PARABOLA A-- Z

EXERCISE - A

1) Find the vertex, focus, length of the latus rectum, the equation of the axis and the the directrix of each of the following parabolas:

a) 3y²= 5x.                   (0,0),(5/12,0),5/3,y= 0, 12x+5= 0

b) 2x² + 3y= 0.         (0,0),(0,-3/8),3/2,x=0, 8y -3= 0

c) (y -3)²= 6(x -2).               (2,3),(7/2,3), 6,y=3, x= 1/2

d) (x-3)²+ 8(y+1)=0.      (3,1),(3,-3), 8,x=3, y= 1

e) y²+12 = 4x+ 4y.   (2,2),(3,2), 4,y=2, x= 1

f) 4y² - 20x - 8y + 39= 0.         (7/4,1),(3,1), 5,y=2, x= 1/2

g) y= x²- 3x +4.          (3/2,7/4),(3/2,2), 1, 2x-3= 0, 2y-3= 0

h) 2y²- 4y +5 = x.       (3,1),(25/8,2), 1/2 ,y=1, 8x -23= 0

2) a) Find the point on the parabola y²= 12x at which the ordinate is double the abscissa.                 (3,6)

b) Find the point on the parabola y²= -36 x at which the ordinate is three times the abscissa. (-4,-12)

3) Find the coordinates of focus and the length of latus rectum of the parabola y²= 2mx which passes through the point of intersection of the straight line x/a + y/b =1 and x/b + y/a =1.          

4) The parabola x²+ 2py =0 passes through the point (4,-2); find the coordinates of focus and the length of latus rectum.      (0,-2); 8

5) Find the coordinates of focus and the length of latus rectum of the parabola y²= 2ax which passes through the point of intersection of the straight line x/3 + y/2 =1 and x/2 + y/3 =1.       

6) Find the coordinates of focus and the length of latus rectum of the parabola y²= 4ax which passes through the point of intersection of the straight line 3x + y =-5 and x + 3y =1.      

7) The parabola y²= 4ax which passes through the centre of the circle 2x²+ 2y²- 4x +12y-1 =0. Find the coordinates of the focus , length of the latus rectum.    (9/4,0)9, 4x+9=0

8) The parabola y²= 4ax which passes through the centre of the circle x²+ y²+4x -12y-4 =0. Find the length of the latus rectum. 18

9) Find the focal distance of a point on the parabola x²= 8y if the ordinates of the point be 11.              13

10) Find a point on the parabola y²= 8x whose focal distance is 4. If the focal distance of a point on the parabola be 8, then find the co-ordinate of that point. (2,±4),(6,±4√3)

11) The focal distance of a point on the parabola y²=12x is 6; find the coordinates of the point.          (3,6) and (3,-6)

12) find the point on the parabola y²= 16x whose distance from the dielectrix is 6 units. (2,±4√2)

13) ) find the equation of the parabola passing through (8,2) whose vertex is at origin and axis is y-axis. Find the distance of that point from the focus of the parabola.      x²= 32y

14) the parabola x²= 4ay passes through the centre of the circle x²+ y²-8x +4y -3= 0. Find the coordinates of the focus and the length of the latus rectum of the parabola.            (0,-2),8 units

15) The parabola passes through y²= 2ax passes through the centre of the circle 4x²+ 4y² - 8x +12y-7=0. Find the focus, length of the latus rectum and the equation of the directrix of the parabola.     (9/16,0),9/4, 16x+9= 0

16) The parabola y²= 4ax passes through the point of intersection of the line 3x+ y+5= 0 and x+ 3y-1= 0. Find the coordinates of the focus and the length of the latus rectum of the parabola.      (-1/8,0),1/2

17) A parabola passes through the point (3,2) and (-2,-1) and its axis is the x-axis. Find the equation of the parabola. 5y²= 3x +11

EXERCISE - B

1) Find the equation of the parabola whose coordinates of vertex  the focus are (-2, 3) and (1, 3) respectively.           y²- 6y - 12x =15.

2) Find the equation of the parabola whose vertex is at (1,2) and focus is at (-1,2). 

        (y-2)²+8(x- 1) = 0

3) The coordinates of the vertex and focus are respectively (2,3) and (2,-1). Show that the question of the parabola is x²- 4x +16y =44.   

4)  Find the equation of the parabola whose co-ordinates of vertex and focus (0,0) and (3/2,0) respectively.           y²= 6x

5) the vertex of a parabola is at the origin and its focus is (0, -5/4); find the equation of the parabola.       x²+ 5y=0

6) A parabola having vertex at the origin and axis along x-axis passes through (6,-2); find the equation of the parabola.     3y²= 2x

7) The axis of a parabola is along y-axis and vertex is (0,0). If it passes through (-3,2). Find the co-ordinates of its focus.     x²+ 16y=0.

8) Find the equation of the parabola whose vertex is at (-2,3) axis is parallel to x-axis and length of lactus rectum is 12.          (y +3)²= 12(x -2)

9) The coordinates of the vertex and focus of a parabola are (1,2) and (-1,2) respectively; find its equation.       (y -2)²+ 8(x +1)=0

10) Show that the equation of the parabola whose vertex is (2,3) and focus is (2,-1) is x²- 4x + 16y = 44.

EXERCISE - C

1) Determine the positions of the point

a) (3,6) b) (4,3) c) (1,-3) with respect to the parabola y²= 9x.      Outside, Inside, on

2) Examine with reasons the validity of the following statement: "The point (4,3) lies outside the parabola y²= 4x but the point (-4,-3) lies within it".

