List of top Questions asked in KEAM- 2017

The equations \( \lambda x - y = 2 \), \( 2x - 3y = -\lambda \), and \( 3x - 2y = -1 \) are consistent for:

In an experiment with 15 observations on \( x \), the results \( \sum x^2 = 2830 \) and \( \sum x = 170 \) were available. One observation (20) was wrong and was replaced by 30. The corrected variance is:

The value of \( \cos \left( \tan^{-1} \left( \frac{3}{4} \right) \right) \) is:

The angle between the pair of lines \( \frac{x-2}{2} = \frac{y-1}{5} = \frac{z+3}{-3} \) and \( \frac{x+2}{-1} = \frac{y-4}{8} = \frac{z-5}{4} \) is:

The distance of the point \( (3, -5) \) from the line \( 3x - 4y - 26 = 0 \) is:

\( \sin 765^{\circ} \) is equal to:

If the straight line \( y = 4x + c \) touches the ellipse \( \frac{x^2}{4} + y^2 = 1 \), then \( c \) is equal to:

If \( a \) and \( b \) are the non-zero distinct roots of \( x^2 + ax + b = 0 \), then the minimum value of \( x^2 + ax + b \) is:

\( \int_{-1}^{1} \max\{x, x^3\} \, dx \) is equal to:

Let \( a, a + r \) and \( a + 2r \) be positive real numbers such that their product is 64. Then the minimum value of \( a + 2r \) is equal to:

The sum \( S = \frac{1}{9!} + \frac{1}{3!7!} + \frac{1}{5!5!} + \frac{1}{7!3!} + \frac{1}{9!} \) is equal to:

If \( x \in [0, \frac{\pi}{2}] \), \( y \in [0, \frac{\pi}{2}] \) and \( \sin x + \cos y = 2 \), then the value of \( x + y \) is equal to:

The vertex of the parabola \( y^2 - 4y - x + 3 = 0 \) is:

If \( \vec{a}, \vec{b}, \vec{c} \) are vectors such that \( \vec{a} + \vec{b} + \vec{c} = 0 \) and \( |\vec{a}| = 7, |\vec{b}| = 5, |\vec{c}| = 3 \), then the angle between \( \vec{c} \) and \( \vec{b} \) is:

Let \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a > 0 \). The minimum of \( f \) is attained at a point \( q \) and the maximum is attained at a point \( p \). If \( p^3 = q \), then \( a \) is equal to:

For all real numbers \( x \) and \( y \), it is known that the real valued function \( f \) satisfies \( f(x) + f(y) = f(x + y) \). If \( f(1) = 7 \), then \( \sum_{r=1}^{100} f(r) \) is equal to:

The equation of the plane that passes through the points \( (1, 0, 2), (-1, 1, 2) \) and \( (5, 0, 3) \) is:

If the vectors \( 2\hat{i} + 2\hat{j} + 6\hat{k} \), \( 2\hat{i} + \lambda\hat{j} + 6\hat{k} \), \( 2\hat{i} - 3\hat{j} + \hat{k} \) are coplanar, then the value of \( \lambda \) is:

\( \int_{2016}^{2017} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{4033 - x}} \, dx \) is equal to:

The solution of \( \frac{dy}{dx} + y \tan x = \sec x \), given \( y(0) = 0 \), is:

The equation of the hyperbola with vertices \( (0, \pm 15) \) and foci \( (0, \pm 20) \) is:

The distance between \( (2, 1, 0) \) and \( 2x + y + 2z + 5 = 0 \) is:

The area bounded by the curves \( y = -x^2 + 3 \) and \( y = 0 \) is:

The order of the differential equation \( \left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^5 = 0 \) is:

The area of the circle \( x^2 - 2x + y^2 - 10y + k = 0 \) is \( 25\pi \). The value of \( k \) is equal to: