Chemistry-
General
Easy

Question

(Me)2SiCl2 on hydrolysis will produce -

  1. left parenthesis M e right parenthesis subscript 2 end subscript S i left parenthesis O H right parenthesis subscript 2 end subscript    
  2. left parenthesis M e right parenthesis subscript 2 end subscript S i equals O    
  3. open square brackets negative O minus left parenthesis M e right parenthesis subscript 2 S i minus O minus close square brackets subscript n minus    
  4. M e subscript 2 end subscript S i C l left parenthesis O H right parenthesis    

The correct answer is: left parenthesis M e right parenthesis subscript 2 end subscript S i left parenthesis O H right parenthesis subscript 2 end subscript

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General
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If fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A over denominator x plus 1 end fraction plus fraction numerator B over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction then B=

Given,
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A over denominator x plus 1 end fraction plus fraction numerator B over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis plus B left parenthesis x plus 3 right parenthesis plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A left parenthesis x squared plus 3 x plus 2 x plus 6 right parenthesis plus B left parenthesis x plus 3 right parenthesis plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A x squared plus A 5 x plus A 6 plus B x plus B 3 plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A space x squared plus x left parenthesis 5 A plus B right parenthesis plus 6 A plus 3 B plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
N o w comma space o n space c o m p a r i n g space b o t h space s i d e s space w e space c a n space s a y space t h a t comma
A equals 1 comma space 5 A plus B equals 5 space space comma space space space space 6 A plus 3 B plus C equals 1
space space space space space space space space space space space rightwards double arrow 5 plus B equals 5 space space space space space space space rightwards double arrow 6 plus 0 plus C equals 1
space space space space space space space space space space space rightwards double arrow B equals 0 space space space space space space space space space space space space space rightwards double arrow C equals negative 5

If fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A over denominator x plus 1 end fraction plus fraction numerator B over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction then B=

Maths-General
Given,
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A over denominator x plus 1 end fraction plus fraction numerator B over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis plus B left parenthesis x plus 3 right parenthesis plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A left parenthesis x squared plus 3 x plus 2 x plus 6 right parenthesis plus B left parenthesis x plus 3 right parenthesis plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A x squared plus A 5 x plus A 6 plus B x plus B 3 plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
fraction numerator x squared plus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A space x squared plus x left parenthesis 5 A plus B right parenthesis plus 6 A plus 3 B plus C over denominator left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction
N o w comma space o n space c o m p a r i n g space b o t h space s i d e s space w e space c a n space s a y space t h a t comma
A equals 1 comma space 5 A plus B equals 5 space space comma space space space space 6 A plus 3 B plus C equals 1
space space space space space space space space space space space rightwards double arrow 5 plus B equals 5 space space space space space space space rightwards double arrow 6 plus 0 plus C equals 1
space space space space space space space space space space space rightwards double arrow B equals 0 space space space space space space space space space space space space space rightwards double arrow C equals negative 5
General
maths-

The set S : = { 1, 2, 3 .........12} is to be partitioned into three sets A, B, C of equal size. Thus A unionunion C = S, A intersection B = B  C = A intersection C = ϕ. The number of ways to partition S is -

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
fraction numerator table row 12 end table over denominator table row 4 end table table row 4 end table table row 4 end table end fraction×fraction numerator table row 3 end table over denominator table row 3 end table end fraction = fraction numerator table row 12 end table over denominator left parenthesis table row 4 end table right parenthesis to the power of 3 end exponent end fraction

The set S : = { 1, 2, 3 .........12} is to be partitioned into three sets A, B, C of equal size. Thus A unionunion C = S, A intersection B = B  C = A intersection C = ϕ. The number of ways to partition S is -

maths-General
S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
fraction numerator table row 12 end table over denominator table row 4 end table table row 4 end table table row 4 end table end fraction×fraction numerator table row 3 end table over denominator table row 3 end table end fraction = fraction numerator table row 12 end table over denominator left parenthesis table row 4 end table right parenthesis to the power of 3 end exponent end fraction
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A one kilowatt motor is used to pump water from a well 10 m deep. The quantity of water pumped out per second is nearly