3) For what values of a will the point (8,4) be an inside point of the parabola y²= 4ax ?    a> 1/2

EXERCISE - D

1) Find the equation of the parabola whose vertex is at (4,-2), length of the latus rectum is 8 and axis is y+ 2=0.           (y+2)²= ± 8(x -4) 

2) Find the equation of the parabola whose vertex is at (-2,3) and the equation of the directrix is x +7=0.         (y-3)²= 20(x +2)

3) Find the equation of the parabola was vertex is (0,0) and directrix is the line x + 3 = 0.          y²= 12x

4) Find the equation of the parabola whose vertex is at the origin and directrix is the line y - 4 =0.      x²+ 16y=0

5) Find the equation of the parabola whose focus is at the origin and the equation of the directrix is x + y=1.           x²- 2xy + y²+ 2x + 2y -1=0

6) Find the equation of the parabola and the coordinates of its focus if the vertex of the parabola is at (-1,1) and the directrix is x + y + 4=0.     (0,0), x²- 2xy + y² -8x -8y -16 =0

7) Find the vertex, length of the latus rectum and the axis of the parabola whose focus is at the point (3,4) and directrix is 3x + 4y +25=0.       (0,0),20 units, 4x = 3y

8) Find the equation of the parabola whose focus is (2,1) and whose directrix is 3x - y+ 1 = 0.     x²+ 9y²+ 6xy - 46x -18y +49=0.

9) The equation of the directrix of a parabola x= y and the coordinates of its focus are (4,0). Find the equation of the parabola.     x²+ y²+ 2xy - 16x + 32 =0.

10) Find the co-ordinates of vertex and the length of latus of the parabola whose focus is (0,0) and the directrix is the line 2x + y=1.       (1/5,1/10) and 2/√5

*11) Find the equation of the parabola whose coordinates of vertex are (-2,3) and the equation of the directrix is 2x + 3y+8=0.           

*12) t6he directrix of the parabola is x + y + 4 = 0 and vertex is the point (-1,-1). Find 

a) the position of focus.              (0,0)

b) the equation of the parabola.     x²+ y² - 2xy - 8x -8y -16 =0.

EXERCISE - E

1) Find the equation of the parabola whose vertex is (-1,3) and focus is (3,-1).       2{(x -3)²+ (y +1)²}= (x - y +12)²

2) 

) Find the equation of the circle whose diameter is the line segment joining the focus of the parabola y²= 12x and the centre of the circle x²+ y²-18x- 16y+45= 0.           x²+ y²-12x- 8y+27=0

) Find the equation of the circle whose diameter is the latus rectum of the parabola y²-4ax = 0 and prove that the circle passes through the point of intersection of the axis and the directrix of the parabola.   x²+ y²-2ax- 3a²=0

) Find the equation of the circle passing through the origin and through the foci of the two parabolas y²= 8x and x²= 24y.      x²+ y²-2x- 6y=0

) Find two points in the parabola x² = 8y, each of which is at a distance of 4 units from the focus. Find also the equation of the circle whose diameter is the line segment joining the two points.   (4,2),(-4,2) , x²+ y²- 4y-12=0

) A circle is drawn through the vertex and the two ends of the latus rectum of a parabola. If the length of the latus rectum be 2m, find the radius of the circle.        5m/4

) if the vertex and the focus of a parabola are on the x-axis and at distance of a and b respectively from the origin, then show that the equation of the parabola is y²= 4(b - a)(x - a).

) The vertex of a parabola is at (-4,2), axis is parallel to y-axis and length of the latus rectum is 8; find the equation of the parabola.      (x+4)²= 8(y -2) or (x +4)²+8(y -2)=0

) A parabola has its axis parallel to x-axis and passes through the points (2,0),(1,-1) and (-2,-6). Find the equation of the parabola.       15x= y²+16y +30

) Find the equation of the parabola passing through the point (3,0),(- 3,0)( 2,5) and having its axis parallel to the y-axis. Also find the coordinate of its vertex.      x²+ y =9,(0,9)

) The equation of the directrix of a parabola is x= y and the co-ordinates of its focus are (4,0); find the equation of the parabola.      x²+ 2xy + y²-16x + 32=0

) Find the equation of the parabola whose focus is at (5,3) and vertex is at (3,1).     x²- 2xy + y²- 20x - 12y +68 =0

) The coordinates of the focus and the vertex of a parabola are respectively (6,2) and (4,3). Find the equation of the parabola.       x²+ 4xy + 4y²- 60x - 20y +200 =0 

) The vertex of a parabola is at (2,-3) and the equation of the latus rectum is x - y +5=0; find the equation of the parabola.      x²+ 2xy + y²+ 42x -38y -199 =0

) if the coordinates of one end of the latus rectum of a parabola are (4,-1) and the co-ordinate of the point of intersection of the axis in the lactus rectum are (4,3), find the equation of the parabola.       y² -8x -6y+25 =0 or y²+ 8x -6y -39 =0

) Prove that, if the coordinates of the point are x= 1 - 2t, y= 3t²- 2 (t is parameter), the locus of the point is a parabola. Find the vertex and the length of the latus rectum of the parabola.      (1,-2), 4/3 units 

) if k be a variable parameter, show that the locus point x = 2 sec k - 1, y = -2 tan² k is a parabola. Find the vertex and the length of the latus rectum of the parabola.   M(-1,2), 2 units

) A double ordinate of the parabola y²= 4ax subtends a right angle at the vertex. Find the length of that double ordinate.        8a

) If the length of a double ordinate of the parabola 2y²= 3x is 6, find the equation of the circle whose diameter is that double ordinate.      x² + y²- 12x + 27 =0

) Find the equation of the chord of the parabola y²= 4ax passing through the points  (at₁², 2at₁) and (at₂², 2at₂).     2x - y(t₁ + t₂)+ 2at₁t₂ =0.    

