Chapter 1

The photograph of a house occupies an area of \(1.75 \mathrm{~cm}^{2}\) on a \(35 \mathrm{~mm}\) slide. The slide is projected on to a screen and the area of the house on the screen is \(1.55 \mathrm{~m}^{2}\). The linear magnification of the projector-screen arrangement, is (a) \(84.1\) (b) \(96.1\) (c) \(94.1\) (d) \(86.1\)

A highly rigid cubical block \(A\) of small mass \(M\) and side \(L\) is fixed rigidly on to another cubical block of same dimensions and of low modulus of rigidity \(\eta\) such that the lower face of \(A\) completely covers the upper face of \(B\). The lower face of \(B\) is rigidly held on a horizontal surface. \(A\) small force \(F\) is applied perpendicular to one of the side faces of \(A\). After the force is withdrawn, block \(A\) executes small oscillations, the time period of which is given by (a) \(2 \pi \sqrt{M \eta L}\) (b) \(2 \pi \sqrt{\frac{M \eta}{L}}\) (c) \(2 \pi \sqrt{\frac{M L}{\eta}}\) (d) \(2 \pi \sqrt{\frac{M}{\eta L}}\)

In an experiment to measure the height of a bridge by dropping stone into water underneath, if the error in measurement of time is \(0.1 \mathrm{~s}\) at the end of \(2 \mathrm{~s}\), then the error in estimation of height of bridge will be [Kerala CEE 2004] (a) \(0.49 \mathrm{~m}\) (b) \(0.98 \mathrm{~m}\) (c) \(1.96 \mathrm{~m}\) (d) \(2.12 \mathrm{~m}\)

If \(C\) is the restoring couple per unit radian twist and \(I\) is the moment of inertia, then the dimensional representation of \(2 \pi \sqrt{\frac{I}{C}}\) will be (a) \(\left[\mathrm{M}^{0} \mathbf{L}^{0} \mathrm{~T}^{-1}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{LT}^{-1}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)

A cube has a side of length \(1.2 \times 10^{-2} \mathrm{~m} .\) Calculate its volume (a) \(1.7 \times 10^{-6} \mathrm{~m}^{3}\) (b) \(1.73 \times 10^{-6} \mathrm{~m}^{3}\) (c) \(1.70 \times 10^{-6} \mathrm{~m}^{3}\) (d) \(1.732 \times 10^{-6} \mathrm{~m}^{3}\)

The velocity \(v\) of water waves may depend on their wavelength \((\lambda)\), the density of water \((\rho)\) and the acceleration due to gravity \((\mathrm{g})\). The method of dimensions gives the relation between these quantities as (a) \(v^{2} \propto \lambda^{-1} \rho^{-1}\) (b) \(v^{2} \propto g \lambda\) (c) \(v^{2} \propto g \lambda \rho\) (d) \(g^{-1} \propto \lambda\)

A spectrometer gives the following reading when used to measure the angle of a prism. Main scale reading \(=58.5^{\circ}\) Vernier scale reading \(=09\) division Given that 1 division on main scale corresponding to \(0.5^{\circ}\). Total division on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from the above data \(\quad\) [AIEEE 2012] (a) \(58.59^{\circ}\) (b) \(58.77^{\circ}\) (c) \(58.65^{\circ}\) (d) \(59^{\circ}\)

If \(E, m, J\) and \(G\) represent energy, mass, angular momentum and gravitational constant respectively, then the dimensional formula of \(E J^{2} / m^{5} G^{2}\) is (a) \(\left[\mathrm{MLT}^{-2}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{0}\right]\) (d) dimensionless

Let \(\left[\varepsilon_{0}\right.\) ] denote the dimensional formula of the perimitivity of vacuum. If \(M=\) mass, \(L=\) length, \(T\) = Time and \(A=\) electric current, then [JEE Main 2013] (a) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{2} \mathrm{~A}\right]\) (b) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{4} \mathrm{~A}^{2}\right]\) (c) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{2} \mathrm{l}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{-2}\right]\) (d) \(\left[\varepsilon_{0}\right]=\left[\mathrm{M}^{-} \mathrm{l}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{2}\right]\)

Crane is British unit of volume (one crane \(=170.4742\) ). Convert crane into SI units. (a) \(0.170474 \mathrm{~m}^{3}\) (b) \(17.0474 \mathrm{~m}^{3}\) (c) \(0.00170474 \mathrm{~m}^{3}\) (d) \(1704.74 \mathrm{~m}^{3}\)