P equals m g h divided by t space o r space m divided by t equals P divided by g h space o r space m divided by t equals 1000 divided by left parenthesis 10 cross times 10 right parenthesis space k g equals 10 space k g

A one kilowatt motor is used to pump water from a well 10 m deep. The quantity of water pumped out per second is nearly

Physics-General
P equals m g h divided by t space o r space m divided by t equals P divided by g h space o r space m divided by t equals 1000 divided by left parenthesis 10 cross times 10 right parenthesis space k g equals 10 space k g
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Assertion (A) : The Remainder obtained when the polynomial x to the power of 6 4 plus x squared 7 plus 1 is divided by x plus 1 Is 1
Reason (R): If f left parenthesis x right parenthesis is divided by x minus a then the remainder is f(a)

Assertion (A) : The Remainder obtained when the polynomial x to the power of 6 4 plus x squared 7 plus 1 is divided by x plus 1 Is 1
Reason (R): If f left parenthesis x right parenthesis is divided by x minus a then the remainder is f(a)

maths-General
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A ball hits the floor and rebounds after inelastic collision. In this case

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A ball hits the floor and rebounds after inelastic collision. In this case

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By the conservation of momentum in the absence of external force total momentum of the system (ball + earth) remains constant

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Statement I Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.

Statement II Principle of conservation of momentum holds true for all kinds of collisions.

Statement I Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.

Statement II Principle of conservation of momentum holds true for all kinds of collisions.

Physics-General
parallel
General
physics-

A mass M  is lowered with the help of a string by a distance h at a constant acceleration g/2. The work done by the string will be

A mass M  is lowered with the help of a string by a distance h at a constant acceleration g/2. The work done by the string will be

physics-General
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A particle begins at the origin and moves successively in the following manner as shown, 1 unit to the right, 1/2 unit up, 1/4 unit to the right, 1/8 unit down, 1/16 unit to the right etc. The length of each move is half the length of the previous move and movement continues in the ‘zigzag’ manner indefinitely. The co-ordinates of the point to which the ‘zigzag’ converges is –

A particle begins at the origin and moves successively in the following manner as shown, 1 unit to the right, 1/2 unit up, 1/4 unit to the right, 1/8 unit down, 1/16 unit to the right etc. The length of each move is half the length of the previous move and movement continues in the ‘zigzag’ manner indefinitely. The co-ordinates of the point to which the ‘zigzag’ converges is –

Maths-General
General
Maths-

The greatest possible number of points of intersections of 8 straight line and 4 circles is :

 Detailed Solution
The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.

For selecting r objects from n objects can be done by using the formula as follows
C presuperscript n subscript r space equals space fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction
 As mentioned in the question, we have to find the total number of intersection points.
For calculating the points of intersection between two lines, we can use the formula which is mentioned in the hint as follows = C presuperscript 8 subscript 2 space cross times 1 space equals space 28
 For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows = C presuperscript 4 subscript 2 space cross times 2 space equals space 12

 For calculating the points of intersection between one line and one circle, we can use the formula which is mentioned in the hint as follows = C presuperscript 4 subscript 1 space cross times C presuperscript 8 subscript 1 cross times 2 space equals space 64

 Hence, the total number of points of intersection is = 28 + 64 + 12 = 104

The greatest possible number of points of intersections of 8 straight line and 4 circles is :

Maths-General
 Detailed Solution
The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.

For selecting r objects from n objects can be done by using the formula as follows
C presuperscript n subscript r space equals space fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction
 As mentioned in the question, we have to find the total number of intersection points.
For calculating the points of intersection between two lines, we can use the formula which is mentioned in the hint as follows = C presuperscript 8 subscript 2 space cross times 1 space equals space 28
 For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows = C presuperscript 4 subscript 2 space cross times 2 space equals space 12

 For calculating the points of intersection between one line and one circle, we can use the formula which is mentioned in the hint as follows = C presuperscript 4 subscript 1 space cross times C presuperscript 8 subscript 1 cross times 2 space equals space 64

 Hence, the total number of points of intersection is = 28 + 64 + 12 = 104
parallel
General
Maths-

