Mathematics Questions and Answers March 20, 2021

oon Let A 0 0 3 4 5 0 6 7 2 0 0 If B (1A) (1A), what is (1 B) 000 0 , and I is the identity matrix.,If AT Ax x, where 0,x 0 and A is an m xn matrix, show that is also an eigenvalue of AAT. (Hint zero vector cannot be an eigenvector) A is an nxn matrix n, a ER. If Amla 0, Ama 0, prove a, Aa, Aa,..., A1a are linearly independent.

Locate the zeros and poles of g(z) cot(z) and detrmine their order. Also, justify that Res(g,0)

Evaluate S (0) exp (2) dz where C (0) denotes the circle C (0) z z 1 with positive orientation.

Let Q (0, 2) and R(7,5) be given points in the plane. We want to find the point P (x, 0) on the xaxis such that the sum of distances P Q P R is as small as possible. (Before proceeding with this problem, draw a picture) To solve this problem, we need to minimize the following function of x f(x) over the closed interval a, b where a and b .

Find the value of the real number ao of the Fourier series if f(x) x for 0 x 2m.

Prove the following for k 1, 2, 3, .... sin dx cos k ( 0 L

Find the Fourier series of the periodic function f(x), such that , when T x 0 when 0

Let x sin (sin tan (2)). x can be written in the form avb, where a, b and care positive integers, a and care coprime and b is not divisible by the square of any prime. What is the value of a b c Details and assumptions sin and tan 1 represent the inverse of the sin and tan function, and not the reciprocal.

Find the Fourier series to represent eax for x (, ).

How many numbers are there between 99 and 1000, having at least one of their digits 7

Deduce the halfrange cosine series for f(x) kx when 0x 12 and f(x) k(1x) when 12 x 1.

Express the given function f(x) as Fourier sine half range series. ,0x f(x) x I 2 T

Express f(x) sin(xl) as half range cosine series for 0xl.

A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done How many of these committees would consist of 1 man and 2 women

In a small village, there are 87 families, of which 52 families have at most 2 children. In a rural development programme, 20 families are to be chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made

In how many of the distinct permutations of the letters in MISSISSIPPI do the four is not come together

Find the number of permutations of the letters of the word ALLAHABAD.

2. How many 5digit telephone numbers can be constructed using the digits 0 to 9, if each number starts with 67 and no digit appears more than once

If P713 is divisible by 11, find the value of the smallest natural number P?

with How many diagonals n sides 2 are there in a polygon

mice were placed in two experimental groups and one control group, with all groups equally large. In how many ways can the mice be placed into three groups

How many numbers greater than 1000000 can be formed using the digits 1, 2, 0, 2, 4, 2, 4

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girls (ii) at least one boy and one girl (iii) at least three girls

How many points on a chords circle can be drawn through.

of A its polygon sides 2. 2 has 44 diagonals. Find the number

Determine the number of 5 card combinations out of a deck of 52 cards, if there is exactly one ace in each combination.

Randomly select a 7 digit natural number. a) Find the probability that the number has 4 odd digits, 3 even numbers, and no number is repeated. b) Find the probability that the number 2 is present exactly 2 times, the number 3 is present exactly 3 times, the remaining digits are present no more than 1 time.

Evaluate the following integral 2x sec (x 3) tan (x 3) dx

Seven friends picked 7 quarts of blueberries. Three of the friends will share 4 quarts of blueberries equally, and the other 4 friends will share 3 quarts of the blueberries equally. In which group does each friend get more blueberries

Consider the linear of all sequences. such that (xK2) converges for 20,123 Find the dimension basis for the space 8 C(2) KO and a 2 (xn) 0 space cc.

Prove that

p2zi, then

Obtain the Fourier series for the function defined by 0, when 2

Find c for the 2periodic shifted pulse F(x) 1 for sxsh elsewhere in ,

rewrite the given limit so that it is not an indeterminate form. 55 cos x 3x Use lHpitals Rule to evaluate lim x0 lim x0 O A. lim X0 Choose the limit equivalent to the given limit that can be evaluated using limit laws and commonly known limits. O B. lim x0 O c. lim x0 lim x0 OD. lim X0 55 cos x 2 3x 55 cos x 2 3x 55 cos x 2 3x 55 cos x 2 3x lim

minutes The waiting time in restaurant for a customer to be served at the counter has a density function fox) 2e20 x70 x elsewhere 0 Find the average waiting time

Let o be the random Variable that a certain denote the life in hours electronic device. The probability density of Function of the device is fo 20000 X3 ) O x 7100 , elsewhere Find me expected life of the device

Verity is fix) 2 is pdf (1) 0 (11) Find P(OLXZ1) 01 1762 elsewhere

Find the particular solution to the following differential equation ez dy 4 given that y) 3

Find the equation of the line with Slope 1 and passing through (5,2). Write your equation in the form y mx b.y

Differentiate y x

Find the Fourier series for f(x) L 2x if Lx0 if 0 x L onL x L

Find the equation of the tangent at the Point (3, 4) if x y 25

Calculate the Fourier sine series of the function f(x) x for x in (0.2) and show that its differentiated series does not converge at all except for 2 0.

Compute the matrix exponential e in the case that A The characteristic polynomial of the given matrix is 102 H

Compute the matrix exponential e in the case that A The characteristic polynomial of the given matrix is f(x)X(tr A)X det AX6A 10 (A3) 1.

Use the matrix exponential to solve the initial value problem y(t) Ay(t). TIL A

Obtain the constant terms and the coefficients of the first sine and cosine terms in the Fourier series of f (x) as given in the following table. 2 24 X f(x) 0 9 1 18 3 28 4 26 5 20

Given the function y6x48x1 Determine the turning points and their nature.

Determine the asymptotes of me graph of the function fbx) x 4x x 4x3

Find Fourier halfrange even expansion of the function, 0sxsl

Consider the problem Minimize ex subject to Ax 2 b, x 20. Let x be an optimal solution. Suppose that A is decomposed into A and b is decomposed A into such that Ax b and Ax b. Show that x is also an optimal solution to the problem Minimize ex subject to Ax b, x 0, and to the problem Minimize ex subject to Ax b, x 0.

ABCD is a Parallelogram. If angle ABC = 55 degree, then what is the angel of ADC ?

A B 550 D e

A study conducted in 2016 stated that for cats in central Italy about 33 of them are male. A random sample of seven cats is selected. Answer these questions with the help of the binomial function. (a) What is the probability that the sample contains exactly three male cats (b) What is the probability that the sample contains no male cats (c) What

A rectangular box which is open at the top has a capacity of 256 cm. Determine the dimensions of the box such that the least material of construction of the box is required. Use Langrages method of multipliers to obtain the solution

Minimise f(x, y) x y subject to xy10

Find the value of cos(x), if sin(x)=1/2 in [0,pi/2].

Solve 4x +7x+2=0 giving the answer correct to 2 decimal places.

Divide 4a-6ab +5h by 2a-b.

x1 Resolve 23x2 fractions. into partial

Writelog 16log 272log 5 as the logarithm of a single number.

Write (a) log 30 (b) log 450 in terms of log 2, log 3 and log 5 to any base.

Determine the truncation error bound when evaluating 1x using trapezoidal nule if n10 1 dx 1 f

Find the Sep sizeh so that the absolute value of the or by simpsons rule si comect to 4dp when 13 Omor pom 6 xbxb solving ft 2 or solving de 8 mustulos

Use the Gauss Legendre Quadrature formula to Compute the Integrald Ifde when n 3 in the interval 1, 1 kon

Rice weighing 3 3 / 4 pounds was divided equally and placed in 4 containers. How many ounces of rice were in each?

The distance between the projectors of a straight line AB is 60 mm. one end A is 10 mm above HP and 20 mm in front of VP. While the other end B is 40 mm below the HP and 50 mm behind the VP. Draw its projections and mark its traces. Also find its true length and true inclinations with both the reference planes. (b) The top view of a plate, the

Use x dx 72 to evaluate each definite integral without using the Fundamental Theorem of Calculus.,(a) (b) 9 xp J PX 2 dx,xp zx xp zxz y ()

Find the area of the plane region R bounded by x y, yx3, y 3, and the x axis. Write the integral in both orders dx dy and dy dx and compute the area whichever is easier. Convert this Cartesian integral into an equivalent Polar integral.

