In a double slit experiment the distance between the Slits 0.1 cm and the screen is placed at 50 cm from the slit plane. when one slit is covered with a transparent sheet having thickness t and refractive index $n = 1.5$ the central fringe shifts by 0.2 cm. The value of $t$ is

Two spheres having equal mass m, charge q and radius R, are moving towards each other. Both have speed u at an instant when distance between their centers is 4R. Minimum value of u so that they touch each other is
(A) $\sqrt{\frac{q^2}{4\pi\varepsilon_0 mR}}$
(B) $\sqrt{\frac{q^2}{8\pi\varepsilon_0 mR}}$
(C) $\sqrt{\frac{q^2}{16\pi\varepsilon_0 mR}}$
(D) $\sqrt{\frac{q^2}{\pi\varepsilon_0 mR}}$

Compare magnitude of force in different region
(A) $\mathrm{F}_{\mathrm{AB}} > \mathrm{F}_{\mathrm{BC}} > \mathrm{F}_{\mathrm{CD}} > \mathrm{F}_{\mathrm{DE}}$ (B) $\mathrm{F}_{\mathrm{BC}} > \mathrm{F}_{\mathrm{DE}} > \mathrm{F}_{\mathrm{AB}} > \mathrm{F}_{\mathrm{CD}}$ (C) $\mathrm{F}_{\mathrm{BC}} > \mathrm{F}_{\mathrm{DE}} > \mathrm{F}_{\mathrm{CD}} > \mathrm{F}_{\mathrm{AB}}$ (D) $\mathrm{F}_{\mathrm{AB}} > \mathrm{F}_{\mathrm{CD}} > \mathrm{F}_{\mathrm{BC}} > \mathrm{F}_{\mathrm{DE}}$

Determine B at center.

Find moment of inertia about given axis
$$I = \frac{ML^{2}}{3} + \left(\frac{ML^{2}}{12} + M(L)^{2}\right)$$

If equivalent Resistance of circuit between A \& B is $x \Omega$. Determine value of x?

Temperature of 10 mole ideal gas having molar sp. heat capacity at constant pressure $\mathrm{C}_{\mathrm{p}} = 7\mathrm{R}$ in increased by 10 K find increase in internal energy of gas. $(R = \frac{25}{3})$

In a meter bridge two balancing resistances are $30\Omega$ and $20\Omega$. If galvanometer shows zero deflection for the jockey's contact point P. Then find the length A.P.
(A) 70 cm
(B) 60 cm
(C) 40 cm
(D) 30 cm

In a microscope the objective is having focal length $\mathbf{f}_{\mathbf{0}} = \mathbf{20~cm}$ eyepiece is having focal length $\mathbf{f}_{\mathbf{e}} = \mathbf{4~cm}$. The tube length is $\mathbf{32~cm}$. Then magnification produced by this microscope for normal adjustment is $\underline{\quad}$.

Statement-1: Angular fringe width increase if separation between slits and screen increase.
Statement-2: Angular fringe width increase if source of higher wavelength is used.
(A) Statement-1 is true, Statement-2 is True, Statement-2 is a correct explanation for statement-1.
(B) Statement-1 is true, Statement-2 is True, Statement-2 is NOT a correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false
(D) Statement-1 is False, statement-2 is True.

A conducting circular loop of area $1.0\mathbf{~m}^{\mathbf{2}}$ is placed perpendicular to a magnetic field which varies as $\mathrm{B} = \sin(100\mathrm{t})$ tesla. If the resistance of the loop is $100\Omega$ then average thermal energy dissipated in the loop in period is

Let $O$ be the vertex (apex) of a right circular cone such that the radius of the base is 1 and the slant height is 3.
(1) Consider the net of the cone, which consists of a sector and a circle. (The net of a solid is a 2-dimensional shape that can be folded to form that solid.) The central angle of the sector is $\square$ ABC , and the area of the sector is $\square$ D $\pi$.
(2) Take two points A and B on the circumference of the base such that the line segment AB is a diameter. Take a point P on the segment OB and consider a path on the side of this circular cone which starts from the point A , passes through the point P and returns to A . Denote the length of the path by $\ell$.
(i) If $\mathrm { OP } = 2$, then the smallest value of $\ell$ is $\mathbf { E }$. $\mathbf { F }$.
(ii) Let point P be any point on the line segment OB . When $\ell$ is minimized, then $\mathrm { OP } = \frac { \mathbf { G } } { \mathbf { G } }$, and the value of $\ell$ is $\square \sqrt { } \square$.

We are to find a two-digit natural number $a$ such that $a + 9$ is a multiple of 7 and $a + 8$ is a multiple of 13.
First of all, $a + 9$ and $a + 8$ can be represented as
$$a + 9 = \mathbf { M } m , \quad a + 8 = \mathbf { N O } n ,$$
where $m$ and $n$ are natural numbers. From these two equalities, we have
$$\mathbf { M } m - \mathbf { N O } n = \mathbf { P } .$$
Since the pair of $m = \mathbf { Q }$ and $n = \mathbf { R }$ is an integral solution of (1), we have
$$\mathbf { M } ( m - \mathbf { Q } ) = \mathbf { NO } ( n - \mathbf { R } ) .$$
From (2), a natural number $n$ satisfying (1) can be represented as
$$n = \mathbf { S }$$
where $k$ is an integer. Thus
$$a = \mathbf { U V } k + \mathbf { W } ,$$
and since $a$ is a two-digit natural number, $a = \mathbf { X Y }$.

According to information from the Business Association of Aquaculture of Spain, during 2016 gilt-head bream, sea bass and turbot were marketed in Spain for a total of 275.8 million euros. In that report it states that a total of 13740 tonnes of gilt-head bream and 23440 tonnes of sea bass were marketed. As for turbot, 7400 tonnes were sold for a value of 63.6 million euros. Knowing that the kilogram of gilt-head bream was 11 cents more expensive than the kilogram of sea bass, it is requested to calculate the price per kilogram of each of the three types of fish mentioned above.

The first performance in the USA of Mahler's eighth symphony took place in Philadelphia in 1916 with the participation of an orchestra, two choirs with the same number of members, a third children's choir and, in addition, eight guest soloists who did not belong to any of the choirs. One tenth of the total number of performers in the three choirs was 15 units less than the number of orchestra members. The members of each of the two non-children choirs exceeded by 140 units the sum of the children's choir and orchestra members. The number of orchestra members exceeded by 21 units one twelfth of the total number of performers. How many performers did the orchestra and each of the choirs have? How many performers were there in total?

6. Suppose $a , b , c$ are three positive integers. If 25 is the greatest common divisor of $a$ and $b$, and $3, 4, 14$ are all common divisors of $b$ and $c$, which of the following is correct?
(1) $c$ must be divisible by 56.
(2) $b \geq 2100$.
(3) If $a \leq 100$, then $a = 25$.
(4) The greatest common divisor of $a , b , c$ is a divisor of 25.
(5) The least common multiple of $a , b , c$ is greater than or equal to $25 \times 3 \times 4 \times 14$.

11. Decomposing the positive integer 18 into a product of two positive integers gives
$$1 \times 18, 2 \times 9, 3 \times 6$$
three ways. Among these three decompositions, $3 \times 6$ has the smallest difference between the two numbers, so we call $3 \times 6$ the optimal decomposition of 18. When $p \times q ( p \leq q )$ is the optimal decomposition of a positive integer $n$, we define the function $F ( n ) = \frac { p } { q }$. For example, $F ( 18 ) = \frac { 3 } { 6 } = \frac { 1 } { 2 }$. Which of the following statements about the function $F ( n )$ are correct?
(1) $F ( 4 ) = 1$.
(2) $F ( 24 ) = \frac { 3 } { 8 }$.
(3) $F ( 27 ) = \frac { 1 } { 3 }$.
(4) If $n$ is a prime number, then $F ( n ) = \frac { 1 } { n }$.
(5) If $n$ is a perfect square, then $F ( n ) = 1$.

Part Two: Fill-in-the-Blank Questions (45 points)

Instructions: 1. For questions A through I, mark your answers on the "Answer Section" of the answer sheet at the indicated row numbers (12–32).
2. Each completely correct answer is worth 5 points. Wrong answers are not penalized. Incomplete answers receive no points.
A. A sample survey of 1000 families with two children in a certain region obtained the following data, where (boy, girl) represents a family where the first child is a boy and the second child is a girl, and so on.

Family Type	Number of Families
(boy, boy)	261
(boy, girl)	249
(girl, boy)	255
(girl, girl)	235

From this data, the estimated ratio of boys to girls in families with two children in this region is approximately (rounded to the nearest integer).
B

9. In coordinate space, three spheres of radius 1 are placed on the $xy$-plane and are mutually tangent to each other. Let their centers be $A, B, C$ respectively. A fourth sphere of radius 1 is placed above these three spheres and is tangent to all three spheres, maintaining stability. Let the center of the fourth sphere be $P$. Which of the following options are correct?
(1) The plane containing points $A, B, C$ is parallel to the $xy$-plane
(2) Triangle $ABC$ is an equilateral triangle
(3) Triangle $PAB$ has one side of length $\sqrt{2}$
(4) The distance from point $P$ to line $AB$ is $\sqrt{3}$
(5) The distance from point $P$ to the $xy$-plane is $1 + \sqrt{3}$

2. In professional baseball, the ERA (Earned Run Average) is an important statistic for understanding a pitcher's performance. It is calculated as follows: If a pitcher has pitched $n$ innings with a total of $E$ earned runs, then the ERA is $\frac { E } { n } \times 9$. A pitcher previously pitched 90 innings with an ERA of 3.2. In the most recent game, this pitcher pitched 6 innings with no earned runs. After completing this game, the pitcher's ERA becomes
(1) 2.9
(2) 3.0
(3) 3.1
(4) 3.2
(5) 3.3

3. A circular track is divided into inner and outer loops with radii of 30 and 50 meters, respectively. Person A walks at constant speed on the inner loop, and person B runs at constant speed on the outer loop. It is known that for every one lap A completes, B completes exactly two laps. If A walks 45 meters, then during the same time period, B runs
(1) 90 meters
(2) 120 meters
(3) 135 meters
(4) 150 meters
(5) 180 meters

4. License plates in a certain region consist of six characters: the first two are uppercase English letters other than O, and the last four are Arabic numerals from 0 to 9. However, three consecutive 4's are not allowed. For example, AA1234 and AB4434 are valid license plates, while AO1234 and AB3444 are not. The number of license plates with first character A and last character 4 is
(1) $25 \times 9 ^ { 3 }$
(2) $25 \times 9 ^ { 2 } \times 10$
(3) $25 \times 900$
(4) $25 \times 990$
(5) $25 \times 999$

7. Which of the following options contain rational numbers?
(1) 3.1416
(2) $\sqrt{3}$
(3) $\log_{10}\sqrt{5} + \log_{10}\sqrt{2}$
(4) $\frac{\sin 15^{\circ}}{\cos 15^{\circ}} + \frac{\cos 15^{\circ}}{\sin 15^{\circ}}$
(5) The unique real root of the equation $x^{3} - 2x^{2} + x - 1 = 0$

A company has only three members: a manager, a secretary, and a salesperson. If only the secretary receives a 10\% salary increase, the company's total salary expenditure increases by 3\%; if only the salesperson receives a 20\% salary increase, the company's total salary expenditure increases by 4\%. If only the manager's salary is reduced by 15\%, then the company's total salary expenditure will decrease by (12).(13)\%.

A manufacturer produces two types of electric vehicles, Type A and Type B. The costs for producing these two types involve three categories: battery, motor, and others. The costs for each category are shown in the table below (unit: 10,000 yuan):

	Battery Cost	Motor Cost	Other Cost
Type A	56	26	48
Type B	40	20	56

The selling price formula for the two types of electric vehicles is the sum of ``$x$ times the battery cost'', ``$y$ times the motor cost'', and ``$\frac { x + y } { 2 }$ times the other cost'', that is,
Selling Price $=$ Battery Cost $\times x +$ Motor Cost $\times y +$ Other Cost $\times \frac { x + y } { 2 }$ where the multipliers $x, y$ must satisfy ``$1 \leq x \leq 2, 1 \leq y \leq 2$, and the selling prices of both Type A and Type B electric vehicles do not exceed 200 (10,000 yuan)''. To differentiate its products, the manufacturer wants to maximize the price difference between Type A and Type B electric vehicles. Based on the above information, answer the following questions.
(1) Write the selling prices of Type A and Type B electric vehicles (in terms of $x$ and $y$), and explain why ``the selling price of Type A electric vehicles is always higher than that of Type B electric vehicles''. (4 points)
(2) On a coordinate plane, draw the feasible region of $(x, y)$ satisfying the conditions in the problem, and shade the region with diagonal lines. (4 points)
(3) Find the values of multipliers $x$ and $y$ that maximize the price difference between Type A and Type B electric vehicles. What is the maximum price difference in units of 10,000 yuan? (6 points)

When an artist uses single-point perspective to draw spatial scenes on a flat piece of paper, the following principles must be followed: I. A straight line in space must be drawn as a straight line on the paper. II. The relative positions of points on a line in space must be consistent with the relative positions of the points drawn on the paper. III. The $K$ value of any four distinct points on a line in space must be the same as the $K$ value of the four points drawn on the paper, where the $K$ value is defined as follows: For any four ordered distinct points $P_1, P_2, P_3, P_4$ on a line, the corresponding $K$ value is defined as $$K = \frac{\overline{P_1P_4} \times \overline{P_2P_3}}{\overline{P_1P_3} \times \overline{P_2P_4}}$$ An artist follows the above principles to draw a line in space and four distinct points $Q_1, Q_2, Q_3, Q_4$ on that line on paper, where $\overline{Q_1Q_2} = \overline{Q_2Q_3} = \overline{Q_3Q_4}$. If the line drawn on the paper is viewed as a number line and the points on it are represented by coordinates, which of the following sets of four coordinates is most likely to be the coordinates of these four points on the paper?
(1) $1, 2, 4, 8$
(2) $3, 4, 6, 9$
(3) $1, 5, 8, 9$
(4) $1, 2, 4, 9$
(5) $1, 7, 9, 10$