Assertion (A):Iffraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis open parentheses x squared plus 1 close parentheses end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B x plus C over denominator x squared plus 1 end fraction ,then Aequals 1 fifth comma B equals 1 fifth comma C equals negative 2 over 5
Reason (R) : fraction numerator 1 over denominator left parenthesis x minus a right parenthesis open parentheses x squared plus b close parentheses end fraction equals fraction numerator 1 over denominator a squared plus b end fraction open square brackets fraction numerator 1 over denominator x minus a end fraction minus fraction numerator x plus a over denominator x squared plus b end fraction close square brackets

fraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis open parentheses x squared plus 1 close parentheses end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B x plus C over denominator x squared plus 1 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator A left parenthesis x squared plus 1 right parenthesis plus B x plus C left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x squared minus 1 right parenthesis end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator A x squared plus A plus B x squared minus 2 B x plus C x minus 2 C over denominator left parenthesis x minus 2 right parenthesis left parenthesis x squared minus 1 right parenthesis end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator x squared left parenthesis A plus B right parenthesis plus x left parenthesis C minus 2 B right parenthesis plus A minus 2 C over denominator left parenthesis x minus 2 right parenthesis left parenthesis x squared minus 1 right parenthesis end fraction
N o w comma space A plus B equals 0 space space space space space space space space space space space space space space space space space space space space space C minus 2 B equals 0 space space space space space space space space space space space space space space space space space space space space space space space A minus 2 C equals 1
space space space space space space space space rightwards double arrow A equals negative B space space space space space space space space space space space space space space space space space space rightwards double arrow C equals 2 B space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow negative B minus 4 B equals 1
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow negative 5 B equals 1 rightwards double arrow B equals fraction numerator negative 1 over denominator 5 end fraction
t h e r e f o r e space s t a t e m e n t space 1 space i s space n o t space t r u e.                                             fraction numerator 1 over denominator left parenthesis x minus a right parenthesis open parentheses x squared plus b close parentheses end fraction equals fraction numerator 1 over denominator a squared plus b end fraction open square brackets fraction numerator 1 over denominator x minus a end fraction minus fraction numerator x plus a over denominator x squared plus b end fraction close square brackets
R H S equals fraction numerator 1 over denominator a squared plus b end fraction open square brackets fraction numerator x squared plus b minus x squared plus a squared over denominator open parentheses x minus a close parentheses open parentheses x squared plus b close parentheses end fraction close square brackets
space space space space space space space space space equals fraction numerator 1 over denominator a squared plus b end fraction open parentheses fraction numerator a squared plus b over denominator open parentheses x minus a close parentheses open parentheses x squared plus b close parentheses end fraction close parentheses
space space space space space space space space space equals fraction numerator 1 over denominator left parenthesis x minus a right parenthesis open parentheses x squared plus b close parentheses end fraction
L H S equals R H S
t h e r e f o r e space s t a t e m e n t space 2 space i s space t r u e.

Assertion (A):Iffraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis open parentheses x squared plus 1 close parentheses end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B x plus C over denominator x squared plus 1 end fraction ,then Aequals 1 fifth comma B equals 1 fifth comma C equals negative 2 over 5
Reason (R) : fraction numerator 1 over denominator left parenthesis x minus a right parenthesis open parentheses x squared plus b close parentheses end fraction equals fraction numerator 1 over denominator a squared plus b end fraction open square brackets fraction numerator 1 over denominator x minus a end fraction minus fraction numerator x plus a over denominator x squared plus b end fraction close square brackets

Maths-General
fraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis open parentheses x squared plus 1 close parentheses end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B x plus C over denominator x squared plus 1 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator A left parenthesis x squared plus 1 right parenthesis plus B x plus C left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x squared minus 1 right parenthesis end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator A x squared plus A plus B x squared minus 2 B x plus C x minus 2 C over denominator left parenthesis x minus 2 right parenthesis left parenthesis x squared minus 1 right parenthesis end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator x squared left parenthesis A plus B right parenthesis plus x left parenthesis C minus 2 B right parenthesis plus A minus 2 C over denominator left parenthesis x minus 2 right parenthesis left parenthesis x squared minus 1 right parenthesis end fraction
N o w comma space A plus B equals 0 space space space space space space space space space space space space space space space space space space space space space C minus 2 B equals 0 space space space space space space space space space space space space space space space space space space space space space space space A minus 2 C equals 1
space space space space space space space space rightwards double arrow A equals negative B space space space space space space space space space space space space space space space space space space rightwards double arrow C equals 2 B space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow negative B minus 4 B equals 1
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow negative 5 B equals 1 rightwards double arrow B equals fraction numerator negative 1 over denominator 5 end fraction
t h e r e f o r e space s t a t e m e n t space 1 space i s space n o t space t r u e.                                             fraction numerator 1 over denominator left parenthesis x minus a right parenthesis open parentheses x squared plus b close parentheses end fraction equals fraction numerator 1 over denominator a squared plus b end fraction open square brackets fraction numerator 1 over denominator x minus a end fraction minus fraction numerator x plus a over denominator x squared plus b end fraction close square brackets
R H S equals fraction numerator 1 over denominator a squared plus b end fraction open square brackets fraction numerator x squared plus b minus x squared plus a squared over denominator open parentheses x minus a close parentheses open parentheses x squared plus b close parentheses end fraction close square brackets
space space space space space space space space space equals fraction numerator 1 over denominator a squared plus b end fraction open parentheses fraction numerator a squared plus b over denominator open parentheses x minus a close parentheses open parentheses x squared plus b close parentheses end fraction close parentheses
space space space space space space space space space equals fraction numerator 1 over denominator left parenthesis x minus a right parenthesis open parentheses x squared plus b close parentheses end fraction
L H S equals R H S
t h e r e f o r e space s t a t e m e n t space 2 space i s space t r u e.
General
physics-

A force–time graph for a linear motion is shown in figure where the segments are circular. The linear momentum gained between zero and  is

As the area above the time axis is numerically equal to area below the time axis therefore net momentum gained by body will be zero because momentum is a vector quantity

A force–time graph for a linear motion is shown in figure where the segments are circular. The linear momentum gained between zero and  is

physics-General

As the area above the time axis is numerically equal to area below the time axis therefore net momentum gained by body will be zero because momentum is a vector quantity

General
Maths-

How many different nine digit numbers can be formed from the number 2,2,3,3,5,5,8,8,8 by rearranging its digits so that the odd digits occupy even position ?

 Detailed Solution
Here we need to find the total number of nine digit numbers that can be formed using the given digits i.e. 2, 2, 3, 3, 5, 5, 8, 8, 8.
O space e space O space e space O space e space O space e space O
Here, e is for the even places and O is for the odd places of the digit number.
The digits which are even are 2, 2, 8, 8 and 8.

Number of even digits  5
The digits which are odd are 3, 3, 5 and 5.
Number of odd digits 4
We have to arrange the odd digits in even places.

Number of ways to arrange the odd digits in 4 even places = fraction numerator 4 factorial over denominator 2 factorial cross times 2 factorial end fraction = 6

Now, we have to arrange the even digits in odd places.

Number of ways to arrange the even digits in 5 odd places  = fraction numerator 5 factorial over denominator 2 factorial cross times 3 factorial end fraction space equals space 10

Total number of 9 digits number = 6 cross times 10 space equals space 60


 
 
 

How many different nine digit numbers can be formed from the number 2,2,3,3,5,5,8,8,8 by rearranging its digits so that the odd digits occupy even position ?

Maths-General
 Detailed Solution
Here we need to find the total number of nine digit numbers that can be formed using the given digits i.e. 2, 2, 3, 3, 5, 5, 8, 8, 8.
O space e space O space e space O space e space O space e space O
Here, e is for the even places and O is for the odd places of the digit number.
The digits which are even are 2, 2, 8, 8 and 8.

Number of even digits  5
The digits which are odd are 3, 3, 5 and 5.
Number of odd digits 4
We have to arrange the odd digits in even places.

Number of ways to arrange the odd digits in 4 even places = fraction numerator 4 factorial over denominator 2 factorial cross times 2 factorial end fraction = 6

Now, we have to arrange the even digits in odd places.

Number of ways to arrange the even digits in 5 odd places  = fraction numerator 5 factorial over denominator 2 factorial cross times 3 factorial end fraction space equals space 10

Total number of 9 digits number = 6 cross times 10 space equals space 60


 
 
 
parallel

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