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The value of the limit

`underset{x->0}{lim} (e^x-e^(-x)-2x)/(x-sin x)` is

If `f(x)=x^5-5x^4+5x^3-10` has local maximum and minimum at x=p and x=q, respectively, then (p,q) equals

Two digits are selected randomly from the set `{1, 2, 3, 4, 5, 6, 7, 8}` without replacement one by one. The probability that minimum of the two digits is less than 5 is-

If `veca=hati+hatj+hatk , vecb = hati-hatj+hatk , vecc =hati+2hatj-hatk` ,then the value of

`|[(veca.veca)(veca.vecb)(veca.vecc)], [(vecb.veca)(vecb.vecb)(vecb.vecc)], [(vecc.veca)(vecc.vecb)(vecc.vecc)]|` is :

The focal chord of the parabola `(y-2)^2=16(x-1)` is a tangent to the circle `x^2+y^2-14x-4y+51=0`, then slope of the focal chord can be

If the variable line `y=kx+2h` is tangent to an ellipse `2x^2+3y^2=6`, then locus of P(h, k) is a conic C whose eccentricity equals

Let `x_1, x_2, x_3, x_4, .............x_n` be n observation and let `barx` be their arithmetic mean and `sigma^2` be their variance.

Statement-1: Variance of observations `2x_1, 2x_2, 2x_3, ........., 2x_n` is `4sigma^2`

Statement-2: Arithmetic mean of `2x_1, 2x_2, 2x_3,.........., 2x_n` is `4barx`

A number x is chosen at random from the set `{1, 2, 3, 4, .........100}`. Then probability of the event that the chosen number x satisfy the inequality `((x-10)(x-50))/(x-30) >=0` is –

The minimum area of a triangle formed by any tangent to the ellipse `x^2/16+y^2/81=1` and the coordinate axes is

The value of definite integral `int_0^(1/2) (ln(1+2x))/(1+4x^2) dx` equals

Given three points P, Q, R with P(5,3) and R lies on the x-axis. If equation of RQ is x-2y=2 and PQ is parallel to the x-axis, then the centroid of `triangle PQR` lies on the line

If `x=(1+t)/t^3, y=3/(2t^2)+2/t`, then `x(dy/dx)^3-dy/dx` is equal to (where t is a real parameter)

The sum of all the solution(s) of the equation `sin^-1 2x=cos^-1x` is-

If tangent at point (1, 2) on curve `y=ax^2 +bx+7/2` be parallel to normal at (-2, 2) on the curve `y=x^2+6x+10`, then

If line `vecr=(hati-2hatj-hatk)+lambda(2hati+hatj+2hatk)` is parallel to the plane `vecr.(3hati-2hatj-mhatk)=14`, then the value of m is

`text(If) f(x) = {(e^x+a,text(for) x<0),(x-3,text(for) x>=0):}`

is differentiable at x=0, then 'a' equal to

A function y=f(x) satisfies the differential equation `f(x). sin2x-cos x+(1+sin^2 x) f(x)=0` where f(0)=0. Then value of `f(pi/6)` is equal to

The complex numbers `sin x+i cos 2x` and `cos x-i sin 2x` are conjugate to each other, for

The circumference of the circle `x^2+y^2-2x+8y-q=0` is bisected by the circle `x^2+y^2+4x+12y+p=0`, then p+q is equal to

Which of the following pairs are not logically equivalent?

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