/b

EXERCISE - D 

1)  1 + a         b          c

         a       1 +  b       c      = 1+a+b+c

         a          b        1 +c

2)    a          b         0
        0         a          b       = a³+b³
        b         0          a 

3) b+c     a+b        a

     c+a      b+c        b     = a³+b³+c³

     a+b      c+a        c 

4)     a²      2ab        b²

        b²        a²        2ab     = (a³ + b³)²

      2ab        b²         a² 

5)1+a            b            c

     a            1+b          c = = 1+a+b+c

     a               b         1+c 

6) a²+1        ab           ac   

     ab        b²+1         bc.   =1 + a² + b² + c²

     ac          bc         c²+1

EXERCISE --E

1) a+b+2c         a                 b

         c           b+c+2a          b =  2(a+b+c)³

         c               a            c+a+2b 

2) x+y+2z         x          y

      z          y+z+2x     y = 2(x+y+z)³

      z               x      z+x+2y

5) (y+z)²     x²        x²

       y²      (z+x)²     y² = 2xyz(x+y+z)³

      z²          z²      (x+y)²

2) a(1+x)       b             c

        a         b(1+x)       c     = abcx²(x+3)
        a             b       c(1+x) 

3) (b+c)²       b²         c²

        a²     (c+a)²       c². = 2abc(a+b+c)³

        a²       b²       (a+b)² 

4) x+4           x           x

         x         x+4        x   = 16(3x +4)

         x          x         x+4

5) b+c         a           b

    c+a         c            a = (a+b+c)(a - c)²

    a+b          b           c 

6) a+b+2c         a               b

        c            b+c+2a        b.    = 2(a+b+c)³

        c                a         c+a+2b 

7) a²+x       ab         ac

      ab        b²+x       bc   =x²(x+ a²+b² + c²)

      ac          bc       c²+x 

8)  a-b-c           2a          2a

        2b          b-c-a        2b = (a+b+c)³

        2c             2c        c-a-b   

EXERCISE -F

1) 1       a        bc

    1        b       ca        = (a - b)(b - c)(c - a)

    1        c       ab

2) 1      1        1

    a       b        c = (a - b)(b - c)(c - a)

    a²      b²      c²

3) 1       1        1

     a       a²      a³  = ab(a - 1)(b - 1)(b-a)

     b       b²      b³

4)1     x     x³

   1     y     y³.  = (x - y)(y - z)(z - x)(x+y+z)

   1     z     z³

5)   a       b - c     c - b

     a - c      b       c - a

     a - b    b - a       c

= (a+b - c)(b+c - a)(c +a - b)

6)  a      a²      b+c

      b      b²      c+a

      c      c²       a+b

= (b - c)(c - a)(a - b)(a+b+c) 

7) ) 1       1         1

       a       b         c.   = (a-b)(b-c)(c-a)

       a²     b²        c²

8) a      b       c

    a²     b²     c²   = abc(a - b)(b - c)(c - a)

    a³     b³     c³

9) 1     x      x³

     1     y      y³= = (x-y)(y-z)(z-x)(x+y+z)

     1     z      z³

10) 1      a      bc

       1      b      ca.  = (a-b)(b- c)(c - a)

       1      c      ab

11) 1     b+c   b²+ c²

       1     c+a   c²+a².  = (a - b)(b - c)(c -a)

       1     a+b   a²+b²

12) a+b+ c        - c             - b

          - c         a+b+c          - a

          - b            - a          a+b+c

= 2(a+b)(b+c)(c+a).

13) bc          a           a²

       ca          b           b²

       ab          c           c²

= (a-b)(b-c)(c-a)(ab+bc+ca)

14)     a²         bc.      (b+c)²

           b²         ca.      (c+a)²

           c²         ab.       (a+b)²

=(a-b)(b-c)(c-a)(a+b+c)(a²+b²+c²)

15)  x          y- z        z -y

      x-z           y           z-x

     x- y          y-x          z

= (x+y-z)(y+z-x)(z+x-y)

16) a+x              x              x

      x                 b+x            x

      x                   x             c+x

= abc(1 + x/a + x/b + x/c) 

17) sin²A   sinA    cos²A

      sin²B   sinB     cos²B

      sin²C   sinC    cos²C   

= - (sinA-sinB)(sinB- sinC)(sinC- sinA) 