The wavelength associated with a moving particle depends upon power \(p\) of its mass \(m, q\) th power of its velocity \(v\) and \(r\) th power of Planck's constant \(h\). Then the correct set of values of \(p, q\) and \(r\) is (a) \(p=1, q=-1, r=1\) (b) \(p=1, q=1, r=1\) (c) \(p=-1, p=-1, r=-1\) (d) \(p=-1, q=-\mathbf{L} r=1\)

If \(x=a-b\), then the maximum percentage error in the measurement of \(x\) will be (a) \(\left(\frac{\Delta a+\Delta b}{a-b}\right) \times 100 \%\) (b) \(\left(\frac{\Delta a}{a}-\frac{\Delta b}{b}\right) \times 100 \%_{0}\) (c) \(\left(\frac{\Delta a}{a-a}+\frac{\Delta b}{a-b}\right) \times 100 \%\) (d) \(\left(\frac{\Delta a}{a-a}-\frac{\Delta b}{a-b}\right) \times 100 \%\)

If \(X=A \times B\) and \(\Delta X, \Delta A\) and \(\Delta B\) are maximum absolute errors in \(X, A\) and \(B\) respectively, then the maximum relative error in \(X\) is given by (a) \(\Delta X=\Delta A+\Delta B\) (b) \(\Delta X=\Delta A-\Delta B\) (c) \(\frac{\Delta X}{X}=\frac{\Delta A}{A}-\frac{\Delta B}{B}\) (d) \(\frac{\Delta X}{X}=\frac{\Delta A}{A}+\frac{\Delta B}{B}\)

The percentage errors in the measurement of mass and speed are \(2 \%\) and \(3 \%\) respectively. How much will be the maximum error in the estimate of kinetic energy obtained by measuring mass and speed? (a) 1196 (b) \(8 \%\) (c) \(59 \%\) (d) \(1 \%\)

Error in the measurement of radius of sphere is \(2 \%\). The error in the measurement of volume is (a) \(1 \%\) (b) 596 (c) \(3 \%\) (d) 690

There are atomic clocks capable of measuring time with an accuracy of 1 part in \(10^{11} .\) If two such clocks are operated with precision, then after running for \(5000 \mathrm{yr}\), these will record (a) a difference of nearly \(2 \mathrm{~s}\) (b) a difference of 1 day (c) a difference of \(10^{11} \mathrm{~s}\) (d) a difference of \(1 \mathrm{yr}\)

If there is a positive error of \(50 \%\) in the measurement of speed of a body, then the error in the measurement of kinetic energy is (a) \(25 \%\) (b) \(50 \%\) (c) \(100 \%\) (d) \(125 \%\)

The radius of the sphere is \((4.3 \pm 0.1) \mathrm{cm}\). The percentage error in its volume is (a) \(\frac{0.1}{4.3} \times 100\) (b) \(3 \times \frac{0.1 \times 100}{4.3}\) (c) \(\frac{1}{3} \times \frac{0.1 \times 100}{4.3}\) (d) \(3+\frac{0.1 \times 100}{4.3}\)

A public park, in the form of a square, has an area of \((100 \pm 0.2) \mathrm{m}^{2}\). The side of park is (a) \((10 \pm 0.01) \mathrm{m}\) (b) \((10 \pm 0.1) \mathrm{m}\) (c) \((10.0 \pm 0.1) \mathrm{m}\) (d) \((10.0 \pm 0.2) \mathrm{m}\)

The specific resistance \(\rho\) of a circular wire of radius \(r_{1}\) resistance \(R\) and length \(l\) is given by \(\rho=\frac{\pi r^{2} R}{l}\). Given, \(r=0:(24 \pm 0.02) \mathrm{cm}, R=(30 \pm 1) \Omega\) and \(l=(4.80 \pm 0.01) \mathrm{cm}\). The percentage error in \(\rho\) is nearly (a) \(7 \%\) (b) \(9 \%\) (c) 139 (d) \(20 \%\)

The initial temperature of a liquid is \((80.0 \pm 0.1)^{\circ} \mathrm{C}\). After it has been cooled, its temperature is \((10.0 \pm 0.1)^{\circ} \mathrm{C}\). The fall in temperature in degree centigrade is (a) \(70.0\) (b) \(70.0 \pm 0.3\) (c) \(70.0 \pm 0.2\) (d) \(70.0 \pm 0.1\)

A physical quantity is represented by \(X=\mathrm{M}^{a} \mathrm{~L}^{b} \mathrm{~T}^{-c}\). If percentage errors in the measurements of \(M, L\) and \(T\) are \(\alpha \%, \beta \%\) and \(\gamma \%\) respectively, then total percentage error is (a) \((\alpha a+\beta b-\gamma c) \%\) (b) \((\alpha a+\beta b+\gamma c) \%\) (c) \((\alpha a-\beta b-\gamma c) q\) (d) 096

