2024-04-13 07:29
(Maximum Marks: 80)
(Time allowed: Two hours and a half)
Attempt all questions from Section A and any four questions from Section B.All work, including rough work, must be clearly shown and… Read More
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MATHEMATICS MADE EASY FOR STUDENTS
2023-03-26 12:01
Question 1: Evaluate:
Answer:
Question 2: Evaluate:
Answer:
Answer: Read More
2023-03-26 06:00
Question 1: Evaluate each of the following:
Answer:
Question 2: Prove the following results:
Answer:
Answer… Read More
2023-03-25 08:02
Question 1: Evaluate each of the following:
Answer:
Question 2: Evaluate each of the following:
Answer:
Question 3: Evaluate each of the following:
A… Read More
2023-03-24 12:24
Question 1: Find the principal values of each of the following:
Answer:
Question 2: Find the domain of
Answer:
Question 3: Evaluate each of the following:
Answer: Read More
2023-03-21 12:09
Question 1: Find the principal values of each of the following:
Answer:
Answer:
Question 3: For the principal values, evaluate the following:
Answer: Read More
2023-03-20 11:37
Question 1: Find the principal values for each one of the following:
Answer:
Question 2: Find the principal values, evaluate the following:
Answer:
Question 3: Find the domain… Read More
2023-03-20 10:42
Question 1: Find the principal value of each of the following:
Answer:
Question 2: Find the principal values, evaluate each of the following:
Answer:
Question 3: Evaluate each… Read More
2023-03-20 06:32
Question 1: Find the domain of definition
Answer:
Question 2: Find the domain of
Answer:
Question 3: Find the domain of
Answer:
Question 4: Find the principal value… Read More
2023-03-11 05:55
Question 1: Find the principal values of:
Answer:
Question 2:
Answer:
Question 3: Find the domain of each of the following functions:
Answer:
Answer:… Read More
2023-03-10 15:10
Question 1: Let be a binary operation on the set of integers, defined by . Find the value of [CBSE 2011]
Answer:
Given that is a binary operation on the set I of integers.
The operation i… Read More
2023-03-09 12:36
Answer:
Hence, the composition table is as follows:
Answer:
Hence, the composition table is as follows:
Answer:
Hence, the composition table is as follows:
Answer:… Read More
2023-03-08 13:34
Question 1: Let be a binary operation on defined by for all
(i) Show that is both commutative and associative.
(ii) Find the identity element in
(iii) Find the invertible element… Read More
2023-03-07 15:16
Question 1: Find the identity element in the set of all positive integers defined by for all,
Answer:
Given that binary operation is valid for set of all positive integers defined by &n&hell…Read More
2023-03-07 14:43
Question 1: Let be a binary operation on defined by
Answer:
Given that is an operation that is valid on all natural numbers and is defined by
According to the problem, b… Read More
2022-12-27 13:00
Question 1: Determine whether each of the following operations define binary operation on the given set or not:
(i)
(ii)
(iii)
(vi)
Answer:
(i)
Given that is an operatio… Read More
2022-12-19 15:29
Question 1: State with reasons whether following functions have inverse:
Answer:
Therefore, does not have an inverse.
Therefore, does not have an inverse.
Here, different… Read More
2022-12-10 04:23
Answer:
Let us calculate ,
Clearly, the range of is not a subset of the domain of
Let us calculate
Clearly, the range of is a subset of the domain of… Read More
2022-12-10 04:23
Question 1: Find are defined by:
Answer:
Question 2: Let Show that are both defined. Also, find
Answer:
Question 3: Let Show t… Read More
2022-12-10 04:22
Question 1: Give an example of a function
(i) which is one-one but not onto
(ii) which is not one-one but onto
(iii) which is neither one-one nor onto
Answer:
(i)
C… Read More
2022-08-07 02:18
Question 31: Let be a finite set. If is a one-one function, show that is onto also.
Answer:
In order to prove that f is onto function, we will have to show that every element in A… Read More
2022-08-07 02:17
Question 1: Find whether the following functions are one-one or not:
Answer:
Hence, is a one-one function from to itself.
Here, does not provide the unique solution but it pr… Read More
2022-07-09 15:54
Question 1: Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:
&n&hell…Read More
2022-07-09 15:54
Question 1: Show that the relation defined by is an equivalence relation.
Answer:
To prove that relation is equivalence, we need to prove that it is reflexive, symm… Read More
2022-07-09 15:53
Question 1: Three relations are defined on set as follows:
(i)
(ii)
(iii)
Find whether each one of is reflexive, symmetric and transitive.
Answer:
(i)&nb&hell…Read More
2022-03-17 14:39
Question 1: Find the the distance between the points (3, 6) and (0,2).
Answer:
Therefore the distance between the given points
Question 2: Find the distance between the orig… Read More
2022-03-08 15:56
Answer:
If two matrices are equal, their corresponding elements are also equal.
In each case, write the conclusion (if any) that you can draw.
Answer:
(iii) Yes;… Read More
2022-03-08 15:53
Question 1: Find the remainder when is divided by
Answer:
Question 2: Find the value of if the division of by leaves a remainder of 5.
Answer:
The value of at is
Quest… Read More
2022-03-05 07:06
Question 1:
(i) lf find
(ii) If find the value of
Answer:
(i)
[ICSE Board 2002]
Answer:
Question 3: Two numbers are in the ratio 3 : 5. If 8 is added t… Read More
2022-03-02 17:10
Question 1: The triangle and is first reflected in the line onto triangle and then is reflected in the origin onto Write down the co-ordinates of :
(i) … Read More
2022-03-01 15:50
Question 1: Find two natural numbers which differ by 3 and the sum of whose squares is 117.
Answer:
Let the natural numbers be and
Since is not a natural nu… Read More
2022-02-28 14:30
Question 1: Solve:
(i) (ii) (iii)
Answer:
(i)
(ii)
(iii)
Answer:
Que… Read More
2022-02-27 10:38
Question 1: If the replacement set is the set of natural numbers (N), find the solution set of:
(i) (ii)
Answer:
Since, the replacement set = N (set of natural… Read More
2022-02-26 11:22
Question 1: Calculate the money required to buy :
(i) 350, Rs. 20 shares at a premium of Rs. 7
(i) 275, Rs. 60 shares at a discount of Rs. 10
(iii) 50, Rs. 75 shares quoted at Rs. 71.50
Answ… Read More
2022-02-25 13:27
Question 1: Ronit opened a savings bank account with a bank on l6th May, 1997 and deposited Rs. 850. He withdrew Rs.300 on 3rd June, 1997 and after that he neither deposited nor withdrew any… Read More
2022-02-23 15:24
Question 1: A person purchased a pair of shoes costing Rs. 850. Calculate the total amount to be paid by him, if the rate of Sales Tax is 6%.
Answer:
Question 2: Mr. Gupta purchase… Read More
2022-02-22 15:40
Question 1: Calculate the amount and the compound interest on Rs. 7500 in 2 years and at 6% compounded annually.
Answer:
Question 2: Calculate the a… Read More
2022-02-21 16:39
Question 1: Find the amount and the compound interest on Rs. 10,000 at 8 per cent per annum and in 2 years.
Answer:
Question 2:… Read More
2022-02-20 10:23
Maximum Marks: 40Time allowed: One and a half hoursAnswers to this Paper must be written on the paper provided separately.You will not be allowed to write during the first 10 minutes.This ti… Read More
2022-02-19 16:44
Maximum Marks: 40Time allowed: One and a half hoursAnswers to this Paper must be written on the paper provided separately.You will not be allowed to write during the first 10 minutes.This ti… Read More
2022-02-15 16:51
Question 1: Two plants A and B of a factory show following results about the number of workers and the wages paid to them.
In which plant A or B is there greater va… Read More
2022-02-14 17:01
Question 1: Calculate the Mean and Standard Deviation for the following data:
Answer:
Calculation of Mean, Variance and Standard Deviation
Question 2: Calculate the Standar… Read More
2022-02-13 01:43
Question 1: Find the standard deviation for the following distribution:
Answer:
Median value of is 34.5
Calculation of Variance and Standard Deviation
Question 2: Table b… Read More
2022-02-12 05:05
Question 1: Find the mean, variance and standard deviation for the following data:
(i) 2, 4, 5, 6, 8, 17 (ii) 6, 7, 10, 12, 13, 4, 8, 12
(iii) 227, 235, 255, 2… Read More
2022-02-09 13:21
Question 1: Compute the mean deviation from the median of the following distribution:
Class
0-10
10-20
20-30
30-40
40-50
Frequency
5
10
20
5
10
Answer:
Calculation of Mean… Read More
2022-02-07 01:02
Question 1: Calculate the mean deviation from the median of the following frequency distribution:
Heights in inches
58
59
60
61
62
63
64
65
66
No of students
15
20
32
35
53
22
2… Read More
2022-02-06 11:42
Question 1: Calculate the mean deviation about the median of the following observations:
(i) 3011, 2780, 3020, 2354, 3541, 4150, 5000
(ii) 38, 70, 48, 34, 42, 55, 63, 46, 54, 44
(iii) 34, 66… Read More
2022-02-05 11:39
Differentiate the following functions with respect to :
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:… Read More
2022-02-05 02:45
Differentiate the following functions with respect to :
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:… Read More
2022-02-02 13:20
Differentiate the following functions with respect to :
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:… Read More
2022-02-01 12:57
Question 1: Differentiate each of the following from first principle:
… Read More
2022-01-28 13:27
Answer:
We have:
Answer:
We have:
Answer:
We have:
Answer:
We have:
Answer:
We have:
Answer:
We have:
Question 7: Find the derivation of the foll… Read More
2022-01-27 15:25
Note:
Evaluate the following limits:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:… Read More
2022-01-19 00:50
Evaluate the following limits:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer: Read More
2022-01-18 01:18
Note:
Evaluate the following limits
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer… Read More
2021-12-26 11:45
Note:
Evaluate the following limits:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Ans… Read More
2021-12-21 12:34
Evaluate the following limits:
Answer:
Answer:
Answer:
Answer:
This is of the form
Rationalizing the numerator:
Answer:
This is of the form
Rat… Read More
2021-12-12 13:17
Evaluate the following limits:
Answer:
Rationalizing the numerator, we get
Answer:
Rationalizing the numerator, we get
Answer:
Rationalizing the numera… Read More
2021-12-09 00:46
Note:
—————————————————————————&md&hell…Read More
2021-12-06 14:49
Evaluate the following limits:
Answer:
So, is a common factor in numerator and denominator. Factorizing the numerator and denominator, we get
Answer:
So, is a… Read More
2021-12-05 15:31
Evaluate the following limits:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
is a constant function. Its value does not depend on
Answer:
Answ… Read More
2021-12-05 13:22
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:… Read More
2021-12-03 16:54
Question 1: Check the validity of the following statements:
Answer:
(i) p :100 is a multiple of 4 and 5. $
Since, 100 is a multiple of 4 and 5, the statement is true. Hence, it i… Read More
2021-12-03 02:00
Question 1: Write each of the following statements in the form “if p, then q”.
Answer:
(i) You can access the website only if you pay a subscription fee.
If you pay a subscriptio… Read More
2021-12-02 02:33
Question 1: Write the negation of each of the following statements:
(i) For every
(ii) There exists
Answer:
(i) For every
Negation of the given statement: There exists a number such that… Read More
2021-12-01 01:23
Question 1: Find the component statements of the following compound statements:
Answer:
(i) The sky is blue and the grass is green.
The component statements of the given compound statement a… Read More
2021-11-30 03:26
Question 1: Write the negation of the following statements:
Answer:
(i) Bangalore is the capital of Karnataka
It is not true that Bangalore is the capital of Karnataka. OR Bangalore is not t… Read More
2021-11-29 01:41
Question 1: Find out which of the following sentences are statements and which are not. Justify your answer.
(i) Listen to me, Ravi! (ii) Every set is… Read More
2021-11-28 11:56
Question 1:
(a) If A and B be mutually exclusive events associated with a random experiment such that then find:
(i) (ii) (ii… Read More
2021-11-27 00:40
Question 1: Which of the following cannot be valid assignment of probability for elementary events oroutcomes of sample space
Elementary Events
(i)
(ii)
(iii)
(i… Read More
2021-11-20 02:00
Question 1: A coin is tossed. Find the total number of elementary events and also the total number events associated with the random experiment.
Answer:
Note: If there an elements in a set… Read More
2021-11-19 15:23
Question 1: A coin is tossed once. Write its sample space.
Answer:
When a coin is tossed, there are two possibilities. It could either turn up Head (H) or Tail (T).
Hence the sample space wi… Read More
2021-11-19 15:21
Question 1: The vertices of the triangle are The internal bisector of angle meets at Find the coordinates of and the length
Answer:
is the internal bisector of
Therefore divides… Read More
2021-10-13 01:14
Question 1: Find the distance between the following pairs of points:
(i) (ii)
Answer:
(i)
(ii) … Read More
2021-10-09 13:54
Question 1: Name the octants in which the following points lie:
(i) (ii) (iii) (iv) (v) … Read More
2021-10-09 03:20
Question 1: The equation of the directrix of a hyperbola is Its focus is and eccentricity . Find the equation of the hyperbola.
