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Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z. (i) Show that * is both commutative and associative. (ii) Find the identity element in Z (iii) Find the invertible element in Z.

(i) First we have to prove the commutativity of * Let a, b ∈ Z. then, a * b =... View Article

The mass of bullet is 100 g. It is fired from a gun of mass 20 kg with a velocity of 100 m/s. Determine the recoil speed of the gun.

Given Mass of bullet =m=100g=100 /1000kg=1/10kg Velocity of bullet, v=100ms-1 Mass of gun, M=20kg Find out We have to determine... View Article

Two equal negative charges – q are fixed at points (0,-a) and (0,a) on the y-axis. A positive charge Q is released from rest at the point (2a, 0) on the x-axis. The charge Q will (A) Execute simple harmonic motion about the origin (B) Move to the origin and remain at rest (C) Move to infinity (D) Execute oscillatory but not simple harmonic motion

The correct answer is (D) Execute oscillatory but not simple harmonic motion Solution Motion is simple harmonic only if the... View Article

Find the length of the longest altitude of the ∆abc whose sides are 35 cm, 54 cm, and 61 cm respectively.

Given ABC be a triangle in which sides AB = 35 cm, BC = 54 cm, CA = 61 cm... View Article

If the sum of first 6 terms is 9 times to the sum of first 3 terms of the same G.P., then common ratio of the series will be

If in a sequence of terms, each succeeding term is generated by multiplying each preceding term with a constant value,... View Article

The area enclosed by the equation |x| + |y| = 4 is ___

Given equation can be expressed as When x < 0 and y > 0, – x + y = 4... View Article

If an infinite GP has the first term x and the sum 5, then which of the following is correct? a. x < –10 b. –10 < x < 0 c. 0 < x < 10 d. x > 10

The correct answer is (c) 0 < x < 10 Solution Geometric progression If in a sequence of terms, each succeeding... View Article

How can we find out a perfect square of any number?

An integer that can be expressed as the square of another integer is called a perfect square. In other words,... View Article

Find the square root of 2401 by long division method

Division method The division method of the square root is a very familiar and easy method available to get the... View Article

Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.

Given Rectangular dimensions length is 40 cm a diagonal is 41 cm Find out We have to determine the perimeter... View Article

If a,b,c are three distinct positive real numbers, then the number of real roots of ax²+2b∣x∣−c=0 is

Roots of quadratic equation The values of variables satisfying the given quadratic equation are called its roots. In other words,... View Article

Two numbers are in the ratio 3:5. When each of these numbers is increased by 10, their ratio becomes 5:7. Find the greater number.

Two numbers are in the ratio 3:5. When each of these numbers is increased by 10, their ratio becomes 5:7.... View Article

Normals are drawn at point P,Q,R lying on parabola y²=4x which intersect at (3,0) then area of △PQR is

The simplest equation of a parabola is y2 = x when the directrix is parallel to the y-axis. In general, if... View Article

A metal sphere A of radius ‘a’ is charged to potential V. What will be its potential if it is enclosed by a spherical conducting shell B of radius ‘b” and the two are connected by a wire?

Electric Potential Due to a Point Charge The electric potential at a point in an electric field is defined as... View Article

If the distance between the points (2, –2) and (–1, x) is 5, one of the values of x is

Distance formula: d2 = (x2 – x1)2 + (y2 – y1)2 Solution Distance formula: d2 = (x2 – x1)2 + (y2 – y1)2 d = √((x2 – x1)2 + (y2 – y1)2 )... View Article

Discuss motion of a car along a banked road with figure and show that [latex]v_{max}=tan^{-1}frac{v^{2}}{rg}[/latex]

Banking of roads is defined as the phenomenon in which the edges are raised for the curved roads above the... View Article

In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is (A) -4/5 (B) 1/5 (C) 4 (D) none the these

The correct answer is (c) 4 Solution Geometric progression If in a sequence of terms, each succeeding term is generated by... View Article

The potential energy function of a particle in the x−y plane is given by U=k(x+y), where k is a constant. The work done by the conservative force in moving a particle from (1,1) to (2,3):

A conservative force is a force done in moving a particle from one point to another, such that the force... View Article

How much water should be filled in a container 23.5 cm in height, so that it appears half filled when viewed from the top of the container?

Given The total height of the container = 21 cm Find out We have to determine how much water should... View Article

Points P,Q,R are in a vertical line such that PQ=QR.A. A ball at P is allowed to fall freely. What is the ratio of the descent through PQ and QR

Let us assume PR = 2x So, PQ = QR = x The ball begins with zero velocity. Let us... View Article

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