If the letters in the English alphabets A to M represent the number from 1 to 13 respectively and N represents 0 and the letters O to Z correspond from -1 to -12, find the sum of integers for the names given below. For example, Math `rarr` Sum `rarr` 13 + 1 - 6 + 8 = 16: SUCCESS

If the letters in the English alphabets A to M represent the number from 1 to 13 respectively and N represents 0 and the letters O to Z correspond from -1 to -12, find the sum of integers for the names given below. For example, Math rarr Sum rarr 13 + 1 - 6 + 8 = 16: YOUR NAME

Two dice are numbered 1,2,3,4,5,6 and 1,1,2,2,3,3 respectively. They are rolled and the sum of the numbers on them is noted. Find the probability of gettting each sum from 2 to 9 separately.

The plane denoted by P_(1) : 4x+7y+4z+81=0 is rotated through a right angle about its line of intersection with the plane P_(2) : 5x+3y+ 10 z = 25 . If the plane in its new position is denoted by P , and the distance of this plane from the origin is d, then find the value of [k//2] (where [*] represents greatest integer less than or equal to k).

A pair of numbers is picked up randomly (without replacement) from the set {1,2,3,5,7,11,12,13,17,19}. The probability that the number 11 was picked given that the sum of the numbers was even is nearly a. 0. 1 b. 0. 125 c. 0. 24 d. 0. 18

If the points (1,1,p) and (-3,0,1) are equidistant from the plane vecr.(3hati+4hatj-12hatk)+13=0 , then find the value o p.

Two different numbers are taken from the set {0,1,2,3,4,5,6,7,8,9,10}dot The probability that their sum and positive difference are both multiple of 4 is x//55 , then x equals ____.

Let A be an mxxn matrix. If there exists a matrix L of type nxxm such that LA=I_(n) , then L is called left inverse of A. Similarly, if there exists a matrix R of type nxxm such that AR=I_(m) , then R is called right inverse of A. For example, to find right inverse of matrix A=[(1,-1),(1,1),(2,3)] , we take R=[(x,y,x),(u,v,w)] and solve AR=I_(3) , i.e., [(1,-1),(1,1),(2,3)][(x,y,z),(u,v,w)]=[(1,0,0),(0,1,0),(0,0,1)] {:(implies,x-u=1,y-v=0,z-w=0),(,x+u=0,y+v=1,z+w=0),(,2x+3u=0,2y+3v=0,2z+3w=1):} As this system of equations is inconsistent, we say there is no right inverse for matrix A. The number of right inverses for the matrix [(1,-1,2),(2,-1,1)] is

Column I, Column II Four dice (six faced) are rolled. The number of possible out comes in which at least one dice shows 2 is, p. 210 Let A be the set of 4-digit number a_1a_2a_3a_4w h e r ea_1> a_2> a_3> a_4dot then n(A) is equal to, q. 750 The total number of three-digit numbers, the sum of whose digits is even, is equal to, r. 671 The number of four-digit numbers that can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7 so that each number contains digit 1 is, s. 450

To find the sum sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7) , we follow the following method. Put 7theta = 2npi , where n is any integer. Then " " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta This means that sin theta takes the values 0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7) . From Eq. (i), we now get " " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3) Rejecting the value sin theta =0 , we get " " 4 cos theta (1-2 sin^(2) theta ) = 4 sin ^(2) theta - 3 or 16 cos^(2) theta (1-2 sin^(2) theta)^(2) = ( 4sin ^(2) theta -3)^(2) or 16(1-sin^(2) theta) (1-4 sin^(2) theta + 4 sin ^(4) theta) " " = 16 sin ^(4) theta - 24 sin ^(2) theta +9 or " " 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta -7 =0 This is cubic in sin^(2) theta with the roots sin^(2)( 2pi//7), sin^(2) (4pi//7), and sin^(2)(8pi//7) . The sum of these roots is " " sin^(2)""(2pi)/(7) + sin^(2)""(4pi)/(7) + sin ^(2)""(8pi)/(7) = (112)/(64) = (7)/(4) . The value of (tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2)""(3pi)/(7))xx (cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is