Explanation:

The quadratic equation in mathematics is given as follows:

AX2 + BX + C = 0,

where, A, B and C are constants. The solution of the equation comprises two roots as power of X is 2.

R1 = (-B + sqrt (B * B - 4 * A * C)) / 2 * A

R2 = (-B - sqrt (B * B - 4 * A * C)) / 2 * A

The expression B * B - 4 * A * C is known as discriminant (say dis), and on the basis of its value the roots are determined as equal (dis = = 0) imaginary (dis < 0) or unequal (dis > 0) as shown in the above program.

In the program, the value of three constants are taken as output using A, B and C and the value of dis is calculated.

Explanation:
The ASCII/unicode values is from 97 to 122 (inclusive both) for lowercase alphabets and 65 to 90 (inclusive both) for uppercase alphabets. It is checked whether the entered character is within these two ranges using else-if ladder. Similarly ASCII/Unicode values for digits are from 48 to 57 (inclusive both), so character entered is also checked with this range. If all the three conditions fail then the character entered must be a special symbol.

Explanation:
There is no break statement in case 1, so both case 1 and case 2 are assumed to be true and so is the output. In fact, due to the absence of break statement in the second case, rest of the statements are considered part of the second case till a break is not found. Break in case 2 causes control to come out from switch. If break were not in case 2 also then default would have got executed too.

Explanation:
Any +ve number is taken as input. The number n is multiplied by 1 through 10 inside the loop and the loop counter; the number and value are displayed by formatting as to produce the desired output.

Explanation:
Initially the number of elements taken is n. The first number is taken outside the loop and assumed to be maximum; this number is stored in max. The remaining numbers are taken inside the loop. On each iteration, the number is compared with the max. If the max is less than the number taken, the number will be the maximum one. This is checked through if statement. In the end when control comes out from while loop, max is displayed.

Explanation: The logic to reverse a number is quite simple. It can be understood step by step inside the loop:
Suppose orig = > original number entered by programmer.
   rev => reverse of the orig.
   r => remainder
While (orig! =0)
{
   r = orig%10;
   rev = rev *10 +r;
   orig= orig /10;
}

Explanation: A number is prime if it is completely divisible by 1 and itself, e.g., 1, 3, 5, 7, 11, 13, 17, 19, 23, etc. To check whether a number is prime or not, start from a counter c = 2 (every number divides by 1) and continue till c < = num/2 since no number is completely divisible by a number which is more than half of that number. For example, 12 is not divisible by 7, 8, 9, 10, 11 which are greater than 6. So it should be checked whether the number divisible by any number < =num/2 is true; simply print ' Number is not prime ' and exit from the program, using System.exit(0). If num%c = = 0 is never true during the loop then at the end when loop exits maturely print 'Number is prime'.

Explanation: A number is called palindrome if on reversing it is equal to the original number, e.g., 121, 454 and 3443, etc. To check whether an entered number is palindrome or not, simply reverse a number; the original number becomes zero when controls come out from the loop. So, the original number is saved in a variable before processing; in the above program, it is in the save variable.

Explanation:

In the series to be generated, odd numbers are +ve and even numbers are −ve. Take one variable k = 1. k is multiplied by −1 inside the loop in each iteration.

Initially sum = 0 + (1 * 1) gives sum = 1 then k becomes k = 1 * − 1= −1, which is used in the second iteration of the loop. In the second iteration, sumbecomes sum = 1 + (2 * −1) => sum = 1 − 2 = − 1 and value of k changes to 1 again (−1 * −1 = 1) and this continues for n times.

Explanation:
pow is standard library function which returns power of first argument to second number, i.e., pow(2, 3) returns 2 to the power 3, i.e., 8. A number of terms are taken in n and loop is run from 1 to n. On each iteration, 2 was found to the power t and added to sum. The pattern is displayed within the for loop and sum of series is displayed outside the loop.

Explanation:
A number is called perfect if sum of its factor is equal to the number itself, e.g., 6; its factors are 1, 2 and 3 and their sum is 6, which is equal to the number itself, so it is a perfect number. Similarly 28 is a perfect number. In the loop, the sum is initialized to 1 and the loop is run for num/2 as for any number, e.g., t which is more than num/2, num%t will not be zero (excluding num itself).

Explanation: Two for loops have been used in the program. One is to control the number of rows and second is to control the number of columns. Initially assumeline = 5 so outer for loop runs five times. In the first run, row = 1 and inner loop runs only once. The value of value variable is incremented and printed. Control moves to the new line. For row = 2, inner loop runs twice and second row of output is printed. This continues till row < = line.

Explanation: For the first run of outer for loop, ch = 'A'. When first iteration of inner for loop finishes, value of ch is incremented by 1 and becomes B. As one A on first row, two B on second row and so on have to be printed, ch at the end of every iteration of outer for loop is incremented.