\(f(x)=\left\{\begin{array}{cc}\frac{x-4}{|x-4|}+a, & x < 4 \\ a+b, & x=4 \\ \frac{x-4}{|x-4|}+b, & x>4\end{array}\right.\)
If \(f(x)\) given above is continuous at \(x=4\), then find the values of ' \(a\) ' and ' \(b\) '.

A bag contains \(n\) white and \(n\) black balls. Pairs of balls are drawn at random without replacement successively, until the bag is empty. If the number of ways in which each pair consists of one white and one black ball is 14400 , then \(n\) is equal to

If \(f(x)=10 \cos x+(13+2 x) \sin x, \quad\) then \(f^{\prime \prime}(x)+f(x)\) is equal to

A coin is tossed three times. Let A be the event of "getting three heads" and B be the event of "getting a head on the first toss". Then A and B are

In any triangle \(\mathrm{ABC}, a(b \cos C-c \cos B)=\)

\(A(a, 0)\) is a fixed point and \(\theta\) is a parameter such that \(0 < \theta < 2 \pi\). If \(\mathrm{P}(\mathrm{a} \cos \theta, \mathrm{a} \sin \theta)\) is a point on the circle \(\mathrm{x}^2+\mathrm{y}^2=\mathrm{a}^2\) and \(\mathrm{Q}(\mathrm{b} \sin \theta,-\mathrm{b} \cos \theta)\) is a point on the circle \(x^2+y^2=b^2\), then the locus of the centroid of the triangle APQ is

\((3+\sqrt{8})^5+(3-\sqrt{8})^5=\)

Eight different letters of an alphabet are given. Words of four letters from these are formed. The number of such words with at least one letter repeated is :

Three charges \(\mathrm{Q},+\mathrm{q}\) and \(+\mathrm{q}\) are placed at the vertices of a right angled isoscels triangle as shown in the figure. If the net electrostatic potential energy of the system is zero, the value of \(Q\) is

If \(\Delta_r \mathrm{H}^{\ominus}\) and \(\Delta_r \mathrm{~S}^{\ominus}\) are standard enthalpy change and standard entropy change respectively for a reaction, the incorrect option is

\(\lim _{x \rightarrow 0} \frac{\tan (x)+4 \tan (2 x)-3 \tan (3 x)}{x^2 \tan (x)}\) is equal to

The values of \(\lambda\) and \(\mu\) for which the system of equations \(x+y+z=6, x+2 y+3 z=10, x+2 y+\lambda z=\mu\) has infinitely many solutions are

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)