) if the lines joining the vertex with the two points (at₁², 2at₁) and (at₂², 2at₂) on the parabola y²= 4ax are at right angle, prove that t₁t₂ + 4=0.  

) Show that the equation of the chord of the parabola x²= 4ay through the point (x₁, y₁) and (x₂, y₂) on it, is (x - x₁)(x - x₂)=nx²- 4ay.    

) Prove that if the co-ordinates of one end of the focal chord of the parabola y²= 4ax (at², 2at) then the co-ordinates of other end of the chord will be (a/t², - 2a/t).

) A straight line through the focus of the parabola y²= 4ax intersect the parabola at the point (x₁², y₁) and (x₂, y₂) show that  x₁ x₂ =a².    

) A focal chord SE of the parabola y²= 8x passes through the point end point, having positive coordinates, of another EE' : x =4. Find the equation and the length of the chord.         y= 2√2 (x -2), 9 units 

) Q is any point on y²= 4ax, ordinate of the point Q is QN, P is the midpoint of QN; prove that the locus of the point P is a parabola, the length of whose latus rectum is one-fourth of the length of the latus rectum.       

) PN is any ordinate of the parabola y²= 4ax;  the point M of divides PN in the ratio m: n.  Find the locus of M.         (m+ n)²y²= 4an²x

) Show that the locus of the middle point of chords of the parabola y²= 4ax which pass through the vertex is the parabola y²= 2ax.

) If P(3, 3/2) is a fixed point and Q is a variable point on the parabola x²= 6y, find the locus of the point which divides PQ in the ratio 1:2.      x²- 4x - 2y +6=0

) A parabola arch span 60cms and a height of 45 cms. Find the distance from the central axis the point on the arch whose height is 25cms.       20 cms

) Prove that the circle drawn on the focal cord of a parabola as diameter touches its directrix .

) The equation of the axis and the directrix of a parabola are y - 3=0 and x +3=0 respectively and the length of the latus rectum is 8 units. Find the equation and the vertex of the parabola.         (y -3)²= 8(x +1),(-1,3) or (y -3)²+ 8(x +5)= 0, ,(-5,3) 

) Find the equation of the parabola whose focus is at (0,-4), axis is parallel to x-axis and the length of the latus rectum is 16 units.     (y +4)²= 16(x +4), or (y +4)²+16(x -4)=0

) Find the locus of the middle points of those chords of the parabola y²= 4ax which pass through a fixed point (m,n).      y²- ny = 2a(x - m)

) Find the locus of the middle points of those chords of the parabola y²= 4ax which subtends right angles at its vertex .      y²= 2a(x - 4a)

) P, Q, R are three points on the parabola y²= 4ax such that PQ passes through the focus and PR is perpendicular the axis . Show that the locus of the midpoint of QR is y²= 2a(x +a)

Objective and Short Answer Type Questions 

1) If the co-ordinates of the focus and the vertex of a parabola are (4,3)!and (1,-1) respectively, find the point of intersection of axis and directrix and the length of the latus rectum.       (-2,-5), 20 units

2) if the point of intersection of the axis and the directrix of a parabola and the vertex are (6,5) and (4,1) respectively, find the distance of the point (-1,-2) from the focus.   √10 units 

3) Find such a point on the parabola y²= -9x  that the ordinate of the point is 3 times of its abscissa .    (-1,-3)

4) If the parabola x²+ 5py=0 passes through the point (3,-3), find the coordinates of the focus and the length of the latus rectum.         (0,-3/4)

5) What is the distance of the point (-2,6) on the parabola y² + 18x = 0 from the focus ?    13/2 units 

6) Find the equation of the axis of the parabola whose focus is at (3,4) and the equation of the directrix is 3x + 4y+25=0.         4x - 3y=0

7) Find the equation of the parabola whose vertex is at origin, axis is x-axis and which passes through the point (-3,4).         3y²+ 16x =0

8) Find the equation of the parabola whose vertex is at the origin, axis is y-axis and which passes through the point (-3,4).     4x²= 9y

9) Find the equation of the parabola whose focus is at (6,0) and directrix is x= 0.     y²= 12(x -3)

10) If the vertex of a parabola is at the origin and its directrix is 2x+ 5=0, find its equation .       y²= 10x

11) If the vertex of a parabola is at the origin and its directrix is 3y -7=0, find the equation of the parabola.         3x²+ 28y=0

12) Find the length of the latus rectum of the parabola 5x²+ 30x + 2y +59=0.     2/5

13) Find the length of the latus rectum of the parabola y = - 2x²+ 12x -17.     1/2

14) Show that the latus rectum of a parabola is the third proportional of the abscissa and ordinate of a point on the parabola.        

15) A straight line parallel to the axis of a parabola interesects the parabola to one point /two points/more than two points. Which one is correct ?      

16) what type of conic section is the locus of the moving point (at², 2at)?  Find the equation of the locus.        Parabola, y²= 4ax.

VARIATION A- Z

EXERCISE -A

1) A bicycle factory produces 840 bicycle in 6 months. How many bicycles will it produce in 5 months ? 5

2) The cost of 2 dozen oranges is ₹48. What is the cost of 180 oranges?

3)  A man can do pack of 275 boxes in 25 cartons. Find the number of boxes he can pack in 16 cartons.

4) A loaded bus travels 14 km 25 minutes. If the speed remains the same, how far can it travel in 5 hours ?

5)  X and Y very directly and x= 8, when y= 2, find X, when Y= 10.

6) Suppose 2 kg of sugar contain 9 x 10⁶ crystals. How many sugar crystals are there in

A) 5 kg of sugar ?

