JEE Main 2021 (Online) 25th July Morning Shift

Let the foot of perpendicular from a point P(1, 2, \(-\)1) to the straight line \(L:{x \over 1} = {y \over 0} = {z \over { - 1}}\) be N. Let a line be drawn from P parallel to the plane x + y + 2z = 0 which meets L at point Q. If \(\alpha\)...

Let \(f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3\), \(x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]\). Then, f is :

The area (in sq. units) of the region, given by the set \(\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\}\) is :

If b is very small as compared to the value of a, so that the cube and other higher powers of \({b \over a}\) can be neglected in the identity $${1 \over {a - b}} + {1 \over {a - 2b}} + {1 \over {a - 3b}} + ..... + {1 \over {a - nb}} = \alp...

The ratio of the coefficient of the middle term in the expansion of (1 + x)20 and the sum of the coefficients of two middle terms in expansion of (1 + x)19 is _____________.

The term independent of 'x' in the expansion of \({\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}\), where x \(\ne\) 0, 1 is equal to ______________.

Let $$S = \left\{ {n \in N\left| {{{\left( {\matrix{
0 & i \cr
1 & 0 \cr

} } \right)}^n}\left( {\matrix{
a & b \cr
c & d \cr

} } \right) = \left( {\matrix{
a & b \cr
c & d \cr

} } \right)\forall a,b,c,d \...

The value of the definite integral \(\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}}\) is :

Let \(f:[0,\infty ) \to [0,\infty )\) be defined as \(f(x) = \int_0^x {[y]dy}\)where [x] is the greatest integer less than or equal to x. Which of the following is true?

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x < \sqrt 2 ,y(0) = 0\)then, the minimum value of \(y(x),x \in \left( { - \sqrt 2 ,\sqrt 2 } \right)\) is equal to :

Let y = y(x) be solution of the following differential equation \({e^y}{{dy} \over {dx}} - 2{e^y}\sin x + \sin x{\cos ^2}x = 0,y\left( {{\pi \over 2}} \right) = 0\) If \(y(0) = {\log _e}(\alpha + \beta {e^{ - 2}})\), then $$4(\alpha + \b...

Let an ellipse \(E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \({a^2} > {b^2}\), passes through \(\left( {\sqrt {{3 \over 2}} ,1} \right)\) and has eccentricity \({1 \over {\sqrt 3 }}\). If a circle, centered at focus F($$\alp...

Let g : N \(\to\) N be defined asg(3n + 1) = 3n + 2,g(3n + 2) = 3n + 3,g(3n + 3) = 3n + 1, for all n \(\ge\) 0. Then which of the following statements is true?

A spherical gas balloon of radius 16 meter subtends an angle 60\(^\circ\) at the eye of the observer A while the angle of elevation of its center from the eye of A is 75\(^\circ\). Then the height (in meter) of the top most point of the bal...

The locus of the centroid of the triangle formed by any point P on the hyperbola \(16{x^2} - 9{y^2} + 32x + 36y - 164 = 0\), and its foci is :

Let f : R \(\to\) R be defined as$$f(x) = \left\{ {\matrix{
{{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x < 2} \cr
{{e^{{{\tan (x - 2)} \over {x - [x]}}}},} & {x > 2} \cr
{\mu ,} & {x = 2} \c...

The Boolean expression \((p \Rightarrow q) \wedge (q \Rightarrow \sim p)\) is equivalent to :

The values of a and b, for which the system of equations 2x + 3y + 6z = 8x + 2y + az = 53x + 5y + 9z = bhas no solution, are :

Let \(M = \left\{ {A = \left( {\matrix{ a & b \cr c & d \cr } } \right):a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} } \right\}\). Define f : M \(\to\) Z, as f(A) = det(A), for all A\(\in\)M, where z is set of all integers. Then the ...

Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. uni...

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the ...

Let 9 distinct balls be distributed among 4 boxes, B1, B2, B3 and B4. If the probability than B3 contains exactly 3 balls is \(k{\left( {{3 \over 4}} \right)^9}\) then k lies in the set :

The number of real roots of the equation \({e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0\) is :

If \(\alpha\), \(\beta\) are roots of the equation \({x^2} + 5(\sqrt 2 )x + 10 = 0\), \(\alpha\) > \(\beta\) and \({P_n} = {\alpha ^n} - {\beta ^n}\) for each positive integer n, then the value of $$\left( {{{{P_{17}}{P_{20}} + 5\sqrt 2 {P_...

Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of \({{{S_{4n}}} \over {{S_{2n}}}}\) is :

If the value of \({\left( {1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + ....upto\,\infty } \right)^{{{\log }_{(0.25)}}\left( {{1 \over 3} + {1 \over {{3^2}}} + {1 \over {{3^3}}} + ....upto\,\infty } \right)}}\) is \(l\), the...

Consider the following frequency distribution :
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The sum of all values of x in [0, 2\(\pi\)], for which sin x + sin 2x + sin 3x + sin 4x = 0, is equal to :

Let the vectors\((2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k,(1 + b)\widehat i + 2b\widehat j - b\widehat k\) and \((2 + b)\widehat i + 2b\widehat j + (1 - b)\widehat k\), \(a,b,c, \in R\) be co-planar. Then which of ...

Let \(\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k\) and \(\overrightarrow q = \widehat i + 2\widehat j + \widehat k\) be two vectors. If a vector $$\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \wideha...