An optical fibre consists of core of $mu subscript 1 end subscript$ surrounded by a cladding of $mu subscript 2 end subscript less than mu subscript 1 end subscript$ . A beam of light enters from air at an angle $alpha$ with axis of fibre. The highest $alpha$ for which ray can be travelled through fibre is

In steel the young's modulus and the strain at the breaking point are $2 cross times 10 to the power of 11 straight N over straight m squared$ and 0.15 respectively the stress at the breaking point for steel is therefore

Hail stones fall from certain height If only 2% of the mass of the hail stone melt on reaching the ground,, the height from which they fall is ( g = 10 ms^-2, L = 80 cal/gm and J = 4.2J/cal)

maths-

$Sin invisible function application 4530 to the power of ring operator$ =

maths-General

A fish rising vertically up towards the surface of water with speed 3 $m s to the power of negative 1 end exponent$ observes a bird diving vertically down towards it with speed 9 $m s to the power of negative 1 end exponent$ . The actual velocity of bird is

The image of point P when viewed from top of the slabs will be

A cube of side 2 m is placed in front of a concave mirror focal length 1m with its face P at a distance of 3 m and face Q at a distance of 5 m from the mirror. The distance between the images of face P and Q and height of images of P and Q are

Two transparent slabs have the same thickness as shown. One is made of material A of refractive index 1.5. The other is made of two materials B and C with thickness in the ratio 1 : 2. The refractive index of C is 1.6. If a monochromatic parallel beam passing through the slabs has the same number of waves inside both, the refractive index of B is

Maths-

$L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 minus cos space m x over denominator 1 minus cos space n x end fraction comma n not equal to 0$

We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means $fraction numerator 0 over denominator 0 space end fraction o r space fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.$

$L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 minus cos space m x over denominator 1 minus cos space n x end fraction comma n not equal to 0$

Maths-General

maths-

If $log subscript 10 space 100 minus 2 comma log subscript 10 space 101 minus 2.004 comma log subscript 10 space 102 equals 2.0086 comma log subscript 10 space 103 equals 2.0128$ , then $integral subscript 100 superscript 103 log subscript 10 space x d x$ by trapezoidal rule is

maths-General

Maths-

$Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 minus cos space 2 m x over denominator sin squared space n x end fraction left parenthesis m comma n element of Z right parenthesis$

We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means $fraction numerator 0 over denominator 0 space end fraction space o r space fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.$

$Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 minus cos space 2 m x over denominator sin squared space n x end fraction left parenthesis m comma n element of Z right parenthesis$

Maths-General

A wire of length L and radius r is rigidly fixed at one end on stretching the other end of the wire a force F the increase in its lengths is L. If another wire of same material but of length 2 L and radius 2r, is stretched with a force of 2F, the increase in its length will he

Consider the situation shown in figure. Water $open parentheses mu subscript w end subscript equals fraction numerator 4 over denominator 3 end fraction close parentheses$ is filled in a breaker upto a height of 10 cm. A plane mirror fixed at a height of 5 cm from the surface of water. Distance of image from the mirror after reflection from it of an object O at the bottom of the beaker is

One side of a glass slab is silvered as shown. A ray of light is incident on the other side at angle of incidence $i equals 4 5 to the power of o end exponent$ . Refractive index of glass is given as 1.5. The deviation of the ray of light from its initial path when it comes out of the slab is

A plane mirror is placed at the bottom of the tank containing a liquid of refractive index $mu$ . P is a small object at a height h above the mirror. An observer O-vertically above P outside the liquid see P and its image in the mirror. The apparent distance between these two will be