Change the Cartesian integral into Polar integral and evaluate it. 4y x(x y)dx dy

Sketch the region of integration and convert the polar integral to a Cartesian integral. 4 2sece r5 sin0 dr de

For the given integral below. 121 2 1y 112 2 dxdy L 2 1x dydx (a) Sketch the region of integration. (b) Evaluate the above integral by converting double integral in polar form. 10M it into 16M

Here is the region of integration of the integral Cafebo Z Side 1 yx dz dy dx. Top y z 1 1 1 (1, 1, 0) X (1, 1, 0) Rewrite the integral as an equivalent iterated integral in the order a. dy dz dx c. dx dy dz e. dz dx dy. b. dy dx dz d. dx dz dy

Find the volume of the region inside the sphere x y 2 but out side the cylinder x y 1 using spherical coordinates with the order of integration dp do do.

Transform the following triple integral into spherical coordinates in the or der dp do do. 2 4 xy HIT 241 x y x y z2 dzdydx

Let D be the region in xyz space defined by the inequalities 1 x 2, 0xy 2, 0z 1. Evaluate fff(ry 3xyz)dxdydz by applying the D transformation u x, v xy, w 3z and integrating over an appropriate region G in uvwspace.

Evaluate the integral by changing the order of integration in an appropri 4 1 2 6cos(x) dxdydz z ate way 003y

Evaluate into cylindrical coordinates. x y dz dy dx by converting this integral

2T 1 4 Sketch the region of integration and find its volume I SS S dzrdrdo 00 0

Let Let R be the region and inside the find the area. Lueve outside the curve x y x Cy 1 1 Using double integral of the region enclosed by R

Compute the integral of (x,y) (on y5 the region y R bounded by y e and 0 1 over o

Find the area 2 Cuever yx, y (x23 and x 0 (1,1). C(2,01 xaxis. by the y x y(x2) x 2 4x 4 x1 y 1

Evaluate 1 3x e dx dy. (sketch the region of integration)

Convert the given two double integrals into a single double integral and evaluate the single double integral, and also sketch the region neatly 2y 1 S y1 z dz dy t 0 ay1 2 aytl z dz dy Z

Evaluate the double Integral SS 1241) dx dy R where R is the region Cuever x y ay x0 SS12291) dydx R bounded in the the by first quadrant

Evaluate sin (3x1)dx

Find all solutions of y x xy y xy

Use part one of the Fundamental Theorem of Calculus to find the derivative of the following functions H(x) 11dt, g(x) du S.x Salg

Combine terms: 12a + 26b -4b 16a. (a) 4a + 22b. (b) -28a + 30b. (c) -4a + 22b. (d) 28a + 30b. Question no - 2 :- Simplify: (4 5) (13 18 + 2). (a) -1. (b) 2. (c) 1. (d) 2. Question no - 3 :- Factor: 5x 2 15x 20. (a) 5(x-4)(x+1). (b) -2(x-4)(x+5). (c) -5(x+4)(x-1). (d) 5(x+4)(x+1). Question no - 4 :- Solve for x: 2x y = (3/4)x + 6. (a) (y +

Use the right endpoints of the Riemann sums to estimate the area under the graph of f ( x ) = x 2 + 1 over the interval [2, 10], using n = 4 rectangles. Sketch the curve and the representation of the approximation rectangles.

Let X be an arbitrary infinite set. We say that a subset A of X is cofinite if its complement A is a finite subset of X. Let A consists of all the finite and the cofinite subsets of a set X. (a) Show that A is an algebra of subsets of X. (b) Show that A is a oalgebra if and only if X is a finite set.

Solve the following nonlinear programming problem with inequality constraints. Minimize z f(x, x) xx such that, 226 271 2 x 26.

Find the eigenvalues of the matrix A 22

Evaluate the integral S 1 x(x 1) dx

Find the eigenvalues and associated eigenvectors of the matrix A23

Solve the system of linear equations using the GaussJordan elimination method. 3x y 1 7x 2y 1

Solve the system of linear equations using the GaussJordan elimination method. y 8z 9 x 2y 3z 3 7y 5z 12

Solve the system of linear equations using the GaussJordan elimination method. 2x 4y6z 38 x 2y 3z 7 3x 4y 4z 19

Solve the system of linear equations using the GaussJordan elimination method. x2y3z 2 3xy2z1 2x3y5z 3

Use GaussJordan elimination to solve the system 21 2x1 4x1 11x2 21 3x2 6x2 5x 2x5 8x3 5x5 2x3 4x4 25 5 7 5 3x2 2x3 4 2.

If the cost of energy is 0.24 per Kwh how much would it cost to heat a pool of length 12 metres, width 9 metres, depth 1.5 metres from 15 degrees celsius to 27 degrees celsius Hint the specific heat capacity of water is 4,184 Joules