List of practice Questions

If \( A = (2, 3) \) and \( B = (-4, 5) \) are two fixed points, then the locus of a point \( P \) such that the area of \( \triangle PAB \) is 12 square units is

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

If a point \( C \) divides the line segment joining the points with position vectors \( 2\mathbf{i} - 3\mathbf{j} + 2\mathbf{k} \) and \( 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} \) in the ratio \( 2:3 \), then the distance of \( C \) from the point with position vector \( 2\mathbf{i} - \mathbf{j} + \mathbf{k} \) is

Let \( \mathbf{a} = \mathbf{i} - 2\mathbf{j} \), \( \mathbf{b} = 2\mathbf{j} + 3\mathbf{k} \), \( \mathbf{c} = p\mathbf{i} + q\mathbf{j} \) and \( \mathbf{d} = p\mathbf{j} - q\mathbf{k} \) be four vectors. If \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = 3 \) and \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{d} = 0 \), then \( 3p + q = \)

Let \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) and \( \mathbf{b} = \mathbf{i} - 2\mathbf{j} - 3\mathbf{k} \) be two vectors. If \( A_1 \) is the area of the quadrilateral having \( \mathbf{a}, \mathbf{b} \) as its diagonals and \( A_2 \) is the area of the parallelogram having \( \mathbf{a}, \mathbf{b} \) as its adjacent sides, then \( A_1 : A_2 = \)

For some real number \( \lambda \), if the area of the triangle having \( \mathbf{a} = \lambda \mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} + \lambda \mathbf{j} - 3\mathbf{k} \) as two of its sides is \( \frac{\sqrt{195}}{2} \), then the number of distinct possible values of \( \lambda \) is

If the sum of squares of the deviations from the mean of the data \( x_i \) (\( i = 1, 2, \dots, n \)) is \( n\overline{x}^2 \), where \( \overline{x} \) is the mean of \( x_i \)'s, then the sum of squares of \( x_i \)'s is

If \( \cos A + \cos(A + B) + \cos(A + 2B) + \cdots \) upto \( n \) terms \( = \cos \left( \frac{2A + (n-1)B}{2} \right) \frac{\sin \frac{nB}{2}}{\sin \frac{B}{2}} \), then \( \cos \frac{3\pi}{19} + \cos \frac{5\pi}{19} + \cos \frac{7\pi}{19} + \cdots + \cos \frac{17\pi}{19} = \)

In \( \triangle ABC \), if \( r_1 = 2r_2 = 3r_3 \), then

In \( \triangle ABC \), if \( \tan \frac{A}{2} + \tan \frac{C}{2} = \frac{b}{s} \), then \( \sin \left( \frac{A + C}{3} \right) = \)

In \( \triangle ABC \), \( \angle B = 60^\circ \) and \( \angle A = 75^\circ \). If a point \( D \) divides \( BC \) in the ratio \( 2:3 \), then \( \sin \angle BAD : \sin \angle CAD = \)

If \( 3 \cdot {}^5C_0 + 8 \cdot {}^5C_1 + 13 \cdot {}^5C_2 + 18 \cdot {}^5C_3 + 23 \cdot {}^5C_4 + 28 \cdot {}^5C_5 = k \cdot 2^5 \), then \( k = \)

If a seven digit number formed with distinct digits 4, 6, 9, 5, 3, \( x \) and \( y \) is divisible by 3, then the number of such ordered pairs \( (x, y) \) is

In \( \triangle ABC \), if \( \cos A \cos B \cos C = \frac{1}{5} \), then \( \tan A \tan B + \tan B \tan C + \tan C \tan A = \)

If \( \omega \) is a complex cube root of unity, then \( \cos \left( \sum_{k=1}^{2} (k - \omega)(k - \omega^2) \frac{\pi}{175} \right) = \)

Let the two values of \( z = \frac{1 - i}{\sqrt{1 + i}} \) be \( z_1 \) and \( z_2 \). If \( -\frac{\pi}{2}<\text{Arg}(z_1)<\text{Arg}(z_2)<\pi \), then \( \text{Arg}(z_1) + \text{Arg}(z_2) = \)

If \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then the equation whose roots are \( \alpha + \beta \) and \( \frac{1}{\alpha} + \frac{1}{\beta} \) is

If \( 2^n \) divides \( 16! \) and \( 2^{n+1} \) does not divide \( 16! \), then \( n = \)

If \( z_1 = 2 - 3i \) and the roots of the equation \( z^3 + bz^2 + cz + d = 0 \) are \( i, z_1 \) and \( \overline{z_1} \), then \( b + c + d = \)

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \( f(x) = \frac{2x+1}{3} \). If \( \alpha \) is an element in the domain of \( f \) whose image is \( \frac{1}{\alpha} \), then the sum of all possible values of such \( \alpha \) is

What is the intermediate in the following reaction?
RCN + SnCl$_2$ + HCl $\longrightarrow$ RCHO

Which of the following is/are correct?
i. Cr$_2$O$_3$ is an acidic oxide
ii. Mo(VI) and W(VI) are more stable than Cr(VI)
iii. All Cu(II) halides are known except iodide
iv. The E$^\circ$ value for Cr$^{3+}$/Cr$^{2+}$ is much more positive than Mn$^{3+}$/Mn$^{2+}$

Identify the correct statements about order of reaction:
I) It is equal to the sum of the powers of concentration terms in rate equation
II) It is always a whole number
III) It is determined experimentally
IV) It is not applicable for complex reactions