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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

Let a-2b+c = 1. If f(x) is a matrix having entries x+a x+2 x+1 x+b x+3 x+2 x+c x+4 x+3, then

a. f(-50) = 501 b. f(-50) = -1 c. f(50) = 1 d. f(-50) = -501 Solution: Given f(x) = a -2b+c = 1 Applying R1 R1 -2R2+ R3 f(x) = Using... View Article

Let [t] denotes the greatest integer

Let [t] denotes the greatest integer ≤ t and A = 4√(A+5) = 3√(A+1) = √5√(A+21) =5√A = 2Points of discontinuity for f(x) is x = √5Answer: (a)... View Article

Let a function f:[0,5] R, be continuous, f(1) = 3 and F be defined as:

Let a function f:[0,5] R, be continuous, f(1) = 3 and F be defined as: F(x) = Then for the function F, the point x = 1 isa. a point of... View Article

Let a,b ∈R, a ≠ 0, such that the equation, ax²-2bx+5 = 0 has a repeated root

Let a,b ∈R, a ≠ 0, such that the equation, ax2-2bx+5 = 0 has a repeated root α, which is also a root of the equation x2-2bx-10 = 0. If β is the... View Article

In the expansion of….

In the expansion of l2/l1 = 16:1Answer: (a)... View Article

Let f and g be differentiable functions on R, such that fog is the identity function. If for some a, b ∈ R, g’(a) = 5 and g(a) = b, then f'(b) is equal to

a. 2/5b. 5c. 1d. 1/5Solution:f(g(x) = xf’(g(x))g’(x) = 1put x = af’(g(a))g’(a) = 1f’(b) ×5 = 1f’(b) = 1/5Answer: (d)... View Article

If x = 2 sin θ – sin 2 θ and y = 2 cos θ – cos 2 θ, θ ∈ [0, 2π], then d²y/dx² at θ =π is:

a. -3/8b. 3/4c. 3/2d. -3/4Solution:dx/d θ = 2 cos θ -2 cos2 θ dy/d θ = -2 sin θ + 2 sin2 θdy/dx = cot (3... View Article

If 10 different balls has to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is

a. 965/210b. 945/210c. 945/211d. 965/211Solution:Total ways to distribute 10 balls in 4 boxes is = 410 Total ways of placing exactly 2 and 3... View Article

If A = {x∈ R∶ |x| <2} and B = {x∈ R∶ |x-2| ≥3} then

a. A - B = [-1,2]b. B - A = R-(-2,5)c. A⋃B = R-(2,5)d. A∩B = (-2,-1)Solution:A = {x: x ∈ (-2,2)}B = {x: x ∈ (∞,-1] ⋃ [5, ∞)}A∩B = {x: x ∈ (-2,-1]... View Article

The coefficient of x⁴ in the expansion of (1+x+x)¹⁰ is

Solution: General term of the given expression is given by Here, q+2r =4For p = 6, q = 4, r = 0, coefficient = 10!/(6!×4!) = 210For p = 7, q =... View Article

If for x ≥ 0,y = y(x) is the solution of the differential equation (1+x)dy = [(1+x)²+y-3]dx, y(2) = 0, then y(3) is equal to:

Solution: (1+x) dy/dx = [(1+x)2 + (y-3)](1+x) (dy/dx) -y = (1+x)2 -3(dy/dx)-(1/1+x)y = 1+x - 3/(1+x)I.F = y = x(1+x) + 3+c(1+x)At x = 2, y = 0... View Article

The number of distinct solutions of the equation

Solution: sin 2x = ±1/2 Therefore, we have 8 solutions for x ∈ [0,2] Answer: (8)... View Article