JEE Main 2018 (Offline)

The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane,
x + y + z = 7 is :

If L1 is the line of intersection of the planes 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L2 is the line of
intersection of the planes x + 2y - z - 3 = 0, 3x - y + 2z - 1 = 0, then the distance of the origin from the
plane, containing the...

If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is :

Let \(f\left( x \right) = {x^2} + {1 \over {{x^2}}}\) and \(g\left( x \right) = x - {1 \over x}\),
\(x \in R - \left\{ { - 1,0,1} \right\}\).
If \(h\left( x \right) = {{f\left( x \right)} \over {g\left( x \right)}}\), then the local minimu...

Let g(x) = cosx2, f(x) = \(\sqrt x\) and \(\alpha ,\beta \left( {\alpha < \beta } \right)\) be the roots of the quadratic equation 18x2 - 9\(\pi\)x + \({\pi ^2}\) = 0. Then the area (in sq. units) bounded by the curve
y = (gof)(x) and th...

The sum of the co-efficients of all odd degree terms in the expansion of
\({\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5}\), \(\left( {x > 1} \right)\) is

If the tangent at (1, 7) to the curve x2 = y - 6
touches the circle x2 + y2 + 16x + 12y + c = 0, then the value of c is :

If \(\alpha ,\beta \in C\) are the distinct roots of the equation
x2 - x + 1 = 0, then \({\alpha ^{101}} + {\beta ^{107}}\) is equal to :

The value of \(\int\limits_{ - \pi /2}^{\pi /2} {{{{{\sin }^2}x} \over {1 + {2^x}}}} dx\) is

Let y = y(x) be the solution of the differential equation
\(\sin x{{dy} \over {dx}} + y\cos x = 4x\), \(x \in \left( {0,\pi } \right)\).
If \(y\left( {{\pi \over 2}} \right) = 0\), then \(y\left( {{\pi \over 6}} \right)\) is equal to :

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles
of elevation of the top of the tower at P, Q and R are respectively 45\(^\circ\), 30\(^\circ\) and 30\(^\circ\), then the height of ...

Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q.
If these tangents intersect at the
point T(0, 3) then the area (in sq. units) of \(\Delta\)PTQ is :

The integral
\(\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx\)
is equal to

For each t \(\in R\), let [t] be the greatest integer less than or equal to t.
Then $$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \r...

Let S = { t \(\in R:f(x) = \left| {x - \pi } \right|.\left( {{e^{\left| x \right|}} - 1} \right)\)\(\sin \left| x \right|\) is not differentiable at t}, then the set S is equal to

The Boolean expression
\(\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)\) is equvalent to :

If \(\left| {\matrix{ {x - 4} & {2x} & {2x} \cr {2x} & {x - 4} & {2x} \cr {2x} & {2x} & {x - 4} \cr } } \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2}\)
then the ordered pair (A, B) is equal to :

If the system of linear equations
x + ky + 3z = 0
3x + ky - 2z = 0
2x + 4y - 3z = 0
has a non-zero solution (x, y, z), then \({{xz} \over {{y^2}}}\) is equal to

Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the
parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and \(\angle\)CPB =
\(\theta\), then a v...

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and
arranged in a row on a shelf so that the dictionary is always in the middle. The number of such
arrangements is :

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and
this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at
random from the bag, ...

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre
of this triangle, then the radius of the circle having line segment AC as diameter, is :

Let S = { \(x\) \(\in\) R : \(x\) \(\ge\) 0 and
\(2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \right) + 6 = 0\)}. Then S

Let \({a_1}\), \({a_2}\), \({a_3}\), ......... ,\({a_{49}}\) be in A.P. such that
\(\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416\) and \({a_9} + {a_{43}} = 66\).
\(a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m\), then m is equal to

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
12 + 2.22 + 32 + 2.42 + 52 + 2.62 ...........
If B - 2A = 100\(\lambda\), then \(\lambda\) is equal to

Two sets A and B are as under :
A = {(\(a\), b) \(\in\) R \(\times\) R : |\(a\) - 5| < 1 and |b - 5| < 1};
B = {(\(a\), b) \(\in\) R \(\times\) R : 4(\(a\) - 6)2 + 9(b - 5)2 \(\le\) 36 };
Then

If \(\sum\limits_{i = 1}^9 {\left( {{x_i} - 5} \right)} = 9\) and
\(\sum\limits_{i = 1}^9 {{{\left( {{x_i} - 5} \right)}^2}} = 45\), then the standard deviation of the 9 items
\({x_1},{x_2},.......,{x_9}\) is

A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is
the origin and the rectangle OPRQ is completed, then the locus of R is :

If sum of all the solutions of the equation
\(8\cos x.\left( {\cos \left( {{\pi \over 6} + x} \right).\cos \left( {{\pi \over 6} - x} \right) - {1 \over 2}} \right) = 1\)
in [0, \(\pi\)] is k\(\pi\), then k is equal to

Let \(\overrightarrow u\) be a vector coplanar with the vectors \(\overrightarrow a = 2\widehat i + 3\widehat j - \widehat k\) and \(\overrightarrow b = \widehat j + \widehat k\). If \(\overrightarrow u\) is perpendicular to $$\overrigh...