QUESTION 1 It is given that the radius of the semicircle increases at the rate of 0.5 cms¹ when heated. Calculate the rate of change of the area of the semicircle when the radius, r is 10 cm.

Q: Q1 full a Prove the following using the principle of mathematical induction. For n > 1, 1 1 1 1 = +…

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Q: 3 Evaluate: ³₂²₁²[z³(x + y)³] dz dy dx

A: When a triple integral is performed, then the innermost variable is integrated first, then the…

Q: Set up the triple integral that will give the following: = the volume of R using cylindrical…

A: Given R=0≤x≤2, 0≤y≤4-y2, 0≤z≤4-x2-y2 The region is  The down view of the region ( semi circle in xy…

Q: Evaluate the circulation of Gayi + zj+5yk around a square of side 9, centered at the origin, lying…

A: Solution :

Q: Let F(x,y)=ln(x² + y²), then answer the following (4) Find the boundary of the function's domain.…

A: A multivariable function is an expression involving more than one variable. If z=Fx,y, then the…

Q: y"" - 3y" - y' + 3y = 0

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Q: (4) Let K be integer ring module 12 and let I=([4]) and J=([6]) ideals of K. Then [2] belong to ...…

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Q: 4Z is prime ideal of 2Z, but it is not Maximal ideal of Z. To fo

A: To identify given statement is true or false.

Q: If p = a₁ + a₁x + a₂x² and q = bo + b₁x + b₂x² are any 2 vectors in P². Defined function: f(p, q) ==…

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Q: Find the determinant of the matrix [2 4 11 385 16 2 41

A: here we have to calculate the determinant of the matrix…

Q: For every nonempty set A, there exists a relation R which is both an equivalence relation and a…

A: A relation R is said to be equivalence relation if R is Reflexive ,Symmetric and Transitive . R is…

Q: Evaluate: £²³₁ ²[(x²-²²] du dt x+y

A: Given, double integral J=∫-13∫02(x2-y2)2x+ydu dt  we can write it as : ∫-13∫02((x-y)(x+y))2x+ydu dt…

Q: Q2) Select the correct answer for the following: (10 Marks) (1) Let I be a proper ideal of the…

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Q: Find the volume of the solid generated by revolving the region bounded by 2y = e-x², the x-axis and…

A: The region is

Q: het hey2= усаз хуся) - 79 (2) - you tay al soluc: hq = 36t - 242t > " юные '=at усо) =ц' соз су" (o)…

A: As per the instruction, we will be solving part (a). For (a), Let L(y)=y(4)+y(3)-7y(2)-y(1)+6y and…

Q: = r dz dr de where R: 0≤ the solid R. √3x² + 3y² and below the w the solid B

A: To find the given triple integral as,

Q: -3 -4 Evaluate: ₂₁ (4x²y - z³) dz dy dx

A: ∫03∫-24∫10(4x2y-z3) dz dy dx

Q: Graph each function over the indicated interval. X Г

A: 1. (a) Given,                    y=5x;     -2, 2

Q: 3. Let G be the solid region inside the sphere of radius 2 cen- tered at the origin and above the…

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Q: D6 + 3D4 + 3D2 + 1) = 0

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Q: There is an inaccuracy in the following table of a low degree polynomial p(x). x p(x) 0.0 0.000 0.1…

A: The relationship between the data points can be observed using a difference table. Make Newton's…

Q: J2u 0t2 J2u a² дх2

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: (d) V = P₂ (R), W = R³, T(p) = (1,p(1), p(2)), p = P₂ (R); (e) V = P₂ (R), W = R, T(p) = (p(1),…

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Q: Solve each of the following integrals using integration by part: a) 3x² In 2x dx b) [5xsin 3x dx

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Q: Solve the following system by Gauss-Jordan elimination. 2x1 + 7x2 + 33x3 = 29 6x122x2 + 105x3 = 91…

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Q: The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0…

A: Problem is solved using concept of matrix eigenvalue and eigenvector.

Q: 3. Integrate f 10(x+6) x² + 3x-10 2 dx by using partial fraction.

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Q: 4. Suppose a 12th-order homogeneous linear differential equation with constant coefficients yields…

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Q: 3 For which values of a € R the tangent plane to the graph of f(x, y) = 1 + sin(ax + y²) at the…

A: The given graph is                                f(x,y)=1+sin(αx+y2)

Q: So SAF+1° y¹/3 x7 +1 (Hint: Change the order of integration to dy dx.) Y dx dy.

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Q: The volume of the solid E that lies above the cone = sphere p = 2 cos is 21 T 2 C050 (a) dp da de p²…

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Q: Question 1 All statements below are false. Provide a counterexample of f and g each. (a) If f + g…

A: (a) Consider fx=-1x≤11x>1 and gx=1x≤1-1x>1 It is clear that fx and gx are not continuous at…

Q: The definition of Antiderivative states that A. A function P(x) is called an antiderivative of the…

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Q: [2 4 11 Q1/ Find the determinant of the matrix 3 8 5 16 2 41

A: Solution :

Q: Q.9 Let (x) be a solution to the equation [exy'] + xy=0, x=(-∞0,00) dx Given that (-¹)=0,…

A: We can solve using given information

Q: Consider the following Gauss-Jordan reduction: 1 00 01 56 70 0 0 100 56 7 0 →>> 56 7 O 69-69 810 010…

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Q: 5. Use Cartesian coordinates to evaluate Jff ² av JIJ where D is the tetrahedron in the first octant…

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Q: Let V = Rº. Give a definition of addition + or scalar multiplication on V that makes (V,+,-) unable…

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Q: I. Solve the following Linear Systems of Equations by Gaussian Elimination: 1.) 6x + 2y4z = 15 -3x -…

A: Given system is 6x+2y-4z=15-3x-4y+2z=-64x-6y+3z=8 We can write it in matrix form…

Q: Find the area of the region bounded by 2x = and the y-axis using the horizontal strip. 3 y²+1

A: To find the area bounded by curves 2x=3y2+1 and y axis, i.e. x=0

Q: The product of x++ x³ + 2x² and 2x² + 4x + 6 in z, [x]/is: 4x+1 X^2+2 5x^2+ 5x +1 None O

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Q: R B x red area is three times larger than the blue area, find the value of n.

A: We evaluate the region R using integral x dy and the region blue by using the integral y dx. This is…

Q: 10 [ Σ 15m3 n=m =

A: Given summation is ∑m=11015m3 we can write it as 15∑m=110m3         (1)

Q: Any ring R is a Jacobsen radical ring, if it is a simple ring. |--0 LLO O

A: We can solve this using given information

Q: 1. Find the Inverse Laplace Transforms of the following. 3s 2) -335-4 s²-4 2

A: Solution :  Given that  3ss2-4

Q: Find the area of the region bounded by x = and the y-axis using the horizontal strip. 2y²+1

A: The area bounded by two curves: x=f(y) and x=g(y) in the interval: p≤y≤q is computed using the…

Q: Given a shade region, bounded by the line y + x = 5 and the curve x² = y − 3. a. Sketch the line and…

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Q: Find the volume of the solid generated by revolving the region bounded by 2y = e-*², the x-axis and…

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Q: A. Sequences 1. Given that a2 = 6 and a6 = 96 are terms of a geometric sequence, find a4. 2. Given…

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Q: Set-up the iterated double integral in polar coordinates that gives the volume 1++ and under the…

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