Meghalaya Police Service Preliminary Examination 2022 - Multiple Subjects — question paper

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Meghalaya Police Service Preliminary Examination 2022 - Multiple Subjects

Paper #3169 · 17-12-2022 · Multiple subjects · 514 reviewed questions · Showing 361–400 of 514

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The rate of change of the area of a circle with respect to its radius is
1. πr
2. πr²
3. 2πr
4. 2π
On either interval x > 0 or x < 0, an antiderivative of 1/x with respect to x is:
1. ln|x| + C
2. −1/x + C
3. log₁₀|x| + C
4. 1/x + C
With x measured in radians, an antiderivative of sin x with respect to x is:
1. cos x + C
2. −cos x + C
3. csc x + C
4. −csc x + C
Evaluate the definite integral ∫₁² x dx.
1. 1/2
2. -3/2
3. 3/2
4. 5/2
The definite integral of cos x dx from 0 to π/2 is equal to
1. 1
2. 0
3. -1
4. 2
The definite integral of 1/(1+x²) dx from 0 to 1 is equal to
1. 0
2. π/3
3. π/4
4. π/6
The area under the curve y=x² bounded by the x-axis between x=0 and x=2 is
1. 3/8 square units
2. 8/3 square units
3. 4 square units
4. 2 square units
For a continuously differentiable curve y=f(x) on [a,b], where a<b, the arc length between the points with abscissas a and b is
1. Integral from a to b of √[1+(dy/dx)²] dx
2. Integral from a to b of √[1+(dx/dy)²] dy
3. Integral from a to b of y dx
4. Integral from a to b of y² dx
For a sphere of radius a, where a ≥ 0, the volume is
1. (4/3)πa²
2. (1/3)πa²
3. (4/3)πa³
4. (1/3)πa³
For a > 0, the surface area generated by rotating the circle x² + y² = a² about the x-axis is
1. πa²
2. 4πa²
3. 2πa²
4. 8πa²
In three-dimensional Cartesian coordinates, the general equation of a real sphere with positive radius is
1. x²+y²+z²+2ux+2vy+2wz+d=0
2. x²+y²+2gx+2fy+c=0
3. x²+y²+z²=c²
4. None of these
For the line (x+3)/2 = (y−1)/3 = (z+1)/4, the point whose x-coordinate is −1 is
1. (-1,4,3)
2. (-1,3,4)
3. (1,4,3)
4. (1,3,4)
The acute angle between the planes 2x − y + z = 6 and x + y + 2z = 3 is
1. 60 degrees
2. 30 degrees
3. 45 degrees
4. 90 degrees
The equation of the tangent plane at the point (1,1,0) to the sphere x²+y²+z²=2 is
1. x-y=0
2. x+z=2
3. x+y=2
4. x-y=2
The value of f'(3) when f(x)=1/x is
1. 1/9
2. 1/3
3. -1/9
4. -1/3
For a circle of radius a > 0, its curvature magnitude is
1. 1/a
2. a
3. a²
4. 1/a²
If a finite set S has n elements, where n is a non-negative integer, the number of elements in its power set is
1. 2n elements
2. 2^n elements
3. 3^n elements
4. n elements
Let S = {0, 1} and T = {0, −1}. Their Cartesian product S × T is
1. {(0,0),(0,1),(0,-1),(1,-1)}
2. {(0,0),(0,-1),(1,0),(1,-1)}
3. {(0,0),(0,1),(-1,0),(-1,1)}
4. {0,-1}
The product of permutations f = (1 2 3)(1 2) is, with respect to parity,
1. An even permutation
2. An odd permutation
3. An identity permutation
4. None of these
An element e in a ring R is called idempotent if
1. e²=e
2. e²=1
3. e²=0
4. None of these
In the vector space ℝ³, classify the pair {(2, 1, 2), (8, 4, 8)}:
1. Linearly dependent
2. Linearly independent
3. Orthogonal
4. None of these
If W₁ and W₂ are two subspaces of a finite-dimensional vector space V(F), then dim(W₁+W₂) is
1. dim W₁+dim W₂
2. dim W₁+dim W₂+dim(W₁∩W₂)
3. dim W₁+dim W₂-dim(W₁∩W₂)
4. None of these
Let V be a finite-dimensional vector space over a field F, and let W be a subspace of V. The dimension of the quotient space V/W is
1. dim V / dim W
2. dim V+dim W
3. dim V-dim W
4. None of these
In a ring R whose multiplication is not assumed commutative, for a, b ∈ R, (a + b)² equals
1. a²+2ab+b²
2. a²+ab+ba+b²
3. a²+2ba+b²
4. a²+b²
For a group G, if a, b ∈ G, the equation ax = b has how many solutions x ∈ G?
1. A unique solution in G
2. No solution in G
3. Infinite number of solutions in G
4. None of these
Let A and B be finite sets with |A ∩ B| = n. The number of ordered pairs common to A × B and B × A is
1. 2^n
2. 2n
3. n²
4. n
If (2x − 1, −5) = (x, y + 1) as ordered pairs, the value of x is
1. 1
2. 0
3. 1/2
4. -1
Let A and B be subsets of a universal set U, and take complements relative to U. Then B \ Aᶜ equals
1. Bᶜ-A
2. B-A
3. B∩A
4. A-B
The value of the determinant with rows (5²,5³,5⁴), (5³,5⁴,5⁵), (5⁴,5⁶,5⁷) is
1. 5²
2. 0
3. 5^13
4. 5⁷
For all real x, which expression is equivalent to f(x) = 2x when x ≥ 0 and f(x) = 0 when x < 0?
1. f(x)=|x|
2. f(x)=2x
3. f(x)=|x|+x
4. f(x)=2|x|
The set of prime numbers is given by
1. {0,1,2,3,...}
2. {1,2,3,5,7,...}
3. {2,3,5,7,11,...}
4. {2,3,4,5,6,...}
Let U = {1, 2, 3, 4, 5, 6} be the universal set and A = {1, 2, 3, 4}. The complement of A relative to U is
1. {1, 2, 3, 4, 5, 6}
2. {5, 6}
3. {1, 2, 3, 4}
4. {1, 5, 6}
The value of the limit of (1+x)^(1/x) as x tends to 0 is
1. 1
2. 0
3. e
4. 1/e
Let f(x) = cos x − sin x, with x measured in radians. Find f′(π/2).
1. -1
2. 0
3. 1
4. 1/2
A closed cylinder has fixed height h and radius r. The rate of change of its total surface area S with respect to r is
1. 2πr + 2πh
2. 4πr + 4πh
3. 4πr + 2πh
4. 2πr + 4πh
An antiderivative of (x − 1)e^(−x) is
1. -xe^x+C
2. xe^x+C
3. -xe^(-x)+C
4. xe^(-x)+C
For an even integer k ≥ 4, evaluate ∫₀^π sin(kx)/sin x dx.
1. π
2. π/2
3. kπ/2
4. 0
A nonvertical straight line through the origin is written y = mx. For x ≠ 0, eliminating m gives the differential equation
1. y=√(x·dy/dx)
2. dy/dx=y+x
3. dy/dx=y/x
4. dy/dx=x/y
Let log x = ln x. If I₁ = ∫ₑ^{e²} dx/ln x and I₂ = ∫₁² eˣ/x dx, then
1. I₁=I₂
2. 2I₁=I₂
3. I₁+I₂=0
4. I₁=2I₂
The maximum value of a single digit in an octal number system is
1. 8
2. 7
3. 6
4. 5

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