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

A point moves in such a way that sum of squares of its distances from the co-ordinate axis is 36, then distance of then given point from origin are ______.

Let a function y = f(x) is defined by x = e^θsinθ and y = θe^sinθ, where θ is a real parameter, then value of `lim_(θ→0)`f'(x) is ______.

Consider a plane 2x + y – 3z = 5 and the point P(–1, 3, 2). A line L has the equation `(x - 2)/3 = (y - 1)/2 = (z - 3)/4`. The co-ordinates of a point Q of the line L such that `vec(PQ)` is parallel to the given plane are (α, β, γ), then the product βγ is ______.

The number of real values λ, such that the system of linear equations 2x – 3y + 5z = 9, x + 3y – z = –18 and 3x – y + (λ² – |λ|z) = 16 has no solution, is ______.

Let the system of linear equations x + y + az = 2; 3x + y + z = 4; x + 2z = 1 have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, –y*) are collinear points, then the sum of absolute values of all possible values of α is ______.

If a, b, c are non-zero real numbers and if the system of equations (a – 1)x = y + z, (b – 1)y = z + x, (c – 1)z = x + y, has a non-trivial solution, then ab + bc + ca equals ______.

If the system of linear equations x + 2ay + az = 0; x + 3by + bz = 0; x + 4cy + cz = 0 has a non-zero solution, then a, b, c ______.

`lim_(x rightarrow oo) (n^2/((n^2 + 1)(n + 1)) + n^2/((n^2 + 4)(n + 2)) + n^2/((n^2 + 9)(n + 3)) + ... + n^2/((n^2 + n^2)(n + n)))` is equal to ______.

If `lim_(n rightarrow ∞) (1^a + 2^a + ......... + n^a)/((n + 1)^(a - 1)[(na + 2) + ......(na + n)]) = 1/60` for some positive real number a, then a is equal to ______.

f(x) = `int (dx)/(sin^6 x)` is a polynomial of degree

`lim_(n rightarrow ∞) (1^4 + 2^4 + 3^4 + ...n^4)/n^5 - lim_(n rightarrow ∞) (1^3 + 2^3 + 3^3 + ...n^3)/n^5` is ______.

Let (λ, 2, 1) be a point on the plane which passes through the point (4, –2, 2). If the plane is perpendicular to the line joining the points (–2, –21, 29) and (–1, –16, 23), then `(λ/11)^2 - (4λ)/11 - 4` is equal to ______.

If the mirror image of the point (2, 4, 7) in the plane 3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to ______.

Let the tangent to the circle C₁: x² + y² = 2 at the point M(–1, 1) intersect the circle C₂: (x – 3)² + (y – 2)² = 5, at two distinct points A and B. If the tangents to C₂ at the points A and B intersect at N, then the area of the triangle ANB is equal to ______.

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line `vecr = -hatk + λ(hati + hatj + 2hatk)`, λ ∈ R. Then, which of the following points lies on T?

Let P be a plane Ix + my + nz = 0 containing the line, `(1 - x)/1 = ("y" + 4)/2 = ("z" + 2)/3`. If plane P divides the line segment AB joining points A(–3, –6, 1) and B(2, 4, –3) in ratio k:1 then the value of k is equal to ______.

Let ABCD be a square of side of unit length. Let a circle C₁ centered at A with unit radius is drawn. Another circle C₂ which touches C₁ and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C₂ meet the side AB at E. If the length of EB is `α + sqrt(3)β`, where α, β are integers, then α + β is equal to ______.

Let P be a plane passing through the points (1, 0, 1), (1, –2, 1) and (0, 1, –2). Let a vector `vec"a" = αhat"i" + βhat"j" + γhat"k"` be such that `veca` is parallel to the plane P, perpendicular to `(hat"i"+2hat"j"+3hat"k")`and `vec"a".(hat"i" + hat"j" + 2hat"j")` = 2, then (α – β + γ)² equals ______.

The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x² + y² – 2x – 4y + 4 = 0 at two distinct points is ______.

The sum of the squares of the lengths of the chords intercepted on the circle, x² + y² = 16, by the lines, x + y = n, n ∈ N, where N is the set of all natural numbers is ______.