Mathematics MCQs

1

Mathematics • Probability

Consider 10 tosses of a fair die. Let X_i denote the number of times digit i appears. Find P(X_2 = 2 | X_1 = 2). Consider 10 tosses of a fair die. Let X_i denote the number of times digit i appears. Find P(X_2 = 2 | X_1 = 2).

A. C(10,2)(1/6)^2(5/6)^8 C(10,2)(1/6)^2(5/6)^8 B. C(8,2)(1/5)^2(4/5)^6 C(8,2)(1/5)^2(4/5)^6 C. C(8,2)(1/6)^2(5/6)^6 C(8,2)(1/6)^2(5/6)^6 D. (1/6)^2 (1/6)^2

सही उत्तर: B Correct Answer: B

Given X_1 = 2, the remaining 8 outcomes are non-one. For each such outcome, the conditional probability of getting 2 is 1/5. Hence X_2 is binomial with n = 8 and p = 1/5.

2

Mathematics • Probability

We have P(X = x, Y = y) = 1/9 for all x, y in {1, 2, 3}. Find P(X > Y). We have P(X = x, Y = y) = 1/9 for all x, y in {1, 2, 3}. Find P(X > Y).

A. 0 0 B. 1/3 1/3 C. 1 1 D. 1/2 1/2

सही उत्तर: B Correct Answer: B

The ordered pairs satisfying X > Y are (2,1), (3,1), and (3,2). Their total probability is 3 x (1/9) = 1/3.

3

Mathematics • Convex Functions

Let f and g be two real-valued functions defined on [a,b], and h(x) = max(f(x), g(x)) for x in [a,b]. Which of the following is true? Let f and g be two real-valued functions defined on [a,b], and h(x) = max(f(x), g(x)) for x in [a,b]. Which of the following is true?

A. If f and g are differentiable, then so is h; and if f and g are convex, then h is also convex. If f and g are differentiable, then so is h; and if f and g are convex, then h is also convex. B. If f and g are convex, then h is also convex. If f and g are convex, then h is also convex. C. If f and g are differentiable, then so is h. If f and g are differentiable, then so is h. D. If f and g are linear, then h is also linear. If f and g are linear, then h is also linear.

सही उत्तर: B Correct Answer: B

The pointwise maximum of convex functions is convex. Differentiability or linearity need not be preserved at points where the two functions cross.

4

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.

5

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.

6

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.

7

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.

8

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).

9

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.

10

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.

Mathematics MCQs

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