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Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Mathematical discoveries continue to be made today. According to Mikhail B. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. In Latin, and in English until around , the term mathematics more commonly meant "astrology" or sometimes "astronomy" rather than "mathematics"; the meaning gradually changed to its present one from about to This has resulted in several mistranslations.

For example, Saint Augustine 's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians. It is often shortened to maths or, in North America, math. Brouwer , identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems". In formal systems, the word axiom has a special meaning, different from the ordinary meaning of "a self-evident truth".

In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.

The specialization restricting the meaning of "science" to natural science follows the rise of Baconian science , which contrasted "natural science" to scholasticism , the Aristotelean method of inquiring from first principles.

The role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as biology , chemistry , or physics. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions.

Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the other sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics. The opinions of mathematicians on this matter are varied. Many mathematicians [43] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts ; others[ who?

One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created as in art or discovered as in science. It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide.

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In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement , architecture and later astronomy ; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory , a still-developing scientific theory which attempts to unify the four fundamental forces of nature , continues to inspire new mathematics.

But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e. This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what Eugene Wigner has called " the unreasonable effectiveness of mathematics ". For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics.

Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof , such as Euclid 's proof that there are infinitely many prime numbers , and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics.

He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. A theorem expressed as a characterization of the object by these features is the prize. Evaluate the following problems using second fundamental theorem: Evaluate the following Problems using properties of integration.

Find the area of the region bounded by the line Example 7. Find the volume of the solid that result when the region enclosed by the given curves: Find the differential equation that will represent the family of all circles having centres on the x-axis and the radius is unity. Form the differential equation from the following equations. The normal lines to a given curve at each point x.

The curve passes through the point 2.

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Formulate the differential equation representing the problem and hence find the equation of the curve. The temperature T of a cooling object drops at a rate proportional to the difference T.

Here denotes usual multiplication. Prove that the set of all 4th roots of unity forms an abelian group under multiplication.

Construct the truth table for the following statements: Show that the set of all non-zero complex numbers is an abelian group under the usual multiplication of complex numbers. Construct the truth tables for the following statements: Show that the cube roots of unity forms a finite abelian group under multiplication. Show that the set of all 2 X 2 non-singular matrices forms a non-abelian infinite group under matrix multiplication. Three oranges are drawn at random without replacement from this lot.

Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of queens. A discrete random variable X has the following probability distributions. Find the probability mass function. A continuous random variable X has p. Find a and b such that. Two bad oranges are accidentally mixed with ten good ones. A continuous random variable X follows the probability law. Obtain the probability distribution for the number of bad oranges.

Find the expected value of the total number of points shown up. Find his expectation of gain. In a gambling game a man wins Rs. The probability of success of an event is p and that of failure is q. Find the expected number of trials to get a first success. Each question has four options and only one option is correct.

A game is played with a single fair die. In a continuous distribution the p. A student gets 1 mark for a correct answer and loses half mark for a wrong answer. Find the mean and the variance of the probability distribution of the number of successes.

Also find its mean and variance. Find the expected sum of money he can win. Find the mean and variance for the number of aces. While he neither wins nor loses if any other face turns up. What is the expectation of the mark scored by a student if he chooses the answer to each question at random Two cards are drawn with replacement from a well shuffled deck of 52 cards. A player wins Rs. Find the probability distribution of the number of red balls in three draws when a ball is drawn at random with replacement.

Two unbiased dice are thrown together at random. In an entrance examination a student has to answer all the questions. A success is getting an odd number on a toss. Let X be a binomially distributed variable with mean 2 and standard deviation 2.

What is the probability that a person will be exposed to more than 5. If a publisher of non-technical books takes a great pain to ensure that his books are free of typological errors. Find the corresponding 3 probability function. Alpha particles are emitted by a radioactive source at an average rate of 5 in a 20 minutes interval. If persons are inoculated. Find the Mean and Variance for the following probability density functions: Suppose that the probability of suffering a side effect from a certain vaccine is 0.

If getting a doublet is considered a success find the probability of i 4 success ii No success. What is the probability of getting a exactly 2 heads b at least two heads c at most two heads. If 6 candidates appear in the examination what is the probability that at least 5 pass the examination.

