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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 >0 sinθ +cosθ > 1 θ... 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... 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) = 0 a[(-1-3) (-1-α)+(2-3)(2-α)]... 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)→q c) (q→p) ˅~(p→q) d) (~q)˅(... 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... 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... View Article

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

a) y+4x = 2 b) 2y+x = 4 c) x+4y = 8 d) y = 4x+2 Solution: At x =... 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 = \(\begin{array}{l}\begin{bmatrix} 1 & 2 &... 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) -5 b) 10 c) 5 d) -10 Solution: Required plane is \(\begin{array}{l}\begin{vmatrix} x-3 &y-1 & z-1\\ 1& -2& 2\\... View Article

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

a) √6 b) -2√6 c) 6 d) -√6 Solution: (3+2i√54)1/2-(3-i2√54)1/2 = (9+6i2+2×3i√6)1/2-(9+6i2-2×3i√6)1/2 = ((3+√6i)2)1/2 -((3-√6i)2)1/2 = (3+√6i)( 3-√6i) = -2√6i... 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√3 b) 192√3 c) 64√3 d) 256√3 Solution: A: (a cos 300, a sin 300) lies on parabola (a2/4)... 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 = 0 λ = -{1-(1/2)sin22θ} 2(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... 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... 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) 9 b) 5 c) 4 d) 20 Solution: f(1) = 2 f(x+y) = f(x)+f(y) x = y = 1... 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... 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... View Article

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