JEE Main 2022 (Online) 28th June Morning Shift

If two distinct point Q, R lie on the line of intersection of the planes \(- x + 2y - z = 0\) and \(3x - 5y + 2z = 0\) and \(PQ = PR = \sqrt {18}\) where the point P is (1, \(-\)2, 3), then the area of the triangle PQR is equal to :

The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes \(5x + 8y + 13z - 29 = 0\) and \(8x - 7y + z - 20 = 0\) and the points (2, 1, 3) and (0, 1, 2), respectively, is :

Let the plane \(P:\overrightarrow r \,.\,\overrightarrow a = d\) contain the line of intersection of two planes \(\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 6\) and $$\overrightarrow r \,.\,\left( { - 6...

The number of real solutions of \({x^7} + 5{x^3} + 3x + 1 = 0\) is equal to ____________.

Let l be a line which is normal to the curve y = 2x2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to ___________.

The area of the region S = {(x, y) : y2 \(\le\) 8x, y \(\ge\) \(\sqrt2\)x, x \(\ge\) 1} is

If \(\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} }\), where ...

The number of positive integers k such that the constant term in the binomial expansion of \({\left( {2{x^3} + {3 \over {{x^k}}}} \right)^{12}}\), x \(\ne\) 0 is 28 . l, where l is an odd integer, is ______________.

If the tangents drawn at the points \(O(0,0)\) and \(P\left( {1 + \sqrt 5 ,2} \right)\) on the circle \({x^2} + {y^2} - 2x - 4y = 0\) intersect at the point Q, then the area of the triangle OPQ is equal to :

Let the lines \(y + 2x = \sqrt {11} + 7\sqrt 7\) and \(2y + x = 2\sqrt {11} + 6\sqrt 7\) be normal to a circle \(C:{(x - h)^2} + {(y - k)^2} = {r^2}\). If the line \(\sqrt {11} y - 3x = {{5\sqrt {77} } \over 3} + 11\) is tangent to the ...

The number of elements in the set {z = a + ib \(\in\) C : a, b \(\in\) Z and 1 < | z \(-\) 3 + 2i | < 4} is __________.

Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral \(\int\limits_0^1 {[ - 8{x^2} + 6x - 1]dx}\) is equal to :

Let the solution curve \(y = y(x)\) of the differential equation
\(\left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]x{{dy} \over {dx}} = x + \left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]y\)
pa...

Let y = y(x) be the solution of the differential equation \(x(1 - {x^2}){{dy} \over {dx}} + (3{x^2}y - y - 4{x^3}) = 0\), \(x > 1\), with \(y(2) = - 2\). Then y(3) is equal to :

Let a function f : N \(\to\) N be defined by
\(f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right.\)
then, f is

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let \({\pi \over 8}\) and \(\theta\) be the angles of elevation from C to P and A, respectively. If ...

Let the eccentricity of the hyperbola \(H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be \(\sqrt {{5 \over 2}}\) and length of its latus rectum be \(6\sqrt 2\). If \(y = 2x + c\) is a tangent to the hyperbola H, then the value...

Let f : R \(\to\) R be defined as
\(f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.\)
w...

Let p, q, r be three logical statements. Consider the compound statements
\({S_1}:(( \sim p) \vee q) \vee (( \sim p) \vee r)\) and
\({S_2}:p \to (q \vee r)\)
Then, which of the following is NOT true?

If the system of linear equations
\(2x + 3y - z = - 2\)
\(x + y + z = 4\)
\(x - y + |\lambda |z = 4\lambda - 4\)
where, \(\lambda\) \(\in\) R, has no solution, then

Let A be a matrix of order 3 \(\times\) 3 and det (A) = 2. Then det (det (A) adj (5 adj (A3))) is equal to _____________.

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is :

The probability, that in a randomly selected 3-digit number at least two digits are odd, is :

The number of real solutions of the equation \({e^{4x}} + 4{e^{3x}} - 58{e^{2x}} + 4{e^x} + 1 = 0\) is ___________.

Let A1, A2, A3, ....... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = \({1 \over {1296}}\) and A2 + A4 = \({7 \over {36}}\), then the value of A6 + A8 + A10 is equal to

Let A = {1, a1, a2 ....... a18, 77} be a set of integers with 1 < a1 < a2 < ....... < a18 < 77. Let the set A + A = {x + y : x, y \(\in\) A} contain exactly 39 elements. Then, the value of a1 + a2 + ...... + a18 is equal to _____________.

Let R1 and R2 be relations on the set {1, 2, ......., 50} such that
R1 = {(p, pn) : p is a prime and n \(\ge\) 0 is an integer} and
R2 = {(p, pn) : p is a prime and n = 0 or 1}.
Then, the number of elements in R1 \(-\) R2 is _______________...

The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking it was found that, in the observations, 20 was misread as 5. Then, the correct variance is equal to _____________.

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the co-ordin...

If \(\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k\), \(\overrightarrow b = 3\widehat i + 3\widehat j + \widehat k\) and \(\overrightarrow c = {c_1}\widehat i + {c_2}\widehat j + {c_3}\widehat k\) are coplanar vectors and $$...