18) b+c         a        b

       c+a         c        a  =(a+b+c)(a-c)²

       a+b         b        c 

19)

   

EXERCISE - G

17) bc   a²    a²         bc   ab   ca
       b²   ca    b²    =  ab   ca    bc
       c²    c²   ab        ca    bc    ab

2)   bc     a      a²           1      a²      a³

      ca      b     b²   =    1       b²      b³ 

      ab      c     c²          1      c²       c³

3)   1      a      a²          1    a     bc 

      1       b     b²    =    1    b    ca

      1       c     c²          1    c     ab 

4) x - y      1       x         x        1         y

     y - z      1       y    =   y        1        z

     z - x      1       z         z        1        x

5) b+c      a       b         a       b        c
    q+r       p       q    =   p       q        r
    y+z       x       y          x       y        z 

6) b+c   c+a    a+b            a        b        c

     q+r   r+q     p+q = 2     p        q         r

     m+n  n+l     l+m           l        m        n

7)   x - y      1       x             x       1        y

       y - z      1      y     =      y       1         z

       z - x      1     z             z        1         x 

8) a -b+c    a+b-c     a-b-c          a    c      b

     b-c+a     b+c-a     b-c-a = 4.   b    a      c

     c -a+b    c+a-b     c-a-b          c    b     a 

9) x      y       z           1       1       1

    x²     y²      z²    =   x²      y²     z²

   yz     zx      xy         x³     y³      z³

= (x-y)(y-z)(z-x)(xy +yz+ zx) 

9)1     2     3            1      2      3

    2     3    4    =      1      1      1

    3     4    5           1       0     -1

10) a₁x+b₁y+c₁z   b₁   c₁          a₁    b₁    c₃ 

      a₂x+b₂y+c₂z   b₂   c₂ = x     a₂    b₂   c₂ 

      a₃x+b₃y+c₃z   b₃   c₃           a₃    b₃   c₃

21) b+c  c+a   a+b                a       b        c 

       q+r   r+p    p+q    = 2      p      q         r 

      m+n  n+l   l+m                 l       m       n

22) a   b+c   a-b             a       b        c

       b   c+a   b-c  =         b       c        a

       c   a+b   c-a             c       a        b

23)

EXERCISE - H

1)  1          a          a²
      a²        1           a = (a³ - 1)²
      a         a²          1

2).   a²         2ab          b²
       b²           a²         2ab
     2ab          b²           a²
is a perfect square quantity

3)   a²          2a           1
      1             a²         2a
     2a            1           a²
is a perfect square.

4)  a       b         c
      b       c        d     =  ax+b          bx+c
      1      -x        x²        bx+c         cx+d

5) 1+a² - b²         2ab             -2b
         2ab          1-a²+b²          2a
          2b               -2a         1-a²- b²
= (1+ a² + b²)³

6) (1+ 1/a+ 1/b+ 1/c) is a factor of
       1+a       1          1
         1        1+b       1
         1         1         1+c

7) (a - b) is a factor of
         1          1          1
         a           b         c
         a²         b²        c²

8) (a+b+c) is a factor of
         a           b         c
         b           c         a
         c           a         b

9) x³ is a factor of
        x+y        x          x
       6x+4y   2x        4x
       7x+6y   3x        8x

10) (a - 1)² is a factor of

       a+1       2          2
         3        a+2       3
         4          4        a+3 

11) a          b           c

      b          c           a

      c          a            b

= (a+b+c)(ab+ bc+ ca - a² - b² - c²)

11) b+c     a - b        a

      c+a      b - c        b  = 3abc - a³ - b³ - c³

      a+b      c - a        c

EXERCISE -I

1) If a+b+c= s, prove

         s+c     a           b
           c     s+a         b        = 2s³
           c       a         s+b 

2) If p²= a²+b²+c² and q²= ab+ bc + ca, then show that
  p²       q²        q²
  q²       p²        q² = (a³+b³+ c³ - 3abc)²
  q²       q²        p² 

      

3) If x≠ y≠ z and  x      x²       1+x³
                              y      y²      1+y³ = 0
                              z      z²      1+z³
Prove that xyz = - 1

4) If a ≠ b ≠ c and a       a³      a⁴ -1
                                b      b³      b⁴ -1 = 0
                                c      c³      c⁴ -1
Then show that  a b+ c= abc(ab+bc+ca).

