NTA JEE Mains 28th Jan 2026 Shift 1 - Mathematics

Let z be a complex number such that |z - 6| = 5 and |z + 2 - 6i| = 5. Then the value of $$z^{3}+3z^{2}-15z+141$$ is equal to

If $$g(x)=3x^{2}+2x-3, f(0)=-3$$ and $$4g(f(x))=3x^{2}-32x+72$$, then f(g(2)) is equal to:

If the distances of the point (1 , 2, a) from the line $$\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$$ along the lines $$L_{1}:\frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b}$$ and $$L_{2}:\frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}$$ are equal, then a + b + c is equal to

The area of the region $$R=\left\{(x,y):xy\leq 8,1\leq y\leq x^{2},x\geq 0\right\}$$ is

A bag contains 1O balls out of which k are red and (10 - k) are black, where $$0\leq k\leq 10$$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $$x+2\sqrt{2}y=4$$. If the co-ordinates of the vertex A are $$(\alpha, \beta)$$, then the greatest integer less than or equal to $$|\alpha + \sqrt{2}\beta |$$ is

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let x be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let y be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

Let $$y=y(x)$$ be the solution of the differential equation $$x\frac{dy}{dx}-\sin 2y=x^{3}\left(2-x^{3}\right)\cos^{2}y,y\neq 0$$. If y(2) = 0, then tan(y(l)) is equal to

If $$\frac{\tan (A-B)}{\tan A}+\frac{\sin^{2}C}{\sin^{2}A}=1,A,B,C \in \left(0,\frac{\pi}{2}\right)$$, Then

If $$\int_{}^{}\left(\frac{1-5\cos^{2} x}{\sin^{5} x \cos^{2} x}\right)dx=f(x)+C$$, where C is the constant of integration, then $$f\left(\frac{\pi}{6}\right)-f\left(\frac{\pi}{4}\right)$$ is equal to

The value of $$\lim_{x \to 0}\frac{\log_e\!\left(\sec(ex)\cdot \sec(e^{2}x)\cdots \sec(e^{10}x)\right)}{e^{2}-e^{2\cos x}}$$ is equal to

For three unit vectors $$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$$ satisfying $$|\overrightarrow{a}-\overrightarrow{b}|^{2}+|\overrightarrow{b}-\overrightarrow{c}|^{2}+|\overrightarrow{c}-\overrightarrow{a}|^{2}=9$$ and $$|2\overrightarrow{a}+k\overrightarrow{b}+k\overrightarrow{c}|+3$$. the positive value of k is

If $$\alpha, \beta$$, where $$\alpha < \beta$$, are the roots of the quadratic equation  $$\lambda x^{2}-(\lambda + 3)x+3=0$$ and  $$\dfrac{1}{\alpha}-\dfrac{1}{\beta}=\dfrac{1}{3}$$, then the sum of all possible values of $$\lambda$$ is

Let y = x be the equation of a chord of the circle $$C_{1}$$ (in the closed half-plane x c $$\geq$$ 0) of diameter 10 passing through the origin. Let $$C_{2}$$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $$C_{2}$$, which x + ay + b = 0, then a - b is equal to

The common difference of the $$A.P.: a_{1},a_{2},.....,a_{m}$$ is 13 more than the common difference of the $$A.P.:b_{1},b_{2},....,b_{n}$$. If $$b_{31}=-277,b_{43}=-385 \text{ and } a_{78}=327$$ then $$a_{1}$$ is equal to

Let f be a polynomial function such that $$f(x^{2}+1)=x^{4}+5x^{2}+2$$, for all $$x \in R$$. Then $$\int_{0}^{3}f(x)dx$$ is equal to

The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11 , 13, 15, 21, then the mean deviation about the median of all the 10 observations is

Let A, Band C be three $$2\times 2$$ matrices with real entries such that $$B=(I+A)^{-1}$$ and A+C=1. If $$BC=\begin{bmatrix}1 & -5 \\-1 & 2 \end{bmatrix}$$ and $$CB\begin{bmatrix}x_{1}\\ x_{2} \end{bmatrix}=\begin{bmatrix}12\\-6 \end{bmatrix}$$, then $$x_{1}+x_{2}$$ is

The value of $$\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$$ is

Let $$S=\left\{x^{3}+ax^{2}+bx+c:a,b,c, \in N \text{ and }a,b,c \leq 20\right\}$$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $$x^{2}+2$$, is

Let PQR be a triangle such that $$\overrightarrow{PQ}=-2\widehat{i}-\widehat{j}+2\widehat{k}$$ and $$\overrightarrow{PR}=a\widehat{i}+b\widehat{j}-4\widehat{k},a,b \in Z$$. Let S be the point on QR, which is equidistant from the lines PQ and PR. If $$|\overrightarrow{PR}|=9$$ and $$\overrightarrow{PS}=\widehat{i}-7\widehat{j}+2\widehat{k}$$, then the value of 3a - 4b is_______

For some $$\theta \in \left(0,\frac{\pi}{2}\right)$$, let the eccentricity and the length of the latus rectum of the hyperbola $$x^{2}-y^{2}\sec^{2}\theta =8$$ be $$e_{1}$$ and $$l_{1}$$,respectively, and let the eccentricity and the length of the latus rectum of the ellipse $$x^{2}\sec^{2}\theta +y^{2}=6$$ be $$e_{2}$$ and $$l_{2}$$.respectively. If $$e_{1}^{2}=e_{2}^{2}\left(\sec^{2}\theta +1\right)$$, then $$\left(\frac{l_{1}l_{2}}{e_{1}e_{2}}\right)\tan^{2}\theta$$ is equal to_____

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $$\text{R-(a,b)}$$, then $$a^{2}+b^{2}$$ is equal to______

The value of $$\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_{0}^{r}x|\sin \pi x\right)}\right|\right)$$ is ______.

If $$K=\tan\left(\frac{\pi}{4}+\frac{1}{2}\cos^{-1}\left(\frac{2}{3}\right)\right)+\tan\left(\frac{1}{2}\sin^{-1}\left(\frac{2}{3}\right)\right)$$, then the number of solutions of the equation $$\sin^{-1}(kx-1)=\sin^{-1} x-\cos^{-1} x$$ is______.