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?

1

Let g : N $\to$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all n $\ge$ 0. Then wh…

Topic: JEE Main 2021 (25 July Morning Shift)

2

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…

Topic: JEE Main 2021 (25 July Morning Shift)

3

Let y = y(x) be the solution of the differential equation ${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x <…

Topic: JEE Main 2021 (25 July Morning Shift)

4

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 negl…

Topic: JEE Main 2021 (25 July Morning Shift)

5

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

Topic: JEE Main 2021 (25 July Morning Shift)

6

Let f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x …

Topic: JEE Main 2021 (25 July Morning Shift)

7

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 = …

Topic: JEE Main 2021 (25 July Morning Shift)

8

Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of ${{{S_{4n}}} \ove…

Topic: JEE Main 2021 (25 July Morning Shift)

9

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

Topic: JEE Main 2021 (25 July Morning Shift)

1

The values of a and b, for which the system of equations 2x + 3y + 6z = 8, x + 2y + az = 5, 3x + 5y + 9z = b, has no so…

Topic: JEE Main 2021 (25 July Morning Shift)

2

Let 9 distinct balls be distributed among 4 boxes, B1, B2, B3 and B4. If the probability than B3 contains exactly 3 bal…

Topic: JEE Main 2021 (25 July Morning Shift)

3

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

Topic: JEE Main 2021 (25 July Morning Shift)

4

Let an ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$, passes through $\left( {\…

Topic: JEE Main 2021 (25 July Morning Shift)

5

The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 is :

Topic: JEE Main 2021 (25 July Morning Shift)

6

The total number of functions,$f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\}$ such that $f(1)+f(2)=f(3)$, is equal to :

Topic: JEE Main 2021 (25 July Morning Shift)

7

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^{4}+x^{3}+x^{2}+x+1=0$, then $\alpha^{2021}+\beta^{…

Topic: JEE Main 2021 (25 July Morning Shift)

8

For $\mathrm{n} \in \mathbf{N}$, let $\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\ri…

Topic: JEE Main 2021 (25 July Morning Shift)