JEE Main entrance exam Question Bank Solutions

A plane P contains the line x + 2y + 3z + 1 = 0 = x – y – z – 6, and is perpendicular to the plane –2x + y + z + 8 = 0. Then which of the following points lies on P?

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point ______.

The greatest value of c ε R for which the system of linear equations, x – cy – cz = 0, cx – y + cz = 0, cx + cy – z = 0 has a non-trivial solution, is ______.

The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1, 1, 0) is ______.

If x = `a[cosθ + logtan  θ/2]`, y = asinθ then `(dy)/(dx)` = ______.

The point in which the join of (–9, 4, 5) and (11, 0, –1) is met by the perpendicular from the origin is ______.

If the following equations

x + y – 3 = 0 

(1 + λ)x + (2 + λ)y – 8 = 0

x – (1 + λ)y + (2 + λ) = 0

are consistent then the value of λ can be ______.

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 ______.