\(\mathbf{A B}=\mathbf{a}\) and \(\mathbf{A C}=\mathbf{b}\) are the sides of \(\mathbf{a} \triangle A B C . P\) is a point on \(\mathbf{A B}\) and \(Q\) is a point on \(\mathbf{B C}\) such that \(\frac{A P}{P B}=\frac{1}{2}\) and \(\frac{B Q}{Q C}=\frac{1}{2}\). If the point of intersection of \(\mathbf{A Q}\) and \(\mathbf{C P}\) is \(D\) and the area of \(\triangle B C D\) is 7 square units, then the area of the \(\triangle A B C\) (in the same sq units) is

If $A, B, C$ are the angles of a triangle then the system of equations $-x+y\cos C+z\cos B=0, x\cos C-y+z\cos A=0, x\cos B+y\cos A-z=0$ have

The term independent of \(x(x>0, x \neq 1)\) in the expansion of \(\left[\frac{(x+1)}{\left(x^{2 / 3}-x^{1 / 3}+1\right)}-\frac{(x-1)}{(x-\sqrt{x})}\right]^{10}\)

If the equation
\(3 x^2+7 x y+2 y^2+2 g x+2 f y+2=0\) represents
a pair of intersecting lines and the square of the distance of their point of intersection from the origin is \(\frac{2}{5}\), then \(f^2+g^2=\)

If \(y=t^2+t^3\) and \(x=t-t^4\) then \(\frac{d^2 y}{d x^2}\) at \(t=1\) is

If \(\mathbf{a}\) and \(\mathbf{b}\) are two non-zero perpendicular vectors, then a vector \(\mathbf{y}\) satisfying equations \(\mathbf{a} \cdot \mathbf{y}=c\) (where, \(c\) is scalar) and \(\mathbf{a} \times \mathbf{y}=\mathbf{b}\) is

If \(A=(2,3,4)\) and \(B=(-2,3,4)\), then the locus of a point \(P\) such that \(\mathrm{PA}+\mathrm{PB}=4\) is

Which of the following sets are correctly matched?

Metal		Refining process
I)	Hg	Distillation
II)	Cu	Poling
III)	B	Zone refining
IV)	Ti	Liquation

If the shortest distance from \((2,-14)\) to the circle \(x^2+y^2+6 x+4 y-12=0\) is \(d\) and the length of the tangent drawn from the same point to the circle is \(l\), then \(\sqrt{d+l}=\)

When an inductor of inductance \(\frac{6}{\pi} \mathrm{H}\), a capacitor of capacitance \(\frac{50}{\pi} \mu \mathrm{F}\) and resistor of resistance \(R\) are connected in series with an AC supply of rms voltage \(220 \mathrm{~V}\) and frequency \(50 \mathrm{~Hz}\), the rms current through the circuit is \(440 \mathrm{~mA}\). Match the inductive reactance, \(X_L\) the capacitive reactance, \(X_C\) the resistance \(R\) and the impedance \(Z\) of the circuit given in List-I with the corresponding values given in List-II.
\(\begin{array}{llll} \hline & \text { List- I } & & \text { List- II } \\ \hline \text {(A) } & X_L & \text { (i) } & 200 \Omega \\ \hline \text {(B) } & X_C & \text { (ii) } & 300 \Omega \\ \hline \text {(C) } & R & \text { (iii) } & 500 \Omega \\ \hline \text {(D) } & Z & \text { (iv) } & 600 \Omega \\ \hline \end{array}\)

In \(\triangle A B C\), if \(a: b: c=4: 5: 6\), then the ratio between the circumradius and the inradius is

If \(\mathrm{m}, \mathrm{n}\) are respectively the least positive and greatest negative integer values of \(k\) such that \(\left(\frac{1-i}{1+i}\right)^k=-i\), then \(m-n=\)

The acute angle between the lines \(x-y=0\), and \(y=0\) is