NSEJS 2016 Question Paper

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Question 1

Rod AB of radius 2 r is joined with rod BC of radius r . They are of same material and are of same length. The combination carries a current I. Choose the correct statement.

(a) $V_{A B}=4 V_{B C}$
(b) Current per unit in AB and BC are equal
(c) Resistance of AB is greater than of BC
(d) $\mathrm{V}_{\mathrm{BC}}=4 \mathrm{ V}_{\mathrm{AB}}$ Ans. (d) Sol. $\frac{\mathrm{V}_{\mathrm{AB}}}{\mathrm{V}_{\mathrm{BC}}}=\frac{\mathrm{IR}_{\mathrm{AB}}}{\mathrm{IR}_{\mathrm{BC}}}=\frac{\mathrm{I} \rho \ell / \pi(2 \pi)^{2}}{\mathrm{I} \rho \ell / \pi \mathrm{r}^{2}}=\frac{1}{4}$ or $\mathrm{V}_{\mathrm{BC}}=4 \mathrm{ V}_{\mathrm{AB}}$

Question 2

The statement "a is not less than 4 " is correctly represented by

(a) $a<4$
(b) a $>4$
(c) $a \geq 4$
(d) $\mathrm{a} \leq 4$

Question 3

In the figure shown, the current carrying loop is fixed, where as current carrying straight conductor is free to move. Then straight wire will (ignore gravity)

(a) remain stationary
(b) move towards the loop
(c) move away from the loop
(d) rotate about the axis perpendicular to plane of paper

Question 4

Two friends A and B watched a car from the top of their buildings. Angle of depression for A was $10^{\circ}$ more than angle of depression for B , then

(a) A's apartment is taller than B's apartment
(b) B's apartment is taller than A's apartment
(c) A's apartment and B's apartment have same height
(d) We cannot compare the heights of the two apartments

Question 5

A convex mirror of focal length $f$ produces an image of size equal to $\frac{1}{n}$ times the size of the object. Then the object distance is

(a) nf
(b) $\frac{\mathrm{f}}{\mathrm{n}}$
(c) (n + 1)f
(d) (n - 1)f

Question 6

Total surface area of a sphere $S$ with radius $\sqrt{2}+\sqrt{3} \mathrm{ cm}$ is

(a) $400 \pi(5+2 \sqrt{6}) \mathrm{sq} \mathrm{mm}$
(b) $\pi(\sqrt{2}+\sqrt{3})^{2}$ sq cm
(c) $2 \pi(\sqrt{2}+\sqrt{3})^{2} \mathrm{sq} \mathrm{cm}$
(d) $40 \pi(5+2 \sqrt{6}) \mathrm{sq} \mathrm{mm}$

Question 7

The angle between the hour arm and the minute arm of a clock at 2:10 a.m. is

(a) Zero
(b) $4^{\circ}$
(c) $5^{\circ}$
(d) $6^{\circ}$

Question 8

A craft teacher reshapes the wax from a cylinder of candle with section diameter 6 cm and the height 6 cm into a sphere. The radius of this sphere will be

(a) $r=6 \sqrt{3 / 2} \mathrm{ cm}$
(b) $r=6 \mathrm{ cm}$
(c) $\mathrm{r}=3 \sqrt[3]{3 / 2} \mathrm{ cm}$
(d) $\mathrm{r}=6 \sqrt[3]{2} \mathrm{ cm}$

Question 9

A point object O is kept at origin. When a concave mirror $\mathrm{M}_{1}$ placed at $\mathrm{x}=6 \mathrm{ cm}$, image is formed at infinity. When $\mathrm{M}_{1}$ is replaced by another concave mirror $\mathrm{M}_{2}$ at same position, image is formed at $\mathrm{x}=$ 30 cm , then ratio of the focal length of $\mathrm{M}_{1}$ to that of $\mathrm{M}_{2}$ is

(a) $\frac{3}{4}$
(b) $\frac{4}{3}$
(c) 5
(d) $\frac{1}{5}$

Question 10

The number $3^{8}\left(3^{10}+6^{5}\right)+2^{3}\left(2^{12}+6^{7}\right)$ is

(a) A perfect square and a perfect cube
(b) Neither a perfect squre nor a perfect cube
(c) A perfect cube but not a perfect square
(d) A perfect square but not a perfect cube

Question 11

Melting point of a substance is $10^{\circ} \mathrm{C}$. What does this mean?

