If the solution \(y=y(x)\) of the differential equation \(\left(\mathrm{x}^4+2 \mathrm{x}^3+3 \mathrm{x}^2+2 \mathrm{x}+2\right) \mathrm{dy}-\left(2 \mathrm{x}^2+2 \mathrm{x}+3\right) \mathrm{dx}=0\) satisfies \(y(-1)=-\frac{\pi}{4}\), then \(y(0)\) is equal to :

If \(\mathrm{n}\) is the number of solutions of the equation \(2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]\) and \(S\) is the sum of all these solutions, then the ordered pair \((\mathrm{n}, \mathrm{S})\) is :

If all the words (with or without meaning) having five letters, formed using the letters of the word \(SMALL\) and arranged as in a dictionary; then the position of the word \(SMALL\) is :

If \(\sin x=-\frac{3}{5}\), where \(\pi < x < \frac{3 \pi}{2}\) then \(80\left(\tan ^2 x-\cos x\right)\) is equal to :

The function \(f:R \to \left[ { - \frac{1}{2},\frac{1}{2}} \right],\) defined as \(f\left( x \right) = \frac{x}{{1 + {x^2}}}\) is

Let S denote the set of 4-digit numbers abcd such that \( a>b>c>d \) and P denote the set of 5-digit numbers having product of its digits equal to 20. Then \( n(S)+n(P) \) 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 least integral value \(\alpha \) of \(x\) such that \(\frac{{x - 5}}{{{x^2} + 5x - 14}} > 0\) , satisfies

The value of \(\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)\) is

A bag contains \(4\) red and \(6\) black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are re­ turned to the bag. Ifnow a ball is drawn at random from the bag, then the probability that this drawn ball is red, 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 \(P Q\) is :

The sum of all \(3 -\)digit numbers less than or equal to \(500,\) that are formed without using the digit \("1"\) and they all are multiple of \(11 ,\) is ..... .

If the constant term in the binomial expansion of \(\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^9\) is \(-84\) and the Coefficient of \(x^{-3 \ell}\) is \(2^\alpha \beta\), where \(\beta < 0\) is an odd number, Then \(|\alpha \ell-\beta|\) is equal to