B) 1.2 kg of sugar ?

7)  the intrest on a certain sum (principal) at 6% per annum is ₹300. Find the interest on the same sum at 5% per annum for the same period.

8) If the weight of 5 mangoes is 800g, make a table showing the weight of:

A) one mango 

B) 3 mangoes

C) 12 mangoes 

D) 100 mangoes

Also find the number of mangoes win 6.4 kg

EXERCISE -A(1)

 

1) If 6 pens cost ₹45, then the cost of 10 such pen is

A) ₹75 B) ₹72 C) ₹78 D) ₹80

2) If the cost of 10 apples z is ₹90, then the cost of 15 Apple is 

A) ₹145 B) ₹135 C) ₹125 D) ₹150

3) If a man covers a distance of 400 km in 1 hour, then the distance covered in 6 hours is

A) 220km B) 200 km C) 240km D) 280 km

4) If the cost of 5 dozen oranges is ₹150, the cost of 25 such or oranges is

A) 62.50 B) 60 C) ₹65.50 D) ₹70

EXERCISE -B 

1) In a zoo, there was enough foof to 40 animals for 15 days. How long would the food last, if there were 20 more animals in the zoo ?

2) If 104 women can do a piece of work in 70 days, in how many days will 28 women do it ?

3) If 36 workers can do a piece of work in 20 days, in how many days will 15 workers complete the same work ?

4) 6 taps of equal capacity can fill a tub in 30 minutes. How many taps can fill it in 20 minutes ?

5) if 10 persons can dig a well in 20 days, in how many days will 4 persons dig another well of the same dimensions?

6) A 250 m long train is running at a speed of 55 km/h. In how much time will it cross a platform of the length of 520 m?

7) 45 cows can graze a field in 13 days. How many cows will graze the same field in 9 days ?

8) 3 women can do a work in two days. How long will six women take to complete the same work ?

9) In a zoo, there was enough food to feed 60 animals for 20 days. How long would the food last, if there were 40 more animals in the zoo?

10) A school hostel with 40 children has enough provisions to last 15 days. If 10 more children move into the hostel, how many days will the provision last now?

11) Soni has enough money to buy 15 note books worth ₹8 each. If each notebook was to cost ₹4 more, then how many notebooks would shebe able to buy with that amount of money?

12) If 125 people can do a piece of work in 75 days, in how many days will 25 people do it ?

EXERCISE -C

1) If 20 men can do a piece of work in 7 days, then the number of men required to complete the work in 28 days is

A) 4 B) 5 C) 6 D) 8 

2) Mahesh type 540 words in half an hour. How many words would he type in 6 minutes ?

A) 105 B) 1081 C) 110 D) 115

3) if 20 persons can reap a field in 38 days, in how many days will 19 persons reap the same field ?

A) 40 days B) 21 days C) 19 days E) 25 days 

4) 56 women can reap a field in 8 days. If the work is to be completed in 7 days, the extra women needed are

A) 7 B) 8 C) 9 C) 6 

EXERCISE - D

1) If x varies as y, and x= 8 when y= 15, find x when y=10. 

2) if a and b Vera inversely to each other and a =10, when b= 9, find b, when a= 15

A) 3 B) 4 C) 6 D) 8

3) If a and b vary inversely as each other and a=10, when b=9, find b, when a= 15.

A) 4 B) 8 C) 6 D) 10 

4) If P varies inversely as Q and P= 7 when Q= 3, find P when Q=7/3. 

5) A varies as B and C jointly; If A= 2 when B= 3/5 and C= 10/27 , find C when A=54 and B= 3.

6) If x varies as z, and y varies as z, then x± y and √(xy) will each varies as C 

7) If the square of x varies as the cube of y, and x=3 when y= 4, find the value of y when x= 1/√3.

8) If A varies as Bc, then B varies inversely as C/A.

9) If P varies directly as Q and inversely as R; also P= 2/3 when Q= 3/7 and when R= 9/14, find Q when P= √48 and R= √75.

10) If x varies as y, show that x²+ y² varies as x²- y².

11) If y varies as the sum of two quantities, of which one varies directly as x and the other inversely as x, and if y= 6 when x= 4 and y=10/3 when x= 3; find the equation between x and y.

12) If y is equal to the sum of two quantities, one of which varies as x directly, and the other as x² inversely; and if y= 19 when x= 2, or 3; find y in terms of x.= 15, find x when y=10. =10. 

Miscellaneous

1) 6  taps can fill a water tank in 4 hours. If only 2 taps are open, how long will it take to fill the tank ?

2) If 20 men can dig a well in 16 days, how many men will be required to dig the well in 10 days ?

3) Harsha buys 18 bananas spending ₹90, how many more bananas does Monu buy spending ₹135 ?

4) How much would 12 packets of biscuits cost, if it is given that 9 packets of biscuits cost ₹189 ?

5) A bike travels 90 km in 5 hours,

A) How much time does it take to travel 120 km at the same speed ?

B) How much distance does it cover in 2 hours 20 minutes when it travels at the same speed ?

6) if hundred oranges are packed into one carton, 20 such cartons are needed to pack all the oranges.

A) If 80 oranges are packed in one carton, how many such cartons will be needed ?

B) How many oranges can be packed packed in each carton, if 40 identical cartons are there ?

7) if 3 tailers can stitch 10 shirts in 4 days, then how much time will 6 tailors take to stitch 10 shirts?

8) 80 labours can build a building in 16 months. How month will 128 labourers take to build it under the same working conditions ?

PERMUTATIONS AND COMBINATIONS (Short Questions)-2

Mg. A- R.1

1) How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants ?

