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Let \( g(t)=\displaystyle\int\limits_{-\pi/2}^{\pi/2} \cos\!\left(\frac{\pi}{4}\, t+ f(x)\right)\,dx \), where \( f(x)=\log_{e}\!\left(x+\sqrt{x^{2}+1}\right),\; x\in \mathbb{R} \). Then which one of the following is correct?

Let \( P \) be a variable point on the parabola \( y=4x^{2}+1 \). Then, the locus of the mid-point of the point \( P \) and the foot of the perpendicular drawn from the point \( P \) to the line \( y=x \) is:

The value of \( k\in \mathbb{R} \), for which the following system of linear equations
\( 3x-y+4z=3, \)
\( x+2y-3z=-2, \)
\( 6x+5y+kz=-3, \)
has infinitely many solutions, is:

Consider the following three statements:
(A) If \( 3+3=7 \) then \( 4+3=8 \).
(B) If \( 5+3=8 \) then earth is flat.
(C) If both (A) and (B) are true then \( 5+6=17 \).
Then, which of the following statements is correct?

Let \( A, B \) and \( C \) be three events such that the probability that exactly one of \( A \) and \( B \) occurs is \( (1-k )\), the probability that exactly one of \( B \) and \( C \) occurs is \( 1-2k \), the probability that exactly one of \( C \) and \( A \) occurs is \( (1-k) \) and the probability of all \( A, B \) and \( C \) occur simultaneously is \( k^{2} \), where \( 0<k<1\).Then the probability that at least one of \(A\),\(B\) and \(C\) occur is :

For the natural numbers \( m,n \), if \( (1-y)^{m}(1+y)^{n}=1+a_{1}y+a_{2}y^{2}+\ldots +a_{m+n}y^{\,m+n} \) and \( a_{1}=a_{2}=10 \), then the value of \( m+n \) is equal to:

Let in a right angled triangle, the smallest angle be \( \theta \). If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then \( \sin \theta \) is equal to:

In a triangle \( ABC \), if \( |\overrightarrow{BC}| = 3 \), \( |\overrightarrow{CA}| = 5 \) and \( |\overrightarrow{BA}| = 7 \), then the projection of the vector \( \overrightarrow{BA} \) on \( \overrightarrow{BC} \) is equal to:

The lines \( x = ay - 1 = z - 2 \) and \( x = 3y - 2 = bz - 2 \), \( (ab \neq 0) \) are coplanar, if:

Consider the line \( L \) given by the equation \(\displaystyle \frac{x-3}{2} = \frac{y-1}{1} = \frac{z-2}{1} \). Let \( Q \) be the mirror image of the point \( (2, 3, -1) \) with respect to \( L \). Let a plane \( P \) be such that it passes through \( Q \), and the line \( L \) is perpendicular to \( P \). Then which of the following points is on the plane \( P \)?

If \( [x] \) denotes the greatest integer less than or equal to \( x \), then the value of the integral \( \displaystyle\int\limits_{-\pi/2}^{\pi/2} \big[[x]-\sin x\big]\,dx \) is equal to:

If the mean and variance of six observations \( 7, 10, 11, 15, a, b \) are \( 10 \) and \( \dfrac{20}{3} \), respectively, then the value of \( |a-b| \) is equal to:

Let \( f : \mathbb{R} - \left\{ \dfrac{\alpha}{6} \right\} \to \mathbb{R} \) be defined by \( f(x) = \dfrac{5x + 3}{6x - \alpha} \). Then the value of \( \alpha \) for which \( (f \circ f)(x) = x \), for all \( x \in \mathbb{R} - \left\{ \dfrac{\alpha}{6} \right\} \), is:

Let \( y=y(x) \) satisfy the equation \( \dfrac{dy}{dx}-|A|=0 \), for all \( x>0 \), where .If \( y(\pi)=\pi+2 \), then the value of \( y\!\left(\dfrac{\pi}{2}\right) \) is:

The value of \( \tan\!\left(2\tan^{-1}\!\left(\dfrac{3}{5}\right)+\sin^{-1}\!\left(\dfrac{5}{13}\right)\right) \) is equal to:

If the real part of the complex number \( (\,1-\cos\theta+2i\sin\theta\,)^{-1} \) is \( \dfrac{1}{5} \) for \( \theta \in (0,\pi) \), then the value of the integral \( \displaystyle\int\limits_{0}^{\theta}\sin x\,dx \) is equal to:

Let \( r_{1} \) and \( r_{2} \) be the radii of the largest and smallest circles, respectively, which pass through the point \( (-4,1) \) and have their centres on the circumference of the circle \( x^{2}+y^{2}+2x+4y-4=0 \). If \( \dfrac{r_{1}}{r_{2}}=a+b\sqrt{2} \), then \( a+b \) is equal to:

