List of practice Questions

If the points \( (1, 2, 3) \) and \( (2, -1, 0) \) lie on the opposite sides of the plane \( 2x + 3y - 2z = k \), then \( k \) is:

If the area of the triangle on the complex plane formed by the points \( z = x + iy \) and \( z = 1 \) is 200, then the value of \( 3 \times |z| \) must be equal to?

The equation of the chord of the hyperbola \( 25x^2 - 16y^2 = 400 \), which is bisected at the point \( (6, 2) \), is?

If a plane meets the coordinate axes at \( A, B, C \) such that the centroid of the triangle is \( (1, 2, 4) \), then the equation of the plane is?

The volume of the tetrahedron included between the plane \( 3x + 4y - 5z = 60 \) and the coordinate planes is?

\( \int_0^\infty \left( \sin x + \left| \sin x \right| \right) dx \) is equal to?

The value of \( \int_0^\infty \left[ \sqrt{x} \right] dx \), where \( \left[ . \right] \) is the greatest integer function, is?

The negation of \( \sim (p \land q) \lor (p \land \sim q) \) is?

The normals at three points P, Q, and R of the parabola \( y^2 = 4ax \) meet at \( (h, k) \). The centroid of the triangle formed by the points P, Q, and R lies on?

The minimum area of the triangle formed by any tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with the coordinate axes is?

If the line \( lx + my - n = 0 \) will be a normal to the hyperbola, then \( \frac{a^2}{l^2} + \frac{b^2}{m^2} = k \), where \( k \) is equal to?

If \( \cos \alpha + i \sin \alpha = b \), \( c = \cos \gamma + i \sin \gamma \) and \( b + c + a = 1 \), then \( \cos (\beta - \gamma) + \cos (\alpha - \beta) \) is equal to?

The greatest and the least value of \( z = x + iy \), where \( |z| = 1 \) and \( |x| \leq 3 \), are?

The angle between lines joining the origin to the point of intersection of the line \( \sqrt{3}x + y = 2 \) and the curve \( y^2 = x^3 \) is?

The equation of normal to the curve \( y = (1 + x) + \sin^{-1}(\sin x) \) at \( x = 0 \) is?

The value of from the Lagrange's mean value theorem for which \( f(x) = \sqrt{25 - x^2} \) in the interval \( [1,5] \) is?

Two lines \( x - y + 1 = 1 \) and \( x - 3y - k = 2 \) intersect at a point, if \( k \) is equal to?

The minimum value of \( \frac{x}{\log x} \) is?

The triangle formed by the tangent to the curve \( y = x^2 + x + 1 \) at the point \( (1, 3) \) and the coordinate axes lies in the first quadrant. If its area is 2, then the value of \( b \) is?

The statement \( p \rightarrow q \) is equivalent to?

If \( x + y = 2 \cos \theta + 5 \sin \theta \), then \( x^2 + y^2 \) is equal to?

The length of longer diagonal of the parallelogram constructed on \( 5a + 2b \) and \( a - 3b \), if it is given that \( |a| = 2\sqrt{2} \), \( |b| = 3 \), and the angle between \( a \) and \( b \) is \( \frac{\pi}{4} \), is?

If \( r = a \times b \times c + \beta \cdot a + \gamma \cdot b + [a \, b \, c] = 2 \), then \( a + \beta + \gamma \) is equal to?

If \( a \), \( b \), and \( c \) are three non-coplanar vectors and \( p, q, r \) are reciprocal vectors, then \( (p + q + r) \) is equal to?

If the integers \( m \) and \( n \) are chosen at random from 1 to 100, then the probability that a number of the form \( 7m + 7n \) is divisible by 5, equals to?