JEE Main entrance exam Question Bank Solutions

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 ______.

`lim_(x→0) (sin^2x)/(sqrt(2) - sqrt(1 + cos))` equals ______.

The tangent and the normal at a point P on an ellipse `x^2/a^2 + y^2/b^2` = 1 meet its major axis in T and T' so that TT' = a then e²cos²θ + cosθ (where e is the eccentricity of the ellipse) is equal to ______.

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms, are ______.

The eccentricity, foci and the length of the latus rectum of the ellipse x² + 4y² + 8y – 2x + 1 = 0 are respectively equal to ______.

Tangents are drawn from a point on the circle x² + y² = 25 to the ellipse 9x² + 16y² – 144 = 0 then find the angle between the tangents.

The value of `lim_(x→0) (sin(ℓn e^x))^2/((e^(tan^2x) - 1))` is ______.

The equation of the ellipse with its centre at (1, 2), one focus at (6, 2) and passing through the point (4, 6) is ______.

If m = `lim_(x→∞) (ℓ na)^3/(1 + 2lnx + 3(lnx)^2 + 4(lnx)^3` then the value of 8 m is ______.

If f'(1) = 3 and f'(2) = 2, then the value of `lim_(h→0) (f(1 + 2h) - f(1))/(f(2 + 3h) - f(2))` is equal to ______.

If the chord through the points whose eccentric angles are α and β on the ellipse `x^2/a^2 + y^2/b^2` = 1 passes through the focus (ae, 0), then the value of tan `α/2 tan  β/2` will be ______.

Let the ellipse `x^2/a^2 + y^2/b^2` = 1 has latus sectum equal 8 units – if the ellipse passes through   `(sqrt(5), 4)` Then The radius of the directive circle is ______.

The value of `lim_(x→0){(sinx - x + x^3/6)/x^5}` is `1/k`, then k is ______.

The points where the normals to the ellipse x² + 3y² = 37 are parallel to the line 6x – 5y = 2 are ______.