List of practice Questions

If \( \vec{u}, \vec{v}, \vec{w} \) are non-coplanar vectors and \( p, q \) are real numbers, then the equality:

\[ [3\vec{u} \quad p\vec{v} \quad p\vec{w}] - [p\vec{v} \quad \vec{w} \quad q\vec{u}] - [2\vec{w} \quad q\vec{v} \quad q\vec{u}] = 0 \]

holds for:

Statement-I: In the interval \( [0, 2\pi] \), the number of common solutions of the equations

\[ 2\sin^2\theta - \cos 2\theta = 0 \]

and

\[ 2\cos^2\theta - 3\sin\theta = 0 \]

is two.

Statement-II: The number of solutions of

\[ 2\cos^2\theta - 3\sin\theta = 0 \]

in \( [0, \pi] \) is two.

If \( A \) and \( B \) are acute angles satisfying

\[ 3\cos^2 A + 2\cos^2 B = 4 \]

and

\[ \frac{3 \sin A}{\sin B} = \frac{2 \cos B}{\cos A}, \]

Then \( A + 2B = \ ? \)

If

\[ \sin \theta + 2 \cos \theta = 1 \]

and

\[ \theta \text{ lies in the 4\textsuperscript{th} quadrant (not on coordinate axes), then } 7 \cos \theta + 6 \sin \theta =\ ? \]

If

\[ \cos x + \sin x = \frac{1}{2} \]

and

\[ 0 < x < \pi, \text{ then } \tan x = \ ? \]

Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)

Then consider:

Statement-I: \( z \) is a real number

Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)

Then:

If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \) and

\[ \arg(z) - \arg(\omega) = \frac{\pi}{2}, \]

Then the value of \( \overline{z\omega} \) is:

If \( x^a y^b = e^m, \)

and

\[ x^c y^d = e^n, \]

and

\[ \Delta_1 = \begin{vmatrix} m & b \\ n & d \\ \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} a & m \\ c & n \\ \end{vmatrix}, \quad \Delta_3 = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} \]

Then the values of \( x \) and \( y \) respectively (where \( e \) is the base of the natural logarithm) are:

In \(\triangle ABC\), if \(A = 30^\circ\) and

\[ \frac{b}{(\sqrt{3}+1)^2 + 2(\sqrt{2} - 1)}, \quad \frac{c}{(\sqrt{3}+1)^2 - 2(\sqrt{2} - 1)}, \]

Then find the angle \(B\).

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be position vectors of three non-collinear points on a plane. If

\[ \alpha = \left[\mathbf{a} \quad \mathbf{b} \quad \mathbf{c}\right] \text{ and } \mathbf{r} = \mathbf{a} \times \mathbf{b} - \mathbf{c} \times \mathbf{b} - \mathbf{a} \times \mathbf{c}, \]

Then \(\frac{|\alpha|}{|\mathbf{r}|}\) represents:

If

\[ P = (a \times \mathbf{i})^2 + (a \times \mathbf{j})^2 + (a \times \mathbf{k})^2 \]

and

\[ Q = (a \cdot \mathbf{i})^2 + (a \cdot \mathbf{j})^2 + (a \cdot \mathbf{k})^2, \]

Then find the relation between \(P\) and \(Q\).

Given vectors \(\mathbf{a} = \mathbf{i} + \mathbf{j} - 2\mathbf{k}\), \(\mathbf{b} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\), \(\mathbf{c} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}\), and \(\mathbf{r}\) such that

\[ \mathbf{r} \cdot \mathbf{a} = 0, \\ \mathbf{r} \cdot \mathbf{c} = 3, \\ [\mathbf{r} \quad \mathbf{a} \quad \mathbf{b}] = 0, \]

Then find \(|\mathbf{r}|\).