JEE Main entrance exam Question Bank Solutions for Mathematics (JEE Main)

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 ______.

If m be the slope of a tangent to the curve e^2y = 1 + 4x², then ______.

If β is one of the angles between the normals to the ellipse, x² + 3y² = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.

If the tangent to the conic, y – 6 = x² at (2, 10) touches the circle, x² + y² + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.

Let PQ be a focal chord of the parabola y² = 4x such that it subtends an angle of `π/2` at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse E: `x^2/a^2 + y^2/b^2` = 1, a² > b². If e is the eccentricity of the ellipse E, then the value of `1/e^2` is equal to ______.

if `lim_(x→0) (ae^x - bcosx + ce^-x)/(xsinx)` = 2, then a + b + c is equal to ______.

On the ellipse `x^2/8 + "y"^2/4` = 1 let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S' be the foci of the ellipse and e be its eccentricity. If A is the area of the triangle SPS' then, the value of (5 – e²). A is ______.

If the tangents on the ellipse 4x² + y² = 8 at the points (1, 2) and (a, b) are perpendicular to each other, then a² is equal to ______.