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is differentiable at every point of the domain, then the values of \( a \) and \( b \) are respectively:

The equation of one of the straight lines which passes through the point \( (1, 3) \) and makes an angle \( \tan^{-1}(\sqrt{2}) \) with the straight line, \( y + 1 = 3\sqrt{2}\,x \) is:

. If \( \operatorname{Tr}(A) \) denotes the sum of all diagonal elements of the matrix \( A \), then \( \operatorname{Tr}(A) - \operatorname{Tr}(B) \) has value equal to:

The differential equation satisfied by the system of parabolas \( y^2 = 4a(x + a) \) is:

A vector \( \vec{a} \) has components \( 3p \) and \( 1 \) with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, \( \vec{a} \) has components \( p + 1 \) and \( \sqrt{10} \), then a value of \( p \) is equal to:

For the four circles \( M \), \( N \), \( O \) and \( P \), following four equations are given:
Circle \( M \): \( x^2 + y^2 = 1 \)
Circle \( N \): \( x^2 + y^2 - 2x = 0 \)
Circle \( O \): \( x^2 + y^2 - 2x - 2y + 1 = 0 \)
Circle \( P \): \( x^2 + y^2 - 2y = 0 \)
If the centre of circle \( M \) is joined with centre of the circle \( N \), further centre of circle \( N \) is joined with centre of the circle \( O \), centre of circle \( O \) is joined with the centre of circle \( P \) and lastly, centre of circle \( P \) is joined with centre of circle \( M \), then these lines form the sides of a:

The solutions of the equation

The number of integral values of \( m \) so that the abscissa of point of intersection of lines \( 3x + 4y = 9 \) and \( y = mx + 1 \) is also an integer, is:

The integral \( \int \frac{(2x-1)\cos\!\left(\sqrt{(2x-1)^2+5}\right)}{\sqrt{4x^2-4x+6}}\,dx \) is equal to:
(where \( c \) is a constant of integration)

If \( \displaystyle \lim_{x \to 0} \frac{\sin^{-1}x - \tan^{-1}x}{3x^3} \) is equal to \( L \), then the value of \( (6L + 1) \) is:

If the functions are defined as \( f(x)=\sqrt{x} \) and \( g(x)=\sqrt{1-x} \), then what is the common domain of the following functions: \( f+g,\, f-g,\, \frac{f}{g},\, \frac{g}{f} , g-f\) where \( (f \pm g)(x)=f(x)\pm g(x) \), \( (f/g)(x)=\frac{f(x)}{g(x)} \)

The value of \( 3+\cfrac{1}{\,4+\cfrac{1}{\,3+\cfrac{1}{\,4+\cfrac{1}{\,3+\cdots\infty}}}} \) is equal to:

The sum of all the \( 4 \)-digit distinct numbers that can be formed with the digits \( 1 \), \( 2 \), \( 2 \) and \( 3 \) is:

Choose the correct statement about two circles whose equations are given below:
\( x^2 + y^2 - 10x - 10y + 41 = 0 \)
\( x^2 + y^2 - 22x - 10y + 137 = 0 \)

If \( \alpha, \beta \) are natural numbers such that \( 100^\alpha - 199^\beta = (100)(100) + (99)(101) + (98)(102) + \ldots + (1)(199) \), then the slope of the line passing through \( (\alpha, \beta) \) and origin is:

If the equation $a|z|^2 + \overline{\bar{\alpha}z + \alpha\bar{z}} + d = 0$ represents a circle where $a, d$ are real constants then which of the following condition is correct?

Let \( (1 + x + 2x^2)^{20} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{40}x^{40} \). Then, \( a_1 + a_3 + a_5 + \ldots + a_{37} \) is equal to:

Let \( \alpha, \beta, \gamma \) be the real roots of the equation, \( x^3 + ax^2 + bx + c = 0 \), \( (a,b,c \in \mathbb{R} \) and \( a,b \ne 0) \). If the system of equations (in \( u, v, w \)) given by \( \alpha u + \beta v + \gamma w = 0 \); \( \beta u + \gamma v + \alpha w = 0 \); \( \gamma u + \alpha v + \beta w = 0 \) has non-trivial solution, then value of \( \dfrac{\alpha^2}{b} \) is:

The real values function \( f(x) = \dfrac{\text{cosec}^{-1} x}{\sqrt{x - [x]}} \), where \( [x] \) denotes the greatest integer less than or equal to \( x \), is defined for all \( x \) belonging to:

\( \dfrac{1}{3^2 - 1} + \dfrac{1}{5^2 - 1} + \dfrac{1}{7^2 - 1} + \ldots + \dfrac{1}{(201)^2 - 1} \) is equal to:

Let \( z_1, z_2 \) be the roots of the equation \( z^2 + a z + 12 = 0 \) and \( z_1, z_2 \) form an equilateral triangle with origin. Then, the value of \( |a| \) is ____________.

The number of times the digit \( 3 \) will be written when listing the integers from \( 1 \) to \( 1000 \) is ____________.

Let the plane \( a x + b y + c z + d = 0 \) bisect the line joining the points \( (4, -3, 1) \) and \( (2, 3, -5) \) at the right angles. If \( a, b, c, d \) are integers, then the minimum value of \( (a^2 + b^2 + c^2 + d^2) \) is ____________.

Let \( f(x) \) and \( g(x) \) be two functions satisfying \( f(x^2) + g(4 - x) = 4x^3 \) and \( g(4 - x) + g(x) = 0 \), then the value of \( \displaystyle \int_{-4}^{4} f(x^2) \, dx \) is ____________.

If \( f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx , \; (x \geq 0), \; f(0) = 0 \) and \( f(1) = \frac{1}{k} \), then the value of \( K \) is ____________.

A square \( ABCD \) has all its vertices on the curve \( x^2 y^2 = 1 \). The midpoints of its sides also lie on the same curve. Then, the square of area of \( ABCD \) is ______________ .

The mean age of \( 25 \) teachers in a school is \( 40 \) years. A teacher retires at the age of \( 60 \) years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is \( 39 \) years, then the age (in years) of the newly appointed teacher is ____________ .

The equation of the planes parallel to the plane \( x - 2y + 2z - 3 = 0 \) which are at unit distance from the point \( (1, 2, 3) \) is \( ax + by + cz + d = 0 \). If \( (b - d) = K(c - a) \), then the positive value of \( K \) is ______________ .

The number of solutions of the equation \( |\cot x| = \cot x + \frac{1}{\sin x} \) in the interval \( [0, 2\pi] \) is ______________ .

The missing value in the following figure is ______________ .

JEE Main 2021 18th March Morning Shift

Physics Chemistry Mathematics

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The number of integral values of \( m \) so that the abscissa of point of intersection of lines \( 3x + 4y = 9 \) and \( y = mx + 1 \) is also an integer, is: