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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


*form a quadratic equation whose roots are
7+sqrt(3) and
7+sqrt(3)


*find the square root of x^4 -6(x^3) + 19(x^2) -30x + 20 by long division method 

*If A is the matrix
 [1 -1] first row
[2 3]second row
show that [A]^2 - 4A+5I=0


*A matrix has 30 elements.What are the
possible orders it can have?


Set Theory

*Prove A\[B intersection C]={[A\B]union [A\C]}
using  venn diagram [de morgans law of set difference]


*

let A ={0,1,2,3} and B={1,3,5,7,9}
let f:A-->B be a function given by
f(x)=2x+1. Represent this function as
a set of ordered pairs and as a table


*

If universal set,U = {-2,-1,0,1,2,3,4,5,6,7,8,9,10}
A={-2,2,3,4,5,6,7,8,9,10}
B={1,3,5,8,9}
Verify de morgan's law of complementation


Sequence and Series

Find the sum of all 3 digit natural numbers
which are divisible by 8

Coordinate geometry
*Find the area of the triangle formed by the lines
x+y=2, x-y=0,x+2y-6=0


*

Verify the property that diagonals of a
rhombus bisect each other at right angles
for the rhombus with verticesA(0,5),B(-2,-2)
C(5,0),D(7,7)


*The line joining the points A(-2,3) and B(a,5) is parallel
to the line joining C(0,5) and D(-2,1). Find the value of a.


*
P and Q trisect the line segment joining
(2,1) and (5,-8). If P lies on 2x-y+k=0
find the value of k
 



Mensuration
*

A heap of paddy is in the form of a
right circular cone whose diameter
is 4.2 m and height 2.8 m. If the heap
is to be covered exactly by a canvas
to protect it from rain then find
the area of the canvas needed.


*A tent is in the shape of a right circular cylinder  surmounted
 by a cone. The total height and
diameter of the base are 13.5 m
and 28 m. If the height  of the
cylindrical portion is 3 m , find
the total surface area of the tent.



*A cylindrical shaped well of depth 20 m and diameter
14m is dug. The dug out soil is evenly spread  to form
a cuboid platform with base dimension 20m x14m. Find
the height of the platform.


*
An iron right circular cone of diameter 8cm
and height 12 cm is melted and recast into
spherical lead shots each of radius 4mm. How
many lead shots can be made.


 
Statistics
*Find the standard deviation of the first n
natural numbers


LCM and HCF

*The traffic lights at three different road
crossing change after every 48 sec, 72sec
and 108 sec respectively. If they  all change
simultaneously at 8:20:00 hours, when will
they change simultaneously again


*

If the HCF of two numbers is 11 and their
LCM is 7700, and one of the numbers is 275
find the other number
 

Trigonometry

heights and distance
*

A student sitting in a classroom sees a picture on
the black board at a height of 1.5m from the horizontal
level of sight. The angle of elevatin of the picture
is 30 degrees. as the picture is not clear to him he
moves straight towards the black board and sees the
 picture at an angle of elevation 45 degrees. Find the
distance moved by the student



*A ladder leaning against a vertical wall
makes an angle of 60 degrees with the ground.
The foot of the ladder is 3.5 m away from
the wall. Find the length of the ladder


*

From the top of a lighthouse of height 200 feet, the
lighthouse keeper observes a yacht and a barge along the
same line of sight. The angles of depression for the yacht
and the barge are 45 degrees abd 30 degrees respectively.
For safety reasons the two sea vessels should be at least
300 feet apart. If they are less than 300 feet apart, the
keeper has to sound the alarm. Does the keeper have to
sound the alarm



*
A jet fighter at a height of 3000m from the ground
passes directly over another jet fighter . At that instant
the angles of elevation of the two fighters from a
point on the ground are 60 degrees and 45 degrees,
Find the vertical distance between the two jets.


*

The angle of elevation of the top of a hill from
the foot of a tower is 60 degrees and the angle of
elevation of the top of the tower from the foot of the hill
is 30 degrees. If the tower is 50 m high find the
height of the the hill


*
A vertical tree is partially broken by the wind . The top of
the tree still connected to the lower portion
 touches the ground and makes an angle
30 degrees  with the ground. If the top of the tree touches
the ground 30 m away from the foot of the tree, what
was the height of the tree before it broke


*

A person in an helicopter flying at a height of 500 m,
observes two objects lying opposite to each other
on either bank of a river. The angles of depression
of the objects are 30 degrees and 45 degrees. Find
the width of the river.
 


trigonometry standard values
and identities

*
In triangle ABC right angled at C find
the values of cos(A+B) and sin (A+B)


*
If tanA +sinA = m , tanA - sinA = n
show that (m^2)-(n^2)=4sqrt(mn)


Probability

*
A card is drawn from a deck of 52
cards. Find the probability of getting
a king or a heart or a red card.
letA, B, C be the events of getting
a king, a heart, a red card respectively.
 
 
* 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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