Let \(f\) be a twice differentiable function defined on \(R\) such that \(f (0)=1, f ^{\prime}(0)=2\) and \(f ^{\prime}( x ) \neq 0\) for all \(x \in R\). If \(\left|\begin{array}{ll}f(x) & f^{\prime}(x) \\ f^{\prime}(x) & f^{\prime \prime}(x)\end{array}\right|=0,\) for all \(x \in R,\) then the value of \(f (1)\) lies in the interval:

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

If a hyperbola passes through the point \(\mathrm{P}(10,16)\) and it has vertices at \((\pm 6,0),\) then the equation of the normal to it at \(P\) is

Consider two G.Ps. \(2,2^{2}, 2^{3}, \ldots\) and \(4,4^{2}, 4^{3}, \ldots\) of \(60\) and \(n\) terms respectively. If the geometric mean of all the \(60+n\) terms is \((2)^{\frac{225}{8}}\), then \(\sum_{ k =1}^{ n } k (n- k )\) is equal to.

Total number of \(6-\)digit numbers in which only and all the five digits \(1,3,5,7\) and \(9\) appear, is 

Let \(\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4\) be the solution of the equation \(4 x^4+8 x^3-17 x^2-12 x+9=0\) and \(\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m\). Then the value of \(\mathrm{m}\) is ..........

If \(x^{3} d y+x y d x=x^{2} d y+2 y d x ; y(2)=e\) and \(x\) \(>1,\) then \(y (4)\) is equal to

The circle passing through \(( 1, -2)\) and touching the axis of \(x\) at \((3, 0)\) also passes through the point

Let \(y=y(x)\) be the solution of the differential equation \(\sec \mathrm{x} d \mathrm{y}+\{2(1-\mathrm{x}) \tan \mathrm{x}+\mathrm{x}(2-\mathrm{x})\}\) \(\mathrm{dx}=0\) such that \(\mathrm{y}(0)=2\). Then \(\mathrm{y}(2)\) is equal to :

Let \(A\) be a \(3 \times 3\) real matrix such that \(A \left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right) ; A \left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{c}-1 \\ 0 \\ 1\end{array}\right)\) and \(A \left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)\). If \(X =\left( x _{1}, x _{2}, x _{3}\right)^{ T }\) and \(I\) is an identity matrix of order \(3\) , then the system \(( A -2 I ) X =\left(\begin{array}{l}4 \\ 1 \\ 1\end{array}\right)\) has

The sum of the first ten terms of an A.P. is \(160\) and the sum of the first two terms of a G.P. is \(8\). If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

If \(0 < a , b < 1,\) and \(\tan ^{-1} a +\tan ^{-1} b =\frac{\pi}{4},\) then the value of \((a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots\) is ..... .

The equation of circle described on the chord \(3x + y+ 5\, = 0\) of the circle \(x^2 + y^2\, = 16\) as diameter is