What will be the projection of vector \(A=\hat{i}+\hat{j}+\hat{k}\) on vector \(\vec{B}=\hat{i}+\hat{j}\).

Two masses \(m\) and \(\frac{m}{2}\) are connected at the two ends of a massless rigid rod of length \(l\). The rod is suspended by a thin wire of torsional constant \(k\) at the centre of mass of the rodmass system (see figure). Because of torsional constant \(k\), the restoring torque is \(\tau  = k\,\theta \) for angular displacement \(\theta \). If the rod is rotated by \(\theta _0\) and released, the tension in it when it passes through its mean position will be

The electric current in the circuit is given as \(i=i_{ o }(t / T)\). The r.m.s current for the period \(t=0\) to \(t=T\) is ___________

\(N\,moles\) of a diatomic gas in a cylinder are at a temperature \(T\). Heat is supplied to the cylinder such that the temperature remains constant but \(n\,moles\) of the diatomic gas get converted into monoatomic gas. What is the change in the total kinetic energy of the gas ?

Identify the correct statements :
A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.
B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.
C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.
D. For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces.
Choose the correct answer from the options given below.

The two wires \(A\) and \(B\) of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire \(A\) and wire \(B\) is \(20/11\). When the joined wire is kept under certain tension the elongations in the wires \(A\) and \(B\) are equal. If the length of wire \(A\) is \(2.2\) m, then the length of wire \(B\) is _______ m.

Two simple harmonic motions, as shown, are at right angles. They are combined to form Lissajous figures \(x\left( t \right) = A\,\sin \,\left( {at + \delta } \right)\) \(y\left( t \right) = B\,\sin \,\left( {bt} \right)\) Identify the correct match below

The term independent of \(x\) in the expression of \(\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}, x \neq 0\) is

If \(a\) and \(b\) are the roots of equation \(x^2-7 x-1=0\), then the value of \(\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}\) is equal to \(........\).

Let \(f(x)\) be a positive function such that the area bounded by \(y=f(x), y=0\) from \(x=0\) to \(x=a>0\) is \(\mathrm{e}^{-\mathrm{a}}+4 \mathrm{a}^2+\mathrm{a}-1\). Then the differential equation, whose general solution is \(y=c_1 f(x)+c_2\), where \(c_1\) and \(c_2\) are arbitrary constants, is :

A uniform thin rod \(AB\) of length \(L\) has linear mass density \(\mu \left( x \right) = a + \frac{{bx}}{L}\) , where \(x\) is measured from \(A\). If the \(CM\) of the rod lies at a distance of \(\left( {\frac{7}{12}} \right)L\) from \(A\), then \(a\) and \(b\) are related as

The function \(f(x)=x^{3}-6 x^{2}+a x+b\) is such that \(f(2)=f(4)=0\). Consider two statements. \((S_1)\) there exists \(\mathrm{x}_{1}, \mathrm{x}_{2} \in(2,4), \mathrm{x}_{1}<\mathrm{x}_{2}\), such that \(f^{\prime}\left(x_{1}\right)=-1\) and \(f^{\prime}\left(x_{2}\right)=0\) \((S_2)\) there exists \(\mathrm{x}_{3}, \mathrm{x}_{4} \in(2,4), \mathrm{x}_{3}<\mathrm{x}_{4}\), such that \(f\) is decreasing in \(\left(2, x_{4}\right)\), increasing in \(\left(x_{4}, 4\right)\) and \(2 f^{\prime}\left(x_{3}\right)=\sqrt{3} f\left(x_{4}\right)\). Then

If the two metals \(A\) and \(B\) are exposed to radiation of wavelength \(350\,nm\). The work functions of metals \(A\) and \(B\) are \(4.8\,eV\) and \(2.2\,eV\). Then choose the correct option