List of practice Questions

The length and area of cross-section of a copper wire are respectively 30 m and \( 6 \times 10^{-7} \, \text{m}^2 \). If the resistivity of copper is \( 1.7 \times 10^{-8} \, \Omega \, \text{m} \), then the resistance of the wire is
If current of 80 A is passing through a straight conductor of length 10 m, then the total momentum of electrons in the conductor is (mass of electron \( = 9.1 \times 10^{-31} \) kg and charge of electron \( = 1.6 \times 10^{-19} \) C)
If the degrees of freedom of a gas molecule is 6, then the total internal energy of the gas molecule at a temperature of \( 47 \, ^\circ\text{C} \) (in eV) is (Boltzmann constant \( = 1.38 \times 10^{-23} \, \text{J K}^{-1} \))
When a stretched wire of fundamental frequency f is divided into three segments, the fundamental frequencies of these three segments are \(f_1\), \(f_2\) and \(f_3\) respectively. Then the relation among \(f, f_1, f_2, f_3\) and f is (Assume tension is constant) % "and f is" seems redundant
A body of mass 1 kg is suspended from a spring of force constant \( 600 \, \text{N m}^{-1} \). Another body of mass 0.5 kg moving vertically upwards hits the suspended body with a velocity of \( 3 \, \text{m s}^{-1} \) and embedded in it. The amplitude of motion is
When a wire made of material with Young's modulus Y is subjected to a stress S, the elastic potential energy per unit volume stored in the wire is
If the displacement 'x' of a body in motion in terms of time 't' is given by \(x = A\sin(\omega t + \theta)\), then the minimum time at which the displacement becomes maximum is
If the magnitude of a vector \( \vec{p} \) is 25 units and its y-component is 7 units, then its x-component is
A balloon with mass 'm' is descending vertically with an acceleration 'a' (where a \(<\) g). The mass to be removed from the balloon, so that it starts moving vertically up with an acceleration 'a' is
\( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4\sin^2 x} dx = \)
\( \int \frac{13\cos 2x - 9\sin 2x}{3\cos 2x - 4\sin 2x} dx = \)
\( \int \sqrt{x^2+x+1} \ dx \)
If \( y = \tan^{-1}\left(\frac{x}{1+2x^2}\right) + \tan^{-1}\left(\frac{x}{1+6x^2}\right) \), then \( \frac{dy}{dx} = \)
If the tangent drawn at the point \( (x_1,y_1) \), \(x_1,y_1 \in N \) on the curve \( y = x^4 - 2x^3 + x^2 + 5x \) passes through origin, then \( x_1+y_1 = \)
If \( \beta \) is an angle between the normals drawn to the curve \( x^2+3y^2=9 \) at the points \( (3\cos\theta, \sqrt{3}\sin\theta) \) and \( (-3\sin\theta, \sqrt{3}\cos\theta) \), \( \theta \in \left(0, \frac{\pi}{2}\right) \), then
\( \int \left( \sum_{r=0}^{\infty} \frac{x^r 2^r}{r!} \right) dx = \)
If the line of intersection of the planes \(2x+3y+z=1\) and \(x+3y+2z=2\) makes an angle \( \alpha \) with the positive x-axis, then \( \cos \alpha = \)
\( \lim_{n\to\infty} \frac{1}{n^3} \sum_{k=1}^{n} k^2 x = \)
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} a - \frac{\sin[x-1]}{x-1} & , \text{if } x>1
1 & , \text{if } x = 1
b - \frac{\sin([x-1] - [x-1]^3)}{([x-1]^2)} & , \text{if } x<1 \end{cases} \] where \([t]\) denotes the greatest integer less than or equal to t. If f is continuous at \(x=1\), then \(a+b=\)
If the perpendicular distance from the focus of a parabola \(y^2=4ax\) to its directrix is \( \frac{3}{2} \), then the equation of the normal drawn at \( (4a, -4a) \) is
If the distance between the foci of a hyperbola H is 26 and distance between its directrices is \( \frac{50}{13} \), then the eccentricity of the conjugate hyperbola of the hyperbola H is
If Q \( (\alpha, \beta, \gamma) \) is the harmonic conjugate of the point P(0,-7,1) with respect to the line segment joining the points (2,-5,3) and (-1,-8,0), then \( \alpha - \beta + \gamma = \)
If two sides of a triangle are represented by \( 3x^2 - 5xy + 2y^2 = 0 \) and its orthocentre is (2,1), then the equation of the third side is
If \( ax^2 + 2hxy - 2ay^2 + 3x + 15y - 9 = 0 \) represents a pair of lines intersecting at (1,1), then ah =
If the point of contact of the circles \( x^2+y^2-6x-4y+9=0 \) and \( x^2+y^2+2x+2y-7=0 \) is \( (\alpha, \beta) \), then \( 7\beta = \)