If \({ }^{(n-1)} C_r=\left(k^2-3\right){ }^n C_{r+1}\), then an interval containing the values of \(k\), is

The number of positive integers less than 10000 which contain the digit 5 atleast once is

If the normal drawn at a point \(P\) on the curve \(3 y=6 x-5 x^3\) passes through \((0,0)\), then the positive integral value of the abscissa of the point \(P\) is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

\(1-\frac{2}{3}+\frac{2.4}{3.6}-\frac{2.4 .6}{3.6 .9}+\ldots \infty=\)

The minimum value of \(f(x)=\frac{x^2-2 x+3}{x^2-4 x+7}\) is

If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(x^3+3 x^2+4 x\) \(+5=0\), then the cubic equation whose roots are \(1+4 \alpha\), \(1+4 \beta\) and \(1+4 \gamma\) is

\(\begin{aligned} & \frac{x^2+1}{x^4+4}=\frac{A x+B}{x^2-2 x+2}+\frac{C x+D}{x^2+2 x+2} \\ & \Rightarrow 3 A+2 B+3 C=\end{aligned}\)

The area (in square unit) of the triangle formed by \(x+y+1=0\) and the pair of straight lines \(x^2-3 x y+2 y^2=0\) is

Two circles each of radius $5$ units touch each other at $(1,2)$ and $4x+3y=10$ is their common tangent. The equation of that circle among the two given circles, such that some portion of it lies in every quadrant is

Consider the following
Statement-I : Cane sugar is a disaccharide of \(\alpha\)-D-glucose and \(\beta\)-D-fructose
Statement-II : Milk sugar is a diasaccharide of \(\alpha\)-D-glucose and \(\beta\)-D-galactose
The correct answer is

The sides of a triangle are \(3 x+2 y-6=0,2 x-3 y+6=0\) and \(x+2 y+2=0\). If \(P(0, b)\) lies either on the triangle or inside the triangle, then \(b\) lies in the interval

A cylinder of mass \(12 \mathrm{~kg}\) is sliding on plane with an initial velocity \(20 \mathrm{~ms}^{-1}\). If the coefficient of friction between the surface and the cylinder is 0.5 , before stopping, the cylinder describes a distance of