2) A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. 4 persons wish to sit on one particular side and two on the other side. In how many ways can they be seated ?

3) A business man hosts a dinner to 21 guests. He is having 2 round tables which can accommodate 15 and 6 persons each. In how many ways can he arrange the guests ?

4) Find the number of a combinations and permutations of 4 letters taken from the word EXAMINATION.

5) Find the number of ways in which:

A) a selection 

B) an arrangement of, 

4 letters can be made from the letters of the word PROPORTION.

6) How many words can be formed by taking 4 letters at the time from the letters of the word MORADABAD ?

7) How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE.

8) Find the number of permutations of n different things taken r at a time such that two specified things occur together?

9) ²⁰Cᵣ = ²⁰Cᵣ₋₁₀, then ¹⁸Cᵣ is equal to

A) 4896 B) 816 C) 1632 D) none

10) ²⁰Cᵣ= ²⁰Cᵣ₊₄ , then ʳC₃ is equal to

A) 54 B) 56 C) 58 D) none

11) ¹⁵C₃ᵣ = ¹⁵Cᵣ₊₃ , then r is equal to

A) 5 B) 4 C) 3 D) 2

12) ²⁰Cᵣ₊₁ = ²⁰Cᵣ₋₁ , then r is equal to

A) 10 B) 11 C) 19 D) 12

13) If C(n, 12)= C(n, 8), then C(22, n) is equal to

A) 231 B) 210 C) 252 D) 303

14) If ᵐC₁ = ⁿC₂ , then

A) 2m=n B) 2m=n(n+1) C) 2m=n(n -1) D) 2n =m(m -1)

15) ⁿC₁₂ = ⁿC₈ , then n=

A) 20 B) 12 C) 6 D) 30

16) ⁿCᵣ + ⁿCᵣ₊₁ =ⁿ⁺¹Cₓ , then x is

A) r B) r-1 C) n  D) r+1

17) If (a²- a) C 2 = (a² - a) C 4, then a is

A) 2 B) 3 C) 4 D) none

18) ⁵C₁ + ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅ is equal

A) 30 B) 31 C) 32 D) 33 

19) Total number of words found by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equals to 

A) 60 B) 120 C) 7200 D) none

20) There are 12 points in a plane. The number of the straight lines joining any two of them when 3 of them are collinear, is

A) 62 B) 63 C) 64 D) 65

Mg. A- R.2

1) 3 person entre Railway compartment. If there are 5 seats vacant, in how many ways can they take these seats?

A) 60 B) 20 C) 15 D) 125 

2) In how many ways can a committee of 5 be made out of 6 men and 4 Women containing at least one woman ?

A) 246 B) 222 C) 186 D) none of these

3) There are 10 points in a plane and 4 of them are collinear. The number of straight line joining any two of them is.

A) 45 B) 40  C) 39 D) 38 

4) There are 13 players of cricket, out of which 4 are bowlers. In how many ways a team of 11 be selected from them so as to included least two bowlers?

A) 72 B) 78 C) 42 D) none of these

5) If C₀ + C₁ + C₂ +....+ Cₙ = 256, then ²ⁿC₂ is

A) 56 B)120 C) 28 D) 91 

6) The number of ways in which a host lady can invite for a party of 8 out of 12 people of whom two do not want to attend the party together is

A) 2. ¹¹C₇ + ¹⁰C₈ B) ¹⁰C₈ + ¹¹C₇ C) ¹²C₈ - ¹⁰C₆   D) none

7) Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle. Then the number of circles that can be drawn so that each contains at least 3 of the given points is

A) 216 B)156 C) 172 D)  none of these

8) How many different committee of 5 can be formed from 6 men and 4 women on which exact 3 men and 2 women serve?

A) 6  B) 20 C) 60 D) 120

9) If ⁴³Cᵣ₋₆ = ⁴³C₃ᵣ₊₁ , then the value of r is

A) 12 B) 8 C) 6 D) 10 E) 14  

10) The number of diagonals that can be drawn by joining the vertices of an octogon is 

A) 20 B) 28 C) 8 D) 16

11) The value of (⁷C₀ + ⁷C₁)+(⁷C₁+ ⁷C₂) + ......+ (⁷C₆ + ⁷C₇) is

A) 2⁷- 1 B) 2⁸-2 C) 2⁸ -1 D) 2⁸

12) Among 14 players, 5 are bowlers, In how many ways a team of 11 may be formed with at least four bowlers ?

A) 265 B) 263 C) 264 D) 275

13) A lady gives a dinner party for 6 guests,  The number of ways in which they may be selected from among 10 friends if two of the friends will not attend the party together is

A) 112 B) 140 C) 164 D) none

14) If ⁿ⁺¹C₃ = 2.ⁿC₂ ,  then n=

A) 3 B) 4 C) 5 D) 6

15) The number of parallelogram that can be formed from a set of four Parallel Lines intersecting another set of 3 Parallel Lines is

A)  6 B) 9 C) 12 D) 18

16) How many four letter words can be formed using the letters of the word ROAMING

a) 420 b) 210 c) 840 d) 480 e) 620

17) How many three letter words can be formed using the letters of the word PRACTICES?

a) 157 b) 257 c) 357 d) 457 e) 557

18) In a party, each person shook hands with 

LINEAR EQUATION (A- Z)

EXERCISE -A  

SOLVE FOR X:

1) x= 0.                                                0

2) 2x= 0.                                             0

3) 5x= 0.                                             0

4) -3x= 0.                                            0

5) x/2 = 0.                                         0

6) 3x/2= 0.                                        0

7) -5x/3= 0.                                       0

8)  -4x/-3= 0.                                    0

9) 7x/-6= 0.                                       0

10) -4x/8 = 0.                                    0

EXERCISE -B

SOLVE FOR x: 

1) x - 3 = 0.                                        3

2) x + 5 = 0.                                       -5

3) -x - 9 = 0.                                       -9

4) - x + 18 = 0.                                 18

5) 4 - x = 0.                                        4

6) 6 + x = 0.                                      -6

7) -12 - x = 0.                                - 12

8) 27 + x = 0.                                  -27

9) -x -55= 0.                                    -55

10) 32 - x= 0.                                 32

EXERCISE -C

SOLVE FOR X:

1) 4/3 - x = 0.                              4/3

2) 3/7 + x = 0.                              -3/7

3) -5/9 + x = 0.                             5/9

4) -7/8 - x = 0.                              -7/8

5) x - 3/11= 0.                             3/11

6) x + 7/12= 0.                           -7/12

7) - x - 8/19= 0.                        -8/19

CONTINUE........

EXERCISE - D

SOLVE FOR X: 

1) 2x = 1.                                      1/2

2) 3x = 2.                                      2/3

3) -5x = 3.                                     -3/5

4) 7x = -2.                                     -2/7

5) -8x = -3.                                     3/8

6) -3x = 10.                               -10/3

7) -9x = -11.                               11/9

8) 32x= -13.                             -13/32

9) 3x= -22.                                 -22/3

EXERCISE -E 

Solve for x: 

1) x/2 = 3.                                       6

2) x/3 = 2.                                       6

3) - x/4 = 6.                                     -24

4) -x/7 = -3.                                     21

5) x/-3 = 4.                                      -12

6) -x/7 = -13.                                  91

7) x/5 = -4.                                    -20

8) x/-3 = -8.                                    24

EXERCISE -F 

Solve for x:  

1) 2x = 4.                                        2

2) 3x = 9.                                        3

3) 4x = 12.                                        3

4) 

Exercise -G

1) x - 3 = 4.                                               7

2) x - 7 = 6.                                              13

3) x- 21= 5.                                              26

4) x- 17= 8                                              25

5) x - 9 = 5.                                              14

6) x - 31 = 4.                                             35

7) x - 27 = - 6.                                          21

8) x- 21= - 9                                              12

9) x- 17= -12                                             5

10) x - 29 = - 1.                                         28

11) x - 2 =-51.                                         -49

12) x - 9 = -35                                         -26

13) x - 22= -55.                                      -32

14) x- 17= -81                                        -64

15) x - 23 = -51.                                    -28

EXERCISE -H

1) x + 2 = 4.                                                2

2) x + 11 = 16.                                           5

3) x + 21= 25.                                           3

4) x+17= 39                                             22

5) x + 29 = 35.                                          6

6) x +23 = 41.                                           17

7) x + 27 = -16.                                       -43

8) x+21= -15.                                         -36

9) x+ 37= -8                                            45

10) x + 9 = -35.                                     -44

EXERCISE -I

1) 3x - 1/4= 8.     

2) x/7 + x= 16.   

3) 8x= 20+3x

4) 2x/3 +1=7/3

5) 5x/8- 6 = 9

6) 3x - 5/3= x - 3

7) 15x/4 = 5x - 5/2.

8) 3x - 5 = 2x/3 + 9.

9) 32x + 11/4= x/3 +2.    

10) x/3 + x/4 = 14.

11) x/6 + x/4 = x- 7.      

12) 2x/3 + 4x = 42.

EXERCISE J

1) 2x - 5 = 4.                                              9

2) 3x - 17 = 4.                                           7

3) 5x- 20= 5.                                             5

4) 3x- 17= 28                                           15

5) 7x - 9 = 5.                                             2

6) 9x - 23 = 4.                                          3

7) 12x - 6= 6.                                          1

8) 6x- 21= 3.                                           4

9) 8x- 17= 7                                            3

10) 5x - 9 = 106.                                   23

11) x - 40 % of x = 12.  

12) x - 24% of x = 38

EXERCISE -K

1) 3x + 5= 5x -11.   

2) 9x - 7= 6x +14

3) 5x -16= 19- 2x.

4) 5(x +4)= 35.

5) 5x+7= 2x-8

6) 2(x - 5/2) = 0.3

7) 3x - 5 = 2x+ 8.                                   13

8) 5x - 17 = 2x- 8.                                   3

9) 0.6x - 1.9= 0.2x + 0.5.

10) 4x-9= 2x +7.

11) 2a -1= 14- a

12) 7k/5 = k -4.

13) 5y + 18= 11- 2y.

14) x/4 + x/6 = x -7

15) 8x+3= 27+2x

16) 3a+ 2/3 = 2x +1.

EXERCISE-L

1) 5(3-x)= 3(x+4)

2) 9x/(7- 6x)= 15

3) 3x/(5x +2)= -4

4) 2(3x - 2)- 4(2x -5) = 9.

5) 2/3(x -3 )= 5/6 (3x-4)

6) (8x -3)/3x =2

7) (6y -5)/2y = 7/9

8) 7 - 2(5 - 3x)= 4(x -3) +5.

9) 6 - 3(2x- 3)= 5(6-x) -4x.            5

10) 21- 3(x-7) = x +20.

11) 9x +5= 4(x -2)+8

12) 2(x - 5)+ 3(x -2) = (2-5x).   

13) (x - 5) - 16 = 12- 2(x -3). 

14) 3(x - 7)- 2(3x -4) = 8+ 7(x -4). 

15) 6(3x +2)- 5(6x -1) = 3(x -8) - 5(7x -6)+ 9x. 

16) x - (2x + 5)- 5(1- 2x) = 2(3+ 4x) - 3(x -4). 

