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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 the defective, then the probability that it is manufactured by the machine \( C \) is:

and \( Q= PAP^{T} \). If \( P^{T}Q^{2007}P\), then \( 2a+b-3c-4d \) 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 for \( z=\alpha+i\beta \), \( |z+2|=z+4(1+i) \), then \( \alpha+\beta \) and \( \alpha\beta \) are the roots of the equation.

Let \( S_{K}=\dfrac{1+2+\ldots +K}{K} \) and \(\displaystyle \sum_{j=1}^{n} S_{j}^{2}=\dfrac{n}{A}(Bn^{2}+Cn+D) \), where \( A, B, C, D\in \mathbb{N} \) and \( A \) has least value. Then:

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:

The area of the region \( \left\{(x,y): x^2 \leq y \leq 8 - x^2,\; y \leq 7 \right\} \) is:

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 point \( G(10, 10) \) be the centroid of the triangle \( PQR \). If \( c - m = 6 \), then \( (PQ)^2 \) is:

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

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 \( I(x) = \displaystyle \int \frac{(x+1)}{x(1 + x e^{x})^{2}}\,dx,\; x > 0. \) If \( \displaystyle \lim_{x \to \infty} I(x) = 0 \), then \( I(1) \) is equal to:

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:

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together 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:

Negation of \( (p \Rightarrow q) \Rightarrow (q \Rightarrow p) \) is:

Let \( C(\alpha, \beta) \) be the circumcenter 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:

\(\displaystyle \lim_{x \to 0} \left( \left( \frac{(1 - \cos^{2}(3x))}{\cos^{3}(4x)} \right) \left( \frac{(\sin^{3}(4x)}{(\log_{e}(2x+1))^{5}} \right) \right) \) is equal to:

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:

. If \( \left|\operatorname{adj}\!\left(\operatorname{adj}\!\left(\operatorname{adj}(2A)\right)\right)\right| = (16)^n \), then \( n \) is equal to:

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:

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 \( \vec{a}\cdot\vec{c}=-12, \; \vec{c}\cdot(\hat{i}-2\hat{j}+\hat{k})=5 \), then \( \vec{c}\cdot(\hat{i}+\hat{j}+\hat{k}) \) is equal to ________.

If the solution curve of the differential equation \( (y-2\log_e x)\,dx+(x\log_e x^2)\,dy=0,\; x>1 \) passes through the points \( \left(e,\dfrac{4}{3}\right) \) and \( (e^4,\alpha) \), then \( \alpha \) 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 \( 2x+3y-6z+7=0 \). 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 ________.

Let \( [t] \) denote the greatest integer \( \le t \). Then \( \dfrac{2}{\pi}\displaystyle\int_{\pi/6}^{5\pi/6}\big(8[\operatorname{cosec}x]-5[\cot x]\big)\,dx \) is equal to ________.

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 ________.

The largest natural number \( n \) such that \( 3^{n} \) divides \( 66! \) is ________.

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

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 ________.

JEE Main 2023 8th April Morning Shift

Physics Chemistry Mathematics

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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: