JEE Main entrance exam Question Bank Solutions

Let f(x) = (x – a)ⁿg(x) , where g⁽ⁿ⁾(a) ≠ 0; n = 0, 1, 2, 3.... then ______.

The set of values of p for which the points of extremum of the function f(x) = x³ – 3px² + 3(p² – 1)x + 1 lie in the interval (–2, 4), is ______.

A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.

The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.

The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.

Let f(x) = |(x – 1)(x² – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.

The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by

f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`

The minimum value of 2^sinx + 2^cosx is ______.

The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.

Let the line L be the projection of the line: `(x - 1)/2 = ("y" - 3)/1 = ("z" - 4)/2` in the plane x – 2y – z = 3. If d is the distance of the point (0, 0, 6) from L, then d² is equal to ______.

If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.

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 solution. Then the set S ______.

An edge of variable cube is increasing at the rate of 3 cm/s. The volume of the cube increasing fast when the edge is 10 cm long is ______ cm³/s.

If (a, b), (c, d) are points on the curve 9y² = x³ where the normal makes equal intercepts on the axes, then the value of a + b + c + d is ______.

If the curves y² = 6x, 9x² + by² = 16, cut each other at right angles then the value of b is ______.

The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.

Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane is ______.

The curve `(x/a)^n + (y/b)^n` = 2, touches the line `x/a + y/b` = 2 at the point (a, b) for n is equal to ______.

The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.

The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.