Mathematics MCQs

241

Mathematics • Fluid Mechanics

A block of concrete weighs 100 kg in air and weighs 60 kg when immersed in water. The average specific weight of the block is: A block of concrete weighs 100 kg in air and weighs 60 kg when immersed in water. The average specific weight of the block is:

A. 3.2 ton/m^3 3.2 ton/m^3 B. 2.5 ton/m^3 2.5 ton/m^3 C. 2.2 ton/m^3 2.2 ton/m^3 D. 2.1 ton/m^3 2.1 ton/m^3

सही उत्तर: B Correct Answer: B

Loss of weight in water = 100 - 60 = 40 kg, which equals the weight of displaced water. Therefore, relative density = 100/40 = 2.5 and the specific weight is 2.5 ton/m^3.

242

Mathematics • Fluid Mechanics

If the stream function is given by psi = 6x + 12y, then the speed of flow is: If the stream function is given by psi = 6x + 12y, then the speed of flow is:

A. sqrt(180) m s^-1 sqrt(180) m s^-1 B. sqrt(170) m s^-1 sqrt(170) m s^-1 C. 3sqrt(2) m s^-1 3sqrt(2) m s^-1 D. sqrt(0.5) m s^-1 sqrt(0.5) m s^-1

सही उत्तर: A Correct Answer: A

For a two-dimensional stream function, u = partial(psi)/partial(y) = 12 and v = -partial(psi)/partial(x) = -6. Thus speed = sqrt(u^2 + v^2) = sqrt(180) m s^-1.

243

Mathematics • Fluid Mechanics

The continuity equation in Cartesian coordinates is (where the symbols have their usual meanings): The continuity equation in Cartesian coordinates is (where the symbols have their usual meanings):

A. partial(rho)/partial(x) + partial(rho u)/partial(t) + partial(rho v)/partial(y) + partial(rho w)/partial(z) = 0 partial(rho)/partial(x) + partial(rho u)/partial(t) + partial(rho v)/partial(y) + partial(rho w)/partial(z) = 0 B. partial^2(rho)/partial(t)^2 + partial^2(rho u)/partial(x)^2 + partial^2(rho v)/partial(y)^2 + partial^2(rho w)/partial(z)^2 = 0 partial^2(rho)/partial(t)^2 + partial^2(rho u)/partial(x)^2 + partial^2(rho v)/partial(y)^2 + partial^2(rho w)/partial(z)^2 = 0 C. partial^2(rho)/partial(t)^2 + partial(rho u)/partial(x) + partial(rho v)/partial(y) + partial(rho w)/partial(z) = 0 partial^2(rho)/partial(t)^2 + partial(rho u)/partial(x) + partial(rho v)/partial(y) + partial(rho w)/partial(z) = 0 D. partial(rho)/partial(t) + partial(rho u)/partial(x) + partial(rho v)/partial(y) + partial(rho w)/partial(z) = 0 partial(rho)/partial(t) + partial(rho u)/partial(x) + partial(rho v)/partial(y) + partial(rho w)/partial(z) = 0

सही उत्तर: D Correct Answer: D

Conservation of mass gives partial(rho)/partial(t) + div(rho V) = 0, which expands to option D in Cartesian coordinates.

244

Mathematics • Digital Logic

Digital circuit can be made by the repeated use of which gate? Digital circuit can be made by the repeated use of which gate?

A. OR OR B. None of the given options None of the given options C. NOT NOT D. NAND NAND

सही उत्तर: D Correct Answer: D

NAND is a universal gate. Any Boolean function and therefore any digital circuit can be implemented using NAND gates alone.

245

Mathematics • Statics

The Cartesian equation of common catenary is: The Cartesian equation of common catenary is:

A. y = c sinh(x/c) y = c sinh(x/c) B. y = c cosh(x/c) y = c cosh(x/c) C. y = c coth(x/c) y = c coth(x/c) D. y = c tanh(x/c) y = c tanh(x/c)

सही उत्तर: B Correct Answer: B

The curve formed by a uniform flexible chain hanging under its own weight is the catenary y = c cosh(x/c).

246

Mathematics • Fluid Mechanics

For the flow of a viscous Newtonian fluid between two parallel plates located at y = 0 and y = h, the upper plate is moving with velocity U. The flow field is: For the flow of a viscous Newtonian fluid between two parallel plates located at y = 0 and y = h, the upper plate is moving with velocity U. The flow field is:

A. u(y) = Uy/h u(y) = Uy/h B. u(y) = Uy/(4h) u(y) = Uy/(4h) C. u(y) = U(y)/(2h) u(y) = U(y)/(2h) D. u(y) = Uy/(3h) u(y) = Uy/(3h)

सही उत्तर: A Correct Answer: A

For simple Couette flow with no slip, u(0) = 0 and u(h) = U. The velocity profile is linear, so u(y) = Uy/h.

247

Mathematics • Differential Geometry

The orthogonal trajectories on the sphere x^2 + y^2 + z^2 = 1, upon its intersection with the family of planes z = k, -1 <= k <= 1, are given by: The orthogonal trajectories on the sphere x^2 + y^2 + z^2 = 1, upon its intersection with the family of planes z = k, -1 <= k <= 1, are given by:

A. |y| = c|x| and x^2 + y^2 - z^2 = 1 |y| = c|x| and x^2 + y^2 - z^2 = 1 B. |y| = c|z| and x^2 + y^2 + z^2 = 1 |y| = c|z| and x^2 + y^2 + z^2 = 1 C. |z| = c|x| and x^2 + y^2 - z^2 = 1 |z| = c|x| and x^2 + y^2 - z^2 = 1 D. |y| = c|x| and x^2 + y^2 + z^2 = 1 |y| = c|x| and x^2 + y^2 + z^2 = 1

सही उत्तर: D Correct Answer: D

The intersections z = k are latitude circles on the sphere. Their orthogonal trajectories are meridians, represented by planes y = cx through the z-axis, together with the sphere equation.

248

Mathematics • Coordinate Geometry

Find the rectangular coordinates of the point with cylindrical coordinates (2, 2pi/3, 1). Find the rectangular coordinates of the point with cylindrical coordinates (2, 2pi/3, 1).

A. (-1, sqrt(3), 1) (-1, sqrt(3), 1) B. (2, sqrt(3), -2) (2, sqrt(3), -2) C. (1, sqrt(3), 0) (1, sqrt(3), 0) D. (1, sqrt(3), -1) (1, sqrt(3), -1)

सही उत्तर: A Correct Answer: A

Using x = r cos(theta), y = r sin(theta), z = z, we get x = 2 cos(2pi/3) = -1, y = 2 sin(2pi/3) = sqrt(3), and z = 1.

249

Mathematics • Graph Theory

In a graph, the number of vertices with odd degrees is always: In a graph, the number of vertices with odd degrees is always:

A. Odd Odd B. Zero Zero C. Even Even D. Cannot be determined Cannot be determined

सही उत्तर: C Correct Answer: C

By the handshaking lemma, the sum of all vertex degrees is even. Therefore, vertices of odd degree must occur in an even number.

250

Mathematics • Graph Theory

If every point of G has even degree, then G is a: If every point of G has even degree, then G is a:

A. Planar Graph Planar Graph B. Tree Tree C. Eulerian Graph Eulerian Graph D. Pie Chart Pie Chart

सही उत्तर: C Correct Answer: C

A connected graph has an Euler circuit if and only if every vertex has even degree; hence it is Eulerian.

Mathematics MCQs

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