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If α and β are the roots of the equation 2x(2x+1) = 1, then β is equal to:

1) 2α(α-1) 2) -2α(α+1) 3) 2α2 4) 2α(α+1) Solution: 2x(2x+1) = 1 If α and β are the... View Article

If y = ((2/π)x-1) cosec x is the solution of the differential equation, (dy/dx)+p(x)y = ((2/π)-1)cosec x, 0<x,π/2, then the function p(x) is equal to

1) cosec x 2) cot x 3) tan x 4) sec x Solution: y = ((2/π)x-1)cosec x Differentiate w.r.t.x (dy/dx) = (2/π)cosec x-((2x/π)-1)cosec x.cot... View Article

If the tangent to the curve, y = f(x) = xlog_ex, (x>0) at a point (c,f(c)) is parallel to the line-segment joining the points (1,0) and (e,e), then c is equal to

1) e(1/1-e) 2) (e-1)/e 3) 1/(e-1) 4) e(1/e-1) Solution: y = f(x) = x lnx slope of the line joining (1, 0), (e, e) m2 = (e/e-1) m2 = m1 ⇒ ln... View Article

The integral [latex]int_{1}^{2}e^{x}.x^{x}(2+log_{e}x)dx[/latex] equals

1) e(4e-1) 2) e(4e+1) 3) 4e2-1 4) e(2e-1) Solution: Put ex.xx = t Since upper limit = e2.22, Lower Limit = e (ex.xx + ex xx(1+lnx)) dx... View Article

The common difference of the A.P. b₁, b₂,….. b_m is 2 more than the common difference of A.P. a₁, a₂, …a_n.

The common difference of the A.P. b1, b2,..... bm is 2 more than the common difference of A.P. a1, a2, ...an. If a40 = -159, a100 = -399 and b100... View Article

The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5.

The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5. If P (A∪B) = 0.8, P (A∩C) = 0.3, P... View Article

The set of all real values of λ for which the function f(x) = (1-cos² x)(λ+sinx) , x∈(-π/2, π/2), has exactly one maxima and exactly one minima, is

1) (-3/2, 3/2)-{0} 2) (-1/2, ½)-{0} 3) (-3/2, 3/2) 4) (-1/2, 1/2) Solution: f(x) = (1-cos2x) (λ+sinx) f(x) = sin2x... View Article

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies

1) e4+2e2-1 = 0 2) e2+2e-1 = 0 3) e4+e2-1 = 0 4) e2+e-1 = 0 Solution: Equation of normal at (ae, b2/a) (a2x/ae)-b2y/(b2/a) = a2e2... View Article

The integral [latex]int_{0}^{2}left | left | x-1 right |-x right |dx[/latex] is equal to:

Solution: = = = = [(1/2)-0-(1/4)-0]+(1-(1/4))-(1-(1/2))+1 = (1/2)-(1/4)+(3/4)-(1/2)+1 = 3/2 Answer: (3/2)... View Article

If [latex]\lim_{x\to1}\frac{x+x^{2}+x^{3}+…x^{n}-n}{x-1} = 820[/latex] n∈N, then the value of n is equal to:

Solution: Given 1+2+3+…+n = 820 Ʃn = 820 n(n+1)/2 = 820 n = 40 Answer: (40)... View Article

If the letters of the word ’MOTHER’ be permuted and all the words so formed

If the letters of the word ’MOTHER’ be permuted and all the words so formed (with or without meaning) be listed as in a dictionary,... View Article

Let [latex]\vec{a},\vec{b}\; and\: \vec{c}[/latex] be three unit vectors such that [latex]\left | \vec{a}-\vec{b} \right |^{2}+\left | \vec{a}-\vec{c} \right |^{2} =8[/latex].

Let is equal to: Solution: . a2+b2-2a.b+a2+c2-2a.c = 8 2a2+b2+c2-2a.b -2a.c=8 a.b+a.c = -2 Now = 2a2+4b2+4c2+4a.b+4a.c =... View Article

The number of integral values of k for which the line, 3x+4y = k intersects the circle, x²+y²-2x-4y+4 = 0 at two distinct points is

Solution: c: (1,2) and r = 1 cp < r |(3×1+4×2-k)/5| < 1 |11-k| < 5 -5<k-11<5 6<k<16 k = 7,8,9,..,15 So total 9... View Article

The plane passing through the points (1,2,1), (2,1,2) and parallel to the line, 2x = 3y, z = 1 also passes through the point

a) (0,-6,2) b) (0,6,-2) c) (-2,0,1) d) (2,0,-1) Solution: Given lines are 2x = 3y, z = 1 Dr of line (3,2,0) Plane : -2(x-1)+3(y-2)+5(z -1)... View Article

Let y = y(x) be the solution of the differential equation, (2+sinx)/(y+1) (dy/dx) = -cos x, y>0, y(0) = 1. If y(π) = a, and (dy/dx) at x = π is b, then the ordered pair (a,b) is equal to

a) (2,2/3) b) (1,1) c) (2,1) d) (1,-1) Solution: ∫(dy/y+1) = ∫-cos x/(2+sin x)dx ln |y+1| = -ln |2+sin| x +k Put (0,1) k = ln 4... View Article

Let X = {x∈N: 1≤x≤17} and Y = {ax+b:x∈X and a,b∈R, a>0}. If mean and variance of elements of Y are 17 and 216 respectively then a+b is equal to:

a) -27 b) 7 c) -7 d) 9 Solution: X :{1,2,…17} Y: {ax+b:x∈X and a,b∈R, a>0} Given var(Y) = 216 (Ʃy12)/n - mean2 = 216... View Article

Let S be the set of all λ ∈ R for which the system of linear equations

Let S be the set of all λ ∈ R for which the system of linear equations 2x-y+2z = 2 x-2y+λz = -4 x+λy+z=4 has no... View Article

Let α>0, β>0, be such that α³+β² = 4. If the maximum value of the term independent of x in the binomial expansion of (αx^1/9+βx^-1/6)₁₀ is 10k, then k is equal to:

a) 176 b) 336 c) 352 d) 84 Solution: For term independent of x Tr+1 = 10Cr(αx1/9 )10-r(βx-1/6)r =... View Article

If |x| <1, |y| <1, and xy, then the sum to infinity of the following series – (x+y)+(x²+xy+y²)+(x³+x²y+xy²+y³)+… is:

a) (x+y+xy)/(1-x)(1-y) b) (x+y-xy)/(1-x)(1-y) c) (x+y+xy)/(1+x)(1+y) d) (x+y-xy)/(1+x)(1+y) Solution: (x+y)+(x2+xy+y2)+(x3+x2y+xy2+y3)+........ View Article

Area (in sq. units) of the region outside (x/2)+ (y/3) = 1 and inside the ellipse (x²/4)+(y²/9) = 1 is

a) 3(π-2) b) 6(π-2) c) 6(4-π) d) 3(4-π) Solution: Given (x/2)+ (y/3) = 1 A = 4((πab/4)-(1/2)×2×3) =... View Article

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