17) 15(y -4) - 2(y -9)+ 5(y +6)=0

18) 3(5y -7) - 2(9y -11)=  4(8y -13)=0

EXERCISE -M

1) (x -4)(x+4) = (x+4)(x -7)+ 33.

2) (x -2)(x+3) = (x²-4).

3) (6x+7)/(3x+2)= (4x+5)/(2x+3).

4) (3x +5)/(2x+1)= 1/3

5) (2- 9x)/(17- 4x)= 4/5

6) (2 -9x)/(16+ 5x)= 0.

7) (2x +3 )/(3+ x) = 3/2.

8) (2x +1 )/(7x-2) = 3/5

9) (4x +7)/(9-3x) = 1/4

10) (7k +4)/(k+2) = -4/3

11) (8- 3x)/(5x+31) = 2/3

12) (5- 4x )/(3- 2x) = 13/7.

13) (2- 9x )/(17- 4x) = 4/5.

14) (0.5x +4)/(1.2x +6)= 5/3.

15) 7/(x -4 ) = 5/(x +2).

16) (6x -7)/(3x+1) = (2x +1)/(x +5).   

17) (x +1)/(x -2) = (x -2)/(x -3).   

18) (2x -3)/(2x -1)= (3x-1)/(3x +1).

19) (2x -3)/(3x -1)= (2x+3)/(3x +4).

 

20) (2x +3)/(3x+4) = (2x-3)/(3x -2).     -2

21) 2- (3 - x)/(x -1)= (3x+ 4)/(x +1).

22) 2 + (2x -3)/(2x +4)= (3x+ 4)/(x +2).

23) 4/(x -3)+ 2/(x -2)= 6/x.

24) 5/(x +2)- 3/(x -2)= 2/x.           1/2

25) 3/(2x -1)+ 4/(2x +1)= 7/²x. 

26) 1.25+ 9.75/x = 4.5.                  3

27) 2x +1 )/(7x-2) = 3/5

 

Exercise -O

1) (x +5)/2 + x/3 = 20.

2) (5x -3 )/2 - (3x-2)/3 = 2/3.                1

3) (2x +3 )/3 - (3x-2)/4 =1

4) (3x -2 )/7 - (5x-8)/4 =1/14

6) (x +7 )/3 = 1+ (3x- 2)/5.

5) (2x +3 )/3 - (3x-2)/4 =1

6) (3x -2 )/7 - (5x-8)/4 =1/14

7) (a -4 )/5 + (a+ 2)/2 =10

8) (x -1 ) = 3/4 (x +1) - 1/2.

9) 2/3 (3x-2) = 4/5  (2x -3) - 4/3

10) 3/4 (2x -5 ) -5/6 (7- 5x)= 7x/3

11) 4(x + 5/8)/5 -  2(x - 1/4)/3 = 10/9

12)  (x -2 )/3  + 5x/2= 6 -  (x - 5)/6.

13) 12/3  - (3x-4)/5 = (x -)/3.

14) (5x -7)/4  - (2x - 5)/3 = 5x/6.    

15) 2x - (x-8)/6 = 2(2x+19)/9.         52

16) 4- 2(x-4)/3 = (2x +5)/2.

17)  3x/4 - (x - 4)/3 = 5/3.

EXERCISE -P

1) (x +3 )/7 - (3x-5)/5 =(2x -5)/3 -  25.      25

2)(x -2)/3 + (x-3)/4 =(x -1)/2.

 

3) (x -4)/7 - (x +4)/5 = (x +3 )/7 

 

4) (x +5)/6 - (x +1)/9 =  (x + 3)/4.

5) (3x-2)/4 + (2x +3)/3= 2/3 - x.

16) (4x +1)/3 + (2x -1)/2 - (3x -7 )/5 = 6

17) (x +6)/4 - (5x -4)/8 + (x -4 )/5 = 0.

18) x - {2x - (3x-4)7}= (4x-27)/3 - 3

19) 5x -  (x+1)/3 = 6(x +1/30).

20) (y -2)/4  + 1/3 = y - (2y - 1)/3.

21) (7x- 1) - {x - (1- x)/2} = 4x+ 1/2.

22) (5x - 11)/4 + -(3x -7)/2 = (4x-7)/3 + y - 1.

23) 3(7x -1)/4 - {2x - (1-x)/2} = x + 3/2.

24) 2x - {x+ (5x-4)/7}= (4x+7)/3   -7.      5

25) 2x/(x² -16)- 3/(x -4)= 5/(x+4).      4/3

26) 3/(x -2) - 2/(x -3)= 4/(x-3) - 3/(x -1).        

27) (x+3)(x -3)- x(x +5)=6.

28) x(2x +3)- 2x(x -5)= 26.           

29) (x+6)/4 + (x -3)/5= (5x -4)/8.

30) (2x+7)/5 + (3x +11)/2 = (2x +8)/3 - 5

31) (5x-4)/6 =  (4x +1) -  (3x +10)/2

EXERCISE -Q

1) (2x +1)/10 - (x -2)/6  = (3 -2x)/15. And find y, when 1/x + 1/y = 2.      -7/5, 7/19 

2) (2- x)/2 - (x -3)/3 = (1 -x). And find y, when 1/x + 1/y = 2.      

3) If x= (k+1), find the value of k when (5x -3)/2 - (2 + 9k)/e  = 1/4. 

4) If m= (1- 3x)/5, n=  (1- 2x)/3 and 3(n - 2m)+ 1, find x.

