NTA JEE Mains 24th Jan 2025 Shift 2 - Mathematics

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to :

If the system of equations $$\begin{aligned}x + 2y - 3z &= 2, \\2x + \lambda y + 5z &= 5, \\14x + 3y + \mu z &= 33\end{aligned}$$ has infinitely many solutions, then $$\lambda + \mu \text{ is equal to:} $$

Let $$A=\left\{x\in(0,\pi) -\left\{\frac{\pi}{2}\right\} :\log_{(2/\pi)}|\sin x| + \log_{(2/\pi)}|\cos x| = 2 \right\}$$ and $$B=\left\{x\geq0 : \sqrt{x}(\sqrt{x}-4) - 3|\sqrt{x}-2| + 6 = 0 \right\}.$$ Then $$n(A\cup B)$$ is equal to:

The area of the region enclosed by the curves $$y=e^x,\; y=|e^x-1|$$  and the  $$y$$ -axis is:

The equation of the chord of the ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1,$$ whose mid-point is  $$(3,1)$$  is: 

Let the points  $$\left(\frac{11}{2},\alpha\right)$$ lie on or inside the triangle with sides $$x+y=11,\; x+2y=16$$  and $$2x+3y=29.$$ Then the product of the smallest and the largest values of  $$\alpha$$  is equal to:

Let  $$f:(0,\infty)\to R$$ be a function which is differentiable at all points of its domain and satisfies the condition $$x^2 f'(x) = 2x f(x) + 3,$$  with  $$f(1)=4.$$ Then  $$2f(2)$$  is equal to:

$$\text{If }7 = 5 + \frac{1}{7}(5+\alpha) + \frac{1}{7^2}(5+2\alpha)+ \frac{1}{7^3}(5+3\alpha) + \cdots + \infty,\text{ then the value of } \alpha \text{ is:}$$

Let  $$[x]$$  denote the greatest integer function, and let $$m$$  and  $$n$$  respectively be the numbers of the points  where the function $$f(x) = [x] + |x-2|, -2 < x < 3,$$ is not continuous and not differentiable. Then  $$m+n$$  is equal to:

Let  $$A=[a_{ij}]$$  be a square matrix of order 2 with entries either 0 or 1. Let $$E$$  be the event that  $$A$$  is an invertible matrix.  Then the probability  $$P(E)$$ is:

Let the position vectors of three vertices of a triangle be $$4\vec p+\vec q-3\vec r,\;-5\vec p+\vec q+2\vec r$$ and $$2\vec p-\vec q+2\vec r.$$ If the position vectors of the orthocenter and the circumcenter  of the triangle are $$\frac{\vec p+\vec q+\vec r}{4}$$ and $$\alpha\vec p+\beta\vec q+\gamma\vec r$$ { respectively, then $$\alpha+2\beta+5\gamma$$  is equal to:

$$\text{Let }\vec a=3\hat i-\hat j+2\hat k,\quad\vec b=\vec a\times(\hat i-2\hat k)\text{ and } \vec c=\vec b\times\hat k.\text{Then the projection of } (\vec c-2\hat j)\text{ on } \vec a \text{ is:}$$

The number of real solution(s) of the equation $$x^2 + 3x + 2 = \min\{|x-3|,|x+2|\}\text{ is:}$$

$$\text{The function }f:(-\infty,\infty)\to(-\infty,1), \text{ defined by }f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}\text{ is:}$$

In an arithmetic progression, if $$S_{40}=1030$$  and $$S_{12}=57$$, then $$S_{30}-S_{10} $$ is equal to:

Suppose  $$A$$  and  $$B$$  are the coefficients of $$30^{\text{th}}$$  and $$12^{\text{th}}$$  terms respectively in the binomial expansion of  $$(1+x)^{2n-1}.$$ If $$2A=5B,$$  then  $$n$$  is equal to: 

Let $$(2,3)$$  be the largest open interval in which the function $$f(x)=2\log_e(x-2)-x^2+ax+1$$ is strictly increasing and  $$(b,c)$$  be the largest open interval in which the function $$g(x)=(x-1)^3(x+2-a)^2$$ is strictly decreasing. Then  $$100(a+b-c)$$  is equal to:

$$\text{For some } a,b,\text{ let }f(x)=\left|\begin{matrix}a+\dfrac{\sin x}{x} & 1 & b \\a & 1+\dfrac{\sin x}{x} & b \\a & 1 & b+\dfrac{\sin x}{x}\end{matrix}\right|,x\neq 0,\lim_{x\to 0} f(x)=\lambda+\mu a+\nu b,\text{ Then } (\lambda+\mu+\nu)^2 \text{ is equal to:}$$

If the equation of the parabola with vertex $$V\left(\frac{3}{2},3\right)$$ and the directrix $$x+2y=0$$ is $$\alpha x^2+\beta y^2-\gamma xy-30x-60y+225=0$$, then $$\alpha+\beta+\gamma$$ is equal to:

$$\text{If } \alpha > \beta > \gamma > 0,\text{ then the expression}\cot^{-1}\!\left\{\beta+\frac{(1+\beta^2)}{(\alpha-\beta)}\right\} + \cot^{-1}\!\left\{\gamma+\frac{(1+\gamma^2)}{(\beta-\gamma)}\right\} + \cot^{-1}\!\left\{\alpha+\frac{(1+\alpha^2)}{(\gamma-\alpha)}\right\}\text{ is equal to:}$$

Let  $$P$$  be the image of the point  $$Q(7,-2,5)$$ in the line  $$L:\;\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4},$$ and  $$R(5,p,q)$$  be a point on $$L.$$ Then the square of the area of  $$\triangle PQR$$ is  $$\underline{\hspace{2cm}}.$$

If $$\int \frac{2x^2+5x+9}{\sqrt{x^2+x+1}}\,dx=x\sqrt{x^2+x+1}+\alpha\sqrt{x^2+x+1}+\beta\log_e\!\left|x+\frac12+\sqrt{x^2+x+1}\right|+C$$, where $$C$$ is the constant of integration, then $$\alpha+2\beta$$ is equal to $$\underline{\hspace{2cm}}.$$

Let $$y=y(x)$$ be the solution of the differential equation $$2\cos x\,\frac{dy}{dx}= \sin 2x - 4y\sin x,x\in\left(0,\frac{\pi}{2}\right).$$ If $$y\!\left(\frac{\pi}{3}\right)=0$$, then $$y'\!\left(\frac{\pi}{4}\right)+ y\!\left(\frac{\pi}{4}\right)$$ is equal to $$\underline{\hspace{2cm}}.$$

Number of functions $$ f:\{1,2,\ldots,100\}\to\{0,1\} $$ that assign 1 to exactly one of the positive integers less than or equal to 98 is equal to______________

Let $$H_1:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$  and $$H_2:-\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$$ be two hyperbolas having length of latus rectums $$15\sqrt{2}$$ and $$12\sqrt{5}$$ respectively. Let their eccentricities be $$e_1=\sqrt{\frac{5}{2}}$$  and  $$e_2$$ respectively. If the product of the lengths of  their transverse axes is  $$100\sqrt{10},$$ then  $$25e_2^2$$  is equal to $$\underline{\hspace{2cm}}.$$