1. Let f be defined on an open interval (a, b) and assume that f has a local maximum or a local minimum at an interior point c of (a, b). If ƒ has a derivative (finite or infinite) at c, then f'(c) must be 0.

Q: V. Let f(x) be a differentiable function such that for all æ > 1, • f(x) > 1, • f'(x) < f(x) and lim…

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Q: Question 27 Let T : P2 – P2 be defined by T(p(x)) = p(2x + 3) and let B = {1, x, x²}. a If [T]B = d…

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Q: If A = (aij) is the matrix of the quadratic form xi+3x, - x - 6x1x2 + 22x1x3+8x2x3, then the value…

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Q: Question 8 Let B = {b1, b2} be a basis for a vector space V. Let T : V > V be a linear…

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Q: QI) Evaluate the integral dy dx using polar coordinate

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Q: Question 28 Consider the inner product space (P2, (,)) with the inner product (p(t), q(t)) =…

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Q: You are given the data points (xi, Yi) for i = 1, 2, 3 : (2, 3), (1, –8), (2, 9). If y = a + Bx is…

A: We have to apply the least squares method to solve the problem.

Q: (iii) If u = 461 + 362 and [u]c = then the value of r is and the value of S is (iv) If V = 3c1 + 2c2…

A: Given that  b1 = 10c1 +3c2,  b2 =27c1 +8c2 c1 =-8b1 +3b2,   c2=27b1 -10b2

Q: Question 4 Consider the function (a) State the domain of f(r, v). f(I, v) 3zy (b) Find the limit by…

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Q: Let V = {(x, 8) | x € R}. You are given that (V, Đ, O) is a vector space with (x1, 8) O (x2, 8) =…

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Q: 1 + 4 = 5 2 + 5 = 12 3 + 6 = 21 8 + 11 = ?

A: Solution :  Given that  1 + 4 = 5  2 + 5 = 10  3 + 6 = 21  8 + 11 = ?

Q: You are given that B = {1+x, 1 – x², x + 2x?} is a basis far P2 and [p (æ)]g = 3. If p(x) = a + Bx +…

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Q: Saran is a well known golfer in Klang. He would like to enhance his swinging skill to improve his…

A: (a) Given that the table Time (s) 2 4 6 8 10 12 14 Trail 1 (m/s) 17.5 18.5 19.9 21.6 20.3 19.0…

Q: Evaluate cxy)dA av)d4 over the region R bounded by y = 2r, y=x+1 and x = 3. (3, 6) 5.5+ y-2x 4.5+…

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Q: Evaluate the Inverse Laplace Transform 1 F(S)=T51)(s+1)² (s-1) (s+1)²

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Q: (NAS 4) Given, A = x²zî – 2y°z²j + æy?zk Find divAat the point (1, –1, 1)

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Q: A model created by an engineer is represented by the following multivariable functions (a) Find O he…

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Q: (-1)" Given: sin x x2n+1 for x E R 2 (2n + 1)! n=0 (-1)"(2n + 1) 4" (2n + 1)! Compute the exact…

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Q: Determine whether the following series converge or diverge. 4+ cos? n (5 – 4(In n)³)" Σ | 3n – yn…

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Q: Let (V, (,)) be an inner product space and let f, g be vectors in V. If || F|| = 4. ||g|| = 5 and ||…

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Q: Question 11 8. 1 3 7 k 4 Let A = . If the dimension of the eigenspace corresponding 8. 0 0 0 3 to…

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Q: 3- Vn -) Σ Vn3 n=1

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Q: (5) Find the curve y = f(x) in the xy-plane that passes through the point (9, 4) and whose slope at…

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Q: Evaluate the Inverse Laplace Transform of the given functions. s² – 4 1. F(s) (s2 + 4)2 1 2. F(s) s2…

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Q: Question 13 Let T : R? → R*be defined by 1. 8 T X2] x2] .2 If B = is a basis for R´such that T B is…

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Q: Q3/ Evaluate Sz Vsinx dx, using i- Trapezoidal rule ii- Simpson's 3/8 rule Ans. 1.1615

A: 3 Given integral is ∫0π2sinxdx. i We have to evaluate the given integral by using trapezoidal rule.…

Q: Determine whether the following series is convergent or divergent (-1)"3n 4n²–1 (b) ) n=1

A: Using Leibniz alternating series test.

Q: QI) Evaluate the integral dy dx using polar coordinate

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Q: 1.1 Show that if |z| < 1. then lim z" = 0. %3D 1.2 Show that lim Vn = 1.

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Q: Petermine whether the following series is convergent or divergent (-1)"3n² 4n²–1 (b) n=1

A: Given series is ∑n=1∞(-1)n3n24n2-1

Q: If AT = ul v. A and if the vectors U and V satisfy Au 5u and Av = 7v, then True False

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Q: Find out if the improper integral converges or diverges. If the given converges, find its value.…

A: Solution

Q: 3 Through polar coordinates, evaluate S S cos(x² + y²)ā dA, R where R is the region enclosed by…

A: The given question is related with evaluation of a double integral. We have to evaluate ∬R cosx2 +…

Q: II. Consider the sequence (n + 2)! n=1 a.) Show that the given sequence is monotone. b.) Use the…

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Q: Consider the quadr atic form f (x1, 22, æ3) = x} + æ% +7æ% + 22x1æ2. (i) The maximum value of f(x)…

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Q: Given: sinx = (-1)" (2n+1)! -x²n+1for x ER (-1)"(2n+1) y2n+1 Compute the exact value of the sum E=0…

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Q: Write TRUE if the statement is always correct and FALSE if otherwise. (a) If an >0 for all n E N,…

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Q: Through polar coordinates, evaluate ∬cos(x2+y2)3/2dA  where R is the region enclosed by equations…

A: Polar coordinates are related with cartesian coordinates with the following relations x=r cosθy=r…

Q: B-8-8- 2 1 , U3 1 and W =Span{u1, U2, U3}. For 1 Let U1 U2 = 1 |2 -3 in W ? t what value of t is the…

A: Let us consider the vector space V  Spanning set of a vector space : A set of vectors that is used…

Q: f(t) = t2 cos 2t.

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Q: Solve the Euler- Cauchy equation

A: We solve the given differential equation using Euler cauchy method.

Q: Evaluate the Inverse Laplace Transform of the given functions. 1. F(s) s2+2s+1

A: Inverse Laplace transferom

Q: Consider the following Maclaurin series: (-1)"x²n (2n)! cos x = for all x E R. n=0 1. Let f(x) =…

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Q: Question 25 2 1 .Then, the distance 2 - 1 = Span 1 Let W : and y = -21 3 1 11 from y to W is

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Q: 17 37 Consider the orthogonal basis B = -3 10 of R³. -3 1 -9 11] Let V = -16 7 а If vB= |b|, what is…

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Q: 10 Given the matrix G =| 5 51 201 and do =| Then do and d, are not G- %3D conjugate if dį is given…

A: We want to find vector which is not G - conjugate to given matrix and vector.

Q: Q4) Find plane area inside the polar function r = 2 and outside the polar function r = sin (-0) +1…

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Q: -25 Find the characteristic equation of the matrix 3 5 C. 22 -32-25 = 0| A. 22-32+5=0 B. 22 +72+5=D0…

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Q: [3 1 1 1] 1 3 1 1 You are given that the eigenvalues of A = are 1 1 3 1 [1 1 1 3 6, 2, 2, 2,…

A: Symmetric matrix always orthogonally diagonalisable.

Q: Let B = {b1, b2} and C = {C1, c2} be two bases for a vector space V. You are given that b1 = 5c1 +…

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