Mathematics MCQs

311

Advanced Mathematics • Linear Algebra

Obtain the dimension of the dual space for {(x,y,z): x + 2y + 3z = 0}. Obtain the dimension of the dual space for {(x,y,z): x + 2y + 3z = 0}.

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A. 3 3 B. 2 2 C. 4 4 D. 1 1

सही उत्तर: B Correct Answer: B

The given set is the null space of one nonzero linear equation in R^3, so it has dimension 2. A finite-dimensional vector space and its dual have the same dimension.

312

Advanced Mathematics • Linear Algebra

Which one of the following statements is false? Which one of the following statements is false?

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A. For any vector space, a basis set gives a minimal spanning set. For any vector space, a basis set gives a minimal spanning set. B. A basis set for a vector space is not unique. A basis set for a vector space is not unique. C. Any set of vectors containing the null vector is linearly dependent. Any set of vectors containing the null vector is linearly dependent. D. For any linear transformation T:V to V and any basis {v1,...,vk} of V, the set {T(vi): 1 <= i <= k} also gives a basis set. For any linear transformation T:V to V and any basis {v1,...,vk} of V, the set {T(vi): 1 <= i <= k} also gives a basis set.

सही उत्तर: D Correct Answer: D

The image of a basis under an arbitrary linear transformation need not be a basis; this is guaranteed only when the transformation is invertible.

313

Mensuration • Mensuration

A cone with height 32 cm and radius 4 cm is filled with water. How many semi-hemispherical bowls of radius 2 cm can be fully filled using this water? A cone with height 32 cm and radius 4 cm is filled with water. How many semi-hemispherical bowls of radius 2 cm can be fully filled using this water?

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A. 8 8 B. 64 64 C. 16 16 D. 32 32

सही उत्तर: D Correct Answer: D

The cone volume is (1/3)pi(4^2)(32) = 512pi/3. Each hemispherical bowl has volume (2/3)pi(2^3) = 16pi/3. Their ratio is 32.

314

Advanced Mathematics • Mathematical Modelling

A forest can support a herd of 500 deer whose population has a linear growth rate of 20% per year. The number of deer that can be killed by hunting each year to maintain the herd size at 300 is: A forest can support a herd of 500 deer whose population has a linear growth rate of 20% per year. The number of deer that can be killed by hunting each year to maintain the herd size at 300 is:

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A. 28 28 B. 32 32 C. 24 24 D. 20 20

सही उत्तर: C Correct Answer: C

Using logistic growth with carrying capacity 500, annual growth at population 300 is 0.2 x 300 x (1 - 300/500) = 24. Hunting 24 deer per year maintains the population at 300.

315

Advanced Mathematics • Differential Equations

If 1 and e^(-x) are two linearly independent solutions of the equation (1 + x)y'' + xy' - y = (1 + x)^2, then by the method of variation of parameters, the particular integral is: If 1 and e^(-x) are two linearly independent solutions of the equation (1 + x)y'' + xy' - y = (1 + x)^2, then by the method of variation of parameters, the particular integral is:

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A. x + 1 - x^2 x + 1 - x^2 B. x^2 - 1 - e^x x^2 - 1 - e^x C. x^2 + 1 - x x^2 + 1 - x D. x^2 + 1 - e^x x^2 + 1 - e^x

सही उत्तर: C Correct Answer: C

Applying variation of parameters with the two stated complementary solutions gives the particular integral x^2 + 1 - x, as marked in the source paper.

316

Advanced Mathematics • Real Analysis

The function sin(e^x), where x > 0: The function sin(e^x), where x > 0:

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A. Is bounded but does not converge Is bounded but does not converge B. Diverges to infinity as x approaches infinity Diverges to infinity as x approaches infinity C. Is strictly increasing but bounded Is strictly increasing but bounded D. Is bounded and converges to a real number as x approaches infinity Is bounded and converges to a real number as x approaches infinity

सही उत्तर: A Correct Answer: A

The sine function always lies between -1 and 1, but e^x grows without bound and its sine continues to oscillate. Thus the function is bounded but has no limit.

317

Advanced Mathematics • Real Analysis

Let f(x) be a differentiable function on R such that df/dx > 0 for all x and lim f(x) = 2 as x approaches minus infinity. Which of the following is true? Let f(x) be a differentiable function on R such that df/dx > 0 for all x and lim f(x) = 2 as x approaches minus infinity. Which of the following is true?

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A. f(3) > 2 f(3) > 2 B. f(3) >= 2 f(3) >= 2 C. f(3) <= 2 f(3) <= 2 D. f(3) = 2 f(3) = 2

सही उत्तर: A Correct Answer: A

Since f is strictly increasing and approaches 2 at minus infinity, every finite value of f must be strictly greater than 2. Therefore f(3) > 2.

318

Advanced Mathematics • Linear Algebra

Which one of the following is NOT a subspace? Which one of the following is NOT a subspace?

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A. {(x,2x): x belongs to R} over R {(x,2x): x belongs to R} over R B. Q over Q Q over Q C. {(x,y): xy = 0} over R {(x,y): xy = 0} over R D. {(x,y): 2x + 3y = 0} over R {(x,y): 2x + 3y = 0} over R

सही उत्तर: C Correct Answer: C

The set xy = 0 is the union of the coordinate axes and is not closed under addition; for example, (1,0) and (0,1) belong to it but (1,1) does not.

319

Advanced Mathematics • Partial Differential Equations

The integral surface of yp + xq - z = 0 passing through the curve z = x^3, y = 0 is: The integral surface of yp + xq - z = 0 passing through the curve z = x^3, y = 0 is:

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A. z(1 + x^2 - y^2) = (x^2 - y^2)(x + y + z) z(1 + x^2 - y^2) = (x^2 - y^2)(x + y + z) B. 1 + x^2 - y^2 = (x^2 - y^2)(x + y + z) 1 + x^2 - y^2 = (x^2 - y^2)(x + y + z) C. z(1 - x^2 - y^2) = (x^2 - y^2)(x + y + z) z(1 - x^2 - y^2) = (x^2 - y^2)(x + y + z) D. z(1 + x^2 - y^2) = (x^2 + y^2)(x + y + z) z(1 + x^2 - y^2) = (x^2 + y^2)(x + y + z)

सही उत्तर: A Correct Answer: A

Solving the Lagrange first-order PDE by its characteristic equations and imposing z = x^3 on y = 0 yields the surface in option A.

320

Advanced Mathematics • Partial Differential Equations

The particular integral of the equation (2D^2 - D')z = 10e^(x - 3y) is: The particular integral of the equation (2D^2 - D')z = 10e^(x - 3y) is:

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A. 2e^(x - 3y) 2e^(x - 3y) B. e^(x - 3y) e^(x - 3y) C. 2e^(x + 3y) 2e^(x + 3y) D. e^(x + 3y) e^(x + 3y)

सही उत्तर: A Correct Answer: A

For e^(ax+by), replace D by a and D' by b. Here a = 1 and b = -3, so the operator value is 2(1)^2 - (-3) = 5. Thus the particular integral is (10/5)e^(x-3y) = 2e^(x-3y).

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