Mathematics MCQs for JSSC

241

Advanced 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:

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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

Advanced 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:

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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

Advanced 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):

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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

Advanced 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?

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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

Advanced Mathematics • Statics

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

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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

Advanced 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:

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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

Advanced 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:

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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

Geometry • 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).

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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

Advanced 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:

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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

Advanced 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:

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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 for JSSC

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