5) If a≠ b≠ c and  a       a²      a³ - 1
                              b       b²       b³- 1 = 0
                              c       c²      c³ - 1
Show that abc = 1

6) If x³= 1 show that   a       b        c
                                       b      c        a
                                       c      a        b
= (a+ bx+ cx²)   1     b     c
                            x²    c     a
                            x     a      b

7)    a²        a          b
        b²        b         ca
        c²        c         ab
= (ab + bc+ ca)    a²       a         1  
                               b²      b         1
                               c²      c         1 

EXERCISE - J

A) Show the the determinant is independent of x: 

1)      x+1         x+2          x+4
          x+3         x+5          x+8
          x+7        x+10       x+14

2)       x²           (x - 1)²          (x - 2)²
       (x -1)²        (x - 2)²          (x - 3)²
       (x -2)²        (x - 3)²          (x - 4)²

3) x,y,z are different show 1+xyz=0

     x     x²    1+x³

     y     y²    1+y³   = 0

     z     z²   1+z³

4) if a≠b≠c and  a      a³       a⁴-1

                            b       b³      b⁴-1  =0

                            c       c³      c⁴ -1    show a+b+c = abc(ab+bc+ca)

5) (b+c)²    b²       c²

      a²      (c+a)²     c²    =2abc(a+b+c)³

      a²         b²     (a+b)²

26) ˣC₁       ˣC₂         ˣC₃

       ʸC₁        ʸC₂        ʸC₃      evaluate

       ᶻC₁        ᶻC₂         ᶻC₃

EXERCISE - K

SOLVE FOR X:

1)   x      a       b
      a      x        b       = 0
      a      b        x

2)  x      1        1
     1       x        1 = 0
     1       1        x

3) x+1      2          3
      3       x+2        1     = 0
      1         2        x+3

4)     a - x       c         b
           c       b - x      a        = 0
            b        a        c - x

5)   x²      x        1
      a²      a        1 = 0 . (a ≠ b)
      b²      b        1

6) 15 - x         11         10
     11 - 3x      17          16    = 0
       7 - x         14          13

7)  3 - 2x         2            6
      4 - x           4           12   = 0
      1 - x           1             4

8)   x+a    b
        a     x +b   = 0

9)    1      1       1
        p      x       p       = 0
        q      q       x

10) 1 - 5x      2       3
        4x-2       1       2        = 0
        2x-1       3       1

11) x+a        b       c
         c        x+b     a           =0
         a           b     x+c 

12) 

EXERCISE - L

Solve with the help of Cramer's Rule:

Type-1

1) 3x+y =5, x+2y =3.

2) ax+ by=c, a²x +b²y =c²

3) 3/x - 5/y =1 , 2/x +3/y = 7

4) a/x -b/y =a , a/y - b/x =b 

5) x cosβ - y sin β = cosθ ; x sin β+ y cos β = sinθ 

6) (a+b)x - (a-b)y= 4ab;

(a-b)x + (a+b)y = 2(a² -b²) 

7)

Type -2

1) x+y+z=4, 2x-y+2z=5 , x-2y-z=-3

2) 2x +3y - z=1; x + 2y +z= 4 , x - y + z = 5

3) 2y-3z=0, x+3y= -4, 3x+4y =3

4) x+y-6z=0, -3x+y+2z=0, x-y+2z=0

5) 1/x +2/y+ 3/z=2 , 

    2/x +4/y +5/z =3 , 

     3/x +5/y+6/z= 4

6) 2/x +3/y +2=0, 

    5/y - 2/z -4 =0 , 

    3/z +4/x +7=0

       

EXERCISE- M

** determine whether each system is consistent :
1) 2x - 3y + z =1
    x+2y - z =1
    3x - y +2z =6

2) 2x - y + z = 2
    3x + 2y - 4z =1
    x - 4y +6z = 3

3) x +y = 2
    2x - z =1
    2y -3z =1

4) 4x - 3y +1=0 , 7x-8y+10=0,x+y=5

** Find k if:

1) 2x-y+3=0, kx-y+1=0, 5x-y-3=0 are consistent.

2) 2x+3y+4=0, 3x+4y+6=0,  
    4x+5y-k=0 are consistent.