(a) The substance is a liquid at $10^{\circ} \mathrm{C}$.
(b) The substance is a solid at $10^{\circ} \mathrm{C}$.
(c) There is an equilibrium between solid phase and liquid phase at $10^{\circ} \mathrm{C}$
(d) The substance is $50 %$ solid and $50 %$ liquid at $10^{\circ} \mathrm{C}$.

Question 12

Let the number of rectangles formed by 6 horizontal and 4 vertical lines be n and those formed by 5 vertical and 5 horizontal lines be m then we have

(a) $n=m$
(b) $\mathrm{n} \geq \mathrm{m}+1$
(c) $m \geq n$
(d) $m>n+5$

Question 13

The effective resistance between $A$ and $D$ in the circuit shown in the adjacent figure is

(a) $5 \Omega$
(b) $10 \Omega$
(c) $15 \Omega$
(d) $20 \Omega$

Question 14

If ABCD is a rhombus and $\angle \mathrm{ABC}=60^{\circ}$ then

(a) The points $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are concyclic
(b) The quadrilateral has exactly half the area of the square with same sides as ABCD
(c) The quadrilateral has area $\frac{\sqrt{3}}{2} \mathrm{AB}^{2}$
(d) The diagonals of the quadrilateral ABCD are equal and bisect each other at right angle

Question 15

Three bulbs $\mathrm{B}_{1}, \mathrm{ B}_{2}$ and $\mathrm{B}_{3}$ having rated powers $100 \mathrm{ W}, 60 \mathrm{ W}$ and 60 W at 250 V are connected in a circuit as shown in the adjacent figure. If $\mathrm{W}_{1}, \mathrm{ W}_{2}$ and $\mathrm{W}_{3}$ are the output powers of the bulbs $\mathrm{B}_{1}, \mathrm{ B}_{2}$ and $\mathrm{B}_{3}$ respectively, then

(a) $\mathrm{W}_{1}>\mathrm{W}_{2}=\mathrm{W}_{3}$
(b) $\mathrm{W}_{1}>\mathrm{W}_{2}>\mathrm{W}_{3}$
(c) $\mathrm{W}_{1}<\mathrm{W}_{2}=\mathrm{W}_{3}$
(d) $\mathrm{W}_{1}<\mathrm{W}_{2}<\mathrm{W}_{3}$

Question 16

If $\mathrm{a}, \mathrm{b}>0$ then

(a) $a+b \leq \sqrt{a b}$
(b) a + b $>\sqrt{\text { ab }}$
(c) $\mathrm{a}+\mathrm{b} \geq \sqrt{2 \mathrm{ab}}$
(d) None of the above inequalities will hold

Question 17

Following diagram shows refraction of parallel beam of light through a spherical surface. Identify the correct ray diagram

Question 18

Tenth term in the sequence $12,18,20,28, \ldots$ is

(a) 336
(b) 63
(c) 216
(d) 68

Question 19

In the diagram $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ are two plane mirrors at right angles to each other. O is a luminous point object. Consider two images formed due to first reflection at $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$. The area of the triangle formed by the object and two images is

(a) $4 \mathrm{ cm}^{2}$
(b) $2 \mathrm{ cm}^{2}$
(c) $8 \mathrm{ cm}^{2}$
(d) $16 \mathrm{ cm}^{2}$

