List of top Mathematics Questions

If the equation of the hyperbola having foci at \((8,3)\), \((0,3)\) and eccentricity \( \frac{4}{3} \) is \[ \frac{(x-\alpha)^2}{p} - \frac{(y-\beta)^2}{q} = 1, \] then find \( p + q \). Options:

If the equation of the plane passing through the point (2, -1, 3) and perpendicular to each of the planes $3x - 2y + z = 8$ and $x + y + z = 6$ is $lx + my + nz = 1$, then $4m + 2n - 3l$ =
Identify the correct option from the following:

Problem: If one of the lines represented by \(ax^2 + 2hxy + by^2 = 0\) bisects the angle between the positive coordinate axes, then identify the correct relationship between \(a\), \(b\), and \(h\). Identify the correct option from the following:

If the intercepts made by a variable circle on the X-axis and Y-axis are 8 and 6 units respectively, then the locus of the center of the circle is:

Problem: From a point \( P \) on the circle \( x^2 + y^2 = 4 \), two tangents are drawn to the circle \( x^2 + y^2 - 6x - 6y + 14 = 0 \). If \( A \) and \( B \) are the points of contact of those lines, then the locus of the center of the circle passing through the points \( P \), \( A \), and \( B \) is: Identify the correct option from the following:

If the product of the lengths of the perpendiculars drawn from the ends of a diameter of the circle \( x^2 + y^2 = 4 \) onto the line \( x + y + 1 = 0 \) is maximum, then the two ends of that diameter are:

The slope of the non-vertical tangent drawn from the point $(3,4)$ to the circle $x^2 + y^2 = 9$ is:

If the straight line \[ 2x + 3y + 1 = 0 \] bisects the angle between two other straight lines, one of which is \[ 3x + 2y + 4 = 0, \] then the equation of the other straight line is:

If the slopes of both the lines given by \[ x^2 + 2hxy + 6y^2 = 0 \] are positive and the angle between these lines is \[ \tan^{-1} \left(\frac{1}{7}\right), \] then the product of the perpendiculars drawn from the point \((1,0)\) to the given pair of lines is:

Point \(P(6,4)\) lies on the line \(x - y - 2 = 0\). If \(A(\alpha, \beta)\) and \(B(\gamma, \delta)\) are two points on this line lying on either side of \(P\) at a distance of 4 units from \(P\), then find \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\).

A random variable \(X\) follows a binomial distribution in which the difference between its mean and variance is 1. If \(2P(x=2) = 3P(x=1)\), then \(n^2 P(x>1)\) is:

If the transformed equation of the equation \[ 2x^2 + 3xy - 2y^2 - 17x + 6y + 8 = 0 \] after translating the coordinate axes to a new origin \((\alpha, \beta)\) is \[ aX^2 + 2h XY + bY^2 + c = 0, \] then find \(3\alpha + c\).

70% of the total employees of a factory are men. Among the employees of that factory, 30% of men and 15% of women are technical assistants. If an employee chosen at random is found to be a technical assistant, then the probability that this employee is a man is:

If \(A\) and \(B\) are events of a random experiment such that \[ P(A \cup B) = \frac{3}{4}, \quad P(A \cap B) = \frac{1}{4}, \quad P(\overline{A}) = \frac{2}{3}, \] then \(P(\overline{A} \cap B)\) is:

If a discrete random variable \(X\) has the probability distribution \[ P(X = x) = k \frac{2^{2x+1}}{(2x+1)!}, \quad x=0,1,2,\ldots, \] then find \(k\).

A person is known to speak the truth in 3 out of 4 occasions. If he throws a die and reports that it is six, then the probability that it is actually six is:

Two cards are drawn at random from a pack of 52 playing cards. If both the cards drawn are found to be black in colour, then the probability that at least one of them is a face card is:

If \(\vec{a} = 2 \vec{i} - 3 \vec{j} + 4 \vec{k}, \vec{b} = \vec{i} + 2 \vec{j} - \vec{k}, \vec{c} = -3 \vec{i} - \vec{j} + 2 \vec{k}\) and \(\vec{d} = \vec{i} + \vec{j} + \vec{k}\) are four vectors, then evaluate \[ (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = ? \]

If \(\vec{a} = \vec{i} + p \vec{j} - 3 \vec{k}, \vec{b} = p \vec{i} - 3 \vec{j} + \vec{k}, \vec{c} = -3 \vec{i} + \vec{j} + 2 \vec{k}\) are three vectors such that \[ |\vec{a} \times \vec{b}| = |\vec{a} \times \vec{c}|, \] then \(p =\)

The variance of the ungrouped data \(2, 12, 3, 11, 5, 10, 6, 7\) is:

In \(\triangle ABC\), the sum of the lengths of two sides is \(x\) and the product of those lengths is \(y\). If \(c\) is the length of its third side and \(x^2 - c^2 = y\), then the circumradius of that triangle is:

In \(\triangle ABC\), if \(r_1 = 2r_2 = 3r_3\), then find the ratio \(a : b\).

If the area of triangle \(ABC\) is \(4\sqrt{5}\) sq. units, length of the side \(CA\) is 6 units and \(\tan \frac{B}{2} = \frac{\sqrt{5}}{4}\), then its smallest side is of length:

If \(\vec{a}, \vec{b}, \vec{c}\) are three unit vectors such that \[ |\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 15, \] then \[ |\vec{a} - \vec{b} - \vec{c}|^2 - 4(\vec{b} \cdot \vec{c}) = ? \]

Let \(\vec{a}\), \(\vec{b}\) be position vectors of points \(A\) and \(B\) respectively. \(C\) and \(D\) are points on the line \(AB\) such that \(\overrightarrow{AB}, \overrightarrow{AC}\) and \(\overrightarrow{BD}, \overrightarrow{BA}\) are two pairs of like vectors. If \(\overrightarrow{AC} = 3 \overrightarrow{AB}\) and \(\overrightarrow{BD} = 2 \overrightarrow{BA}\), then \(\overrightarrow{CD} =\)