The internal and external diameters of a hollow cylinder are measured with the help of a vernier callipers. Their values are \(4.23 \pm 0.01 \mathrm{~cm}\) and \(3.87 \pm 0.01 \mathrm{~cm}\) respectively. The thickness of the wall of the cylinder is (a) \(0.36 \pm 0.02 \mathrm{~cm}\) (b) \(0.18 \pm 0.02 \mathrm{~cm}\) (c) \(0.36 \pm 0.01 \mathrm{~cm}\) (d) \(0.18 \pm 0.01 \mathrm{~cm}\)

The density of the material of a cube is measured by measuring its mass and length of its side. If the maximum errors in the measurement of mass and the length are \(3 \%\) and \(2 \%\) respectively, the maximum error in the measurement of density is (a) \(1 \%\) (b) \(5 \%\) (c) \(7 \%\) (d) \(9 \%\)

When the planet Jupiter is at a distance of \(824.7\) million \(\mathrm{km}\) from the earth, its angular diameter is measured to be \(35.72^{\prime \prime}\) of arc. The diameter of Jupiter can be calculated as (a) \(1329 \times 10^{7} \mathrm{~km}\) (b) \(1429 \times 10^{5} \mathrm{~km}\) (c) \(929 \times 10^{5} \mathrm{~km}\) (d) \(1829 \times 10^{5} \mathrm{~km}\)

In an experiment, we measure quantities \(a, b\) and \(c\). Then \(x\) is calculated from the formula, \(x=\frac{a b^{2}}{c^{3}}\). The percentage errors in \(a, b, c\) are \(\pm 1 \%, \pm 3 \%\) and \(\pm 2 \%\) respectively. The percentage error in \(x\) can be (a) \(\pm 10\) (b) \(\pm 4 \%\) (c) \(7 \%\) (d) \(\pm 139\)

The time dependence of a physical quantity \(P\) is given by \(P=P_{0} e^{-a t^{2}}\), where \(\alpha\) is a constant and \(t\) is time. Then constant \(\alpha\) is (a) dimensionless (b) dimension of \(t^{-2}\) (c) dimensions of \(\boldsymbol{P}\) (d) dimension of \(t^{2}\)

The least count of a stop watch is \(0.2 \mathrm{~s}\). The time of 20 oscillations of a pendulum is measured to be \(25 \mathrm{~s}\). The percentage error in the measurement of time will be (a) \(8 \%\) (b) \(1.8 \%\) (c) \(0.8 \%\) (d) \(0.1 \%\)

The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate by using the formula \(p=\frac{F}{l^{2}}\).If the maximum errors in the measurement of force and length are \(4 \%\) and \(2 \%\) respectively, then the maximum error in the measurement of pressure is (a) \(1 \%\) (b) \(2 \%\) (c) \(8 \%\) (d) \(10 \%\)

Given, potential difference \(V=(8 \pm 0.5) \mathrm{V}\) and current \(I=(2 \pm 0.2) \mathrm{A}\). The value of resistance \(R\) is (a) \(4 \pm 16.259\) (b) \(4 \pm 6.25 \%\) (c) \(4 \pm 10 \%\) (d) \(4 \pm 896\)

The length, breadth and thickness of a block is measured to be \(50 \mathrm{~cm}, 2.0 \mathrm{~cm}\) and \(1.00 \mathrm{~cm}\). The percentage error in the measurement of volume is (a) \(0.89\) (b) \(8 \%\) (c) 1096 (d) \(12.5 \%\)

Given \(\pi=3.14\). The value of \(\pi^{2}\) with due regard for significant figures is (a) \(9.86\) (b) \(9.859\) (c) \(9.8596\) (d) \(9.85960\)

One side of a cubical block is measured with the help of a vernier callipers of vernier constant \(0.01 \mathrm{~cm}\). This side comes out to be \(1.23 \mathrm{~cm}\). What is the percentage error in the measurement of area? (a) \(\frac{1.23}{0.01} \times 100\) (b) \(\frac{0.01}{1.23} \times 100\) (c) \(2 \times \frac{0.01}{1.23} \times 100\) (d) \(3 \times \frac{0.01}{1.23} \times 100\)

