NTA JEE Mains 1st Feb 2024 Shift 2 - Mathematics

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$px^2 + qx - r = 0$$, where $$p \neq 0$$. If $$p$$, $$q$$ and $$r$$ be the consecutive terms of a non-constant G.P and $$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}$$, then the value of $$(\alpha - \beta)^2$$ is:

If $$z$$ is a complex number such that $$|z| \leq 1$$, then the minimum value of $$\left|z + \frac{1}{2}(3 + 4i)\right|$$ is:

Let $$S_n$$ denote the sum of the first n terms of an arithmetic progression. If $$S_{10} = 390$$ and the ratio of the tenth and the fifth terms is 15 : 7, then $$S_{15} - S_5$$ is equal to:

Let $$m$$ and $$n$$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $$\left(\frac{1}{3}x^{1/3} + \frac{1}{2x^{2/3}}\right)^{18}$$. Then $$\left(\frac{n}{m}\right)^{1/3}$$ is:

The number of solutions of the equation $$4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$$; $$x \in [-2\pi, 2\pi]$$ is:

Let the locus of the mid points of the chords of circle $$x^2 + (y-1)^2 = 1$$ drawn from the origin intersect the line $$x + y = 1$$ at P and Q. Then, the length of PQ is:

Let P be a point on the ellipse $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Let the line passing through P and parallel to y-axis meet the circle $$x^2 + y^2 = 9$$ at point Q such that P and Q are on the same side of the x-axis. Then, the eccentricity of the locus of the point R on PQ such that $$PR : RQ = 4 : 3$$ as P moves on the ellipse, is:

Let $$f(x) = \begin{cases} x-1, & x \text{ is even} \\ 2x, & x \text{ is odd} \end{cases}$$, $$x \in N$$. If for some $$a \in N$$, $$f(f(f(a))) = 21$$, then $$\lim_{x \to a^-} \left\lfloor \frac{x^3}{a} \right\rfloor - \left\lfloor \frac{x}{a} \right\rfloor$$, where $$\lfloor t \rfloor$$ denotes the greatest integer less than or equal to $$t$$, is equal to:

Consider 10 observations $$x_1, x_2, \ldots, x_{10}$$, such that $$\sum_{i=1}^{10}(x_i - \alpha) = 2$$ and $$\sum_{i=1}^{10}(x_i - \beta)^2 = 40$$, where $$\alpha, \beta$$ are positive integers. Let the mean and the variance of the observations be $$\frac{6}{5}$$ and $$\frac{84}{25}$$ respectively. Then $$\frac{\beta}{\alpha}$$ is equal to:

Consider the relations $$R_1$$ and $$R_2$$ defined as $$aR_1b \Leftrightarrow a^2 + b^2 = 1$$ for all $$a, b \in R$$ and $$(a,b)R_2(c,d) \Leftrightarrow a + d = b + c$$ for all $$(a,b,c,d) \in N \times N$$. Then

Let the system of equations $$x + 2y + 3z = 5$$, $$2x + 3y + z = 9$$, $$4x + 3y + \lambda z = \mu$$ have infinite number of solutions. Then $$\lambda + 2\mu$$ is equal to:

If the domain of the function $$f(x) = \frac{\sqrt{x^2 - 25}}{4 - x^2} + \log_{10}(x^2 + 2x - 15)$$ is $$(-\infty, \alpha) \cup [\beta, \infty)$$, then $$\alpha^2 + \beta^3$$ is equal to:

Let $$f(x) = |2x^2 + 5|x| - 3|$$, $$x \in R$$. If $$m$$ and $$n$$ denote the number of points where $$f$$ is not continuous and not differentiable respectively, then $$m + n$$ is equal to:

The value of $$\int_0^1 (2x^3 - 3x^2 - x + 1)^{1/3} dx$$ is equal to:

If $$\int_0^{\pi/3} \cos^4 x \, dx = a\pi + b\sqrt{3}$$, where $$a$$ and $$b$$ are rational numbers, then $$9a + 8b$$ is equal to:

Let $$\alpha$$ be a non-zero real number. Suppose $$f: R \to R$$ is a differentiable function such that $$f(0) = 1$$ and $$\lim_{x \to -\infty} f(x) = 1$$. If $$f'(x) = \alpha f(x) + 3$$, for all $$x \in R$$, then $$f(-\log_e 2)$$ is equal to:

