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If R = {(x,y):x,y ∈ Z, x²+3y² ≤ 8} is a relation on the set of integers Z, then the domain of R^-1 is

a) {-1, 0, 1} b) {-2, -1, 1, 2} c) {0,1} d) {-2,-1,0,1,2} Solution: 3y2 ≤ 8-x2 R:{(0,1), (0,-1), (1,0), (-1,0), (1,1), (1,-1), (-1,1),... View Article

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in

a) (-∞, -9]∪[3, ∞) b) [-3, ∞) c) (-∞, -9] d) (-∞, -3]∪[9, ∞) Solution: (a/r).a.ar = 27 a = 3... View Article

If a function f(x) defined by

If a function f(x) defined by be continuous for some a, b, c ∈ R and f’(0)+f’(2) = e, then the value of a is: a)... View Article

The domain of the function f(x) = sin^-1[(|x|+5)/(x²+1)] is (-∞, -a]U[a,∞). Then a is equal to:

a) (√17-1)/2 b) √17/2 c) (1+√17)/2 d) (√17/2)+1 Solution: -1≤ (|x|+5)/x2+1 ≤1 -x2-1≤|x|+5≤ x2+1 Case 1... View Article

A line parallel to the straight line 2x-y = 0 is tangent to the hyperbola (x²/4)-(y²/2) = 1 at the point (x₁, y₁).

A line parallel to the straight line 2x-y = 0 is tangent to the hyperbola (x2/4)-(y2/2) = 1 at the point (x1, y1). Then x12+5y12 is equal to: a)... View Article

If the line, 2x-y+3 = 0 is at a distance 1/√5 and 2/√5 from the lines 4x-2y+α = 0 and 6x-3y+β = 0, respectively, then the sum of all possible values of α and β is

Solution: L1 : 2x-y+3 = 0 L2 : 4x-2y+α = 0 L3 : 6x-3y+β = 0 |(α/2)-3|/√5 = 1/√5 ⇒ (α/2)-3 = 1, -1 ⇒... View Article

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ’SYLLABUS’ such that two letters are distinct and two letters are alike, is

Solution: SS, Y, LL, A, B, U ABCC type words Answer: 240... View Article

Let f(x) = x[x/2], for -10<x<10, where [t] denotes the greatest integer function. Then the number of points of discontinuity of f is equal to

Solution: f(x) = x[x/2] x∈ (-10, 10) x/2 ∈(-5,5) → 9 integers Check continuity at x = 0 f(0) = 0 f(0+) = 0 f(0-) = 0 continuous at... View Article

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is

Solution: P(at 2, 3 or 4) = 4C2 (2/6)2(4/6)2+ 4C3 (2/6)3(4/6)1+ 4C4 (2/6)4 = 6×(1/9)×(4/9)+4×(1/27)×(2/3)+(1/81) =... View Article

The natural number m, for which the coefficient of x in the binomial expansion of (x^m+1/x²)²² is 1540 is

Solution: Tr+1 = 22Cr (xm)22-r(1/x2)r = 22Cr (x)22m-mr-2r Given 22Cr = 1540 = 22C19 ⇒ r = 19 Since 22m-rm-2r = 1 ⇒ m =... View Article

If α is positive root of the equation, p(x) = x²-x-2 = 0,

If α is positive root of the equation, p(x) = x2-x-2 = 0, then is equal to: a) 1/2 b) 3/√2 c) 3/2 d) 1/√2 Solution:... View Article

If the co-ordinates of two points A and B are (√7, 0) and (-√7, 0) respectively and P is any point on the conic, 9x²+16y²= 144, then PA+PB is equal to

a) 6 b) 16 c) 9 d) 8 Solution: (x2/16)+(y2/9) = 1 e = √(1-9/16) = √7/4 F1(√7, 0), F2(-√7, 0) PF1+PF2 = 2a PA+PB... View Article

The negation of the Boolean expression x ↔ y is equivalent to

a) (x˄y) ˄(∼x˅∼y) b) (x˄y) ˅ (∼x˄∼y) c) (x˄∼y) ˅ (∼x˄∼y) d) (∼x˄y) ˅ (∼x˄∼y) Solution: As we know p... View Article

If [latex]int left ( e^{2x}+2e^{x}-e^{-x}-1 right )e^{(e^{x}+e^{-x})}dx=g(x)e^{(e^{x}+e^{-x})}+c[/latex] where c is a constant of integration, then g(0) is equal to

a) 2 b) e c) 1 d) e2 Solution: ⇒ g(0) = 2 Answer: a... View Article

If the four complex numbers z, [latex]bar{z}[/latex] , [latex]bar{z}-2Rebar{z}[/latex] and z-2Re(z) represent the vertices of a square of side 4 units in the Argand plane, then |z| is equal to

a) 2 b) 4 c) 4√2 d) 2√2 Solution: Let z = x+iy CA2 = AB2+BC2 22x2+22y2 = 32 x2+y2 = 8 √(x2+y2 ) = 2√2 Answer: d... View Article

If the function f(x) is twice differentiable

If the function is twice differentiable, then the ordered pair (k1, k2) is equal to: a) (1,1) b) (1,0) c) (1/2, -1) d) (1/2, 1) Solution:... View Article

If (a, b, c) is the image of the point (1,2,-3) in the line, (x+1)/2 = (y-3)/-2 = z/-1 then a+b+c is

a) 2 b) 3 c) -1 d) 1 Solution: Vector PM and ( ) are perpendicular. ⇒ (2λ-2)2+(1-2λ)(-2)+(3-λ) (-1) = 0 ⇒... View Article

The value of [latex]\int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}\frac{1}{1+e^{\sin x}}dx[/latex] is:

a) π/2 b) π/4 c) π d) 3π/2 Solution: I = = ⇒ 2I = π I = π/2 Answer: a... View Article

If S is the sum of the first 10 terms of the series tan^-1(1/3) + tan^-1(1/7) + tan^-1(1/13) + tan^-1(1/21) +…, then tan(S) is equal to

a) 5/11 b) 5/6 c) -6/5 d) 10/11 Solution: S = Tr = Tr = tan-1(r+1)-tan-1r T1 = tan-12-tan-11 T2 = tan-13-tan-12 T3 = tan-14-tan-13... View Article

If y = y(x) is the solution of the differential equation ((5+e^x)/(2+y))(dy/dx)+e^x = 0 satisfying y(0) = 1, then a value of y(log_e13) is

a) 1 b) 0 c) 2 d) -1 Solution: ((5+ex)/(2+y))(dy/dx) = -ex ∫dy/(2+y) = ∫-e-x/(ex+5) dx ln (y+2) = -ln(ex+5) + C (y+2) (ex+5) = C... View Article