The sum of two positive numbers is 12. Find the minimum possible value..
The sum of two positive numbers is 12. Find the minimum possible value of the sum of the cubes of these numbers. => 66 or 432 or 72 or 1728..
Matrices and Determinants
>If any two rows or columns of a determinant are equal, then its value is zero. If each element of a row or column of a determinant multiplied by k, then its value is multiplied by k. If two rows or columns of determinant are proportional, the value..
>If any two rows or columns of a determinant are equal, then its value is zero. If each element of a row or column of a determinant multiplied by k, then its value is multiplied by k. If two rows or columns of determinant are proportional, the value..Binomial Theorem Summary
of (a + b) n and is given by - If n is an odd natural number, then there are two middle terms in the expansion of (a + b)n and are given by i) The sum of all binomial coefficients in the expansion of (1+x) n is 2 n . ii) The sum of all even binomial..
of (a + b) n and is given by - If n is an odd natural number, then there are two middle terms in the expansion of (a + b)n and are given by i) The sum of all binomial coefficients in the expansion of (1+x) n is 2 n . ii) The sum of all even binomial..To find the sum of a number of terms in Arithmetical Progression:
Let a=first term, d = common difference, l=t n =last term, s = required sum. Then, Writing the series in the reverse order, Adding together the two series, ..
Let a=first term, d = common difference, l=t n =last term, s = required sum. Then, Writing the series in the reverse order, Adding together the two series, ..General Series
1. To find the sum of first n natural numbers. 2. To find the sum to squares of first n natural numbers. 3. To find the sum to the cubes of first n natural numbers. 4. ..
Sigma Notation
The Greek letter S (read as sigma) denotes the sum. When written before the n t h term of series, implies, the sum of all terms obtained by giving to n the different values 1,2,3..
Question 1
Question: Show that the sum of the cubes of any number of consecutive integers is divisible by the sum of those integers. Answer: Let the consecutive integers be (n + 1), (n + 2), (n + 3),....,(n + m) Sum of the cubes..
Question: Show that the sum of the cubes of any number of consecutive integers is divisible by the sum of those integers. Answer: Let the consecutive integers be (n + 1), (n + 2), (n + 3),....,(n + m) Sum of the cubes..Addition of Matrices
If A and B are 2 matrices of the same order, then A + B is the sum of the 2 matrices where each element is got by adding corresponding elements of A and B. ..
If A and B are 2 matrices of the same order, then A + B is the sum of the 2 matrices where each element is got by adding corresponding elements of A and B. ..Sigma Notation
The Greek letter S (read as sigma) denotes the sum. When written before the n t h term of series, implies, the sum of all terms obtained by giving to n the different values 1,2,3n. Thu..
The Greek letter S (read as sigma) denotes the sum. When written before the n t h term of series, implies, the sum of all terms obtained by giving to n the different values 1,2,3n. Thu..Suggested answer:
Find 'a.c ', the product of the coefficient of x 2 and the last term. Find all the factor pairs of 12. (1,12), (3,4), (2,6) are the factor pairs of 12. The factor pair (3,4) is such that 3 + 4 = 7and 3 x 4 = 12 i...
Find 'a.c ', the product of the coefficient of x 2 and the last term. Find all the factor pairs of 12. (1,12), (3,4), (2,6) are the factor pairs of 12. The factor pair (3,4) is such that 3 + 4 = 7and 3 x 4 = 12 i...See what our Users say :
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