BINOMIAL DISTRIBUTION

1) The probability distribution of a discrete variable x is given by:

X:        1       2      3

F(X):  2p      p     4p  find

A) E(x).                                      2.29

B) V(x).                                      0.78

C) Pr(x> 1).                                5/7

2) A random variable x has the following Probability distribution:

x:      0  1   2   3    4    5     6     7      8

P(x): k 2k 4k  6k  8k 10k12k 14k 16k

Find

A) E(x).                                         5.59

B) V(x).                                        4.27

C) p(x <3).                                 7/73

D) p(x ≥ 3).                               66/73

E) p(0< x < 5).                          20/73 

3) If a coin is tossed 3 times, obtain the p.m.f. of the number of heads. Hence obtain its probability distribution. 

4) A box contains 4 and 3 white balls. If 2 balls are drawn from this box, find the p.m.f. of the number of red balls and hence obtain its probability distribution.

5) The mean of a Binomial Distribution is 4 and the standard deviation is 3 -- This statement can not be true, why ?

6) For a Binomial Distribution, the mean is 3 and variance is 2. Find the value of n and p. Hence find the probability that X (the variable value) is 5.          9, 1/3, 224/2187

7) If the mean and the variance of a Binomial distribution are respectively 4 and 8/3, find the values of n and p.              12, 1/3 

8) The mean of a Binomial Distribution is 40 and standard deviation is 6. Calculate n, p and q.    400, 1/10, 9/10 

9) In a Binomial distribution consisting of 5 independent trials, Probabilities of 1 and 2 success are 0.4096 and 0.2048 respectively. Find the parameter p of the distribution.                       1/5

10) With usual notations, find p for a Binomial random variable X if n= 6, and if P(X= 4) = P(X= 2).      1/4

11) An experiment succeeds twice as often as it fails. What is the probability that in the next 5 trials there will be 

A) three success.            80/243

B) at least three success.    64/81

12) An experiment succeed twice as many as times as it fails. Find the chance that in 6 trials, there will be at least five success.    256/729

13) Four coins are tossed simultaneously. What is the probability of getting 2 heads and 2 tails.                                        31/81

14) Eight coins are thrown simultaneously. Find the chance of obtaining.

A) atleast 6 heads.               37/256

B) no heads.                            1/256

C) all heads.     1/256

15) If ten fair coins are tossed, what is the probability that are n of getting to tell you fire point are not more than three heads.         11/64

16) A dice thrown three times. If getting a six is considered a success, find the probability of getting atleast two success.    2/27

17) If 10 coins are tossed 100 times. How many times would you except 7 coins to fall head upward ?    12

18) What is the probability that if a fair coin is tossed 6 times we will get:

A) exactly two heads ?          15/64

B) at least 2 heads.                57/64

19) An unbiased cubic die is tossed 4 times. What is the probability of getting

A) no six.                            625/1296

B) least one six.                 671/1296

C) all odd numbers.                  1/16

D) atleast one even number.   15/16

20) If a die is thrown six times, calculate the probability that:

A) a score of 3 or less occurs on exactly 2 throws.                      15/64

B) a score of more than 2 occurs on exactly 3 throws.                160/729

C) a score of 5 or less occurs atleast once.             46655/46656

D) a score of two or less occurs on at least five occasions.     13/729

21) Assume that on an average one telephone out of 10 is busy. 6 telephone numbers are randomly selected and called. Find the probability that four of them will be busy.                                  0.001215

22) The probability that a college student will be graduate is 0.4. Determine the probability that out of 5 students

A) none.                                      0.08

B) one.                                        0.26

C)  atleast one will be graduate.   0.92

23) A machine produces 2% of  defectives on the average. If 4 articles, are chosen randomly, what is the probability that there will be exactly 2 defective articles.  0.0023

24) 25% of the inhabitants in a large town are bespectacled. What is the probability that a randomly selected group of 6 inhabitants will included Atmost 2 bespectacled persons.    1701/2048

25) The overall percentage of failures in a certain examination is 40. What is the probability that out of a group of 6 candidates atleast 4 passed the examination ?    1701/3125

26) Find the probability that in a family of 5 children there will be

A) at least one boy.       

B) at least one boy and one girl. (Assume that the probability of a female birth is 1/2).     31/32, 15/16

27) In a shooting competition, the probability of a man hitting a target is 1/5. If he fires 5 times, What is the probability of hitting the target atleast twice ?                  821/3125

28) The incidence of occupational disease in an industry is such that the workers have a 20% chance of suffering from it.  What is the probability that out of 6 workers, 4 or more will contact the disease ?     53/3125

29) In 10 independent throws of a defective die, the probability that an even number will appear 5 times is twice the probability that an even number will appear 4 times. Find the probability that an even number will not appear at all in 10 independent throws of the die.   (3/8)¹⁰

30) Suppose that half the population of a town are consumers of rice. 100 investigators are appointed to find out its truth. Each investigator interview 10 individual. How many investigators do you expect to report that three or less of the people interviewed are consumers of rice.                                          17

31) If on the average rainfalls on 12 days in every 30 days, find the probability

A) that the first four days of a given week will be fine, and the remainder wer.                   648/78125

B) that rain will fail on just 3 days of a given week.         4536/15625

32) A coin is tossed 8 times. Find the probability of getting 

A) three times head.                7/32

B) first three times head.     1/256

33) A die is tossed 5 times. Find the probability of getting face 6.

A) 3 times.                        125/3888

B) at least three times.       23/648

C) atmost three times.    3875/3888

D) first three times.            25/7776

34) A machine produces 25% defective items. From a day's production two samples of size 7 and 5 are chosen at random. Find the probability that two samples together will contain more than 25% defective items.                         0.35

35) Three fair coins are tossed 3000 times. Find the frequency distribution of number of heads,  calculate mean and standard deviations of the distribution.    1500, 27.38