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

`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 ______.

If `lim_(x→0) (int_0^x((cos2t - 1)(cost - e^(-t^2))t^-n)dr)/(cosx - 1)` is a finite non-zero number, Then the integer value for n is ______.

The normal to the ellipse `x^2/a^2 + y^2/b^2` = 1 at a point P(x₁, y₁) on it, meets the x-axis in G. PN is perpendicular to OX, where O is origin. Then value of ℓ(OG)/ℓ(ON) is ______.

The point on the ellipse x² + 2y² = 6 closest to the line x + y = 7 is (a, b). The value of (a + b) will be ______.

The ratio of the area of the ellipse and the area enclosed by the locus of mid-point of PS where P is any point on the ellipse and S is the focus of the ellipse, is equal to ______.

Let the eccentricity of an ellipse `x^2/a^2 + y^2/b^2` = 1, a > b, be `1/4`. If this ellipse passes through the point ```(-4sqrt(2/5), 3)`, then a² + b² is equal to ______.

If P₁ and P₂ are two points on the ellipse `x^2/4 + y^2` = 1 at which the tangents are parallel to the chord joining the points (0, 1) and (2, 0), then the distance between P₁ and P₂ is ______.

If `lim_(x rightarrow 0) (ax - (e^(4x) - 1))/(ax(e^(4x) - 1))` exists and is equal to b, then the value of a – 2b is  ______.