Question 20

The probability of a point within an equilateral triangle with side 1-unit lying outside its in-circle (inscribed circle) is

(a) $1-\frac{1}{2(\sqrt{3})}$
(b) $1-\frac{\pi}{3 \sqrt{3}}$
(c) $1-\frac{\pi}{2 \sqrt{3}}$
(d) $1-\frac{2 \pi}{3 \sqrt{3}}$

Question 21

A fisherman of height $h$ is standing on the bank of a lake. A fish in the water perceives his height as $h$ '. Then

(a) $\mathrm{h}^{\prime}>\mathrm{h}$
(b) $h^{\prime}<h$
(c) $h^{\prime}=h$
(d) $h^{\prime}>h$ or $h^{\prime}<h$ depending on position of fish

Question 22

A triangle has perimeter 316 and its sides are of integer length. The maximum possible area for such a triangle is achieved for

(a) Single triangle
(b) Two triangles
(c) Three triangles
(d) More than three triangle

Question 23

Object A is completely immersed in water. True weight of object A is $\mathrm{W}_{\mathrm{A}}$. Weight of water with beaker is $\mathrm{W}_{\mathrm{B}}$. Let B be the buoyant force. $\mathrm{W}_{1}$ and $\mathrm{W}_{2}$ are scale readings of spring balance and weighing scale respectively.

(a) $\mathrm{W}_{1}=\mathrm{W}_{\mathrm{A}}$
(b) $\mathrm{W}_{1}=\mathrm{W}_{\mathrm{A}}+\mathrm{B}$
(c) $\mathrm{W}_{2}=\mathrm{W}_{\mathrm{B}}$
(d) $\mathrm{W}_{2}=\mathrm{W}_{\mathrm{B}}+\mathrm{B}$

Question 24

Number of numbers less than 40 having exactly four divisors is

(a) 15
(b) 12
(c) 11
(d) 14

Question 25

Velocity of a particle moving along a straight line varies with time as shown in the figure. Net forces acting on the aprticle are $\mathrm{F}_{1}, \mathrm{ F}_{2}, \mathrm{ F}_{3}, \mathrm{ F}_{4}$ and $\mathrm{F}_{5}$ in the intervals $\mathrm{OA}, \mathrm{AB}, \mathrm{BC}, \mathrm{CD}$ and DE respectively. Indentify the correct statement

(a) $\mathrm{F}_{1}$ increases with time
(b) $\mathrm{F}_{5}$ is initially positive and then becomes negative
(c) $F_{1}$ and $F_{2}$ are in opposite directions
(d) $\mathrm{F}_{3}$ is negative

Question 26

If set $X$ consists of three elements then the number of elements in the power set of power set of $X$ is

(a) $3^{3}$
(b) $2^{3}$
(c) $3^{8}$
(d) $2^{8}$

Question 27

A wooden block $(\mathrm{W})$ is suspended by using a cord from a heavy steel ball (B). The entire system is dropped from a height. Neglecting air resistance, the tension in the cord is

(a) Zero
(b) The difference in the masses of B and W
(c) The differences in the weights of $\mathbf{B}$ and $\mathbf{W}$
(d) The weights of $B$

Question 28

In a n - sided regular polygon, the radius of the circum-circle is equal in length to the shortest diagonal. the number of values of $\mathrm{n}<60$ for which this can happen is

(a) 0
(b) 1
(c) 10
(d) 2

Question 29

A circus performer of weight W is standing on a wire as shown in the figure. The tension in the wire is

(a) Approximately $\frac{\mathrm{W}}{4}$
(b) Approximately $\frac{\mathrm{W}}{2}$
(c) Much more than $\frac{W}{2}$
(d) Much less than $\frac{W}{2}$

Question 30

Number of integers $n$ such that the number $1+n$ is a divisor of the number $1+n^{2}$ is :

(a) 0
(b) 1
(c) 4
(d) 2

Question 31

In the following diagrams O is point object and I is its image formed by a concave mirror. Identify the diagram. In which position of image I is nearly correct.

