NTA JEE Mains 22nd Jan 2025 Shift 1 - Mathematics

Let $$a_1,a_2,a_3,...$$ be a G.P. of increasing positive terms. If $$a_1a_5 = 28$$ and $$a_2+a_4 = 29$$, then $$a_6$$ is equal to:

Let $$x=x(y)$$ be the solution of the differential equation$$ y^{2}dx+\left( x-\frac{1}{y}\right)dy=0 $$ . If $$x(1)=1$$, then $$\frac{1}{2}$$ is :

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $$\frac{m}{n}$$, where $$gcd(m,n)=1$$, then $$m+n$$ is equal to :

The product of all solutions of the equation $$ e^{5(\log_e x)^{2}+3}=x^{8},x>0 $$, is :

Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line $$x+2y=2$$. If the centroid of $$ \triangle PQR $$ is the point $$ (\alpha, \beta) $$, then $$ 15(\alpha - \beta) $$ is equal to :

$$ \text{Let for }f(x)=7 \tan^{8}x + 7\tan^{6}x-3\tan^{4}x-3\tan^{2}x \text{ } I_1=\int_{0}^{\pi/4}f(x)dx \text{ and }I_2=\int_{0}^{\pi/4}xf(x)dx. \text{ Then } 7I_1+12T_2 \text{ is equal to :} $$

Let the parabola $$ y=x^{2}+px-3 $$, meet the coordinate axes at the points P, Q and R . If the circle C with centre at (-1, -1) passes through the points P, Q and R, then the area of $$ \triangle PQR $$ is :

Let $$ L_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \text{ and } L_2: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5} $$ be two lines. Then which of the following points lies on the line of the shortest distance between $$L_1 \text{ and } L_2 $$ ?

Let $$f(x)$$ be a real differentiable function such that $$f(0)=1$$ and $$f(x+y)=f(x)f^{'}(y)+f^{'}(x)f(y)$$ for all $$x,y \in \mathbb{R}.$$ Then $$\sum_{n=1}^{100} \log_e f(n)$$ is equal to :

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $$16\left(\left(\sec^{-1}x\right)^{2}\left(\cosec^{-1}x\right)^{2}\right) \text{is :} $$

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a twice differentiable function such that $$f(x+y)=f(x)f(y)$$ for all $$x,y \in R.$$ If $$f^{'}(0)=4a$$ and $$f$$ satisfies $$f^{''}(x)-3af^{'}(x)-f(x)=0,a>0$$, then the area of the region $$R= \left\{(x,y) \mid 0\leq y\leq f(ax), 0\leq x \leq2 \right\}$$ is

$$ \text{The area of the region, inside the circle }(x-2\sqrt{3})^{2}+y^{2}=12 \text{ and outside the parabola } y^{2}=2\sqrt{3}x \text{ is :} $$

Let the foci of a hyperbola be $$(1, 4)$$ and $$(1, -12).$$ If it passes through the point $$(1, 6)$$, then the length of its latus-rectum is :

If $$\sum_{r=1}^n T_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$$ then $$\lim_{n \rightarrow \infty} \sum_{r=1}^n\left( \frac {1}{T_r}\right)$$ is equal to:

A coin is tossed three times. Let  $$X$$ denote the number of times a tail follows a head. If $$\mu$$ and $$\sigma^{2}$$ denote the mean and variance of $$X$$, then the value of $$64(\mu+\sigma^{2})$$ is :

The number of non-empty equivalence relations on the set $$\left\{1, 2, 3\right\}$$ is :

A circle $$C$$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $$ r$$ be the radius of a circle that has centre at the point  (2, 5)  and intersects the circle $$ C $$ at exactly two points. If the set of all possible values of r is the interval $$(\alpha, \beta), \text{ then } 3\beta - 2\alpha \text{ is equal to :} $$

$$ \text{Let } A=\left\{1, 2, 3,....,10\right\} \text{ and }B=\left\{ \frac {m}{n},n \in A,m < n \text{ and }gcd(m,n)=1\right\}.$$ Then n(B) is equal to:

Let $$ z_1,z_2 \text{ and } z_3$$ be three complex numbers on the circle $$ \mid z \mid = 1 $$ with $$ arg(z_1)=\frac{-\pi}{4},arg(z_2)=0 \text{ and } arg(z_3)=\frac{\pi}{4}$$.  If $$\mid z_1\overline{z}_2+z_2\overline{z}_3+z_3\overline{z}_1 \mid^{2}= \alpha+ \beta \sqrt{2}, \alpha, \beta \in Z$$, then the value of $$ \alpha^{2}+\beta^{2} \text{ is :} $$

Let $$ A $$ be a square matrix of order 3 such that $$det(A)=-2 \text{ and }det(3adj(-6adj(3A)))=2^{m+n}\cdot3^{mn}$$, $$m>n. \text{ Then } 4m+2n\text{ is equal to } $$_______

$$ \text{If } \sum_{r=0}^5 \frac{^{11}C_{2r+1}}{2r+2}=\frac{m}{n},gcd(m, n)=1, \text{ then }m - n \text{ is equal to } $$ _______

Let $$\overrightarrow{c}$$ be the projection vector of $$\overrightarrow{b}=\lambda\widehat{i}+4\widehat{k}$$, $$\lambda > 0$$, on the vector $$\overrightarrow{a}=\widehat{i}+2\widehat{j}+2\widehat{k}$$. If $$|\overrightarrow{a}+ \overrightarrow{c}|= 7$$, then the area of the parallelogram formed by the vectors $$\overrightarrow{b}$$ and $$\overrightarrow{c}$$ is ______

Let the function, $$f(x)=\begin{cases}-3ax^{2}-2, & x < 1\\a^{2}+bx, & x \geq 0\end{cases}$$ be differentiable for all  $$x \in R,  $$ where $$ a>1, b \in R$$.  If the area of the region enclosed by $$ y=f(x) \text{and the line } y= -20 \text{ is } \alpha+\beta\sqrt{3},\alpha, \beta \in Z$$,  then the value of  $$\alpha + \beta \text{ is } $$_______

Let $$L_1:\frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$$ and $$L_2:\frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}$$, $$\alpha \in R$$, be two lines, which intersect at the point $$B$$. If $$P$$ is the foot of perpendicular from the point $$A(1,1,-1)$$ on $$L_2$$, then the value of $$26\alpha(PB)^{2}$$ is ______