NTA JEE Mains 23rd Jan 2026 Shift 1 - Mathematics

The value of the integral $$\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1+\sqrt[3]{\tan2x}}$$ is:

The sum of all possible values of $$n\epsilon N$$, so that the coefficients of $$x,x^{2}\text{ and }x^{3}$$ in the expansion of $$(1+x^{2})^{2}(1+x)^{n}$$, are in arithmetic progression is:

Number of solutions of $$\sqrt{3}\cos2\theta+8\cos\theta+3\sqrt{3}=0,\theta\epsilon[-3\pi,2\pi]$$ is:

Let $$S=\left\{z:3\leq|2z-3(1+i)|\leq7\right\}$$ be a set of complex nwnbers. Then $$\min_{Z \epsilon S}\left|\left(z+\frac{1}{2}(5+3i)\right)\right|$$ is equal to :

Let $$\overrightarrow{a}=-\widehat{i}+\widehat{j}+2\widehat{k},\overrightarrow{b}=\widehat{i}-\widehat{j}-3\widehat{k},\overrightarrow{c}=\overrightarrow{a} \times \overrightarrow{b}\text{ and }\overrightarrow{d}=\overrightarrow{c}\times\overrightarrow{a}$$. Then $$\large (\overrightarrow{a}-\overrightarrow{b}).\overrightarrow{d}$$ is equal to:

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Let the domain of the function $$f(x)=\log_{3}\log_{5}\log_{7}(9x-x^{2}-13)$$ be the interval (m, n). Let the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ have eccentricity $$\frac{n}{3}$$ and the length of the latus rectum $$\frac{8m}{3}$$. Then $$b^{2}-a^{2}$$ is equal to:

The value of $$\frac{100_{C_{50}}}{51}+\frac{100_{C_{51}}}{52}+...+\frac{100_{C_{100}}}{101}$$ is:

The vertices B and C of a triangle ABC lie on the line $$\frac{x}{1}=\frac{1-y}{-2}=\frac{z-2}{3}$$ The coordinates of A and B are (1, 6, 3) and (4, 9, $$\alpha$$) respectively and C is at a distance of 10 units from B. The area (in sq. units) of $$\triangle$$ABC is :

Let $$f(x) = \left\{\begin{array}{l l}\frac{ax^{2}+2ax+3}{4x^{2}+4x-3} ,& x\neq\quad -\frac{3}{2},\frac{1}{2}\\b, & \quad x=-\frac{3}{2},\frac{1}{2}\\\end{array}\right.$$

be continuous at $$x=-\frac{3}{2}$$. If $$fof(x)=\frac{7}{5}$$ then x is equal to:

A rectangle is formed by the lines x= O, y = O, x=3 and y = 4. Let the line L be perpendicular to 3x +y + 6 = 0 and divide the area of the rectangle into two equal parts. Then the distance of the point $$\left(\frac{1}{2},-5\right)$$ from the line L is equal to :

Let $$\alpha$$ and $$\beta$$ respectively be the maximum and the minimum values of the function $$f(\theta)=4\left(\sin^4\left(\frac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\left(\frac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right),\ \ \theta\in\ R$$. Then $$\alpha+2\beta$$ is equal to:

Let the direction cosines of two lines satisfy the equations : 4l + m - n =0 and 2mn +10nl +3lm= 0. Then the cosine of the acute angle between these lines is :

If $$\alpha$$ and $$\beta$$ ($$\alpha < \beta$$) are the roots of the equation $$(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0,x\geq0\text{ then }\sqrt{\frac{\beta}{\alpha}}+\sqrt{\alpha\beta}$$ is equal to:

Let the mean and variance of 8 numbers - 10, - 7, - 1, x, y, 9, 2, 16 be $$\frac{7}{2}\text{ and }\frac{293}{4}$$ respectively.
Then the mean of 4 numbers x, y, x + y + 1, |x-y| is:

Among the statements :
I: If $$ \begin{vmatrix}1 & \cos\alpha & \cos\beta \\\mathbf{\cos\alpha} & 1 & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & 1\end{vmatrix}=\begin{vmatrix}0 & \mathbf{\cos\alpha}&\mathbf{\cos\beta} \\\mathbf{\cos\alpha} & 0 & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & 0\end{vmatrix}$$, then $$\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=\frac{3}{2}$$, and 

II: $$\begin{vmatrix}x^{2}+x & x+1 & x-2 \\2x^{2}+3x-1 & 3x & 3x-3 \\x^{2}+2x+3 & 2x-1 & 2x-1\end{vmatrix} = px + q$$, then $$p^{2}=196q^{2}$$

Let the line y - x = l intersect the ellipse $$\frac{x^{2}}{2}+\frac{y^{2}}{1}=$$ at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:

Let $$f(x)=\int\frac{(2-x^{2}).e^{x}}{(\sqrt{1+x})(1-x)^{\frac{3}{2}}}dx$$. If f(0)=0, then $$f\left(\frac{1}{2}\right)$$ is equal to:

Let A= {- 2, - 1, 0, 1, 2, 3, 4}. Let R be a relation on A defined by xRy if and only if $$|2x + y| \leq 3$$. Let l be the number of elements in R. Let m and n be the minimun number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l+ m + n is equal to:

Let y = y(x) be the solution of the differential equation $$x^{4}dy+(4x^{3}y+2\sin x)dx=0,x > 0,y\left(\frac{\pi}{2}\right)=0$$. Then $$\pi^{4}y\left(\frac{\pi}{3}\right)$$ is equal to :

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ___ .

Let f be a twice differentiable non-negative function such that $$(f(x))^{2}=25+\int_{0}^{x}\left((f(t))^{2}+(f'(t))^{2}\right)dt$$. Then the mean of $$f(\log_{e}{(1)}),f(\log_{e}{(2)}),.....,f(\log_{e}{(625)})$$ is equal to:

Let |A|=6, Where A is a $$3\times3$$ matrix. If $$|adj(3adj(A^{2}\cdot adj(2A)))|=2^{m}\cdot3^{n},m,n\epsilon N$$, then m+n is equal to:

From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that $$a-b \geq 10$$ is $$\frac{m}{n}$$, gcd (m, n) = 1, then m + n is equal to______.

Let the area of the region bounded by the curve y= max $${\sin x, \cos x}$$, lines x = O, $$x=\frac{3\pi}{2}$$, and the x-axis be A. Then, A+$$A^{2}$$ is equal to_____.