Find approximately the probability that i atmost 1 person suffer. Using Poisson distribution find the probability that there will be i 2 emission ii at least 2 emission in a particular 20 minutes interval.

Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using i Binomial distribution ii Poisson distribution. A pair of dice is thrown 10 times. Marks in an aptitude test given to students of a school was found to be normally distributed. Find the number of students scored between 40 and If pairs are issued. If the height of students are normally distributed with mean Find the work done. Find the inverse of the following matrices: What are the d.

Find the angle between the following lines. Find the direction cosines of the line joining 2. Find the centre and radius of each the following spheres. A force given by 2. Find the modulus or the absolute value of Example 3. Express the following complex numbers in polar form. Find the equation of the hyperbola whose transverse axis is parallel to y.

Find the equation of the parabola if the curve is open leftward. Find the equation of the parabola if the curve is open upward. The luminous intensity I candelas of a lamp at varying voltage V is given by: Find the equation of the following parabola with indicated focus and directrix. Determine the voltage at which the light is increasing at a rate of 0. Find the equation of chord of contact of tangents from the point 2.

Find the equation of the parabola whose vertex is 1. Find the equation of the parabola if i the vertex is 0. Find the equation of the parabola if the curve is open rightward. Prove that Example 5. What is the maximum error in using this value of the radius to compute the volume of the sphere?

The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of atmost 0. Form the differential equation from the equations. Find n. Find the mean and variance of the number of successes. If the sum of mean and variance of a Binomial Distribution is 4. What is the probability that he will knock down less than 2 hurdles.

Is this statement true or false? Prove that the total probability is one. Find the mean and the standard deviation of ships returning safely out of a total of ships. Let Z be a standard normal variate.

Calculate the following probabilities. The difference between the mean and the variance of a Binomial distribution is 1 and the difference between their squares is Find the value of c in the following problems. Every row of A which has all its entries 0 occurs below every row which has a non-zero entry.

In the system of 3 linear equations with three unknowns. A has atleast one minor of r which does not vanish and all higher order minors vanish.

Which of the following is not elementary transformation? The first non-zero entry in each non-zero row is 1. The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.

BTAT 3. ATBT 2. A-1 2. Every homogeneous system linear 1. Which of the following statement is correct regarding homogeneous system? The value of a is 1. The length of the perpendicular from the origin to the plane r. Then the coordinates of B is 1. The vector equation of a plane passing through the line of intersection of the planes r. The complex number form of Modulus of the product of the complex numbers is equal to the sum of their moduli 3.

Sum of the moduli of two complex numbers is equal to their modulus of the sum 2. Which of the following is not true? Arguments of the product of two complex numbers is the product of their arguments. Arguments of the product of two complex numbers is equal to sum of their arguments. Multiplying a complex number by. If z 1 and z 2 are complex numbers then which of the following is meaningful?

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Which of the following is incorrect? The number of distinct roots is n 2. Which of the following are true? Dividing a complex number by i is equivalent to rotating the number clockwise about the origin. Indeterminate 2 UNIT: The foci of the ellipse x 2 4 If the centre of the ellipse is 4.

The equation of chord of contact of tangents from the point 5.

The equation of chord of contact of tangents from the point The equation of the tangent at 3. The equation of the tangent at The equation of chord of contact of tangents from the point 2. Two tangents and 4 normals can be drawn to an ellipse from a point. Two tangents and 4 normals can be drawn to an rectangular hyperbola from a point. Two tangents and 4 normals can be drawn to an hyperbola from a point. Two tangents and 3 normals can be drawn to a parabola from a point.

Directrix 3. The time when the angular acceleration zero is 1. Then the initial velocity and initial acceleration respectively are 2. The speed of the car is 1. The acceleration of the 2 9. Then the maximum height reached by the missile is 1. Then 1. Then f is 1. The law of the mean can also be put in the form 1.

Every identity function is a decreasing function 1. If the motion is upward. Every identity function is an increasing function d. Law of mean 4. If the motion is horizontal. Every constant function is an increasing function b. The extreme value theorem 2. Every constant function is a decreasing function c. Lagranges law of mean is a particular case of generalized law of mean Cauchy c.

Which of the statements are correct? The differential of x tan x is 1. If n is odd. Then the volume of the solid is d 1. If n is even. The order and degree of 1. The order and degree of the differential equation 1. The order and degree of the differential equation d y dx 1. The degree of the differential equation is the degree of the highest order derivative which occurs in it.

The order of a differential equation is the order of the highest order derivative occurring in it. The derivatives are free from radicals and fractions. Which of the following are statements? TTTT 3. Negation of a negation of a statement is the statement itself. FFFT 3. FTTF 4. Which of the following are not statements? TFTF 2 is an integer.

TTFT Which of the following are not statements? If the last column of its truth table contains only T then it is tautology. TFTF 4. TFFT 3. TFFT 2. FTFF 2. FTFT 4.

FTFT 3. If the last column of its truth table contains only F then it is contradiction. TFTF 2. TTTF 2. Which of the following are binary operations on R? For a standard normal distribution the mean and variance are 1. Which of the following is or are correct regarding normal distribution curve? A continuous random variable takes 1. F x is a constant function If X is a continuous random variable then which of the following is incorrect? Mean and variance of binomial distribution are 1.

A discrete random variable takes 1. A discrete random variable X has probability mass function p x. Which of the following is not true regarding the normal distribution? NO ANS. The equation having 4. The length of the latus rectum of the parabola y. If The axis of the parabola y2. The rate at which the diameter is decreasing when the diameter is 10 cms is 1 - 5. The product of the perpendiculars drawn from the point 8. Then the acceleration is 1 4. What is the rate of increase of its area when the side is 15 cm?

The maximum area of the rectangle is 1 2 2 4 3 8 4 16 In a given semi circle of diameter 4 cm a rectangle is to be inscribed. Which of the following curves is concave down? The least possible perimeter of a rectangle of area m2 is 1. The area between the ellipse 4 2 3 The differential equation corresponding to the above statement is k is negative 1.

If a compound statement is made up of three simple statements, then the number of rows in the truth table is 1 8 2 6 3 4 4 2 If p is T and q is F, then which of the following have the truth value T?

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An element of a group can have more than one inverse. If every element of a group is its own inverse, then the group is abelian. The set of all 2x2 real matrices forms a group under matrix multiplication.

A random variable X has the following probability mass function as follows: Its mean is 2. Then the value of n and p are 1. In 5 throws of a die. Then the value of its parameter p is 1.

The mean number of successes is 1. If 3 balls are drawn at random. Then the variance of the successes is 1. If students got more marks above If the mean and standard deviation of a binomial distribution are 12 and 2. Solve the system of linear equations by determinant method. Find the vector and Cartesian equation of the plane through the point 1.

Solve by determinant method. If P represents the variable complex number z. P represents the variable complex number z. With usual symbol. Find the order of all elements of the group Z6. Find k. Even Function: Quotient rule: Gamma Integral: Two rows can have same number of zeros before the first non-zero entry In the system of 3 linear equations with three unknowns. Chord AB is a diameter of the sphere Which of the following statements is correct?

Indeterminate 2 4. The locus of the point of intersection of perpendicular tangents to the ellipse 1. Which of the following statements are correct m 1 and m 2 are slopes of two lines 4. Then the volume of the solid is d d 1. In the group G,. Choose the most suitable answer from the given four alternatives 1. The value of Show that diameter of a sphere subtends a right angle at a point on the surface.

B NOTE: Answer any ten questions. If A is a matrix of order 3.

With usual notations prove that If the magnitude of moment about the point Question No. C NOTE: A kho-kho player in a practice session while running realises that the sum of the distances from the two kho-kho poles from him is always 8m. Evaluate lim x sin x Find the co-ordinates of the point of contact. If I is unit matrix of order n. Then the value of its parameter p is 2nd 1 Resistance to motion F.

Find the area of the region bounded by the ellipse The arch of a bridge is in the shape of a semi —ellipse having a horizontal span of 40 ft and 16 ft high at the centre. Find the height of the bridge at 10 ft. Find the perimeter of the circle with radius a. Flag for inappropriate content. Related titles. Jump to Page.Obtain the probability distribution for the number of bad oranges. The equation having 4. Find the vector and Cartesian equation of the sphere whose centre is 1.

Physics Textbook Part - 1 for Class - 11 - This has resulted in several mistranslations.