List of practice Questions

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

Let \( X \) denote the sum of the numbers obtained when two fair dice are rolled. The variance and standard deviation of \( X \) are?

A four digit number is formed by the digits 1, 2, 3, 4 with no repetition. The probability that the number is odd is?

The vertices of a triangle are \( A(0,4,1) \), \( B(2,-3,-1) \), and \( C(4,5,0) \), then the orthocenter of ABC is?

Which of the following isomerism is shown by ethyl acetoacetate?

The final product obtained in the reaction,