List of top Mathematics Questions

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
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 \( A_1 : A_2 = \)
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 \( 3p + q = \)
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 \( x^3 + 27y^3 = \)
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 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 \( \frac{x^4 - 6x^3 + 9x^2 + 5x - 20}{x^2 - x - 2} = f(x) + \frac{a}{x + p} + \frac{b}{x + q} \), then \( 2(a + b) = \)
In \( \triangle ABC \), \( \angle B = 60^\circ \) and \( \angle A = 75^\circ \). If a point \( D \) divides \( BC \) in the ratio \( 2:3 \), then \( \sin \angle BAD : \sin \angle CAD = \)
In \( \triangle ABC \), if \( r_1 = 2r_2 = 3r_3 \), then
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 \( \tan \frac{A}{2} + \tan \frac{C}{2} = \frac{b}{s} \), then \( \sin \left( \frac{A + C}{3} \right) = \)
The coefficient of \( x^r \) in the expansion of \( \frac{1}{\sqrt{(1 - 2x)^3}} \) 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
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
\( \sin^4 \frac{\pi}{8} + \sin^4 \frac{3\pi}{8} + \sin^4 \frac{5\pi}{8} + \sin^4 \frac{7\pi}{8} = \)
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 the roots of the equation \( 16x^3 - 44x^2 + 36x - 9 = 0 \) are in harmonic progression, then its greatest root is
If \( 2^n \) divides \( 16! \) and \( 2^{n+1} \) does not divide \( 16! \), then \( n = \)
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 \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 - 4x + 3 = 0 \), then \( \frac{2(\alpha^4 + \beta^4) + 3(\alpha^2 + \beta^2)}{\alpha + \beta} = \)