Question 32

If for $\mathrm{x}, \mathrm{y}>0$ we have $\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=2$ then the minimum value of xy is

(a) 2
(b) 1
(c) 4
(d) $\sqrt{2}$

Question 33

Two bodies $A$ and $B$ are charged with equal magnitude of charge but $A$ with positive charge and $B$ with negative. If $M_{A}$ and $M_{B}$ are masses before charging and $M^{I}{ }_{A}$ and $M^{I}{ }_{B}$ are the masses after charging the ( $\mathrm{m}_{0}$ is some constant mass)

(a) $\mathrm{M}_{\mathrm{A}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{A}}+\mathrm{m}_{0}$ and $\mathrm{M}_{\mathrm{B}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{B}}-\mathrm{m}_{0}$
(b) $\mathrm{M}_{\mathrm{A}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{A}}-\mathrm{m}_{0}$ and $\mathrm{M}_{\mathrm{B}}^{\mathrm{I}}=\mathrm{M}_{\mathrm{B}}+\mathrm{m}_{0}$
(c) $M^{I}{ }_{A}=M^{I}{ }_{B}$
(d) $M^{I}{ }_{A}=M_{A}-\frac{m_{0}}{2}$ and $M_{B}^{I}=M_{B}+m_{0}$

Question 34

The number of natural numbers $\mathrm{n} \leq 30$ for for which $\sqrt{\mathrm{n}+\sqrt{\mathrm{n}+\sqrt{\mathrm{n}+\ldots . .}}}$ is a natural number is

(a) 30
(b) Zero
(c) 6
(d) 5

Question 35

A conductor of length $L$ has a varying cross section with area $2 A$ at $P$ and $A$ at $Q$ as shown in the figure. If it carries a steady current I, then

(a) Net charge per unit volume near P is more than net charge per unit volume near Q .
(b) Net charge per unit volume near Q is less than net charge per unit volume.
(c) Current per unit area near P is more than current per unit area near Q .
(d) Current per unit area near P is less than current per unit area near Q .

Question 36

The number of natural numbers $n \leq 30$ for which $\sqrt{n+\sqrt{n+\sqrt{n+\ldots \ldots . .}}}$ is a prime number is

(a) Three
(b) Zero
(c) Nine
(d) Two

Question 37

Vessels A and B are made of conducting material. Both contain water. Vessel A floats in B. Vessel B is now heated at a uniform rate, then

(a) Water in A boils first.
(b) Water in A boils some time after water in B starts boiling.
(c) Water in both A and B start boiling simultaneously.
(d) Water in A does not boil.

Question 38

The number of squares formed by 5 vertical and 4 horizontal lines (all are equispaced) is

(a) 60
(b) 20
(c) 40
(d) 46

Question 39

Every major city in India has a pollution control board to monitor air and water pollution. The following data is from three different localities in Bangalore city from the year 2015.

ppb stands for parts per billion and ppm stands for parts per million. These are different units to express concentration. They are very similar to percentage (which is actually parts per hundred). Based on the above data, which place will you choose to live in?

(a) All localities are equally polluted, so I have no preference.
(b) P is more polluted than Y and Z , hence I will live in either Y or Z .
(c) Locality Y is least polluted, hence I will live in Y .
(d) Z and Y are more polluted than P , hence I will live in P .

Question 40

A body thrown vertically up reaches a maximum height and returns back. Its acceleration is

(a) Downward during both ascent and descent.
(b) Downward at all positions except at the highest point, where it is zero.
(c) Upward during both ascent and descent.
(d) Downward during ascent and upward during descent.

Question 41

The number of integers $a, b, c$ for which $a^{2}+b^{2}-8 c=3$ is

(a) 2
(b) Infinite
(c) 0
(d) 4
RMO 1992, question no. 4 →
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