3) x+y-3=0, (1+k)x +(2+k)y -8 =0
    kx -(1 +k)y +2 =0

4) Find the relation that must exist between a,b,c, if the are consistent
ax+by+c=0, bx+cy+a=0, cx+ay+b=0

5) If the following equatiin are consistent, prove that a=b=c,
    (x-y)(y-b)=ab,
    (x-b)(y-c)=bc,
    (x-c)(y-a)=ca.

6) Obtain the condition of consistency in the form of,
2x+3y-8=0, 7x-5y+3=0, 4x - 6y+β=0 and hence find the value of β.

7) If the equations are consistence and have more than one solution, find the values of β.
u+v = -(βv +1),
u+2v= -β(v-1) +1
3u +8v = β +2

  

Mg. A- R.1

** Find the value of:

1) 23     17     16

     27     20     78

     54     40    157

a) 0 b) 1 c) -1 d) 132 e) 12

2) 29     26       22

     25     31       27

     63     54       46

a) 0 b) 1 c) -1 d) 132 e) 12 

3) a     h      g

     h     b      f

     g     f      c 

a) 0 b) 1 c) -1 d) 132 e) none 

4) b+ c       a - b       a 

    c+a         b - c       b 

   a+ b        c- a        c 

a) 0 b) 1 c) -1 d) 132 e) none 

5) 1         1            1

   -a         -b           -c

  b+ c    c+ a       a+ b 

a) 0 b) 1 c) -1 d) 132 e) 12

6) a⁻¹      a         bc

    b⁻¹       b         ca

    c⁻¹       c         ab 

a) 0 b) 1 c) -1 d) 132 e) 12

7)   x           x+ a         x+ 2a 

   x+ 2a           x           x+ a 

    x+ a         x+ 2a         x 

a) 0 b) 1 c) -1 d) 13 e) none 

Mg. A- R.2 

1) (x+1)(x+2)         x+2          1

    (x+2)(x+3)          x+3         1

     (x+3)(x+4)         x+4          1

a) 9 b) 9a² c) a+ x d) 9a²(a+ x) e) -2

2) a      a+ b           a+ b + c 

    2a    3a+ 2b      4a+ 3b + 2c

    3a    6a+ 3b      10a+ 6b + 3c

a) a b) a² c) a³ d) -a e) -a²

3) 2a - b- c     3b                    3c

       3a          2b - c - a            3c

       3a           3 b              2c - a - b 

a) (a+ b+c) b) (a+ b+c)² c) (a+ b+c)³ d) none 

4) If x+ y+ z=0, then 1       1     1

                                   x        y      z

                                   x³      y³     z³ is 

a) 0 b) 1 c) -1 d) xyz e) none 

5) y+ z     x         x

       y      z+x       y

       z       z       x+ y 

a) xyz b) -xyz c) 2xyz d) -2xyz e) 4xyz 

6) b²+ c²        a²            a² 

         b²       c²+ a²         b² 

         c²          c²        a²+ b²

a) abc b) a²b²c² c) 2a²b²c² d) 4a²b²c²

7) solve : x     3      7

                 2     x      2 = 0

                 7     6      x 

a) -9 b) 2 c) 7 d) all of these

8) x+ a         b           c 

      a          x+ b         c  = 0 . Solve for x

      a            b         x+ c 

a) 0,0,(a+b+c) b) 0,1, a c) 0, 1 b d) 0,0,- (a+b+c)

Mg. A- R.3 

1) solve by Cramer's Rule: 3x+ 4y =5, x - y = -3.

2) a+1        a         a(a²+1) 

    b+1         b        b(b²+1) 

    c+1         c        c(c²+1) 

= (a- b)(b - c)(c- a)(a+ b+ c)

3) (b+c)²          a²           a²

      b²            (c+a)²       b²

      c²               c²        (a+b)²

= 2abc(a+ b+ c)³

4) 1- x        a         a²

      a         a²- x     a³ 

      a²         a³       a⁴ - x

= x²(1+ a²+ a⁴) - x³.

5) Solve for x :.   x³- a³      x²     x 

                             b³- a³      b²    b = 0

                             c³- a³     c²     c       

                                                  b, c, a³/bc 

6)      x          c+ x         b+ x 

       c+ x         x            a+ x = 0

       b+ x      a+ x           x 

                   2abc/(a²+b²+c²- 2bc-2ca-2ab)