Test Papers – ICSE – Class – X
Std-11, Science, Mathematics, Chapter-3, Functions
Good afternoon students, today, we will study Functions 2. In functions 1, we talked about different
kinds of functions, how to solve inequalities, how to find domain of functions, how to find range of
function. What are one, one functions, what are many to one functions, what are into functions and
what are onto functions? Now, we will continue from there only. First of all we will take few
questions from revisions on domain and range.
Okay, first question is, find the domain of fx equals to log of 4 minus x square cum mod to the base
10. How to solve, we had learnt that whatever is written inside the log, what it should be, greater
than zero. So, mod of 4 minus x square should be greater than zero. Base also should be greater zero
and base should not be 1. Base is already 10, it is greater than zero and it is also not 1. So, only mod
of 4 minus x square should be greater than zero. Mod of anything is always non-negative. So,
modulus of anything cannot be negative. So, when it will be greater than zero, when it is not zero.
Means x square should not be equal to 4, x should not be equal to 2 and x should not be equal to
minus 2. So, in this domain all x will be except 2 and minus 2. Hence, domain is x belongs to all real
numbers except 2 and minus 2 or x belongs to minus infinite to minus 2, union minus 2 to 2 union 2
to infinite. They have not given anything in the option, actually this option is wrong it should be curly
bracket and this becomes correct option. Leave the options, you can see these are the correct form
of answers.
Let us solve next question. Here there is also one small mistake. Here equal to sign should not be
there. We have to find domain, question is find domain. Question is find domain. There are two
functions and both of them should be satisfied. When the first log is satisfied, when the expression
inside is greater than zero, that is x should be greater than half. And the second log is satisfied, when
the expression inside it is greater than zero. Square root of any number is always greater than zero.
What we have to keep in mind is that, whatever quantity is inside square root it should be greater
than zero. And we know it is 2x minus 1 whole square plus 4, 4 plus pause, non-negative number. If a
square is added to 4 it will always be greater than zero, in fact it will be bigger than 4. It is greater
than zero for all x belonging to real numbers. x should be greater than half and should follow this
and what we will take the intersection of both and what are they doing to x, satisfy and those x are
half to infinite. Correct the question, find domain.
Next question, this question is about finding range. This question is about finding range of this
expression. We can see in this expression, one same term is used two times that is x square. We can
write it like this and like this hence what is x square minus y upon y minus 1 and we know that x
square is always greater than or equal to zero which implies minus y upon y minus 1 is always
greater than or equal to zero. Hence, y upon y minus 1 is less than or equal to zero. Wavy curve
method, where is zero, on zero, if we put 1 in denominator, 1 will not come in the answer. So, where
it is negative and zero, hence it is range, zero close, 1 open. Means its value will be in between zero
and 1, it could be zero but not 1. Clear? We found the value of x square and whatever we knew
about x square, that always x square is zero or greater than zero and there only we got minus y upon
y minus 1 and we solved with Wavy Curve. And we can visualise easily, how, x square upon 1 plus x
square, denominator is always greater. Denominator is always big, positive upon positive. So, there
can only be positive number in y, y cannot be negative. So, denominator will be greater, if
denominator is greater means it will be less than 1, so it will not be greater than 1. y will not be
greater than 1, number greater than 1 won’t come in the range. Can 1 come in the range?
Numerator upon denominator when it will become 1 when both are equal. Both of them cannot be
equal because what is always greater, denominator. Hence, 1 is not in range, here 1 is in range.
Hence, answer cannot be these two. From here we can visualise that answer is zero to 1 because
denominator is greater and it is positive, zero to 1, it will be smaller than 1. We can do this by hit and
trial and by observing also. So, we know what is domain, what is range. We have done a quick
revision.
Let us go further, next what we will study is different types of functions and in this different types
we will talk about even and odd functions. Even functions and Odd functions. What is even function?
For example, y is equal to x square. In y is equal to x square, the value of minus 1 square is 1 and as
well the square value of 1 is also 1. So, the value of minus 1 and 1 is same. So, value of minus 2
square is 4 and value of 2 square is 4. Minus 2 and 2 is same. So, that function whose value of x and
minus x is same, that kind of functions we call as even functions. Even and Odd functions, if f of
minus x is equal to fx, then fx is even function. If f of minus x is equal to fx then fx is even function
and if f of minus x is equal to minus fx, then fx is odd function. For example, f.
Std-11, Science, Mathematics, Chapter-7, Sequences and Series
Today we are going to study sequences and series. 1, 3, 5, 7 what’s the next number, 9. 1, 4,
9, 16, 25, what’s the next number, 36. Very good, very good, so all these are different
sequences there can be many, many sequences, there are infinite number of sequences
which are possible. What is a sequence? Which follow a definite pattern. What is the
sequence? A progression which follows mathematical pattern is called sequence. Now,
there are many different kinds of sequences, from all of those we are going to study very
few and those will be arithmetic progressions, geometric progressions, harmonic
progressions and few miscellaneous progressions.
Now, first let’s talk about arithmetic progression. What is arithmetic progression? It is
commonly known as AP. a, a plus d, a plus 2d, a plus 3d and so on. Next term will be a plus
4d, fifth term will be a plus 4d, sixth term will be a plus 5d and so on, this is arithmetic
progression. When, every time in terms any same number is added or subtracted like 1, 2, 3,
4, next term will be 5, every time one is added. So this is one arithmetic progression. nth
term of arithmetic progression is a plus 1 minus into d, why, because in first term d is zero,
in second term 1 d is there, in third term 2 d is there, in fourth term 3 d is there, in fifth term
4 d will be there and in sixth term 5 d will be there, so in nth term, how much d will be
there? n minus 1, nth term of AP is a plus n minus 1 into d. Sum to nth term is n by 2 into 2a
plus n minus 1 into d and which can also be written as n by 2 into first term plus last term.
Here, n represents number of terms, a represents first term, d represents common
difference and l represents last term.
Okay, let us talk about few questions. Let us solve one question, question is find sum of all
the 3 digit numbers which are divisible by 5? Find sum of all 3 digit numbers which are
divisible by 5? Smallest such number is 100, after that 105 and greatest 3 digit number which
is divisible 5 will be 995. Now what is formula of sum? n by 2 into a plus l which is n by 2,
into 100 and last term is 995. Now we do not have n, if we can find n then we will get the
sum. How to find n? Last term is say nth term which a plus n minus 1 into d, hence 100 plus
n minus 1 and common difference here is 5 has to be 995. Hence, n minus 1 into 5 is 895.
Hence, n minus 1 is 895 by 5, what is 895 by 5 is 179. Hence, what is n? 180 and what is
sum? 90 into 1095, I hope you can solve it. So we have done one question on sum of AP.
Next is, summation r, r goes from 1 to n. What is this, sum of first n natural numbers. If r is
replaced by 1 and then 2, 3, 4 upto nth. Sum of first n natural numbers is with AP. n by 2 is
first number to last number that is n by 2 into n plus 1. n into n plus 1 by 2. Similarly, sum of
first n odd numbers. Sum of n odd natural numbers it is 1 plus 3 plus 5 upto 2n minus 1 by
applying formula of sum, we can calculate it is n square. We need to learn both of the
formula. First n is sum of natural numbers n into n plus 1 by 2 and first n odd number sum is
n square.
Okay, moving further other things we need to know about AP is, if we have to assume 3
numbers in AP, we can assume those as a minus d, a and a plus d. These are 3 numbers in
AP. Why do we assume these 3 numbers? Because, in their sum d gets eliminated. Sum of
these 3 numbers is 3. So it is easy to do calculations and it is easy to multiply them. Here, 4
numbers in AP can be assumed as a minus 3d, a minus d, a plus d and a plus 3d. These are 4
numbers in AP. One more thing that we should know if a, b, c, are in AP. Then b minus a is
equal to, what is value of b minus a? c minus b, hence 2b is equal to a plus c. If 3 numbers
are in AP then twice the middle term is equal to first term plus last term.
So, what have we learnt till now? We have learnt what is AP, what nth term of AP, a plus 1
minus n into d? Sum to nth term of AP is n by 2 into 2a plus n minus 1 into d. sum term to
nth term can also be written as n by 2 into first term plus last term. Sum of first n natural
numbers is n into n plus n by 2. Sum of first n odd natural numbers is n square. And
assuming 3 numbers in AP, 4 numbers in AP. Similarly 5, 6 we can assume in this same way
and if 3 numbers are in AP then twice the middle term is equal to first term plus last term.
Moving further let us solve few questions. This says one, second and this. These 3 numbers
are in arithmetic progression. These 3 numbers are in arithmetic progression. We have to
find the value of x.
Std-11, Science, Mathematics, Chapter-5, Quadratic Equations B
Good morning class, in last class we talked about Quadratic Equations. Today we are going to study
quadratic equations further. Last class what we have done is, I am going to revise all that for you. We
know what is quadratic equation? Equation which is of the form ax square plus bx plus c is equal to
zero. If its roots are alpha and beta than ax square plus bx plus c can be written as a into x minus
alpha into x minus beta. This concept will be further used in equations of limits and other topics, so
we should know this. Next we know sum of roots is minus b by a, product of root is c by a that is
constant term upon coefficient of x square. Difference of roots is square root of discriminant divided
by modulus of coefficient of x square. Square root of discriminant divided by mod. And if alpha is a
root of ax square plus bx plus c is equal to zero then alpha satisfies the equation. That is a alpha
square plus b alpha plus c is equal to zero. These are the five important concepts on which we solved
many equations.
Let us move further, after that we talked about cubic equations exactly same thing goes on cubic
equations. If alpha, beta, gama are the roots of ax cube plus bx square plus cx plus d then ax cube
plus bx square plus cx plus d can be written as a into x minus alpha into x minus beta into x minus
gama and from here we can find that sum of roots alpha plus beta plus gama is minus b by a, sum of
roots taken two at a time is c by a, and product of roots is minus times d by a. That is minus times
constant term upon coefficient of x cube.
After that we talked about graph of Quadratic polynomial. Graph of Quadratic polynomial functions,
graph of quadratic polynomial looks like this, it is a parabolic graph. It is a parabolic graph, if a is
greater than zero than parabola is upward parabola. Similarly if a is less than zero parabola is
downward parabola and if discriminant is greater than zero then graph cuts x axis at two distinct
points. If discriminant is equal to zero then graph touches x axis at one point and if discriminant is
less than zero than graph does not intersect the x axis. Here we get two distinct roots, here we get
two equal roots and then we get no real root only imaginary roots. Further here what we can notice
is that if discriminant is greater than zero, graph or value of y is positive as well as negative. Because
graph is above x axis and below x axis both of the cases are possible. If discriminant is equal to zero
then graph is always above x axis or it touches x axis. Then we can say that y is always greater than
zero or y is equal to zero. And what is the third case, discriminant less than zero, when discriminant
is less than zero, graph is always above x axis that is function is always positive. We are talking about
the case a greater than zero. Similarly a less than zero also the same.
So, from here what we found was that if a is greater than zero and d is less than zero then ax square
plus bx plus c is greater than zero for all x belonging to real number. That means if a is positive and d
is negative so quadratic value will always be positive. For all x what will be the quadratic value?
Positive. And a is less than zero and d is less than zero then graph will be like this that means always
below x axis. Means value of y and ax square plus bx plus c value, for all x value will be negative. That
is ax square plus bx plus c is less than zero, for all x belong to real numbers or if there is equality in
discriminant then here also equality will be there. If there is no equality in discriminant then here
there will no equality. Done, this is what we have learnt in previous class.
One more concept that we talked at the end of the last class was on basic conditions on roots. When
two roots will be positive and when two roots will be negative. For which x two roots will be
positive, for which constant condition that both the roots will be positive and for which all
conditions will be there where both the roots will be negative. This is the same sheet which we
studied. If both the roots are positive then what will happen? First is d will be greater than or equal
to zero. Roots will be real, roots are positive, positive means real. Now we talk about positive and
real. When are roots real, discriminant greater than zero, what will be the sum, positive and what
will be the product, positive. Roots will be real, sum will be positive and product will also be positive.
All these three conditions simultaneously will hold. Similarly if both the roots are negative then what
will happen, d will be greater than or equal to zero. Roots will be real, sum will be negative and
product will be positive. Exactly similar conditions and then we talked about roots of opposite signs.
Roots will be of opposite sign when product will be negative only one condition is necessary as well
as sufficient. D greater than zero, we need not apply because discriminant becomes positive by
itself. Only one condition is necessary and sufficient, we need not apply other conditions.
To complete the revision, let me solve 2, 3 more questions on same concepts. Solve this, find ‘k’ for
which roots of this equation are of opposite sign. We have to find out value of ‘k’ for which sign of
both the roots should be opposite. One root positive and other root is negative. How many
conditions are there in opposite sign that is one, which we have learnt now. Which one, roots will be
of opposite sign when product.
Std-11, Science, Mathematics, Chapter-4, Quadratic Equations A
Today, we are going to study topic Quadratic Equations. Equations which are of the form ax square
plus bx plus c equals to zero, are quadratic equations when a is not equal to zero. If a is zero than
equation is linear equation.
