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Let [t] denote the greatest integer ≤ t. Then the equation in x, [x]²+2[x+2]-7 = 0 has:

a) exactly four integral solutions. b) infinitely many solutions. c) no integral solution. d) exactly two solutions. Solution: [x]2+2[x+2]-7... View Article

The value of [latex]\sum_{r=0}^{20} \: ^{50-r}C_{6}[/latex] is equal to

a) 51C7-30C7 b) 51C7+30C7 c) 50C7-30C7 d) 50C6-30C6 Solution: = 50C6+49C6+48C6+…+31C6+30C6 Add and subtract 30C7 Using nCr+ nCr-1 =... View Article

Let y = y(x) be the solution of the differential equation, xy’-y = x²(xcosx+sinx),x > 0. If y(π) = π, then y’’(π/2)+y(π/2) is equal to:

a) 2+(π/2)+(π2/4) b) 2+(π/2) c) 1+(π/2) d) 1+(π/2)+ (π2/4) Solution: xy’-y = x2(x cosx + sinx) x > 0, y(π) =π... View Article

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^thA.M. is equal to 2^nd G.M., then m is equal to

Solution:3, …., 243 (m A.M)d = (b-a)/(n+1)= (243-3)/(m+1)= 240/(m+1)4th AM = 3+4d= 3+4(240/m+1)3, …, 243 (3G.M)243 = 3r4r = 32nd G.M = ar2 =... View Article

Let R₁ and R₂ be two relations defined as follows

Let R1 and R2 be two relations defined as follows:R1 = {(a,b) ∈ R2: a2+b2 ∈ Q} andR2 = {(a,b) ∈ R2: a2+b2 ∉Q}, where Q is the set... View Article

If the surface area of a cube is increasing at a rate of 3.6 cm²/sec, retaining its shape; then the rate of change of its volume (in cm³/sec), when the length of a side of the cube is 10cm, is

a) 9b) 10c) 18d) 20Solution:A = 6a2A is the side of cube.dA/dt = 6×2a da/dt⇒ 3.6 = 12×10 (da/dt)⇒ (da/dt) = 3/100V = a3dV/dt =... View Article

Let p, q, r be three statements such that the truth value of (p˄q) -> (∼q˅r) is F. Then the truth values of p, q, r are respectively:

a) F, T, Fb) T, F, Tc) T, T, Fd) T, T, TSolution:(p˄q) -> (∼q˅r) possible when p˄q -> T∼q˅r -> Fp -> T ; p˄q⇒ Tq -> T ; ∼q˅r ->... View Article

If the sum of the series [latex]20+19frac{3}{5}+19frac{1}{5}+18frac{4}{5}+…[/latex] upto n^thterm is 488 and the n^th term is negative, then:

a) n = 60b) n = 41c) nth term is -4d) nth term is Sn = 488⇒ (n/2)(2×20+(n-1)(-2/5)) = 488⇒ 20n-(n2/5)+(n/5) = 488⇒ 100n-n2+n =... View Article

If [latex]\int \sin ^{-1}\left ( \sqrt{\frac{x}{1+x}} \right )dx= A(x)\tan ^{-1}(\sqrt{x})+B(x)+C[/latex] , where C is a constant of integration, then the ordered pair (A(x), B(x)) can be:

a) (x+1, -√x)b) (x-1, -√x)c) (x+1, √x)d) (x-1, √x)Solution: Given ∫tan-1√x. 1 dxHere tan-1√x is the first function and 1 dx is the second... View Article

The probability that a randomly chosen 5 digit number is made from exactly two digits is

a) 134/104b) 121/104c) 135/104d) 50/104Solution:Total case = 9(104)First case : Choose two non-zero digits = 9C2Now, number of 5-digit numbers... View Article

Let x_i be ten observations of a random variable X, then the standard deviation of these observations is

Let xi (1≤ i ≤10)be ten observations of a random variable X. If where 0 ≠ p εR , then the standard deviation of these observations... View Article

[latex]\lim_{x\to a}\frac{(a+2x)^{\frac{1}{3}}-(3x)^{\frac{1}{3}}}{(3a+x)^{\frac{1}{3}}-(4x)^{\frac{1}{3}}}[/latex] is equal to

a) (2/9)4/3b) (2/3)4/3c) (2/3)(2/9)1/3d) (2/9)(2/3)1/3Solution:Apply L hospital rule⇒... View Article

The plane which bisects the line joining the points (4,-2,3) and (2,4,-1) at right angles also passes through the point

a) (0,-1,1)b) (4,0,1)c) (4,0,-1)d) (0,1,-1)Solution:a = 2, b = -6c = 4Equation of plane is2(x-3)+(-6)(y -1)+ 4(z-1) = 0⇒ 2x-6y+4z = 4 passes... View Article

Let the latus rectum of the parabola y² = 4x be the common chord to the circles C₁and C₂ each of them having radius 2√5. Then, the distance between the centres of the circles C₁ and C₂is

a) 8b) 8√5c) 4√5d) 12Solution:C1C2 = 2C1A(C1A)2 + 4 = ( 2√5)2C1A = 4C1C2 = 8Answer: a... View Article

The set of all real values of λ for which the quadratic equations, (λ²+1)x²-4λx+2 = 0 always have exactly one root in the interval (0,1) is

a) (-3,-1)b) (2,4)c) (1,3)d) (0,2)Solution:f(0) f(1) ≤ 0⇒ (2) [λ2-4λ +3] ≤0(λ-1)(λ- 3) ≤0⇒ λ... View Article

If the value of the integral [latex]int_{0}^{frac{1}{2}}frac{x^{2}}{(1-x^{2})^{frac{3}{2}}}dx[/latex] is k/6, then k is equal to

a) 2√3+πb) 3√2+πc) 3√2-πd) 2√3-πSolution: ⇒ (1/√3)-(π/6) = k/6(2√3-π)/6 = k/6k = 2√3-πAnswer: d... View Article

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