A physical quantity \(P\) is related to four observables \(a, b, c\) and \(d\) are as follows \(P=a^{3} b^{2} / \sqrt{c} d\) The percentage errors of measurement in \(a, b, c\) and \(d\) are \(1 \%, 3 \%, 4 \%\) and \(2 \%\) respectively. What is the percentage error in the quantity \(P\), if the value of \(P\) calculated using the above relation turns out to be 3.763, to what value should you round-off the result? [NCERT] (a) \(13 \%\) and \(3.8\) (b) \(1.3 \%\) and \(0.38\) (c) \(1.3 \%\) and \(3.8\) (d) \(3.896\) and 13

Length is measured in metre and time in second as usual. But a new unit of mass is so chosen that \(G=1\). This new unit of mass is equal to (a) \(1.5 \times 10^{7} \mathrm{~kg}\) (b) \(1.5 \times 10^{10} \mathrm{~kg}\) (c) \(6.67 \times 10^{-11} \mathrm{~kg}\) (d) \(6.67 \times 10^{-8} \mathrm{~kg}\)

The length, breadth and thickness of a metal block is given by \(l=90 \mathrm{~cm}, b=8 \mathrm{~cm}, t=2.45 \mathrm{~cm}\). The volume of the block is (a) \(2 \times 10^{2} \mathrm{~cm}^{3}\) (b) \(1.8 \times 10^{2} \mathrm{~cm}^{3}\) (c) \(1.77 \times 10^{2} \mathrm{~cm}^{3}\) (d) \(1.764 \times 10^{2} \mathrm{~cm}^{2}\)

The focal length of a mirror is given by \(\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\) where \(u\) and \(v\) represent object and image distances respectively. The maximum relative error in \(f\) is (a) \(\frac{\Delta f}{f}=\frac{\Delta u}{u}+\frac{\Delta v}{v}\) (b) \(\frac{\Delta f}{f}=\frac{1}{\Delta u / u}+\frac{1}{\Delta v / v}\) (c) \(\frac{\Delta f}{f}=\frac{\Delta u}{u}+\frac{\Delta v}{v}+\frac{\Delta(u+v)}{u+v}\) (d) \(\frac{\Delta f}{f}=\frac{\Delta u}{u}+\frac{\Delta v}{v}+\frac{\Delta u}{u+v}+\frac{\Delta v}{u+v}\)

The measured mass and volume of a body are \(23.42 \mathrm{~g}\) and \(4.9 \mathrm{~cm}^{3}\) respectively with possible error \(0.01 \mathrm{~g}\) and \(0.1 \mathrm{~cm}^{3} .\) The maximum error in density is nearly (a) \(0.2 \%\) (b) \(2 \%\) (c) \(5 \%\) (d) \(10 \%\)

The velocity of transverse wave in a string is \(v=\sqrt{\frac{T}{M}}\) where \(T\) is the tension in the string and \(M\) is mass per unit length. If \(T=3.0 \mathrm{kgf}\), mass of string is \(2.5 \mathrm{~g}\) and length of string is \(1.00 \mathrm{~m}\), then the percentage error in the measurement of velocity is (a) \(0.5\) (b) \(0.7\) (c) \(2.3\) (d) \(3.6\)

The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Á. \(1 \AA\) \(10^{-10} \mathrm{~m}\). The size of the hydrogen atom is about \(0.5 \mathrm{~A}\). The total atomic volume in \(\mathrm{m}^{3}\) of a mole of hydrogen atoms would be [NCERT] (a) \(3.15 \times 10^{-7} \mathrm{~m}^{3}\) (b) \(3.0 \times 10^{-8} \mathrm{~m}^{3}\) (c) \(3.85 \times 10^{-7} \mathrm{~m}^{3}\) (d) \(2.85 \times 10^{-7} \mathrm{~m}^{3}\)

The relative density of the material of a body is the ratio of its weight in air and the loss of its weight in water. By using a spring balance, the weight of the body in air is measured to be \(5.00 \pm 0.05 \mathrm{~N}\). The weight of the body in water is measured to be \(4.00 \pm 0.05 \mathrm{~N}\). Then, the maximum possible percentage error in relative density is (a) \(11 \%\) (b) 1096 (c) \(9 \%_{6}\) (d) \(7 \%\)

The length \(l\), breadth \(b\) and thickness \(t\) of a block are measured with the help of a metre scale. Given \(l=15.12 \pm 0.01 \mathrm{~cm}, \mathrm{~b}=10.15 \pm 0.01 \mathrm{~cm}, t=5.28 \pm 0.01 \mathrm{~cm}\) The percentage error in volume is (a) \(0.64 \%\) (b) \(0.28 \%\) (c) \(0.37 \%\) (d) \(0.489\)