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Less than r/2. 

Just visualize a uniform cylinder of radius r and height r. Its center of mass will be at r/2 from the center of the base circle. Since the given hemisphere can be made by carving out masses from the upper parts the center of mass of the remaining hemisphere will be less than r/2.

Yes. If we start from any two particles of this system, the center of mass (say A) of these two particles will lie on the line joining them. This line will be fully inside the cube. Next, these two particles can be replaced with their total mass at this center of mass A. Now the center of mass of third particle and previous two particles will lie on the line joining the third particle and A, Say this center of mass as B. Again this line joining A and third particle is fully inside the cube, so B is also inside the cube. The total mass of three particles can be assumed to be at B. In this way we can show that the center of mass of all the particles of a system that lies in a cube is necessarily in the cube.

The earth does attract each and every particle. The resultant of all these forces of attraction is Mg which acts at the center of mass (where M is the total mass). That is why weight Mg of an extended body is generally shown in a diagram to act through the center of mass.

(a) Yes. Because the charges are of two different type, positive and negative, unlike mass which is all similar type. For example, suppose that there is 2q charge at the origin and -q at 2 units along the x-axis. Its center of charge will be at a distance 

= {1/(-q+2q)}*(-q*2+2q*0) =-2q/q =-2. 

Clearly, the center of charge, in this case, lies outside the line segment joining the two charges. 

(b) Since the z-coordinate of all the charges, in this case, is zero so Σ_iq_iz_i =0. Hence z-coordinate of center of charge =(1/Q)Σ_iq_iz_i =0. It means the center of charge necessarily lies in X-Y plane. 

(c) No. As we have seen in (a) that center of charge of two charges may lie outside the line segment joining the two charges, hence the center of charge, in this case, may lie outside the cube.

(b) both the bodies move after collision
(c) the moving body comes to rest and the stationary body starts moving

Explanation: 

Since one of the bodies has some velocity before collision hence total momentum of the bodies is nonzero. Since in the absence of external force total momentum is conserved, both the bodies cannot come to rest. Because then the total momentum will become zero. (a) is not true. 

In both the options (b) and (c) it is possible that final momentum is equal to the initial momentum. 

The option (d) is not possible because the total momentum will not remain the same as before the collision.

In a noninertial frame both the bodies will experience equal acceleration in the horizontal direction. So the experiment result will change in the accelerating car. 

After applying pseudo forces the motion can be analyzed as if in an inertial frame. Since the sum of external forces are not zero now, hence the equations (i) "Velocity of separation = Velocity of approach" and (ii) "final momentum = initial momentum" do not remain valid in the accelerating car.

Technically it is not proper to say that the force generated by the engine accelerates the car because the engine does not generate a force along the acceleration. Even if it did so internal reaction force would have balanced it. 

Since the movement of the piston of the engine makes a rotational motion in the wheels which in turn is used to push back the road and due to frictional force (External force) by the road, the car accelerates, hence in common language, such statement is made.

(c) does not shift horizontally

Explanation: 

Since the body breaks due to internal forces, the center of mass of all parts taken together will remain the same and will not shift horizontally.

(c) does not shift horizontally

Explanation: 

Since the body breaks due to internal forces, the center of mass of all parts taken together will remain the same and will not shift horizontally.

(d) both reflection and refraction of light.

Correct Answer is: (a) Cell wall

(a) Protoplast fusion

When the object is at the focal point.

Correct option is  A) at focus

Correct option is  A) at focus

At focus F image from

Correct option is  C) 30 cm

The object distance and image distance are same. So the object is kept at twice the focal length. So the focal length of the convex lens is 10 cm.

B) between optic centre and focus

A) focal length

C) same side of the lens

(x,y) = (0.54 m, 0.36 m)

\(\overrightarrow E\) and \(\overrightarrow B\) are in x-y plane and are mutually perpendicular. Given v = 30 MHz = 30 × 10⁶ Hz, c = 3 × 10⁸ ms^-1

∴ \(\frac{c}{v}=\frac{3\times10^8}{30\times10^6}\) ^{= 10m}

The speed in vacuum is the same for all

c = 3 × 10^-8 ms^-1

O^2- ions form cubic close packed lattice

Number of O^2- = 8 x \(\frac{1}{8}\) = 1 

Number of Fe³⁺ = \(\frac{2}{3}\) octahedral voids = \(\frac{2}{3}\)

Formula of the compound = \(Fe^{3+} + \frac{2}{3}O^{2-} = Fe_2O_3\)

(1) Quotient 36, Remainder 13 

(2) Quotient 116, Remainder 54 

(3) Quotient 151, Remainder 29 

(4) Quotient 3153, Remainder 15 

(5) Quotient 40, Remainder 4 

(6) Quotient 1709, Remainder 7

(1) 33,288 

(2) 1,23,552 

(3) 32,10,442 

(4) 3,11,547 

(5) 36,975 

(6) 28,371

Digit 1 is in lakhs place, its place value is 1,00,000

Digit 8 is in ten thousands place, its place value is 80,000

Digit 8 is in thousands place, its place value is 8,000

Digit 9 is in hundreds place, its place value is 900

Digit 1 is in ten place, its place value is 10

Digit 9 is in units place, its place value is 9

i. 16,410

ii. 20,880

• 4,09,138
• 3,17,045
• 1,20,645
• 9,72,860
• 6,54,302

(1) Washi has the greatest population. Population of Washi is 92,173 

(2) Moregaon has the greater population. 

(3) Gaganbawada has the smallest population. Its population is 35,777

i. Seven crore, fifty-four lakh, twenty-one thousand, six hundred and seven.

ii. Five crore, sixteen lakh, thirty-six thousand, eight hundred and fifty four.

Smallest number : (i) 10,689 (ii) 12358

Biggest number : (i) 98,610 (ii) 85321

Correct option is : (a) 45, 798

Biggest five digit number is 75,430 Smallest five digit number is 30,457

Rupees ninety-four lakh, twenty-nine thousand, four hundred and eight.

Rupees sixty-one lakh, seven thousand, one hundred and eighty-seven

Rupees forty-six lakh, fifty-three thousand, five hundred and seventy.

Rupees forty-five lakh, forty-three thousand one hundred and fifty-nine.

Rupees thirty-seven lakh, one thousand, two hundred and eighty-two.

Rupees twenty-seven lakh, seventy two thousand, three hundred and forty-eight.

Rupees fifty-eight lakh, forty-nine thousand, six hundred and fifty-one.

Smallest number : 35,789

Biggest number : 39,785

i. Digit 3 is in ten thousands place, its place value is 30,000

Digit 5 is in thousands place, its place value is 5,000 

Digit 7 is in hundreds place, its place value is 700 

Digit 0 is in ten place, its place value is 0 

Digit 5 is in units place, its place value is 5

ii. Digit 7 is in lakhs place, its place value is 7.0. 000

Digit 8 is in ten thousands place, its place value is 80,000 Digit 2 is in thousands place, its place value is 2,000 

Digit 8 is in hundreds place, its place value is 800 

Digit 9 is in ten place, its place value is 90 

Digit 9 is in units place, its place value is 9

iii. Digit 8 is in ten lakhs place, its place value is 80,00,000

Digit 2 is in lakhs place, its place value is 2.0. 000

Digit 7 is in ten thousands place, its place value is 70,000 

Digit 4 is in thousands place, its place value is 4,000 

Digit 5 is in hundreds place, its place value is 500 

Digit 0 is in ten place, its place value is 0 

Digit 8 is in units place, its place value is 8

(1) < 

(2) > 

(3) > 

(4) < 

(5) < 

(6) <

i. Rupees seventy-five lakh, fourteen thousand, three hundred and sixty-five.

ii. Rupees thirty-nine lakh, forty-two thousand, one hundred and eighty.

(1) 8,00,000 

(2) 40,000 

(3) 2,000 

(4) 5 

(5) 90 

(6) 600 

(7) 30,00,000

(1) B 

(2) C 

(3) profit of company B > D > A > C

(1) < 

(2) < 

(3) > 

(4) >

(1) Patel

(2) Shinde

(3) 34,67,008 < 37,51,386 < 43,51,239 < 48,00,173

(1) Seven lakh, sisxty-five thousand, two hundred and thirty-four. 

(2) Four lakh, seventy-three thousand, two hundred and twenty-five. 

(3) Three lakh, twenty-seven thousand and one 

(4) Eight lakh seventy-five thousand three hundred and seventy-five 

(5) One lakh fifty thousand four hundred and thirty seven 

(6) Two lakh three thousand one hundred and seventy-four 

(7) Six lakh forty seven thousand eight hundred and fifty-one 

(8) Nine lakh nine hundred and ninety-nine 

(9) Five lakh seventy-five thousand and ten 

(10) Four lakh three thousand and five.

(1) 59,07,017 

(2) 22,10,500 

(3) 52,25,415 

(4) 30,30,030