Calculus ⚫Introduction of Limits ⚪ Consider a function f(x) . Now, if x approaches a value c , and if for any number a > 0, we find a number b > 0 such that |f(x) − l | < a whenever 0 < |x − c| < b , then l is called the limit of function f(x). ⚪ Limits are used to define continuity, derivatives and integrals. ⚪ Limits are denoted as ⚪ Left-Hand and Right-Hand Limits If the values of a function f(x) at x = c can be made as close as desired to the number L 1 at points closed to c and on the left of c, then L 1 is called left-hand limit. If the values of a function f(x) at x = c can be made as close as desired to the number L 2 at points on the right of c and close to c, then L 2 is called right-hand limit. ⚪ Properties of Limits Some of the important uses formula of limits are: ⚪ L's Hospital Rule Question based on L's Hospital Rule (Gate 2017) : The value of : Solution...

Linear pair of equation in two variable We have a pair of linear equation in two variable a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 Solution of two variable linear equation is the value of x and y which satisfy both the equation. Enter the value of coefficient of both the equation in following bracket: Equation 1 : x + y + = 0 Equation 2 : x + y + = 0 calculate

Vector ⚫Introduction ⚪ Those quantities which have both magnitudes and direction are called Vector . Example : velocity ,accerelation , weight , force are some vector quantities. ⚫ Representation of vectors : ⚪ Geometrically a vector is represented by a line segment. For example , . Here A is called the initial point and B is terminal point. Magnitude or modulus of a is expressed as ⚫ Types of Vector ⚪ Zero vector or null vector : A vector whose magnitude is zero is called zero or null vector. ⚪ Unit Vector : A vector whose magnitude is unity (1). The unit vector in the direction of a vector a is denoted by . ⚪ Collinear or Parallel Vector : Vectors having the same or parallel supports are called collinear or parallel vectors ⚪ Co-initial Vector : Vectors having the same initial point. ⚪ Coplanar Vector : A system of vectors is said to be coplanar, if their support are p...

CSAT Practice Question Question no. 21 What is the least four digit number when divided by 3 , 4 , 5 and 6 leaves a remainder 2 in each case ? [CSAT 2020] looks_one 1012 looks_two 1022 looks_3 1122 looks_4 1222 Answer Option looks_two 1022 Solution Solution : Question no. 22 The recurring decimal representation 1.272727.... is equivalent to [CSAT 2020] looks_one 13/11 looks_two 14/11 looks_3 127/99 looks_4 137/99 Answer Option looks_two 14/11 Solution Solution : Question no. 23 What is the greatest length x such that and are integral multiplers of x ? [CSAT 2020] looks_one looks_two looks_3 looks_4 Answer Option looks_4 Solution Solution : Question no. 24 Consider the following data : Year birth rate death rate 1911 - 1921 48.1 35.5 1921 - 1931 46.4 36.3 1931 - 1941 45.2 31.2 1941 - 1951 39.9 27.4 1951 - 1961 ...

Linear Algebra Question no. 1 What are the eigen values of ? looks_one 1,4 looks_two 2,3 looks_3 0,5 looks_4 1,5 Answer Option looks_3 0,5 Solution Solution : ( 4 - λ)(1 - λ) - 4 = 0 λ 2 - 5λ + 4 - 4 = 0 λ 2 - 5λ = 0 λ(λ - 5) = 0 λ = 0 , 5 Question no. 2 What is the eigenvalues of the matrix ? looks_one 1, 4 ,4 looks_two 1 , 4 ,-4 looks_3 3 , 3 ,3 looks_4 1 , 2 ,6 Answer option looks_one 1, 4 ,4 Solution Solution : ( 3 - λ )[(3-λ) 2 -1] + 1(-3 + λ -1 ) - 1(1+3 -λ) = 0 ( 3 - λ )(9 + λ 2 - 6λ -1 ) + 2( λ - 4 ) = 0 ( 3 - λ )( λ 2 - 6λ + 8 ) + 2( λ - 4 ) = 0 ( 3 - λ )( λ - 4)( λ - 2) + 2( λ - 4 ) = 0 (λ - 4) [( 3 - λ )( λ - 2) + 2 ] = 0 (λ - 4) (λ 2 - 5λ + 4) = 0 (...