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
If \( \mathbf{a} = (2x + y)\mathbf{i} + 3\mathbf{j} + 9\mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} + \mathbf{j} - (x - y)\mathbf{k} \) are two collinear vectors, then \( x^3 + 27y^3 = \)
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
In a committee of 25 members, each member is proficient either in Mathematics or in Statistics or in both. If 19 of them are proficient in Mathematics and 16 of them are proficient in Statistics, then the probability that a person selected at random from the committee is proficient in both 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 \( \cos \frac{3\pi}{19} + \cos \frac{5\pi}{19} + \cos \frac{7\pi}{19} + \cdots + \cos \frac{17\pi}{19} = \)
If two angles \( \alpha, \beta \) are such that \( 0<\alpha, \beta<\frac{\pi}{4} \), \( \sqrt{1 + \cos 2\alpha} = \frac{3}{\sqrt{5}} \) and \( \frac{\sqrt{1 - \cos 2\beta}}{\sqrt{1 + \cos 2\beta}} = \frac{1}{7} \), then \( (2\alpha + \beta) = \)
If \( \cosh \alpha + \sinh \alpha = e^x \) and \( \sinh x = \frac{\alpha}{\alpha + 1} \), then \( \tanh x = \)
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 \( \frac{x^4 - 6x^3 + 9x^2 + 5x - 20}{x^2 - x - 2} = f(x) + \frac{a}{x + p} + \frac{b}{x + q} \), then \( 2(a + b) = \)
The coefficient of \( x^r \) in the expansion of \( \frac{1}{\sqrt{(1 - 2x)^3}} \) is
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 \( k = \)
The least value of \( n \) such that \( {}^{n-1}C_6 + {}^{n-1}C_7<{}^nC_8 \) is
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
\( \sin^4 \frac{\pi}{8} + \sin^4 \frac{3\pi}{8} + \sin^4 \frac{5\pi}{8} + \sin^4 \frac{7\pi}{8} = \)
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 \( \cos(\theta - \alpha), \cos \theta \) and \( \cos(\theta + \alpha) \) are in harmonic progression, then \( 2 \tan^2 \theta = \)
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 \( \text{Arg}(z_1) + \text{Arg}(z_2) = \)