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The smallest positive integral value of \( a \), for which all the roots of \( x^4 - a x^2 + 9 = 0 \) are real and distinct, is equal to:

The largest value of \( n \), for which \( 40^n \) divides \( 60! \), is:

Let \( P=[p_{ij}] \) and \( Q=[q_{ij}] \) be two square matrices of order \( 3 \) such that \( q_{ij}=2^{(i+j-1)}p_{ij} \) and \( \det(Q)=2^{10} \). Then the value of \( \det(\operatorname{adj}(\operatorname{adj}P)) \) is:

Let \( a_1,a_2,a_3,a_4 \) be an A.P. of four terms such that each term of the A.P. and its common difference \( l \) are integers. If \( a_1+a_2+a_3+a_4=48 \) and \( a_1 a_2 a_3 a_4 + l^4 = 361 \), then the largest term of A.P. is equal to:

Let \( f \) be a function such that \( 3 f(x) + 2 f\!\left( \frac{m}{19x} \right) = 5x, \, x \neq 0 \), where \( m = \sum_{i=1}^{9} (i^2) \). Then \( f(5)-f(2) \) is equal to:

Let \(\vec {a} = 2\hat{i} - 5\hat{j} + 5\hat{k} \quad \text{and} \quad \vec{b} = \hat{i} - \hat{j} + 3\hat{k} \). If \( \vec{c} \) is a vector such that \( 2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0} \) and \( (\vec{a} - \vec{b}) \cdot \vec{c} = -97 \), then \( \left|\vec{c} \times \hat{k}\right|^{2} \) is equal to:

If the domain of the function \( f(x)=\sin^{-1}\!\left(\dfrac{1}{x^{2}-2x-2}\right) \) is \( (-\infty,\alpha] \cup [\beta,\gamma] \cup (\delta,\infty) \), then \( \alpha+\beta+\gamma+\delta \) is equal to:

Let the length of the latus rectum of an ellipse \( \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ (a>b), \) be \( 30 \). If its eccentricity is the maximum value of the function \( f(t)=-\dfrac{3}{4}+2t-t^{2} \), then \( \left(a^{2}+b^{2}\right) \) is equal to:

\( f(x)=\displaystyle \int \dfrac{7x^{10}+9x^{8}}{(1+x^{2}+2x^{9})^{2}} \, dx,\ x>0, \ \lim_{x\to 0} f(x)=0 \) and \( f(1)=\dfrac{1}{4} \). . and \( B=\operatorname{adj}(\operatorname{adj}A) \) be such that \( |B|=81 \), then \( \alpha^{2} \) is equal to:

Let the image of parabola \( x^{2}=4y \) in the line \( x-y=1 \) be \( (y+a)^{2}=b(x-c) \), \( a,b,c \in \mathbb{N} \). Then \( a+b+c \) is equal to:

The sum of all values of \( \alpha \), for which the shortest distance between the lines \( \frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha} \) and \( \frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha} \) is \( \sqrt{2} \), is:

Let \( \vec{a}=2\hat{i}-\hat{j}-\hat{k},\ \vec{b}=\hat{i}+3\hat{j}-\hat{k} \) and \( \vec{c}=2\hat{i}+\hat{j}+3\hat{k} \). Let \( \vec{v} \) be the vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \), such that the length of its projection on the vector \( \vec{c} \) is \( \frac{1}{\sqrt{14}} \). Then \( |\vec{v}| \) is equal to:

Let \( f(\alpha) \) denote the area of the region in the first quadrant bounded by \( x=0, x=1, y^{2}=x \) and \( y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2} \). Then \( \left(f(0)+f(1)\right) \) is equal to:

Consider the following three statements for the function \( f:(0,\infty)\to \mathbb{R} \) defined by \( f(x)=|\log_e x|-|x-1| \):
(\( \mathrm{I} \)) \( f \) is differentiable at all \( x>0 \).
(\( \mathrm{II} \)) \( f \) is increasing in \( (0,1) \).
(\( \mathrm{III} \)) \( f \) is decreasing in \( (1,\infty) \).
Then:

\( \left( \frac{1}{3} + \frac{4}{7} \right) + \left( \frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2} \right) + \left( \frac{1}{3^3} + \frac{1}{3^2} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^2}{7^2} + \frac{4^3}{7^3} \right) + \ldots \) upto infinite terms, is equal to:

Let \( X = \{ x \in N : 1 \le x \le 19 \} \) and for some \( a, b \in R \), \( Y = \{ ax + b : x \in X \} \). If the mean and variance of the elements of \( Y \) are \( 30 \) and \( 750 \), respectively, then the sum of all possible values of \( b \) is:

Let the angles made with the positive \( x \)-axis by two straight lines drawn from the point \( P(2,3) \) and meeting the line \( x+y=6 \) at a distance \( \sqrt{\frac{2}{3}} \) from the point \( P \) be \( \theta_1 \) and \( \theta_2 \). Then the value of \( (\theta_1+\theta_2) \) is:

Let \( y=y(x) \) be a differentiable function in the interval \( (0,\infty) \) such that \( y(1)=2 \), and \( \lim_{t\to x} \left(\dfrac{t^{2}y(x)-x^{2}y(t)}{x-t}\right)=3 \) for each \( x>0 \). Then \( 2y(2) \) is equal to:

The letters of the word “UDAYPUR” are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word “UDAYPUR” is:

Let \([t]\) denote the greatest integer less than or equal to \( t \). If the function

is continuous at \( x=0 \), then \( a^{2}+b^{2} \) is equal to:

Let \( z = (1+i)(1+2i)(1+3i)\ldots(1+ni) \), where \( i=\sqrt{-1} \). If \( |z|^{2}=44200 \), then \( n \) is equal to ________ .

Let \( S \) be a set of \( 5 \) elements and \( P(S) \) denote the power set of \( S \). Let \( E \) be an event of choosing an ordered pair \( (A,B) \) from the set \( P(S)\times P(S) \) such that \( A\cap B=\phi \). If the probability of the event \( E \) is \( \dfrac{3^{p}}{2^{q}} \), where \( p,q\in\mathbb{N} \), then \( p+q \) is equal to ____________ .

The number of elements in the set \( \{\, x \in [0,180^\circ] : \tan(x+100^\circ) = \tan(x+50^\circ)\tan x \tan(x-50^\circ) \,\} \) is _______________.

If \( f(x) \) satisfies the relation \( f(x)=e^x+\int_{0}^{1}(y+x e^x)\,f(y)\,dy \), then \( e+f(0) \) is equal to _______________.

Let \( (h,k) \) lie on the circle \( C:x^{2}+y^{2}=4 \) and the point \( (2h+1,\,3k+2) \) lie on an ellipse with eccentricity \( e \). Then the value of \( \dfrac{5}{e^{2}} \) is equal to ____________.

JEE Main 2026 24th Jan Evening Shift

Mathematics Physics Chemistry

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If \( f(x) \) satisfies the relation \( f(x)=e^x+\int_{0}^{1}(y+x e^x)\,f(y)\,dy \), then \( e+f(0) \) is equal to _______________.