JEE Main 2019 (Online) 11th January Morning Slot

The plane containing the line \({{x - 3} \over 2} = {{y + 2} \over { - 1}} = {{z - 1} \over 3}\) and also containing its projection on the plane 2x + 3y \(-\) z = 5, contains which one of the following points ?

The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle \({\pi \over 4}\) with the plane y \(-\) z + 5 = 0 are :

The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = \(\left\{ {x\, \in R:{x^2} + 30 \le 11x} \right\}\) is :

The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is :

The sum of the real values of x for which the middle term in the binomial expansion of \({\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}\) equals 5670 is :

The value of r for which 20Cr 20C0 + 20Cr\(-\)1 20C1 + 20Cr\(-\)2 20C2 + . . . . .+ 20C0 20Cr  is maximum, is

The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :

A square is inscribed in the circle x2 + y2
– 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes.
Then the distance of the vertex of this square which is nearest to the origin is :

Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :

Let \({\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}\left( {i = \sqrt { - 1} } \right),\,\,\) where x and y are real numbers, then y \(-\) x equals :

The value of the integral \(\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx\) (where [x] denotes the greatest integer less than or equal to x) is

If y(x) is the solution of the differential equation \({{dy} \over {dx}} + \left( {{{2x + 1} \over x}} \right)y = {e^{ - 2x}},\,\,x > 0,\,\) where \(y\left( 1 \right) = {1 \over 2}{e^{ - 2}},\) then

If  xloge(logex) \(-\) x2 + y2 = 4(y > 0), then \({{dy} \over {dx}}\) at x = e is equal to :

If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

Let f : R \(\to\) R be defined by f(x) = \({x \over {1 + {x^2}}},x \in R\).   Then the range of f is :

Let fk(x) = \({1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\) for k = 1, 2, 3, ... Then for all x \(\in\) R, the value of f4(x) \(-\) f6(x) is equal to

Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is :

If  \(\int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}}\) dx = A(x)\({\left( {\sqrt {1 - {x^2}} } \right)^m}\) + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))m equals :

Let [x] denote the greatest integer less than or equal to x. Then $$\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( {\left| x \right| - \sin \left( {x\left[ x \right]} \right)} \right)}^2}} \over {{x^2}}...

Let \(f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.\) and
\(g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).\)
Then, in the ...

If q is false and p \(\wedge\) q \(\leftrightarrow\) r is true, then which one of the following statements is a tautology ?

If the system of linear equations
2x + 2y + 3z = a
3x – y + 5z = b
x – 3y + 2z = c
where a, b, c are non zero real numbers, has more one solution, then :

Let A = \(\left( {\matrix{ 0 & {2q} & r \cr p & q & { - r} \cr p & { - q} & r \cr } } \right).\)   If  AAT = I3,   then   \(\left| p \right|\) is :

Two integers are selected at random from the set {1, 2, ...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x2 – c2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is

Let a1, a2, . . . . . ., a10 be a G.P.    If \({{{a_3}} \over {{a_1}}} = 25,\) then \({{{a_9}} \over {{a_5}}}\) equals

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is \({{27} \over {19}}\).Then the common ratio of this series is :

The outcome of each of 30 items was observed; 10 items gave an outcome \({1 \over 2}\) – d each, 10 items gave outcome \({1 \over 2}\) each and the remaining 10 items gave outcome \({1 \over 2}\)+ d each. If the variance of this outcome dat...

Let  \(\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,\) \(\overrightarrow b = \widehat i + \lambda \widehat j + 4\widehat k\) and $$\overrightarrow c = 2\widehat i + 4\widehat j + \left( {{\lambda ^2} - 1} \right)\widehat k$...