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Let and be two events such that the probability that exactly one of them occurs is 2/5 and the probability that or occurs is 1/2, then the probability of both of them occur together is

a. 0.10 b. 0.20 c. 0.01 d. 0.02 Solution: P(A or B) = P(A U B) = 1/2 P(exactly one of A or B)= 2/5 ⇒ P(A) − P(A ∩ B) + P(B) − P(A ∩ B) =... View Article

If the distance between the plane, 23x-10y-2z+48 = 0 and the plane containing the lines

If the distance between the plane, 23x-10y-2z+48 = 0 and the plane containing the lines k = 3Answer: (3)Answer: (14)... View Article

The number of terms common to the A.P.’s 3,7,11,…407 and 2,9,16,…709 is

Solution:First common term is 23Common difference = LCM(7,4) = 2823+(n−1)28 ≤ 407n−1 ≤ 13.71n = 14... View Article

If the curves x² -6x +y² +8 = 0 and x²-8y+y²+16-k = 0, (k>0) touch each other at a point, then the largest value of k is

Solution:Two circles touch each other if C1C2 = ǀr1± r2ǀ√k +1 = 5 or ǀ√k-1ǀ = 5k = 16 or 36Maximum value of k is 36.Answer: (36)... View Article

If C_r = ²⁵C_r and C₀+5⋅C₁+9⋅C₂ +⋯+101. C₂₅ =2²⁵ .k then k is equal to

Solution:S = 25C0 + 525C1 + 925C2 + ….+9725C24 +10125C25 = 225k ..(i)Reverse and apply property nCr = nCn-r in all coefficientsS = 10125C0 +... View Article

If vector a is perpendicular to vector b times vector c, then vector a times vector b times vector c is equal to

Let is equal to Solution: . Answer : (30)... View Article

What is the ordered pair (λ, f(θ))

If I = (2/(1+k))-1)dkI = 2 lnǀ1+kǀ -k+cI = 2 lnǀ1+tan θǀ -tan θ +cGiven I = tan θ+ 2 log f(θ) + cλ= -1, f(θ) = ǀ1+... View Article

A random variable X has the following probability distribution:

A random variable X has the following probability distribution:X12345P(X)K22KK2K5K2Then P(X>2) is equal to:a. 7/12b. 23/36c. 1/36d.... View Article

Let an be the nth term of a G.P. of positive terms. If sum from n=1 to 100 for 2n+1 terms and sum for 2n terms = 100

a. 300 b. 175 c. 225 d. 150 Solution: an is a positive term of GP. Let GP be a, ar, ar2,..... = a3+a5+a7+…+a201 200 = ar2 + ar4 +... View Article

If one end of focal chord AB of the parabola y²= 8x is at A(1/2,-2) , then the equation of tangent to it at B is

a. x+2y+8 = 0b. 2x-y-24 = 0c. x-2y+8 = 0d. 2x+y-24 = 0Solution:Let PQ be the focal chord of the parabola y2 = 8xP(t1) = (2t12, 4t1) and Q(t2) =... View Article

If [latex]\frac{dy}{dx}= \frac{xy}{x^{2}+y^{2}}[/latex] , y(1) = 1, then a value of x satisfying y(x)= e is

a. √3eb. (1/2)√3ec. √2ed. e/√2Solution:Let y = vxdy/dx = v+x(dv/dx)v+x(dv/dx) = vx2/(x2(1+v2) = v/(1+v2)x(dv/dx) = -v3/(1+v2)(1/x)dx =... View Article

If [latex]x = \sum_{n=0}^{\infty }(-1)^{n}\tan ^{2n}\theta[/latex] and [latex]y = \sum_{n=0}^{\infty }\cos ^{2n}\theta[/latex] where 0 < θ < π/4, then:

a. y(1+x) = 1 b. x(1-y) = 1 c. y(1-x) = 1 d. x(1+y) = 1 Solution: y = 1+cos2 θ +cos4 θ + … 1/y = sin2 θ x = 1-tan2 θ + tan4 θ-… x... View Article

The length of minor axis (along y-axis) of an ellipse of the standard form is 4/√3. If this ellipse touches the line x + 6y = 8, then its eccentricity is

a. (1/2)(√5/3)b. (1/2)√(11/3)c. √(5/6)d. (1/3)√(11/3)Solution:If 2b = 4/√3b = 2/√3Comparing y = (-x/6)+(8/6) with y = mx ±√(a2m2+b2)m = -1/6 and... View Article

If p -> (p ∧~ q) is false. Then the truth values of p and q are respectively

a. F, Tb. T, Fc. F, Fd. T, TSolution:Given p ->(p ∧~ q)Truth table:pq~q(p ∧~ q)p->(p ∧~ q)TTFFFTFTTTFTFFTFFTFTp→(p ∧~ q) is false when p is true... View Article

The following system of linear equations:7x+6y-2z = 0, 3x+4y+2z = 0 and x-2y-6z = 0, has

a. infinitely many solutions, (x,y,z) satisfying y = 2zb. infinitely many solutions (x,y,z) satisfying x = 2zc. no solutiond. only the trivial... View Article

Find the area (in sq. units) of the region bounded by the curves y = f(x) and y = g(x) between the lines 2x = 1 to 2x = √3 is

Given: and g(x)= (x− 1/2)2, x ∈ R. Then the area (in sq. units) of the region bounded by the curves y = f(x) and y = g(x) between the lines 2x =... View Article