List of practice Questions

The magnitude of the external force acting on moving bodies can be known directly by

A body of mass 10 g is tied to a string of length 0.4 m and it is whirled in a horizontal plane with a speed of 6 m/s. Then the tension in the string is

Three particles of each mass \( m \) are kept at the three vertices of an equilateral triangle. If \( I_1 \) is the moment of inertia of the system of the particles about an axis along one side of the triangle and \( I_2 \) is the moment of inertia of the system of the particles about an axis along the perpendicular bisector of a side, then \( I_1 : I_2 \) is

If \( u(n) = \int_0^{\frac{\pi}{2}} (1 + \sin t)^n \sin 2t \, dt,\; n \in \mathbb{N} \), then \( u(4) = \)

If a curve passes through (1, 2) and has the slope of its tangent \(1 - \frac{1}{x^2}\) at a point \((x, y)\), then the equation of that curve is

The differential equation having \( y = (a + b)e^{cx+d} \) as its general solution, where \( a, b, c, d \) are arbitrary constants, is

The integrating factor of the linear differential equation \( \frac{dy}{dx} = \frac{1}{4x + 3y} \) is

If \(\displaystyle \int \frac{3x+2}{4x^2+4x+5} \, dx = A \log(4x^2+4x+5) + B \tan^{-1} \left(x + \frac{1}{2}\right) + C\), then \((A, B) =\)

\(\displaystyle \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3} dx =\)

If \(g(x)\) is the inverse of the function \(f(x)\) and \(f'(x) = \dfrac{1}{h(x)}\), then what is the value of \(g'(x)\)?

If the slope of the tangent on a curve at any point \((x, y)\) is equal to \[ \frac{y^2 - x^2}{2xy} \] then the equation of the normal at the point \(\left(1, \frac{\sqrt{3}}{2}\right)\) is

Let \(A(1,15),\ B(3,-12),\ C(6,12)\) be three consecutive turning points of a continuous curve \(y = f(x)\). If \(f(x) = 0\) only for \(x = \alpha\) and \(x = \beta\), then \[ |\beta - \alpha| < ? \]

Evaluate the integral \[ \int \frac{\sin^{-1} \sqrt{x} - \cos^{-1} \sqrt{x}}{\sqrt{x} \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right)} \, dx \]

Let the transverse axis of a hyperbola \( H \) be parallel to the X-axis and \( x^2 + y^2 - 2x - 4y + 3 = 0 \) be the equation of the auxiliary circle of \( H \). If the asymptotes of \( H \) are at right angles, then the equation of the hyperbola is:

Assertion (A): Every rational function is continuous at every real number \(x\) at which it is defined.
Reason (R): Every rational function is a quotient of two polynomials.

If \(x^2 \tan^{-1}\left(\frac{y}{x}\right) - y^2 \tan^{-1}\left(\frac{x}{y}\right) = k\), then \(\left(\frac{dy}{dx}\right)_{(1,1)} =\) ?

If a focal chord of the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} = 1\) meets its minor axis at the point \((0, 3)\), then the perpendicular distance from the centre of the ellipse to this focal chord is:

If \( PS \) is the median of the triangle with vertices \( P(2, 2) \), \( Q(6, -1) \), and \( R(7, 3) \), then the equation of the line passing through \( (1, -1) \) and parallel to \( PS \) is:

Two sides of a rhombus are along the lines \( x - y + 1 = 0 \) and \( 7x - y - 5 = 0 \). If its diagonals intersect at \( (-1, -2) \), then one of the vertices of this rhombus is:

If \(2x^2 - 5xy + 2y^2 = 0\) represents two sides of a triangle whose centroid is \((1,1)\), then the equation of the third side is:

A point on the plane passing through the points \((\sqrt{2},1,4), (0,-1,0)\) and \((0,0,1)\) is:

A box contains 5 white balls and 7 black balls. In how many ways can 3 balls be drawn from the box such that at least one black ball is included?}

Two dice are rolled. If \( A \) denotes the event that the same number shows on each die and \( B \) denotes the event that the sum of the numbers on both dice is greater than 7, then \( P(A \mid B) \) and \( P(B \mid A) \) respectively are:

The circumcentre of the triangle with vertices \( (-2,3), (2,-1), (4,0) \) is:

Let \[ \vec{a} = 2\hat{i} + \hat{j} - \hat{k} \quad \text{and} \quad \vec{b} = \hat{i} + 3\hat{j} - 5\hat{k} \] be two vectors, and \(\vec{r}\) be a vector along the vector \(3\vec{a} - 2\vec{b}\) such that \(|\vec{r}| = \sqrt{74}\). If the direction of \(\vec{r}\) is opposite to that of \(3\vec{a} - 2\vec{b}\), then \(\vec{r} =\)