Consider a $$\triangle ABC$$ where $$A(1, 3, 2)$$, $$B(-2, 8, 0)$$ and $$C(3, 6, 7)$$. If the angle bisector of $$\angle BAC$$ meets the line BC at D, then the length of the projection of the vector $$\vec{AD}$$ on the vector $$\vec{AC}$$ is:

If the mirror image of the point $$P(3, 4, 9)$$ in the line $$\frac{x-1}{3} = \frac{y+1}{2} = \frac{z-2}{1}$$ is $$(\alpha, \beta, \gamma)$$, then $$14(\alpha + \beta + \gamma)$$ is:

Let P and Q be the points on the line $$\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$$ which are at a distance of 6 units from the point $$R(1, 2, 3)$$. If the centroid of the triangle PQR is $$(\alpha, \beta, \gamma)$$, then $$\alpha^2 + \beta^2 + \gamma^2$$ is:

Let Ajay will not appear in JEE exam with probability $$p = \frac{2}{7}$$, while both Ajay and Vijay will appear in the exam with probability $$q = \frac{1}{5}$$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:

The lines $$L_1, L_2, \ldots, L_{20}$$ are distinct. For $$n = 1, 2, 3, \ldots, 10$$ all the lines $$L_{2n-1}$$ are parallel to each other and all the lines $$L_{2n}$$ pass through a given point P. The maximum number of points of intersection of pairs of lines from the set $$\{L_1, L_2, \ldots, L_{20}\}$$ is equal to:

If three successive terms of a G.P. with common ratio $$r$$ $$(r > 1)$$ are the length of the sides of a triangle and $$\lfloor r \rfloor$$ denotes the greatest integer less than or equal to r, then $$3\lfloor r \rfloor + \lfloor -r \rfloor$$ is equal to:

Let ABC be an isosceles triangle in which A is at $$(-1, 0)$$, $$\angle A = \frac{2\pi}{3}$$, $$AB = AC$$ and B is on the positive x-axis. If $$BC = 4\sqrt{3}$$ and the line BC intersects the line $$y = x + 3$$ at $$(\alpha, \beta)$$, then $$\frac{\beta^4}{\alpha^2}$$ is:

Let $$A = I_2 - 2MM^T$$, where M is real matrix of order $$2 \times 1$$ such that the relation $$M^TM = I_1$$ holds. If $$\lambda$$ is a real number such that the relation $$AX = \lambda X$$ holds for some non-zero real matrix X of order $$2 \times 1$$, then the sum of squares of all possible values of $$\lambda$$ is equal to:

If $$y = \frac{\sqrt{x+1}x^2 - \sqrt{x}}{x\sqrt{x} + x + \sqrt{x}} + \frac{1}{15}(3\cos 2x - 5\cos 3x)$$, then $$96y'\left(\frac{\pi}{6}\right)$$ is equal to:

Let $$f: [0, \infty) \to R$$ and $$F(x) = \int_0^x tf(t) \, dt$$. If $$F(x^2) = x^4 + x^5$$, then $$\sum_{r=1}^{12} f(r^2)$$ is equal to:

Three points $$O(0,0)$$, $$P(a, a^2)$$, $$Q(-b, b^2)$$, $$a > 0$$, $$b > 0$$, are on the parabola $$y = x^2$$. Let $$S_1$$ be the area of the region bounded by the line PQ and the parabola, and $$S_2$$ be the area of the triangle OPQ. If the minimum value of $$\frac{S_1}{S_2}$$ is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to:

The sum of squares of all possible values of $$k$$, for which area of the region bounded by the parabolas $$2y^2 = kx$$ and $$ky^2 = 2(y - x)$$ is maximum, is equal to:

If $$\frac{dx}{dy} = \frac{1 + x - y^2}{y}$$, $$x(1) = 1$$, then $$5x(2)$$ is equal to:

Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = -\hat{i} - 8\hat{j} + 2\hat{k}$$ and $$\vec{c} = 4\hat{i} + c_2\hat{j} + c_3\hat{k}$$ be three vectors such that $$\vec{b} \times \vec{a} = \vec{c} \times \vec{a}$$. If the angle between the vector $$\vec{c}$$ and the vector $$3\hat{i} + 4\hat{j} + \hat{k}$$ is $$\theta$$, then the greatest integer less than or equal to $$\tan^2 \theta$$ is: