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The set of all possible values of θ in the interval (0, π) for which the points (1, 2) and (sin θ, cos θ ) lie on the same side of the line x+y = 1 is

a) (0, π/4)b) (0, π/2)c) (0, 3π/4)d) (π/4, 3π/4)Solution:L11×L22 >0sinθ +cosθ > 1θ ∈(0, π/2)Answer: (b)... View Article

Let S be the sum of the first 9 terms of the series

Let S be the sum of the first 9 terms of the series:{x+ka}+{x2+(k+2)a}+{x3+(k+4)a}+{x4+(k+6)a}+... where a ≠ 0 and a ≠ 1. If S =... View Article

Let f(x) be a quadratic polynomial such that f(-1)+f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other roots lies in

a) (0,1)b) (1,3)c) (-1,0)d) (-3,-1)Solution:Let f(x) = a (x-3) (x-α)f(-1)+f(2) = 0a[(-1-3) (-1-α)+(2-3)(2-α)] = 0a[4+4α-2+α]=05α+2 = 0α =... View Article

Which of the following is a tautology from given options

Which of the following is a tautology?a) (p→q) ˄( q→p)b) (~p) ˄(p ˅q)→qc) (q→p) ˅~(p→q)d) (~q)˅( p˄q)→qSolution:pq~p(p ˅q)~p ˄(p˅q)~p... View Article

Let f: (-1, ∞)→R be defined by f(0) = 1 and f(x) = (1/x)log_e(1+x), x ≠ 0. Then the function f

a) increases in (-1, ∞)b) decreases in (-1,0) and increases in (0, ∞)c) increases in (-1,0) and decreases in (0, ∞)d) decreases... View Article

Consider a region R = {(x,y)∈R² : x² ≤y ≤ 2x}

Consider a region R = {(x,y)∈R2 : x2 ≤y ≤ 2x}. If a line y = α divides the area of region R into two equal parts, then which of the... View Article

The equation of the normal to the curve y = (1+x)^2y +cos²(sin ^-1x) at x = 0 is

a) y+4x = 2b) 2y+x = 4c) x+4y = 8d) y = 4x+2Solution:At x = 0, y = 1+cos2(0) = 2P;(0,2)y = (1+x)2y+cos2(sin-1x)y =... View Article

Let A = {X= (x,y,z)^T: PX = 0 and

Let A = {X= (x,y,z)T: PX = 0 and x2+y2+z2 = 1}, where P = then the set A:a) contains more than two elements.b) is a singleton.c) contains... View Article

A plane passing through the point (3,1,1) contains two lines whose direction ratios are (1, -2, 2) and (2, 3, -1) respectively. If this plane also passes through the point (α,-3,5), then α is equal to

a) -5b) 10c) 5d) -10Solution:Required plane is -4(x-3)+5(y-1)+7(z-1) = 04x-5y-7z = 0(α, -3,5) lies on 4x-5y-7z = 0α = 5Answer: (c)... View Article

The imaginary part of (3+2√-54)^1/2-(3-2√-54)^1/2 can be

a) √6b) -2√6c) 6d) -√6Solution:(3+2i√54)1/2-(3-i2√54)1/2=... View Article

The area (in sq. units) of an equilateral triangle inscribed in the parabola y² = 8x, with one of its vertices on the vertex of this parabola, is

a) 128√3b) 192√3c) 64√3d) 256√3Solution:A: (a cos 300, a sin 300) lies on parabola(a2/4) = 8×a√3/2a =... View Article

If the equation cos⁴θ+sin⁴θ+λ = 0 has real solutions for θ, then λ lies in the interval

a) (-1/2, -1/4]b) [-1,-1/2]c) [-3/2, -5/4]d) (-5/4, -1)Solution: cos4θ+sin4θ+λ = 0λ = -{1-(1/2)sin22θ}2(λ+1) = sin22θ0 ≤2(λ+1) ≤10 ≤λ+1... View Article

Let E^C denote the complement of an event E.

Let EC denote the complement of an event E. Let E1, E2 and E3 be any pair wise independent events with P(E1)>0 and P(E1 ∩ E2 ∩ E3) = 0. Then... View Article

If the sum of first 11 terms of an A.P

If the sum of first 11 terms of an A.P, a1,a2,a3… is 0 (a1 ≠ 0) then the sum of the A.P, a1, a3 , a5…a23 is ka1, where k is... View Article

Let f:R -> R be a function which satisfies f(x+y) = f(x)+f(y) for all x, y ∈ R. If f(1) = 2 and g(n) = [latex]sum_{k=1}^{n-1}f(k)[/latex], n ∈ N then the value of n for which g(n) = 20 is

a) 9b) 5c) 4d) 20Solution:f(1) = 2f(x+y) = f(x)+f(y)x = y = 1f(2) = 2+2 = 4x = 2, y = 1f(3) = 4+2 = 6g(n) = f(1)+f(2)+.........+f(n-1)= 2+4+6+... View Article

If the tangent to the curve, y = e^x at a point (c, e^c) and the normal to the parabola,y² = 4x at the point (1,2) intersect at the same point on the x-axis, then the value of c is

Solution:Tangent at (c, ec)y-ec = ec(x-c) ....(i)normal to parabola y-2 = -1(x-1)x+y = 3 ...(ii)At x-axis y = 0in (i), x = c-1in (ii), x = 3c-1 =... View Article

For a positive integer n, (1+1/x)ⁿ is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2:5:12, then n is equal to

Solution: Let 3 consecutive coefficients are nCr-1 : nCr : nCr+1::2:5:12 nCr-1/ nCr = 2/5 and nCr/ nCr+1 = 5/12 r/(n-r+1) = 2/5 and... View Article

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