The sum of all the local minimum values of the twice differentiable function \( f:\mathbb{R}\to\mathbb{R} \) defined by \( f(x)=x^{3}-3x^{2}-\dfrac{3\,f''(2)}{2}\,x+f'(1) \) is:

If \( f:\mathbb{R}\to\mathbb{R} \) is given by \( f(x)=x+1 \), then the value of \( \displaystyle \lim_{n \to \infty} \dfrac{1}{4} \left[ f(0) + f\left(\dfrac{5}{n}\right) + f\left(\dfrac{10}{n}\right) + ... + f\left(\dfrac{5(n - 1)}{n}\right) \right],\) is:

If sum of the first \(21\) terms of the series \( \displaystyle \log_{9^{1/2}} x + \log_{9^{1/3}} x + \log_{9^{1/4}} x + \dots, \), where \( x>0 \), is \(504\), then \(x\) is equal to:

If the point on the curve \( y^{2}=6x \) nearest to the point \( \left(3,\dfrac{3}{2}\right) \) is \( (\alpha,\beta) \), then \( 2(\alpha+\beta) \) is equal to _____.

Let \( A=\{a_{ij}\} \) be a \(3\times3\) matrix, where

Then \( \det\!\big(3\,\operatorname{Adj}(2A^{-1})\big) \) is equal to ________.

For \( p>0 \), a vector \( \vec{v}_{2}=2\hat{i}+(p+1)\hat{j} \) is obtained by rotating the vector \( \vec{v}_{1}=\sqrt{3}\,p\,\hat{i}+\hat{j} \) by an angle \( \theta \) about origin in counter clockwise direction. If \( \tan\theta=\dfrac{(\alpha\sqrt{3}-2)}{(4\sqrt{3}+3)} ,\) then the value of \( \alpha \) is equal to ______.

For \( k\in\mathbb{N} \), let \( \dfrac{1}{\alpha(\alpha+1)(\alpha+2)\ldots(\alpha+20)}=\displaystyle\sum_{k=0}^{20}\dfrac{A_{k}}{\alpha+k} \), where \( \alpha>0 \). Then the value of \( 100\left(\dfrac{A_{14}+A_{15}}{A_{13}}\right)^{2} \) is equal to ________.

Let a function \( g : (0,4] \to \mathbb{R} \) be defined as
, then the number of points in the interval \( (0,4) \) where \( g(x) \) is NOT differentiable, is ____________.

Let \( \{a_n\}_{n=1}^{\infty} \) be a sequence such that \( a_1=1,\, a_2=1 \) and \( a_{n+2}=2a_{n+1}+a_n \) for all \( n\ge 1 \). Then the value of \( 47\displaystyle\sum_{n=1}^{\infty}\frac{a_n}{2^{3n}} \) is equal to ____________.

Let a curve \( y=y(x) \) be given by the solution of the differential equation \( \cos\!\left(\dfrac{1}{2}\cos^{-1}\!\left(e^{-x}\right)\right)dx=\sqrt{e^{2x}-1}\,dy \). If it intersects y-axis at \( y=-1 \), and the intersection point of the curve with x-axis is \( (\alpha,0) \), then \( e^{\alpha} \) is equal to ______.

Consider a triangle having vertices \( A(-2,3) \), \( B(1,9) \) and \( C(3,8) \). If a line \( L \) passing through the circum-center of triangle \( ABC \), bisects line \( BC \), and intersects y-axis at point \( \left(0,\dfrac{\alpha}{2}\right) \), then the value of real number \( \alpha \) is ________.

The number of solutions of the equation \( \log_{(x+1)}\!\left(2x{^{2}}+7x+5\right)+\log_{(2x+5)}(x+1)^{2}-4=0, \; x>0, \) is ______.

If \( \displaystyle\lim_{x\to 0}\frac{\alpha x e^{x}-\beta \log_{\alpha}(1+x)+\gamma x^{2}e^{-x}}{x\sin^{2}x}=10, \; \alpha,\beta,\gamma \in \mathbb{R}, \) then the value of \( \alpha+\beta+\gamma \) is ____________.

JEE Main 2021 20th July Evening Shift

Physics Chemistry Mathematics

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For the natural numbers \( m,n \), if \( (1-y)^{m}(1+y)^{n}=1+a_{1}y+a_{2}y^{2}+\ldots +a_{m+n}y^{\,m+n} \) and \( a_{1}=a_{2}=10 \), then the value of \( m+n \) is equal to: