NTA JEE Mains 27th Jan 2024 Shift 1 - Mathematics

If $$S = \{z \in \mathbb{C} : |z - i| = |z + i| = |z - 1|\}$$, then $$n(S)$$ is:

The number of common terms in the progressions $$4, 9, 14, 19, \ldots$$ up to $$25^{th}$$ term and $$3, 6, 9, 12, \ldots$$ up to $$37^{th}$$ term is :

If $$A$$ denotes the sum of all the coefficients in the expansion of $$(1 - 3x + 10x^2)^n$$ and $$B$$ denotes the sum of all the coefficients in the expansion of $$(1 + x^2)^n$$, then :

$${}^{n-1}C_r = (k^2 - 8) \; {}^{n}C_{r+1}$$ if and only if :

The portion of the line $$4x + 5y = 20$$ in the first quadrant is trisected by the lines $$L_1$$ and $$L_2$$ passing through the origin. The tangent of an angle between the lines $$L_1$$ and $$L_2$$ is :

Four distinct points $$(2k, 3k), (1, 0), (0, 1)$$ and $$(0, 0)$$ lie on a circle for $$k$$ equal to :

If the shortest distance of the parabola $$y^2 = 4x$$ from the centre of the circle $$x^2 + y^2 - 4x - 16y + 64 = 0$$ is $$d$$, then $$d^2$$ is equal to :

The length of the chord of the ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$, whose mid point is $$(1, \frac{2}{5})$$, is equal to:

If $$a = \lim_{x \to 0} \frac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4}$$ and $$b = \lim_{x \to 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$$, then the value of $$ab^3$$ is :

Let $$a_1, a_2, \ldots, a_{10}$$ be 10 observations such that $$\sum_{k=1}^{10} a_k = 50$$ and $$\sum_{\forall k < j} a_k \cdot a_j = 1100$$. Then the standard deviation of $$a_1, a_2, \ldots, a_{10}$$ is equal to :

Let $$S = \{1, 2, 3, \ldots, 10\}$$. Suppose $$M$$ is the set of all the subsets of $$S$$, then the relation $$R = \{(A, B) : A \cap B \neq \phi; \; A, B \in M\}$$ is :

Consider the matrix $$f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$$. Given below are two statements : Statement I: $$f(-x)$$ is the inverse of the matrix $$f(x)$$. Statement II: $$f(x) f(y) = f(x + y)$$. In the light of the above statements, choose the correct answer from the options given below

The function $$f : \mathbb{N} - \{1\} \rightarrow \mathbb{N}$$; defined by $$f(n) =$$ the highest prime factor of $$n$$, is :

Consider the function $$f(x) = \begin{cases} \frac{a(7x - 12 - x^2)}{b|x^2 - 7x + 12|}, & x < 3 \\ 2^{\frac{\sin(x-3)}{x - [x]}}, & x > 3 \\ b, & x = 3 \end{cases}$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$. If $$S$$ denotes the set of all ordered pairs $$(a, b)$$ such that $$f(x)$$ is continuous at $$x = 3$$, then the number of elements in $$S$$ is :

If $$\int_0^1 \frac{1}{\sqrt{3+x} + \sqrt{1+x}} \, dx = a + b\sqrt{2} + c\sqrt{3}$$, where $$a, b, c$$ are rational numbers, then $$2a + 3b - 4c$$ is equal to :

If $$(a, b)$$ be the orthocentre of the triangle whose vertices are $$(1, 2), (2, 3)$$ and $$(3, 1)$$, and $$I_1 = \int_a^b x \sin(4x - x^2) \, dx$$, $$I_2 = \int_a^b \sin(4x - x^2) \, dx$$, then $$36 \frac{I_1}{I_2}$$ is equal to :

Let $$x = x(t)$$ and $$y = y(t)$$ be solutions of the differential equations $$\frac{dx}{dt} + ax = 0$$ and $$\frac{dy}{dt} + by = 0$$ respectively, $$a, b \in \mathbb{R}$$. Given that $$x(0) = 2$$; $$y(0) = 1$$ and $$3y(1) = 2x(1)$$, the value of $$t$$, for which $$x(t) = y(t)$$, is :

