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The temperature \( T(t) \) of a body a time \( t = 0 \) is \( 160^\circ F \) and it decreases continuously as per the differential equation \( \dfrac{dT}{dt} = -K (T - 80) \), where \( K \) is a positive constant. If \( T(15) = 120^\circ F \), then \( T(45) \) is equal to:

If the function \( f:(-\infty,-1)\to (a,b] \) defined by \( f(x) = e^{x^3-3x+1} \) is one – one and onto, then the distance of the point \( P(2b+4,a+2) \) from the line \( x+e^{-3}y=4 \) is:

If \( a=\sin^{-1}(\sin 5)) \) and \( b=\cos^{-1}(\cos 5)) \), then \( a^{2}+b^{2} \) is equal to:

Let \( P \) be a parabola with vertex \( (2,3) \) and directrix \( 2x+y=6 \). Let an ellipse \( E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1, a>b \), of eccentricity \( \dfrac{1}{\sqrt{2}} \) pass through the focus of the parabola \( P \). Then, the square of the length of the latus rectum of \( E \), is;

The number of solutions, of the equation \( e^{\sin x}-2e^{-\sin x}=2 \), is:

Let the mean and the variance of \( 6 \) observations \( a, b, 68, 44, 48, 60 \) be \( 55 \) and \( 194 \), respectively. If \( a > b \), then \( a + 3b \) is:

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed \( 3 \) times, then the probability of getting two tails and one head is:

If for some \( m ,n\) : \( {^6}C_m + 2({^6}C_{m+1}) + {^6}C_{m+2} > {^8}C_3 \) and \( {^{\,n-1}}P_3 : {^{\,n}}P_4 = 1:8 \) then \( {^{\,n}}P_{m+1} + {^{\,n+1}}C_m \) is equal to:

Let \( f:\mathbb{R}\to (0,\infty) \) be strictly increasing function such that \( \displaystyle\lim_{x\to\infty}\dfrac{f(7x)}{f(x)}=1 \). Then, the value of \( \displaystyle \lim_{x\to\infty}\left[\dfrac{f(5x)}{f(x)}-1\right] \) is equal to:

Let \( A(a,b) \), \( B(3,4) \) and \( C(-6,-8) \) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point \( P(2a+3,7b+5) \) from the line \( 2x+3y-4=0 \) measured parallel to the line \( x-2y-1=0 \) is:

Let \( f,g:(0,\infty)\to \mathbb{R} \) be two functions defined by \( f(x)=\displaystyle\int_{-x}^{x} (|t|-t^{2})e^{-t^{2}}\,dt \) and \( g(x)=\displaystyle\int_{0}^{x^{2}} t^{1/2}e^{-t}\,dt \). Then, the value of \( 9\big(f(\sqrt{\log_e 9})+\sqrt{\log_e 9})\big) \) is equal to:

The shortest distance, between lines \( L_1 \) and \( L_2 \), where \( L_1:\; \dfrac{x-1}{2}=\dfrac{y+1}{-3}=\dfrac{z+4}{2} \) and \( L_2 \) s the line, passing through the line, passing through the points \( A=(-4,4,3), B=(-1,6,3) \) and perpendicular to the line \( \dfrac{x-3}{-2}=\dfrac{y}{3}=\dfrac{z-1}{1} \) is:

The number of ways in which \( 21 \) identical apples can be distributed among three children such that each child gets at least \( 2 \) apples, is:

Let \( (\alpha,\beta,\gamma) \) be the mirror image of the point \( (2,3,5) \) in the line \( \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} \). Then \( 2\alpha+3\beta+4\gamma \) is equal to:

The area of the region enclosed by the parabolas \( y=4x-x^2 \) and \( 3y=(x-4)^2 \) is equal to:

Let \( 2^{\text{nd}},8^{\text{th}} \) and \( 44^{\text{th}} \) terms of a non-constant A.P. be respectively the \( 1^{\text{st}},2^{\text{nd}} \) and \( 3^{\text{rd}} \) terms of a G.P.. If the first term of the A.P. is \( 1 \), then the sum of its first \( 20 \) terms is equal to:

Let \( z_1 \) and \( z_2 \) be two complex numbers such that \( z_1+z_2=5 \) and \( z_1^3+z_2^3=20+15i \). Then, \( \left|z_1^4+z_2^4\right| \) equals:

Consider the function \( f:(0,\infty)\to \mathbb{R} \) defined by \( f(x)=e^{-\lvert \log_{e} x \rvert} \). If \( m \) and \( n \) be respectively the number of points at which \( f \) is not continuous and \( f \) is not differentiable, then \( m+n \) is:

Let a variable line passing through the centre of the circle \( x^{2}+y^{2}-16x-4y=0 \), meet the positive co-ordinate axes at the points \( A \) and \( B \). Then the minimum value of \( OA+OB \), where \( O \) is the origin, is equal to:

Let \( A \) be a \( 3\times 3 \) real matrix such that
.Then, the system has.

Let \( a,b,c \) be the lengths of three sides of a triangle satisfying the condition \( (a^{2}+b^{2})x^{2}-2b(a+c)x+(b^{2}+c^{2})=0 \). If the set of all possible values of \( x \) is the interval \( (\alpha,\beta) \), then \( 12(\alpha^{2}+\beta^{2}) \) is equal to ______.

Let \( A=\{1,2,3,\ldots,100\} \). Let \( R \) be a relation on \( A \) defined by \( (x,y)\in R \) if only if \( 2x=3y \). Let \( R_{1} \) be a symmetric relation on \( A \) such that \( R\subset R_{1} \) and the number of elements in \( R_{1} \) is \( n \). Then the minimum value of \( n \) is __________.

Let \( \vec{a}=3\hat{i}+2\hat{j}+\hat{k} \) and \( \vec{b}=2\hat{i}-\hat{j}+3\hat{k} \) and \( \vec{c} \) be a vector such that \( (\vec{a}+\vec{b})\times \vec{c}=2(\vec{a}\times \vec{b})+24\hat{j}-6\hat{k} \) and \( (\vec{a}-\vec{b}+\hat{i})\cdot \vec{c}=-3 \). Then \( \lvert \vec{c} \rvert^{2} \) is equal to ______.

\(\displaystyle \Big|\frac{120}{\pi^{2}} \int_{0}^{\pi} \dfrac{x^{2}\sin x \cos x}{\sin^{4}x+\cos^{4}x}\,dx\Big| \) is equal to ________.

If \(\displaystyle \lim_{x\to 0}\frac{ax^{2}e^{x}-b\log_{e}(1+x)+cxe^{-x}}{x^{2}\sin x}=1 \), then \( 16\left(a^{2}+b^{2}+c^{2}\right) \) is equal to ________.

Let \( A(-2,-1), B(1,0), C(\alpha ,\beta ) \) and \( D(\gamma ,\delta ) \) be the vertices of a parallelogram \( ABCD \). If the point \( C \) lies on \( 2x-y=5 \) and the point \( D \) lies on \( 3x-2y=6 \), then the value of \( \left|\alpha +\beta +\gamma +\delta \right| \) is equal of ________.

A line passes through \( A(4,-6,-2) \) and \( B(16,-2,4) \). The point \( P(a,b,c) \), where \( a,b,c \) are non‑negative integers, on the line \( AB \) lies at a distance of \( 21 \) units from the point \( A \). The distance between the points \( P(a,b,c) \) and \( Q(4,-12,3) \) is equal to ________.

Let the coefficient of \( x^{r} \) in the expansion of \( (x+3)^{n-1}+ (x+3)^{n-2}(x+2)+ (x+3)^{n-3}(x+2)^{2}+\cdots +(x+2)^{n-1} \) be \( \alpha_{\gamma} \). If \(\displaystyle \sum_{r=0}^{n}a_{\gamma}=\beta ^{n}-\gamma ^{n},\ \beta ,\gamma \in \mathbb{N} \), then the value of \( \beta ^{2}+\gamma ^{2} \) equals ________.

Let \( A \) be \( 3 \times 3 \) matrix and \( \det(A) = 2 \). If \( n = \det\dfrac{(\operatorname{adj}(\operatorname{adj}(\ldots(\operatorname{adj} A))) )}{ 2024 \text{-times} }\) remainder when \( n \) is divided by \( 9 \) is equal to ______.

Let \( Y = Y(x) \) be the solution of the differential equation \( \sec^{2}x \, dx + \left(e^{2y}\tan^{2}x + \tan x\right) dy = 0, \, 0 < x < \dfrac{\pi}{2}, \, y\left(\dfrac{\pi}{4}\right)=0 \). If \( y\!\left(\dfrac{\pi}{6}\right)=\alpha \), then \( e^{8\alpha} \) is equal to ______.

If two vectors \( \vec{A} \) and \( \vec{B} \) having equal magnitude \( R \) are inclined at an angle \( \theta \), then:

JEE Main 2024 31st Jan Evening Shift

Mathematics Physics Chemistry

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The number of ways in which \( 21 \) identical apples can be distributed among three children such that each child gets at least \( 2 \) apples, is: