Inspector of Statistics • Paper-II
Which index is used to calculate inflation?
A)
Consumer Price Index (CPI)
B)
Index of Industrial Production (IIP)
C)
Human Development Index (HDI)
D)
Purchasing Managers’ Index (PMI)
CPI is the principal index used to measure consumer-price inflation.
Inspector of Statistics • Paper-II
Which of the following districts of Mizoram have Autonomous District Councils?
A)
Mamit, Kolasib, Aizawl
B)
Lawngtlai, Siaha
C)
Aizawl, Saitual, Hnahthial
D)
Champhai, Serchhip, Khawzawl
In Mizoram, Autonomous District Councils (ADCs) are established under the Sixth Schedule of the Constitution of India to protect the interests of tribal communities. There are currently three ADCs in the state: 1. Lai Autonomous District Council (LADC) located in Lawngtlai district, 2. Mara Autonomous District Council (MADC) located in Siaha district, and 3. Chakma Autonomous District Council (CADC) which operates within the Lawngtlai district. Therefore, the districts that host these councils are Lawngtlai and Siaha. Options a, c, and d list districts that do not have Autonomous District Councils.
Inspector of Statistics • Paper-II
With reference to the International Union for Conservation of Nature and Natural Resources (IUCN) and the Convention on International Trade in Endangered Species of Wild Fauna and Flora (CITES), which of the following statements is/are correct?
I. IUCN is an organ of the United Nations and CITES is an international agreement between governments
II. IUCN runs thousands of field projects around the world to better manage natural environments
III. CITES is legally binding on the States that have joined it, but this Convention does not take the place of national laws
Select the correct answer using the code given below:
A)
I only
B)
II and III only
C)
I and III only
D)
I, II and III
IUCN is not a UN organ; statements II and III are the correct ones.
Inspector of Statistics • Paper-II
What does ‘marginal utility’ in economics signify?
A)
Small utility
B)
Additional utility
C)
Minimum utility
D)
Satisfied utility
Marginal utility is the additional utility from consuming one more unit.
Inspector of Statistics • Paper-III
The complete integral of the differential equation p^2 = zq is
A)
z = be^(ax + a^2y)
B)
z = be^(x + a^2y)
C)
z = be^(x + ay)
D)
None of these
Using z=e^u gives u_y=u_x²; a complete integral is u=ax+a²y+ln b, hence z=b e^(ax+a²y).
Inspector of Statistics • Paper-III
In the General quadrature formula, Simpson’s 1/3rd Rule is obtained by putting
A)
n = 1
B)
n = 2
C)
n = 2 & 4
D)
n = 4
Simpson’s 1/3 rule is obtained with n=2 subintervals.
Inspector of Statistics • Paper-III
If a and b are constant vectors, then
A)
∇.(a.b) = a
B)
∇.(a.b) = b
C)
∇.(a.b) = 0
D)
None of these
Option C is correct: ∇.(a.b) = 0.
Inspector of Statistics • Paper-III
The radius of the sphere x^2 + y^2 + z^2 - 2x + 4y - 6z = 2 is
A)
5
B)
4
C)
3
D)
2
Completing squares gives (x−1)²+(y+2)²+(z−3)²=16, so the radius is 4.
Inspector of Statistics • Paper-III
The centre of gravity of a uniform semi-circular lamina of radius a lies on the radius bisecting the lamina at a distance of
A)
4a/3π from the centre
B)
4a/2π from the centre
C)
a/3π from the centre
D)
a/2π from the centre
The centroid of a semicircular lamina is 4a/(3π) from the circle centre along its axis of symmetry.
Inspector of Statistics • Paper-III
While solving an IPP, any non-integer variable in the solution is picked up to
A)
enter the solution
B)
leave the solution
C)
obtain the cut constraint
D)
all of the above
Option C is correct: obtain the cut constraint.
Inspector of Statistics • Paper-III
The real root of the equation cosx = 3x - 1 lies between
A)
0 and 1
B)
0 and π
C)
0 and 2π
D)
0 and π/2
f(0)=2 and f(1)=−2, so continuity places the root between 0 and 1.
Inspector of Statistics • Paper-III
Suppose X is binomially distributed with parameters n and p. If E[X] = 5 and Var[X] = 4, then
A)
n = 25 and p = 1/5
B)
n = 20 and p = 1/5
C)
n = 25 and p = 2/5
D)
n = 20 and p = 2/5
For a binomial variable, np=5 and np(1−p)=4, giving p=0.2 and n=25.
Inspector of Statistics • Paper-III
For maximization LPP, the objective function coefficient for an artificial variables is
A)
+ M
B)
- M
C)
+ 1
D)
zero
In the Big-M method for a maximization problem, artificial variables receive a large negative coefficient (−M) so they are driven out of the basis.
Inspector of Statistics • Paper-III
The solution of transportation problem with m-sources and n-destinations is feasible, if the number of allocations are
A)
m + n - 1
B)
m + n + 1
C)
m + n
D)
m x n
Option A is correct: m + n - 1.
Inspector of Statistics • Paper-III
For any primal and its dual:
A)
optimum value of the objective function is same
B)
both primal and dual cannot be feasible
C)
primal will have an optimum solution if and only if dual does too
D)
all the above
Strong duality gives equal optimal objective values for a feasible bounded primal/dual pair; the standard theorem also links existence of an optimum. Option C is the best statement here.
Inspector of Statistics • Paper-III
The least velocity with which a body can be projected to have horizontal range R is
A)
g/R
B)
R/g
C)
gR
D)
None of these
For a projectile, R=u² sin2θ/g. Maximum range occurs at 45°, so the least speed for a given R is √(gR); that exact formula is not listed.
Inspector of Statistics • Paper-III
If a body is in equilibrium, under the action of three equal non-parallel coplanar forces, then the angle between any two of these forces is
A)
45°
B)
60°
C)
90°
D)
120°
Three equal coplanar forces in equilibrium must be symmetrically spaced by 120°.
Inspector of Statistics • Paper-III
The integrating factor of the differential equation ydx - xdy = 0 is
A)
x^2
B)
x
C)
1/x
D)
1/x^2
Multiplying by 1/x² makes the differential equation exact; hence 1/x² is an integrating factor.
Inspector of Statistics • Paper-III
The position of a moving point at time t referred to rectangular axes is given by x = acosωt, y = asinωt. The path of the particle is a
A)
straight line
B)
circle of radius a
C)
cycloid
D)
parabola of latus rectum 4a
From x=a cos(ωt) and y=a sin(ωt), x²+y²=a², which is a circle of radius a.
Inspector of Statistics • Paper-III
Let V be the field of real numbers R^3. Which of the following is not a subspace of V?
A)
{ (x, 2y, 3z) : x, y, z in R }
B)
{ (x, y, z) : x, y, z in R }
C)
{ (x, y, z) : x, y, z in Q } where Q is the field of rational numbers
D)
{ (x+4y, y, x+2y) : x, y, z in R }
The rational-coordinate set Q³ is not a vector subspace of R³ because it is not closed under multiplication by arbitrary real scalars.
Inspector of Statistics • Paper-III
For the 3x3 matrix [[3, 1, -1], [2, 2, -1], [2, 2, 0]], the space of eigen vectors associated with the eigen value 2 has dimension:
A)
0
B)
1
C)
2
D)
3
Solving (A−2I)v=0 leaves one free variable, so the eigenspace dimension is 1.
Inspector of Statistics • Paper-III
The general solution of the Lagrange's equation 2yzp + zxq = 3xy is:
A)
phi(x^2 - y^2, y^2 - z^2) = 0
B)
phi(x^2 - 2y^2, 3y^2 - z^2) = 0
C)
phi(x^2 - 2y^2, y^2 + z^2) = 0
D)
phi(x^2 + y^2, 3x^2 - z^2) = 0
Characteristic equations/integrals give x²−2y²=c1 and 3y²−z²=c2, leading to the stated general solution form.
Inspector of Statistics • Paper-III
The limit as x -> 0 of (e^(2x) - 1) / 2x is:
A)
0
B)
1/2
C)
1
D)
2
Using e^{2x}=1+2x+o(x), the limit is 1.
Inspector of Statistics • Paper-III
Let z = r(cos theta + i sin theta), then:
A)
z^n = r^n(cos n theta - i sin n theta)
B)
z^n = r^n(cos n theta + i sin n theta)
C)
z^n = r^n(cos theta + i sin theta)^n
D)
z^n = r(cos n theta + i sin n theta)
De Moivre's theorem gives z^n=r^n[cos(nθ)+i sin(nθ)].
Inspector of Statistics • Paper-III
The linear second order partial differential equation x^2 u_xx + xy u_xy - y u_xx = 0 is:
A)
Parabolic
B)
Elliptic
C)
Hyperbolic
D)
Circular
As written the PDE has a discriminant indicating a hyperbolic classification; the item appears to contain a typographical coefficient, but C is the intended classification.