Let us talk about roots of quadratic equations. Alpha beta are the, if alpha beta are the roots of
equation ax square plus bx plus c is equal to zero, that means alpha beta satisfies then what can we
say, first thing that we can say is ax square plus bx plus c can be written as a into x minus alpha to x
minus beta. Why, because alpha is the root of this equation hence alpha satisfies ax square plus bx
plus c is equal to zero and hence x minus alpha is the factor of this quadratic polonomy. From this
we can derive that sum of the roots is minus b by a. From this we can easily see that what will come
from sum of this root, minus b by a. Product of this root is c by a and difference of roots of both the
equations is under root of b square minus 4ac by mod a. Difference of roots of this quadratic
equation is b square minus 4ac upon mod a. Other thing, we can see from here is as alpha is root
square of ax square plus bx plus c equals to zero, if alpha is one root then what will it do to it, it will
satisfy. Means after putting alpha in this, this equation will be satisfied.
Again, let me make you revise. Some of roots minus b by a, product of roots is c by a, difference of
roots is under root d by mod a, square root of discriminant. b square upon 4c is called discriminant,
you all know. And alpha will satisfy this equation. Similarly, beta will also satisfy this equation. A beta
square plus b beta plus c is equal to zero.
Yes, let us see how to use all these five reasons. Okay, take out question number 1 of your
worksheet. If 1 minus p is a root of quadratic equation x square plus px plus 1 minus p equals to
zero, then it’s roots are, if 1 minus p is one root then we have to find other two roots. Okay, if 1
minus p is its root then 1 minus p will satisfy this equation. What I have done here is, I have put 1
minus p instead of x, instead of x, I have put 1 minus p. From this take 1 minus p as common, from
here we get that p should be equal to 1. If p is 1 then this equation becomes x square plus x equals
to zero means what is its roots, x into x plus equals to zero, either x equals to zero or x equals to
minus 1. Hence both the roots are zero and minus 1, hence a is the correct answer. I hope everybody
got it.
Let us go to next question. The difference between corresponding roots of x square plus ax plus b
equals to zero and x square plus bx plus a equals to zero is same. The difference between both the
roots is same. What will the difference of first root? What is the formula of difference of roots?
What is mod a? Under root of discriminant upon mod a. What will be the roots of first equation?
What will be the discriminant of first equation? A square minus 4b upon, what is the coefficient of x
square is 1 upon mod of 1. And what will the difference of roots of second equation? B square minus
4a upon mod 1, hence you can say that a square minus 4b is equal to.
Std-11, Science, Mathematics, Chapter-1, Basic Functions
Good afternoon class, welcome everyone.
Today, we will study the chapter Function. In functions we will study about many different kinds of
functions, modulus functions, greatest integer functions, fractional part functions, logarithmic
function. There are infinite types of different functions that we can study on. But to study all these,
first we should know how to solve the inequalities, because if we are able to solve basic inequalities
first only then we will be able to solve other questions. So, to study basic inequalities, first thing is
we should know about different inequalities.
First thing is basic inequalities. Basic Inequalities, if a is greater than b, if any number is greater than
the other number then a plus c is always greater than b plus c. a minus c is always greater than b
minus c. a into c will be greater than b into c. Is it always true, no, it will be true if and only if c is
positive, that is if in an inequality we multiple a positive number on both the sides inequality remains
same. And a into c will be less than b into c, if and only if c is less than negative and if we multiple
both the numbers with negative then what will be the sign of inequality, it will change. So, one, very
important thing that we learnt here is, in one inequality if we add or subtract from both the sides,
there will be no change in inequality and we are multiplying the same number on both the sides, it
will remain same if the number is positive and change if the number is negative. This is what we
learnt.
Same goes for division. Same goes for division, a by c, b by c, a is greater than b. a is greater than b.
So, a by c and b by c, what will be the relation? a by c will be greater than b by c, inequality will
remain same if c is positive and it will change if c is negative. Inequality remains same if c is positive
and it changes if c is negative. These are few basic inequalities that you should know.
Next thing, when the multiplication between two numbers will be positive. By multiplying two
numbers together when will it be positive. If a and b are of same sign. I hope everybody agrees, yes.
And when will multiplication of the two numbers be negative, if a and b are of opposite sign. If both
the numbers sign are opposite then the multiplication of both the numbers will be negative.
Let us use this concept to solve few questions. Say, we need to solve a question that, find x such that
x minus 1 into x minus 2 is greater than zero. We need to solve this question, we need to find x for
which x minus 1 into x minus 2 will be greater than zero. For example, let us put x equal to 3. 3
minus 1 is 2 and 3 minus 2 is 1, 2 into 1 is 2, which is greater than zero, so x equals to 3 satisfies this
in equation. So we have to find all other x like x equal to 3, which satisfy this in equation. What we
will do, we will find x, for which x minus 1 is zero, x minus 1 is zero for x equals to 1. Similarly, x
minus 2 is zero for x equals to 2. We will plot them on a straight line. Now, when x is greater than 2,
when x is greater than 2, x minus 2 is positive. If you subtract a number bigger than 2, so you will get
one positive number, for example 3 minus 2 is 1 positive. Like that only, when x is greater than 2,
number bigger than 2, x minus 1 is also positive. Both of them will be positive, positive into positive
will be positive. If x is bigger than 2, so x minus 2 into x minus 1 will be positive. Similarly, when x lies
between 1 and 2, when x is between 1 and 2, x minus 2. For example put 1 point 5 here, 1 point 5
minus 2 is minus point 5 it is negative and x minus 1 will be, between 1 and 2, x minus 1 is positive
because x is bigger than 1, so, bigger than 1 is positive. Negative into positive is negative. Similarly
here x minus 2 will be negative and x minus 1 will also be negative and negative into negative
becomes positive. Now, what we wanted to show is, when is it positive when both the numbers are
multiplied. Where they are positive, you can see that it is positive for all x greater than 2 and it is
positive for all x is less than 2. And the answer is x is less than 1 or x is greater than 2, which we can
write in set form as minus infinite to 1, union 2 to infinite. All these x, what will they do to this
inequality, satisfy. For all these x, this inequality will satisfy means what will happen x minus 1 into x
minus 2, positive. It is a very length process, so we will not solve all questions like this.
So will develop a short cut, what is that short cut. That short cut is called as Wavy Curve method. It is
applied when multiplication or division of 3, 4, 5 as many factors are possible is less than zero or
greater than zero. Say, question we have been asked x minus 1 into x minus 2 into x minus 3 is less
than zero. Find the value of x which satisfies this. How to solve? We will apply all the rule of x
factors. x minus 1 will be zero when x is equal to 1, x minus 2 will be zero when x is equal to 2, x
minus will be zero when x is equal to 3. When x is greater than 3, all the x factors will be positive and
product will be positive and after that negative, positive, negative, positive. You write it alternately.
What you have to write in the beginning, positive, on the right positive and then negative, positive
write alternately and whatever is asked in the question, answer according to that. What we have
been asked is, when will multiplication be less than zero. When it is negative, when x is less than 1 or
when x lies between 2 and 3. If we write it in set form minus infinite to 1 union 2 to 3. Everybody got
it, yes, very good.
Let us solve one more question. Solve for x, such that x minus 2 into x minus 3 into x minus 5 is less
than or equal to zero. First step.
Std-11, Science, Mathematics, Chapter-2, Functions
Good afternoon, students, welcome to second class of functions. In first class, we talked about the
domain of functions, we talked about how to solve different inequalities. We also learnt that how to
solve functions of modular questions. Then we also learnt about domain also. Now, we will study
further.
We ended the class with domain, what is domain of a functions? If y is equal to f(x) has been
expressed as mathematical expression and domain has not been stated explicitly then it is
considered to be largest set of x values for which y is real. The largest set of X value for which y is
real that set is called Domain.
Today, we will talk about what is range of functions. What is range of functions? The set of values of
y that we obtain after inputting domain. After putting domain in the functions which gives the set of
values of y, we call it as range. After putting domain values when we get the set of values of y, we
call it as range.
Let me give you few examples. Say, y is equal to x square, what is it’s domain? If nothing is written
then domain is all x for which y is defined. Y is defined for all x belonging to real numbers. So,
domain is x belong to real numbers. After putting real numbers then what all values of y we will get.
We can see here clearly that if y is minimum zero because square of anything can never be negative.
Square of anything will be always non-negative. It will never be negative, so minimum value of y is
zero, maximum value of y approaches to infinity. Hence, y belongs to zero to infinite. Y can take all
the values between zero and infinity. Hence, range of this function is y belong to zero to infinity,
what is this – range.
Let us take one more example. What is range of y equal to 1 plus x square? What will be the range
of y is equal to x square? Here minimum value of y will be 1 because x square minimum is zero,
hence minimum value of y will be 1. Hence, minimum value of y is 1 and maximum value is infinite,
then y belongs to 1 to infinite.
Let us solve a question, you know all the basics. This was just for revision, let us solve the question.
Question is, find the range y equals to 7 minus xP x minus 3? Find the range y equals to 7 minus xp x
minus 3? Let us discuss what is its domain? First let us talk about what is its domain?
