List of top Questions asked in KEAM

The centre of the circle whose radius is \( 5 \) and which touches the circle \( x^2 + y^2 - 2x - 4y - 20 = 0 \) at \( (5, 5) \) is:

The slope of the straight line joining the centre of the circle \( x^2 + y^2 - 8x + 2y = 0 \) and the vertex of the parabola \( y = x^2 - 4x + 10 \) is:

The shortest distance between the circles \( (x-1)^2 + (y+2)^2 = 1 \) and \( (x+2)^2 + (y-2)^2 = 4 \) is:

A straight line perpendicular to the line \( 2x + y = 3 \) is passing through \( (1,1) \). Its y-intercept is:

If \( p \) and \( q \) are respectively the perpendiculars from the origin upon the straight lines whose equations are \( x\sec\theta + y\csc\theta = a \) and \( x\cos\theta - y\sin\theta = a\cos 2\theta \), then \( 4p^2 + q^2 \) is equal to:

The points \( (2, 5) \) and \( (5, 1) \) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line \( y = 2x + k \), then the value of \( k \) is:

The number of points \( (a, b) \), where \( a \) and \( b \) are positive integers, lying on the hyperbola \( x^2 - y^2 = 512 \) is:

The circumcentre of the triangle with vertices \( (8, 6), (8, -2) \) and \( (2, -2) \) is at the point:

The ratio by which the line \( 2x + 5y - 7 = 0 \) divides the straight line joining the points \( (-4, 7) \) and \( (6, -5) \) is:

If \( p \) is the length of the perpendicular from the origin to the line whose intercepts with the coordinate axes are \( \frac{1}{3} \) and \( \frac{1}{4} \), then the value of \( p \) is:

The value of \( \tan(1^\circ) + \tan(89^\circ) \) is equal to:

If \( \sin \theta + \csc \theta = 2 \), then the value of \( \sin^6 \theta + \csc^6 \theta \) is equal to:

If \( 0 < x < \pi \), then \( \frac{\sin 8x + 7\sin 6x + 18\sin 4x + 12\sin 2x}{\sin 7x + 6\sin 5x + 12\sin 3x} = \)

The value of \( \sec^2(\tan^{-1} 3) + \csc^2(\cot^{-1} 2) \) is equal to:

\( \cos^{-1}\left(\cos\left(\frac{7\pi}{5}\right)\right) = \)

Let \( s_n = \cos\left(\frac{n\pi}{10}\right) \), \( n=1,2,3, \ldots \). Then the value of \( \frac{s_1s_2\cdots s_{10}}{s_1+s_2+\cdots+s_{10}} \) is equal to:

If \( ab < 1 \) and \( \cos^{-1}\left(\frac{1-a^2}{1+a^2}\right) + \cos^{-1\left(\frac{1-b^2}{1+b^2}\right) = 2 \tan^{-1} x \), then \( x \) is equal to:}

The value of \( \sin^{-1} \left( \frac{2\sqrt{2}}{3} \right) + \sin^{-1 \left( \frac{1}{3} \right) \) is equal to:}

The truth values of \( p, q \) and \( r \) for which \( (p \wedge q) \vee (\sim r) \) has truth value F are respectively:

Let \( \theta \in [0, \frac{\pi}{2}] \). Which one of the following is true?

\( \sim [(\sim p) \wedge q] \) is logically equivalent to:

The number of solutions for the system of equations \( 2x + y = 4 \), \( 3x + 2y = 2 \), and \( x + y = -2 \) is:

Let \( p, q \) and \( r \) be any three logical statements. Which one of the following is true?

The shaded region shown in the figure is given by the inequations: [height=5cm]{Screenshot 2026-05-01 102102.png}

The number of solutions of the inequation \( |x - 2| + |x + 2| < 4 \) is: