List of top Mathematics Questions

Let \( A(1, 6, 3) \) and point \( B \) and \( C \) lie on the line \[ \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \] where \( B(4, 9, \alpha) \) and point \( C \) is 10 units from \( B \). Find the area of \( \triangle ABC \):

The locus of point of intersection of tangent drawn to the circle \( (x - 2)^2 + (y - 3)^2 = 16 \), which subtends an angle of \( 120^\circ \) is

Let \( O \) be the vertex of the parabola \( y^2 = 16x \). The locus of centroid of \( \triangle OPA \) when \( P \) lies on the parabola and \( A \) lies on the x-axis and \( \angle OPA = 90^\circ \).

If \( O \) is the vertex of the parabola \( x^2 = 4ay \), \( Q \) is the point on the parabola. If \( C \) is the locus of the point which divides \( OQ \) in ratio 2:3, the equation of the chord of \( C \) which is bisected at point \( (1, 2) \) is

If the point of intersection of the ellipses \[ x^2 + 2y^2 - 6x - 12y + 23 = 0 \] \[ 4x^2 + 2y^2 - 20x - 12y + 35 = 0 \] lie on a circle of radius \( r \) and centre \( (a,b) \), then the value of \( ab + 18r^2 \) is:

Given triangle $OAB$ where $O$ is the origin, $A=(0,-\sqrt{3}a)$ and $B=(-\sqrt{2}b,0)$. Let the circumradius of $\triangle OAB$ be $4$ units. If the locus of the centroid of $\triangle OAB$ is a circle, then its radius is:

If $P(h,k)$ is a variable point on $x^2 + y^2 = 4$ and $Q(2h+1,\,3k+3)$ always lies on an ellipse, if eccentricity of the ellipse is $e$, then $\dfrac{5}{e^2}$ is equal to

If \( L_1 \) and \( L_2 \) are two parallel lines and \( \triangle ABC \) is an equilateral triangle, then the area of triangle ABC is

Let \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) and \( y = x + 1 \) intersect each other at points A & B, then \( \angle AOB \) (where O is the centre of the ellipse) is:

Let one end of a focal chord of the parabola \( y^{2} = 20x \) be \( (20, -20) \). If \( P(\alpha, \beta) \) divides the chord internally in the ratio \( 2 : 3 \), find the minimum value of \( \alpha + \beta \).

An equilateral triangle \( OAB \) is inscribed in the parabola \( y = 4x^2 \) whose vertex is \( O \). Find the least distance of the circle described on \( AB \) as diameter from the origin.

Orthocentre of equilateral \(\Delta ABC\) is at the origin. If side \(BC\) lies along \(x + 2\sqrt{2}y = 4\). If coordinates of vertex \(A\) are (a,b). Find the value of \(\lfloor |a + \sqrt{2}b| \rfloor\), where \([ \cdot ]\) denotes G.I.F. :

If \( PQ \) is a chord perpendicular to the transverse axis of \[ \frac{x^2}{4} - \frac{y^2}{b^2} = 1 \] of eccentricity \( \sqrt{3} \) such that \( \triangle OPQ \) is an equilateral triangle (where \( O \) is the origin), then the area of \( \triangle OPQ \) is:

If $P(h,k)$ is a variable point on $x^2 + y^2 = 4$ and $Q(2h+1,\,3k+3)$ always lies on an ellipse, if eccentricity of the ellipse is $e$, then $\dfrac{5}{e^2}$ is equal to

The value(s) of \( \alpha \) for which the line \( \alpha x + 2y = 1 \) never touches the hyperbola \[ \frac{x^2}{9} - \frac{y^2}{1} = 1 \] is/are:

If \( \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n \) and \( a_{10} = 59,\; a_6 = 7a_1 \), then find \( \alpha + \beta \):

If \(6\int_{1}^{x} f(t)dt = 3x f(x) + x^3 - 4, x \geq 1\) then value of (f(2)-f(3)) is :

Let \( A = \{1,2,3,\ldots,9\} \) and \( R \subset A \times A \) be defined by \[ (x,y) \in R \iff |x-y| \text{ is a multiple of } 3. \] Consider the following statements:
S\(_1\): Number of elements in \( R \) is 36.
S\(_2\): \( R \) is an equivalence relation.
Which of the following is correct?

If \( f(3) = 18, f'(3) = 0 \) and \( f''(3) = 4 \), then the value of \[ \lim_{x \to 3} \ln \left( \frac{f(x+2)}{f(3)} \right)^{\frac{18}{(x-3)^3}} \] is equal to

If \[ I(x) = 3\int \frac{dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}, \quad I(0) = \frac{\sqrt{3}}{4}, \] then find \( I(1) \):

If \[ \lim_{x \to 0} \frac{e^{a(x-1)} + 2\cos(bx) + e^{-x}(c - 1)}{x \cos x - \ln(1 + x)} = 2, \] Then the value of \( a^2 + b^2 + c^2 \) is:

If \( \alpha, \beta \) are roots of the equation \( 12x^2 - 20x + 3 = 0 \), \( \lambda \in \mathbb{R} \). If \( \frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2} \), then the sum of all possible values of \( \lambda \) is:

If \( \cos(\alpha + \beta) = -\frac{1}{10} \) and \( \sin(\alpha - \beta) = \frac{3}{8} \), where \[ 0<\alpha<\frac{\pi}{3} \quad \text{and} \quad 0<\beta<\frac{\pi}{4}, \] and \[ \tan(2\alpha) = \frac{3\left(1 - \sqrt{55}\right)}{\sqrt{11} \left(s + \sqrt{5}\right)}, \] and \( r, s \in \mathbb{N} \), then \( r^2 + s \) is:

If sum of first 4 terms of an A.P. is 6 and sum of first 6 terms is 4, then sum of first 12 terms of an A.P. is

Find possible no. of triplets (b, c, d), such that \(x^2 + 2\) is divisor of \(x^3 + bx^2 + cx + d\) & b, c, d \( \le \) 20 & b, c, d \( \in \) N :