List of top Mathematics Questions

If \( \beta \) is the angle made by the perpendicular drawn from origin to the line \( L = x + y - 2 = 0 \) with the positive X-axis in the anticlockwise direction. If \( a \) is the X-intercept of the line \( L = 0 \) and \( p \) is the perpendicular distance from the origin to the line \( L = 0 \), then \( \tan \beta + p^2 = \):
If the ratio of the perpendicular distances of a variable point \( P(x,y,z) \) from the X-axis and from the YZ-plane is 2:3, then the equation of the locus of \( P \) is:
If the direction ratios of two lines are \( (3,0,2) \) and \( (0,2,k) \), and \( \theta \) is the angle between them, and if \( |\cos \theta| = \frac{6}{13} \), then \( k = \)
If \( x - 2y - 6 = 0 \) is a normal to the circle \( x^2 + y^2 + 2gx + 2fy - 8 = 0 \) and the line \( y = 2 \) touches this circle, then the radius of the circle can be:
If the ratio of the distances of a variable point \(P\) from the point \((1,1)\) and the line \(x - y + 2 = 0\) is \(1/\sqrt{2}\), then the equation of the locus of \(P\) is:
(-2, -1), (2,5) are two vertices of a triangle and \( \left(2, \frac{5}{3} \right) \) is its orthocenter. If \( (m, n) \) is the third vertex of that triangle, then \( m+n = \):
If \(A(-2,4,a)\), \(B(1,3,b)\), \(C(0,4,c)\), and \(D(-5,6,1)\) are collinear points, then \(a+b+c=\)
When the origin is shifted to the point \( (2,b) \) by translation of axes, the coordinates of the point \( (4,4) \) have changed to \( (6,8) \). When the origin is shifted to \( (a,b) \) by translation of axes, if the transformed equation of \( x^2 + 4xy + y^2 = 0 \) is \( X^2 + 2HXY + Y^2 + 2GX + 2FY + C = 0 \), then \( 2H(G + F) = \) ?
The slope of a line L passing through the point \((-2, -3)\) is not defined. If the angle between the lines L and \( ax - 2y + 3 = 0 \) (where \( a>0 \)) is 45°, then the angle made by the line \( x + ay - 4 = 0 \) with positive X-axis in the anticlockwise direction is:
The lengths of two equal sides of an isosceles triangle are given by \(L_1 = 2x + y - 3 = 0\) and \(L_2 = ax + by + c = 0\). If \(L_3 = x + 2y + 1 = 0\) is the third side of this triangle and \((5, 1)\) is a point on \(L_2\), then \(b^2/|ac|\) is:
The direction cosines of two lines are connected by the relations \[ l - m + n = 0, \quad 2lm - 3mn + nl = 0. \] If \( \theta \) is the angle between these two lines, then \( \cos \theta \) is:
If the angle between the pair of lines given by the equation \(ax^2 + 4xy + 2y^2 = 0\) is \(45^\circ\), then the possible values of 'a' are
If \( (2, -1) \) is the point of intersection of the pair of lines \[ 2x^2 + axy + 3y^2 + bx + cy - 3 = 0 \quad \text{then} \quad 3a + 2b + c = \]
P and Q are the extremities of a focal chord of the parabola \( y^2 = 4ax \). If \( P = (9,9) \) and \( Q = (p, q) \), then \( p - q \) is:
If the distance from a variable point \( P \) to the point \( (4,3) \) is equal to the perpendicular distance from \( P \) to the line \( x + 2y - 1 = 0 \), then the equation of the locus of the point \( P \) is:
If \( Q \) and \( R \) are the images of the point \( P(2,3) \) with respect to the lines \( x - y + 2 = 0 \) and \( 2x + y - 2 = 0 \) respectively, then \( Q \) and \( R \) lie on
If the line \( 4x - 3y + p = 0 \) (where \( p + 3>0 \)) touches the circle \( x^2 + y^2 - 4x + 6y + 4 = 0 \) at the point \( (h,k) \), then the value of \( h - 2k \) is:
If \(P\left( \frac{\pi}{4} \right)\), \(Q\left( \frac{\pi}{3} \right)\) are two points on the circle \(x^2 + y^2 - 2x - 2y - 1 = 0\), then the slope of the tangent to this circle which is parallel to the chord \(PQ\) is:
The pole of the line \(x - 5y - 7 = 0\) with respect to the circle \(S \equiv x^2 + y^2 - 2x - 2y + 1 = 0\) is \(P(a,b)\). If \(C\) is the centre of the circle \(S = 0\) then \(PC =\):
P and Q are the points of trisection of the line segment joining the points \( (3, -7) \) and \( (-5,3) \). If line PQ subtends a right angle at a variable point R, then the locus of R is:
If \( (4,3) \) and \( (1,-2) \) are the endpoints of a diagonal of a square, then the equation of one of its sides is:
The area of the parallelogram formed by the lines \( L_1: \lambda x + 4y + 2 = 0 \), \( L_2: 3x + 4y - 3 = 0 \), \( L_3: 2x + \mu y + 6 = 0 \), and \( L_4: 2x + y + 3 = 0 \), where \( L_1 \) is parallel to \( L_2 \) and \( L_3 \) is parallel to \( L_4 \), is
If \( (h, k) \) is the internal center of similitude of the circles \( x^2 + y^2 + 2x - 6y + 1 = 0 \) and \( x^2 + y^2 - 4x + 2y + 4 = 0 \), then find \( 4h \).
(a,b) is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation \(2x^2 - 3xy + 4y^2 + 5y - 6 = 0\). If the angle by which the axes are to be rotated in positive direction about the origin to remove the \(xy\)-term from the equation \(ax^2 + 23abxy + by^2 = 0\) is \( \theta \), then \(2\theta =\)
If a circle passing through the point \((1,1)\) cuts the circles \(x^2 + y^2 + 4x - 5 = 0\) and \(x^2 + y^2 - 4x + 3 = 0\) orthogonally, then the center of that circle is: