List of top Mathematics Questions

The range of the expression: \[ \frac{1}{\sin^2x + 3\sin x \cos x + 5\cos^2x} \] is:

If \( \cosh x = \frac{5}{4} \), then \( \tanh 3x = \)

In \( \triangle ABC \), if \( \sin B = \sin C \) and \( 3 \cos B = 2 \cos C \), then \( \triangle ABC \) is:

If \( \triangle ABC \) is a right-angled isosceles triangle and \( \angle C = 90^\circ \), then \( r = \frac{1}{5} \) is:

In \( \triangle ABC \), if \[ a \cos^2 \frac{C}{2} + \cos^2 \frac{A}{2} = \frac{3b}{2}, \] then \( a + c : b \) is:

If \[ \frac{x + 2}{x^2 - 3} \text{ is one of the partial fractions of } \frac{3x^3 - x^2 - 2x + 17}{x^4 + x^2 - 12}, \text{ then the other partial fraction of it is:} \]

If \( 7i - 4j - 5k \) is the position vector of vertex A of a tetrahedron ABCD and \( -i + 4j - 3k \) is the position vector of the centroid of the triangle BCD, then the position vector of the centroid of the tetrahedron ABCD is:

Let \( \mathbf{OA = i + 2j - 2k} \) and \( \mathbf{OB = -2i - 3j + 6k} \) be the position vectors of two points A and B. If C is a point on the bisector of \( \angle AOB \) and \( OC = \frac{\sqrt{42}}{2} \), then \( OC = \)

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}| = \sqrt{14} \), \( |\vec{b}| = \sqrt{14} \), and \( \vec{a} \cdot \vec{b} = -7 \), then \[ \frac{|\vec{a} \times \vec{b}|}{|\vec{a} \cdot \vec{b}|} \] is:

Let \( (a, b) \) denote the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} = 2i + 3j + 6k \), \( |\mathbf{a}| = 4 \), and \( (\mathbf{a}, \mathbf{b}) = \cos^{-1} \left( \frac{4}{21} \right) \), then \( \mathbf{a} + \mathbf{b} = \)

The distance of a point \( \vec{a} \) from the plane \( r \cdot m = q \) is given by \( \frac{| \vec{a} \cdot m - q |}{|m|} \). If the distance of the point \( i + 2j + 3k \) from the plane \( \vec{r} \cdot (2i + 6j - 9k) = -1 \) is \( p \) and the distance of the origin from this plane is \( q \), then \( p - q = \dots \)

The mean of 5 observations is 4.4 and their variance is 8.24. If three of those observations are 1, 2, and 6, then the other two observations are:

Three screws are drawn at random from a lot of 50 screws containing 5 defective ones. Then the probability of the event that all 3 screws drawn are non-defective, assuming that the drawing is (a) with replacement (b) without replacement respectively is:

A coin is tossed three times. Let A be the event of "getting three heads" and B be the event of "getting a head on the first toss". Then A and B are:

From a collection of eight cards numbered 1 to 8, if two cards are drawn at random, one after the other with replacement, then the probability that the product of numbers that appear on the cards is a perfect square is:

If A and B are events of a random experiment with \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.3 \), then the probability that neither A nor B occurs is:

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96 is:

If \( P(X = x) = k \left( \frac{3}{8} \right)^x \), where \( x = 1, 2, 3, \dots \) is the probability distribution function of a discrete random variable X, then \( k = \)

If \( A(4, 0) \) and \( B(-4, 0) \) are two points, then the locus of a point \( P \) such that \( PA - PB = 4 \) is:

If \( P(X = x) = k \left( \frac{3}{8} \right)^x \), where \( x = 1, 2, 3, \dots \) is the probability distribution function of a discrete random variable X, then \( k = \)

The diagonals AC and BD of a rhombus ABCD intersect at the point (3, 4). If \( BD = \frac{2}{\sqrt{2}} \), \( A = (1, 2) \), \( A = (\alpha, \beta) \), \( D = (\gamma, \delta) \), and \( \alpha<\delta<\gamma<\beta \), then \( \beta + \gamma - \delta = \dots \)

The lines \( p(x^2 + 1) + x - y + q = 0 \) and \( (p^2 + 1)x^2 + (p^2 + 1)y + 2q = 0 \) are perpendicular to a line L. Then the equation of the line L is:

If \( 2x^2 - 3xy + y^2 = 0 \) represents two sides of a triangle and \( x + y - 1 = 0 \) is its third side, then the distance between the orthocenter and the circumcenter of that triangle is:

Assertion (A): The difference of the slopes of the lines represented by \( y^2 - 2xy \sec^2 \alpha + (3 + \tan^2 \alpha) \left( 1 + \tan^2 \alpha \right) \cos^2 \theta = 0 \) is 4. Reason (R): The difference of the slopes represented by \( ax^2 + 2hxy + by^2 = 0 \) is \( \frac{2\sqrt{h^2 - ab}}{|b|} \).

If \( A(3, -1, 1), B(0, 2, 3), C(4, 8, 11) \) are three points, then the coordinates of the foot of the perpendicular drawn from the point A to the line joining the points B and C is: