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

The Coefficient of \(x ^{-6}\), in the expansion of \(\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9\), is \(........\).

A plane which bisects the angle between the two given planes \(2x -y + 2z -4 = 0\) and \(x + 2y + 2z -2 = 0\), passes through the point

The equation of a normal to the curve \(\sin \,y = x\,\sin \,\left( {\frac{\pi }{3} + y} \right)\) at \(x\, = 0\), is 

Let [ ] denote the greatest integer function and \( f(x)=lim_{n\rightarrow\infty}\frac{1}{n^{3}}\sum_{k=1}^{n}[\frac{k^{2}}{3^{x}}] \) Then \( 12\sum_{j=1}^{x}f(j) \) is equal to ........... .

If \(\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\ldots \frac{1}{2^{10} \cdot 3}=\frac{K}{2^{10} \cdot 3^{10}}\), then the remainder when \(K\) is divided by \(6\) is

Let \(3,6,9,12, \ldots\) upto \(78\) terms and \(5,9,13,17, \ldots\) upto \(59\) terms be two series. Then, the sum of the terms common to both the series is equal to

Let \(y=y(x)\) be a solution of the differential equation \((x \cos x) d y+(x y \sin x+y \cos x-1) d x=0\), \(0 < x < \frac{\pi}{2}\). If \(\frac{\pi}{3} y\left(\frac{\pi}{3}\right)=\sqrt{3}\), then \(\left|\frac{\pi}{6} y^{\prime \prime}\left(\frac{\pi}{6}\right)+2 y^{\prime}\left(\frac{\pi}{6}\right)\right|\) is equal to \(.........\).

Let, \(\alpha, \beta\) be the distinct roots of the equation \(\mathrm{x}^2-\left(\mathrm{t}^2-5 \mathrm{t}+6\right) \mathrm{x}+1=0, \mathrm{t} \in \mathrm{R}\) and \(\mathrm{a}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\). Then the minimum value of \(\frac{\mathrm{a}_{2023}+\mathrm{a}_{2025}}{\mathrm{a}_{2024}}\) is

Let he sum of the coefficient of first three terms in the expansion of \(\left(x-\frac{3}{x^2}\right)^n ; x\ne 0, n \in N\) be 376 . Then, the coefficient of \(x^4\) is equal to:

If the area of the region \(S=\left\{(x, y): 2 y-y^2 \leq x^2 \leq 2 y, x \geq y\right\}\) is equal to \(\frac{ n +2}{ n +1}-\frac{\pi}{ n -1}\), then the natural number \(n\) is equal to \(...............\).

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

The number of functions \(f\), from the set\(A=\left\{x \in N: x^{2}-10 x+9 \leq 0\right\}\) to the set \(B=\left\{n^{2}: n \in N\right\}\) such that \(f(x) \leq(x-3)^{2}+1\), for every \(x \in A\), is.