Answer:
Let be the focus and be a point on the Hyperbola… Read More
2021-09-28 13:18
Note: Also plotted the ellipse equations. Students should have a feel of what the equation looks like.
Question 1: Find the equation of the ellipse whose focus is the directrix and eccentr… Read More
2021-09-08 01:00
Question 1: Find the equation of the parabola whose:
(i) focus is and the directrix is
(ii) focus is and the directrix is
(iii) focus is and the directrix &n&hell…Read More
2021-02-20 01:20
Note: If the end points of the diameter is and , then the equation of the circle is given by:
Question 1: Find the equation of the circle, the end points of whose diameter are and ; Find… Read More
2021-02-18 00:49
Note: The general equation of circle is where the center is and radius
Question 1: Find the coordinates of the center and radius of each of the following circles :
i) … Read More
2021-02-15 00:34
Note: The general equation of a circle is where is the center of the circle and is the radius of the circle.
Question 1: Find the equation of the circle with:
(i) Centre and radius  &hell…Read More
2021-02-11 01:08
Question 1: Find the equation of a straight line through the point of intersection of the lines and and parallel to .
Answer:
The equation of a straight line passing through the point of i… Read More
2021-02-06 05:24
Note: We know that the equations of two lines passing through and making and angle with the given line are
Question 1: Find the equation of the straight lines passing t… Read More
2021-02-04 02:20
Question 1: Prove that the area of the parallelogram formed by the lines:
is
sq. units. Deduce the condition for these lines to form a rhombus.
Answer:
Let be a parallelogram the eq… Read More
2021-02-03 01:03
Question 1: Determine the distance between the following pair of parallel lines :
(i) and (ii) and
(iii) and (iv) and
Answer:
(i) … Read More
2021-02-02 01:04
Question 1: Find the distance of the point from the straight line .
Answer:
Given equation:
Comparing with we get
Therefore perpendicular distance from point from
Thus the… Read More
2021-02-01 01:27
Question 1: Find the values of so that the point lies inside or on the triangle formed by the lines and .
Answer:
Let the triangle be . Let,
… … … i)
&nb&hell…Read More
2021-01-31 00:07
Question 1: Find the angles between each of the following pairs of straight lines:
(i) and
(ii) and
(iii) and
(iv) and
v) and
Answer:
(i) Let and be the s… Read More
2021-01-30 00:47
Note: If is a line, then the equation perpendicular to this line is given by where .
If is a line, then the equation parallel to this line is given by where .
Question 1: Find the… Read More
2021-01-24 01:11
Question 1: Prove that the following sets of three lines are concurrent:
i) and
ii) and
iii) and
Answer:
i) Given lines are , and
We have:… Read More
2021-01-23 01:21
Note: If and intersect at a point , then and
Question 1: Find the point of intersection of the following pairs of lines :
i) and ii) and
i… Read More
2021-01-18 01:10
Question 1: Reduce the equation to:
(i) slope-intercept form and find slope and y-intercept (ii) intercept form and find intercept on the axes; (… Read More
2021-01-17 01:28
Note: The equation of a straight line passing through and making an angle with the positive x-axis is given by , where is the distance of the point on the line from the point .
The… Read More
2021-01-13 01:20
Note: The equation of the line whose length of the perpendicular from the origin is and the angle made by the perpendicular with the positive x-axis is given by is given by
Question 1: Fi… Read More
2021-01-12 01:03
Note: We know that the equation of the line is where is the x-intercept and is the y-intercept.
Question 1: Find the equation to the straight line:
(i) cutting off intercepts and from t… Read More
2021-01-11 01:09
Note: Equation of a like is also defined as
Question 1: Find the equation of the straight through the following pair of points:
i) and ii) and
iii) an… Read More
2021-01-10 01:04
Note: When slope and a point is given, the equation of the line is
Question 1: Find the equation of the straight line passing through the point and having slope .
Answer:
Here and… Read More
2021-01-09 00:35
Note: The equation of a line having slope of and as is given by
Question 1: Find the equation of a line making an angle of with the x-axis and cutting off an intercept from y-axi… Read More
2021-01-08 00:58
Question 1: Find the equation of the line parallel to x-axis and passing through .
Answer:
Equation of a line parallel to x-axis is
The line passes through
Hence the equation of the line… Read More
2021-01-07 01:00
Question 1: Find the slopes of the lines which make the following angles with the positive direction of x-axis:
i) ii) &n&hell…Read More
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