JEE Main 2021 (Online) 26th February Morning Shift

Consider the three planesP1 : 3x + 15y + 21z = 9,P2 : x \(-\) 3y \(-\) z = 5, and P3 : 2x + 10y + 14z = 5Then, which one of the following is true?

If (1, 5, 35), (7, 5, 5), (1, \(\lambda\), 7) and (2\(\lambda\), 1, 2) are coplanar, then the sum of all possible values of \(\lambda\) is :

Let (\(\lambda\), 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 $${\left( {{\lambda \over {11...

Let f be any function defined on R and let it satisfy the condition : \(|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R\)If f(0) = 1, then :

The maximum slope of the curve \(y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x\) occurs at the point :

The area bounded by the lines y = || x \(-\) 1 | \(-\) 2 | is ___________.

The maximum value of the term independent of 't' in the expansion of \({\left( {t{x^{{1 \over 5}}} + {{{{(1 - x)}^{{1 \over {10}}}}} \over t}} \right)^{10}}\) where x\(\in\)(0, 1) is :

Let m, n\(\in\)N and gcd (2, n) = 1. If $$30\left( {\matrix{
{30} \cr
0 \cr

} } \right) + 29\left( {\matrix{
{30} \cr
1 \cr

} } \right) + ...... + 2\left( {\matrix{
{30} \cr
{28} \cr

} } \right) + 1\left(...

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ \(\bot\) OB. Then, the area of the triangle PQB (in square units) is :

The value of the integral \(\int\limits_0^\pi {|{{\sin }\,}2x|dx}\) is ___________.

The value of \(\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx\) is :

The value of \(\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} }\), where [ x ] is the greatest integer \(\le\) x, is :

The difference between degree and order of a differential equation that represents the family of curves given by \({y^2} = a\left( {x + {{\sqrt a } \over 2}} \right)\), a > 0 is _________.

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000...

If y = y(x) is the solution of the equation \({e^{\sin y}}\cos y{{dy} \over {dx}} + {e^{\sin y}}\cos x = \cos x\), y(0) = 0; then $$1 + y\left( {{\pi \over 6}} \right) + {{\sqrt 3 } \over 2}y\left( {{\pi \over 3}} \right) + {1 \over {\sqr...

If \({{{{\sin }^1}x} \over a} = {{{{\cos }^{ - 1}}x} \over b} = {{{{\tan }^{ - 1}}y} \over c}\); \(0 < x < 1\), then the value of \(\cos \left( {{{\pi c} \over {a + b}}} \right)\) is :

The value of \(\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right)} \over {\sqrt 3 h\left( {\sqrt 3 \cosh - \sinh } \right)}}} \right\}\) is :

The number of solutions of the equation log4(x \(-\) 1) = log2(x \(-\) 3) is _________.

The value of \(\left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & 1 \cr {(a + 3)(a + 4)} & {a + 4} & 1 \cr } } \right|\) is :

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is :

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is :

The sum of 162th power of the roots of the equation x3 \(-\) 2x2 + 2x \(-\) 1 = 0 is ________.

In an increasing geometric series, the sum of the second and the sixth term is \({{25} \over 2}\) and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to :

The sum of the infinite series \(1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{22} \over {{3^5}}} + ......\) is equal to :

Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, \(-\)1) is the set :

The intersection of three lines x \(-\) y = 0, x + 2y = 3 and 2x + y = 6 is a :

If \(\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1\), the number of solutions of the given equation when \(x \in \left[ {0,{\pi \over 2}} \right]\) is __________.

The number of integral values of 'k' for which the equation \(3\sin x + 4\cos x = k + 1\) has a solution, k\(\in\)R is ___________.

If \(\overrightarrow a\) and \(\overrightarrow b\) are perpendicular, then $$\overrightarrow a \times \left( {\overrightarrow a \times \left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \rig...