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questions

Integration

* Integrate e^{x^(1/3)}dx

* evaluate integral of [e^(-x) +1] ^2

*evaluate ∫ (sin 3x)^4 dx with limits 0 to pi

*the integral of (1+x) /[ x *sqrt(x-2)]

* evaluate the integral of sqrt(tan  x)

* evaluate integral of { sqrt{1+(x^2)} } / {x^2}
using integration by parts


*∫ secx dx using integration by parts

*integral of (e^(ax) )  cosbx using integration by parts

*∫ arc(tan4x) using integration by parts

*evaluate the integral of coshx cosx by integration by part method

*evaluate .∫ (x) sin x  dx using integration by parts

*evaluate .∫ x cos(mx) dx using integration by parts

*integral of log(sin x) from 0  to pi/2 (where the log is of base e)




Area using integral

* evaluate the area of an ellipse (x /a) + (y/b) =1 using integration

* find the area between y = x and x =y

* find the area bounded b y =x and y = 3x

Differential equation

*solve x(dy/dx) + (1+x)y =2





Differentiation
differentiation from first principles
*differentiate sqrt(x) from first principle

*if y = x sinx , find dy/dx from first principles

*find the derivative of f(x) =1/x at x=3 using the
lim {f(x)-f(a)} / {x-a} as x---->a



*derivative of ln(x + sqrt(x^2-1))

Implicit differentiation

*find dy/dx if sqrt(x) +sqrt(y) = 8

*if  xy + y^2 =1 , find dy/dx

*if √(xy) = x - 2y, find dy/dx using implicit differentiation

Differentiation using trigonometric substitution 
*differentiate. arctan √ [(1-x)/(1+x)] w.r.t..x using trigonometric substitution



Application of differential calculus
* power series expansion for y = arctan(x) in powers of x

*find the maclaurin series for sec x



Boolean Algebra
*show that (p v q) → r ≡ (p → r) ^ (q→ r)

Mathematical Induction


*Prove that [1/(1x2) ]+[1/(2x3)]+...+ [1/( n(n+1) )] = n/n+1
using mathematical induction)


*prove by mathematical induction
 that 1^3 +2^3 + 3^3 +...+ n^3 = [ ( n (n+1) ) /2 ] ^2


Partial fraction
*put into partial fraction 1 / { (x^3) + 1 }

Algebra
*When a polynomial p(x) is divided by (x+1), the remainder is 4.
And when it is divided by (x-2), it's remainder is 3.
What is the remainder  when p(x) is divided by x-x-2


*Express  the recurring decimal 0.484848... as a fraction
 using the concept of geometric series

*If the sum of two consecutive odd integers is 20,
find the numbers


*Factorise 2(x^3) - 3 (x^2)  - 3x +2


*Person A takes 6 days less than the time taken by the person
B to finish a particular piece of work. If both A and B work
together , they can finish it in 4 days. Find the time that B
 would take to finish this work by himself


*form a quadratic equation whose roots are
3 and 4



* Complex Number

Notes on complex numbers, concept of the imaginary unit i, powers of i,
general form of a complex number, modulus, amplitude etc.

* Progression
Arithmetic progression(A.P.), Geometric progression (G.P), Harmonic progression (H.P) ,
 nth term , sum of n terms, of A.P. and G.P sum of infinite number of terms of a G.P
, sum of the first n natural numbers,
sum of the squares of the first n natural numbers and
sum of the cubes of the first n natural numbers


* Trigonometry formula
trigonometry identity, trigonometry formulae, trigonometric
ratio/value of standard angle





*Analytical Geometry
The distance formula, section formula, midpoint formula, various equations of
straight line like point slope,two point, intercept form, equation of circle, condition
for orthogonal circle, equation of tangent to a circle are given.

*Vector
vector dot product, cross product,magnitude or modulus of a vector, unit vector
scalar triple product [S.T.P], vector triple product[V.T.P]


*list of differentiation formula
list of differentiation formula like power rule, trigonometric function


*integral calculus integration formula
list of formula used for integration of functions, integral calculus

*correlation coefficient equation of regression line

Carl pearson correlation coefficient, equation of the line of regression
of y on x and the regression line of x on y
 
*Indeterminate forms, LHospital's rule, Permutation and Combination
the list of indeterminate forms, how to evaluate limits of the form 0/0 and infinity/ infinity  using
l hospital's rule, the concept of permutation and combination


*square root without using calculator by using long division method
method of finding the square root by long division without using calculator

*Hyperbolic function
Definition of hyperbolic functions sinh x, cosh x ,tanh x , coth x, cosech x, sech x
and some identities of hyperbolic functions.

*Laplace Transform Inverse laplace transform
Laplace transform and inverse laplace transform of some standard functions
for b.tech, be students.

* Graph Theory definitions
This page contains some of the most basic definitions from the subject of
 graph theory, which might be useful to the students of computer science.


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