If $$\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$$, $$\vec{b} = 3(\hat{i} - \hat{j} + \hat{k})$$ and $$\vec{c}$$ be the vector such that $$\vec{a} \times \vec{c} = \vec{b}$$ and $$\vec{a} \cdot \vec{c} = 3$$, then $$\vec{a} \cdot ((\vec{c} \times \vec{b}) - \vec{b} - \vec{c})$$ is equal to

The distance, of the point $$(7, -2, 11)$$ from the line $$\frac{x-6}{1} = \frac{y-4}{0} = \frac{z-8}{3}$$ along the line $$\frac{x-5}{2} = \frac{y-1}{-3} = \frac{z-5}{6}$$, is :

If the shortest distance between the lines $$\frac{x-4}{1} = \frac{y+1}{2} = \frac{z}{-3}$$ and $$\frac{x-\lambda}{2} = \frac{y+1}{4} = \frac{z-2}{-5}$$ is $$\frac{6}{\sqrt{5}}$$, then the sum of all possible values of $$\lambda$$ is :

If $$\alpha$$ satisfies the equation $$x^2 + x + 1 = 0$$ and $$(1 + \alpha)^7 = A + B\alpha + C\alpha^2$$, $$A, B, C \geq 0$$, then $$5(3A - 2B - C)$$ is equal to _______.

If $$8 = 3 + \frac{1}{4}(3 + p) + \frac{1}{4^2}(3 + 2p) + \frac{1}{4^3}(3 + 3p) + \ldots \infty$$, then the value of $$p$$ is _______.

Let the set of all $$a \in \mathbb{R}$$ such that the equation $$\cos 2x + a \sin x = 2a - 7$$ has a solution be $$[p, q]$$ and $$r = \tan 9° - \tan 27° - \frac{1}{\cot 63°} + \tan 81°$$, then $$pqr$$ is equal to _______.

Let $$A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$$, $$B = [B_1 \; B_2 \; B_3]$$, where $$B_1, B_2, B_3$$ are column matrices, and $$AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$, $$AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$$, $$AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$. If $$\alpha = |B|$$ and $$\beta$$ is the sum of all the diagonal elements of $$B$$, then $$\alpha^3 + \beta^3$$ is equal to _______.

Let $$f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3), \; x \in \mathbb{R}$$. Then $$f'(10)$$ is equal to _______.

Let for a differentiable function $$f : (0, \infty) \rightarrow \mathbb{R}$$, $$f(x) - f(y) \geq \log_e\left(\frac{x}{y}\right) + x - y, \; \forall x, y \in (0, \infty)$$. Then $$\sum_{n=1}^{20} f'\left(\frac{1}{n^2}\right)$$ is equal to _______.

Let the area of the region $$\{(x, y) : x - 2y + 4 \geq 0, \; x + 2y^2 \geq 0, \; x + 4y^2 \leq 8, \; y \geq 0\}$$ be $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime numbers. Then $$m + n$$ is equal to _______.

If the solution of the differential equation $$(2x + 3y - 2)dx + (4x + 6y - 7)dy = 0$$, $$y(0) = 3$$, is $$\alpha x + \beta y + 3\log_e|2x + 3y - \gamma| = 6$$, then $$\alpha + 2\beta + 3\gamma$$ is equal to _______.

The least positive integral value of $$\alpha$$, for which the angle between the vectors $$\alpha\hat{i} - 2\hat{j} + 2\hat{k}$$ and $$\alpha\hat{i} + 2\alpha\hat{j} - 2\hat{k}$$ is acute, is _______.

A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required and let $$a = P(X = 3)$$, $$b = P(X \geq 3)$$ and $$c = P(X \geq 6 \mid X > 3)$$. Then $$\frac{b + c}{a}$$ is equal to _______.