NTA JEE Main 8th April2023 Shift 1 - Mathematics

Let $$\alpha, \beta, \gamma$$ be the three roots of the equation $$x^3 + bx + c = 0$$ if $$\beta\gamma = 1 = -\alpha$$ then $$b^3 + 2c^3 - 3\alpha^3 - 6\beta^3 - 8\gamma^3$$ is equal to

If for $$z = \alpha + i\beta$$, $$|z + 2| = z + 4(1+i)$$, then $$\alpha + \beta$$ and $$\alpha\beta$$ are the roots of the equation

The number of arrangements of the letters of the word 'INDEPENDENCE' in which all the vowels always occur together is

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is

Let $$S_K = \dfrac{1+2+\ldots+K}{K}$$ and $$\sum_{j=1}^n S_j^2 = \dfrac{n}{A}(Bn^2 + Cn + D)$$ where $$A, B, C, D \in N$$ and $$A$$ has least value, then

If the coefficients of three consecutive terms in the expansion of $$(1+x)^n$$ are in the ratio 1:5:20 then the coefficient of the fourth term is

Let $$C(\alpha, \beta)$$ be the circumcentre of the triangle formed by the lines $$4x + 3y = 69$$, $$4y - 3x = 17$$, and $$x + 7y = 61$$. Then $$(\alpha - \beta)^2 + \alpha + \beta$$ is equal to

Let $$R$$ be the focus of the parabola $$y^2 = 20x$$ and the line $$y = mx + c$$ intersect the parabola at two points P and Q. Let the points G(10, 10) be the centroid of the triangle PQR. If $$c - m = 6$$, then $$PQ^2$$ is

$$\lim_{x \to 0} \left(\left(\dfrac{1-\cos^2(3x)}{\cos^3(4x)}\right)\left(\dfrac{\sin^3(4x)}{(\log_e(2x+1))^5}\right)\right)$$ is equal to

Negation of $$(p \to q) \to (q \to p)$$ is

Let the number of elements in sets $$A$$ and $$B$$ be five and two respectively. Then the number of subsets of $$A \times B$$ each having at least 3 and at most 6 elements is

Let $$\begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$. If $$|adj(adj(adj(2A)))| = (16)^n$$, then $$n$$ is equal to

Let $$P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix}$$, $$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ and $$Q = PAP^T$$. If $$P^TQ^{2007}P = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ then $$2a + b - 3c - 4d$$ is equal to

Let $$f(x) = \dfrac{\sin x + \cos x - \sqrt{2}}{\sin x - \cos x}$$, $$x \in [0, \pi] - \{\dfrac{\pi}{4}\}$$, then $$f\left(\dfrac{7\pi}{12}\right) f''\left(\dfrac{7\pi}{12}\right)$$ is equal to

Let $$I(x) = \int \dfrac{x+1}{x(1+xe^x)^2} dx$$, $$x > 0$$. If $$\lim_{x \to \infty} I(x) = 0$$ then $$I(1)$$ is equal to

The area of the region $$\{(x,y): x^2 \le y \le 8-x^2, y \le 7\}$$ is

If the points with position vectors $$\alpha\hat{i} + 10\hat{j} + 13\hat{k}$$, $$6\hat{i} + 11\hat{j} + 11\hat{k}$$, $$\dfrac{9}{2}\hat{i} + \beta\hat{j} - 8\hat{k}$$ are collinear, then $$(19\alpha - 6\beta)^2$$ is equal to

The shortest distance between the lines $$\dfrac{x-4}{4} = \dfrac{y+2}{5} = \dfrac{z+3}{3}$$ and $$\dfrac{x-1}{3} = \dfrac{y-3}{4} = \dfrac{z-4}{2}$$ is

If the equation of the plane containing the line $$x + 2y + 3z - 4 = 0 = 2x + y - z + 5$$ and perpendicular to the plane $$\vec{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})$$ is $$ax + by + cz = 4$$ then $$(a - b + c)$$ is equal to

In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found defective then the probability that it is manufactured by the machine C is

The largest natural number $$n$$ such that $$3n$$ divides 66! is ______.

Let $$[t]$$ denote the greatest integer $$\le t$$. If the constant term in the expansion of $$\left(3x^2 - \dfrac{1}{2x^5}\right)^7$$ is $$\alpha$$ then $$[\alpha]$$ is equal to ______.

Consider a circle $$C_1: x^2 + y^2 - 4x - 2y = \alpha - 5$$. Let its mirror image in the line $$y = 2x + 1$$ be another circle $$C_2: 5x^2 + 5y^2 - 10fx - 10gy + 36 = 0$$. Let $$r$$ be the radius of $$C_2$$. Then $$\alpha + r$$ is equal to ______.

Let the mean and variance of 8 numbers x, y, 10, 12, 6, 12, 4, 8 be 9 and 9.25 respectively. If $$x > y$$, then $$3x - 2y$$ is equal to ______.

Let $$A = \{0, 3, 4, 6, 7, 8, 9, 10\}$$ and $$R$$ be the relation defined on $$A$$ such that $$R\{(x,y) \in A \times A: x-y$$ is odd positive integer or $$x-y = 2\}$$. The minimum number of elements that must be added to the relation $$R$$, so that it is a symmetric relation, is equal to ______.

If $$a_\alpha$$ is the greatest term in the sequence $$a_n = \dfrac{n^3}{n^4 + 147}$$, $$n = 1, 2, 3, \ldots$$, then $$\alpha$$ is equal to ______.

Let $$[t]$$ denote the greatest integer $$\le t$$. Then $$\dfrac{2}{\pi} \int_{\pi/6}^{5\pi/6} (8[\csc x] - 5[\cot x]) dx$$ is equal to ______.

If the solution curve of the differential equation $$(y-2\log_e x)dx + (x\log_e x^2)dy = 0$$, $$x \gt 1$$ passes through the points $$(e, \dfrac{4}{3})$$ and $$(e^4, \alpha)$$, then $$\alpha$$ is equal to ______.

Let $$\vec{a} = 6\hat{i} + 9\hat{j} + 12\hat{k}$$, $$\vec{b} = \alpha\hat{i} + 11\hat{j} - 2\hat{k}$$ and $$\vec{c}$$ be vectors such that $$\vec{a} \times \vec{c} = \vec{a} \times \vec{b}$$. If $$\vec{a} \cdot \vec{c} = -12$$, and $$\vec{c} \cdot (\hat{i} - 2\hat{j} + \hat{k}) = 5$$ then $$\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k})$$ is equal to ______.

Let $$\lambda_1, \lambda_2$$ be the values of $$\lambda$$ for which the points $$\left(\dfrac{5}{2}, 1, \lambda\right)$$ and $$(-2, 0, 1)$$ are at equal distance from the plane $$2x + 3y - 6z + 7$$. If $$\lambda_1 > \lambda_2$$ then the distance of the point $$(\lambda_1 - \lambda_2, \lambda_2, \lambda_1)$$ from the line $$\dfrac{x-5}{1} = \dfrac{y-1}{2} = \dfrac{z+7}{2}$$ is ______.