Maths-
General
Easy

Question

The coefficient of  x to the power of n in fraction numerator left parenthesis 1 plus x right parenthesis left parenthesis 1 plus 2 x right parenthesis left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis 1 minus x right parenthesis left parenthesis 1 minus 2 x right parenthesis left parenthesis 1 minus 3 x right parenthesis end fraction is

  1. 12 minus 30.2 to the power of n plus 20.3 to the power of n
  2. 12 plus 30.2 to the power of n plus 20.3 to the power of n
  3. 12 plus 30.2 to the power of n minus 20.3 to the power of n
  4. 12 minus 30.2 to the power of n minus 20.3 to the power of n

The correct answer is: 12 minus 30.2 to the power of n plus 20.3 to the power of n

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(Me)2SiCl2 on hydrolysis will produce -

(Me)2SiCl2 on hydrolysis will produce -

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fraction numerator 1 over denominator x to the power of 4 plus 1 end fraction equals

fraction numerator 1 over denominator x to the power of 4 plus 1 end fraction equals

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An elastic string of unstretched length  and force constant  is stretched by a small length . It is further stretched by another small length . The work done in the second stretching is

An elastic string of unstretched length  and force constant  is stretched by a small length . It is further stretched by another small length . The work done in the second stretching is

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A shell of mass  moving with velocity  suddenly breaks into 2 pieces. The part having mass  remains stationary. The velocity of the other shell will be

A shell of mass  moving with velocity  suddenly breaks into 2 pieces. The part having mass  remains stationary. The velocity of the other shell will be

physics-General
General
Maths-

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
parallel
General
Physics-

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
General
maths-

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
General
physics-

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

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

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

physics-General

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

parallel
General
Physics-

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
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
General
Maths-

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
parallel
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
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

parallel

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