Mid term class 10th paper

Section A

(Section A consists of 20 questions of 1 mark each)

1. If two positive integers $a$ and $b$ are written as $a = x^3 y^2$ and $b = x y^3$, where $x, y$ are prime numbers, then $\text{HCF}(a, b)$ is: (a) $xy$ (b) $x y^2$ (c) $x^3 y^3$ (d) $x^2 y^2$
2. The number of zeroes that the polynomial $p(x) = (x - 2)^2 + 4$ can have is: (a) 1 (b) 2 (c) 0 (d) 3
3. The pair of equations $x + 2y + 5 = 0$ and $-3x - 6y + 1 = 0$ has: (a) a unique solution (b) exactly two solutions (c) infinitely many solutions (d) no solution
4. Which of the following is a quadratic equation? (a) $x^2 + 2x + 1 = (4 - x)^2 + 3$ (b) $-2x^2 = (5 - x)\left(2x - \frac{2}{5}\right)$ (c) $(k + 1)x^2 + \frac{3}{2}x = 7$, where $k = -1$ (d) $x^3 - x^2 = (x - 1)^3$
5. In an AP, if $d = -4$, $n = 7$, and $a_n = 4$, then $a$ is equal to: (a) 6 (b) 7 (c) 20 (d) 28
6. In $\triangle ABC$, $DE \parallel BC$ such that $AD = x$, $DB = x - 2$, $AE = x + 2$, and $EC = x - 1$. The value of $x$ is: (a) 4 (b) 3 (c) 2 (d) 5
7. The distance of the point $P(-6, 8)$ from the origin is: (a) 8 (b) $2\sqrt{7}$ (c) 10 (d) 6
8. If $\sin \theta + \cos \theta = \sqrt{2}\cos \theta$, then the value of $\tan \theta$ is: (a) $\sqrt{2}-1$ (b) $\sqrt{2}+1$ (c) $\frac{1}{\sqrt{2}}$ (d) $\sqrt{3}$
9. If a pole 6 m high casts a shadow $2\sqrt{3}\text{ m}$ long on the ground, then the sun’s elevation is: (a) 60° (b) 45° (c) 30° (d) 90°
10. If the perimeter and the area of a circle are numerically equal, then the radius of the circle is: (a) 2 units (b) $\pi$ units (c) 4 units (d) 7 units
11. From an external point $P$, tangents $PA$ and $PB$ are drawn to a circle with centre $O$. If $\angle APB = 70^\circ$, then $\angle AOB$ is: (a) $110^\circ$ (b) $100^\circ$ (c) $120^\circ$ (d) $90^\circ$
12. If $\triangle ABC \sim \triangle DEF$, with $AB = 4\text{ cm}$, $DE = 6\text{ cm}$, and the perimeter of $\triangle ABC = 16\text{ cm}$, then the perimeter of $\triangle DEF$ is: (a) 24 cm (b) 18 cm (c) 32 cm (d) 20 cm
13. The mode and mean of a data are 18 and 24 respectively. The median of the data is: (a) 22 (b) 21 (c) 24 (d) 20
14. Which of the following cannot be the probability of an event? (a) $\frac{2}{3}$ (b) $-1.5$ (c) 15% (d) 0.7
15. If $\alpha$ and $\beta$ are the zeroes of the polynomial $f(x) = x^2 - 5x + k$ such that $\alpha - \beta = 1$, the value of $k$ is: (a) 6 (b) 4 (c) 5 (d) 8
16. The mid-point of the line segment joining the points $A(-2, 8)$ and $B(-6, -4)$ is: (a) $(-4, 2)$ (b) $(2, -4)$ (c) $(-4, -2)$ (d) $(4, 2)$
17. A card is drawn from a well-shuffled deck of 52 playing cards. The probability of getting a black face card is: (a) $\frac{3}{13}$ (b) $\frac{3}{26}$ (c) $\frac{1}{26}$ (d) $\frac{3}{52}$
18. If $x = 3$ is one root of the quadratic equation $x^2 - 2kx - 6 = 0$, then the value of $k$ is: (a) 1 (b) $\frac{1}{2}$ (c) 2 (d) $-\frac{1}{2}$

Direction for Q19 & Q20: In question numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:

• (a) Both (A) and (R) are true and (R) is the correct explanation of (A).
• (b) Both (A) and (R) are true but (R) is not the correct explanation of (A).
• (c) (A) is true but (R) is false.
• (d) (A) is false but (R) is true.

19. Assertion (A): The HCF of two numbers is 5 and their product is 150, then their LCM is 30. Reason (R): For any two positive integers $a$ and $b$, $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$.
20. Assertion (A): The point $(0, 4)$ lies on the y-axis. Reason (R): The x-coordinate of any point on the y-axis is zero.

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Section B

(Section B consists of 5 questions of 2 marks each)

21. Prove that $3 + 2\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is irrational.

22. In $\triangle ABC$, $D$ and $E$ are points on sides $AB$ and $AC$ respectively such that $DE \parallel BC$. If $AD = 2.4\text{ cm}$, $AE = 3.2\text{ cm}$, and $EC = 4.8\text{ cm}$, find the length of $AB$.

23. Find the coordinates of the point which divides the line segment joining the points $(4, -3)$ and $(8, 5)$ internally in the ratio $3:1$. (OR) Find a relation between $x$ and $y$ such that the point $(x, y)$ is equidistant from the points $(3, 6)$ and $(-3, 4)$.

24. Evaluate: $\frac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

25. A quadrilateral $ABCD$ is drawn to circumscribe a circle. Prove that $AB + CD = AD + BC$. (OR) Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

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Section C

(Section C consists of 6 questions of 3 marks each)

26. Find the zeroes of the quadratic polynomial $p(x) = 6x^2 - 3 - 7x$ and verify the relationship between the zeroes and the coefficients.
27. Solve the following pair of linear equations: $2x + 3y = 11$

$2x - 4y = -24$

Hence, find the value of $m$ for which $y = mx + 3$. 28. The sum of the 4th and 8th terms of an AP is 24, and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP. (OR) If the sum of the first $n$ terms of an AP is given by $S_n = 3n^2 + 5n$, find the $n$-th term of this AP. Hence, find its 15th term. 29. Prove the identity: $\frac{\sin \theta - 2\sin^3 \theta}{2\cos^3 \theta - \cos \theta} = \tan \theta$

30. Prove that the lengths of tangents drawn from an external point to a circle are equal. (OR) In two concentric circles, prove that a chord of the larger circle which touches the smaller circle is bisected at the point of contact.
31. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears: (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5

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Section D

(Section D consists of 4 questions of 5 marks each)

32. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers. (OR) A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
33. State and prove Basic Proportionality Theorem (Thales Theorem).
34. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point. (OR) As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships ($\text{use } \sqrt{3} \approx 1.732$).
35. The median of the following data is 525. Find the values of $x$ and $y$, if the total frequency is 100:

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Section E

(Section E consists of 3 Case-Based Questions of 4 marks each)

36. Case Study 1 (Coordinate Geometry)

To conduct sports day activities in a rectangular shaped school ground $ABCD$, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along $AD$, as shown in the layout below.

Niharika runs $\frac{1}{4}\text{th}$ the distance $AD$ on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}\text{th}$ the distance $AD$ on the 8th line and posts a red flag. Take the origin $A(0,0)$, the line $AB$ along the x-axis, and $AD$ along the y-axis.

1. Write the coordinates of the green flag posted by Niharika. (1 Mark)
2. Write the coordinates of the red flag posted by Preet. (1 Mark)
3. Find the shortest distance between both the flags. (2 Marks) (OR) If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, at what distance and on which line should she post her flag? (2 Marks)

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37. Case Study 2 (Arithmetic Progression)

A manufacturer of laptop batteries produced 600 units in the third year and 700 units in the seventh year. Assuming that the production increases uniformly by a fixed number every year:

1. Find the fixed increase in production each year (common difference). (1 Mark)
2. Find the production in the first year. (1 Mark)
3. Find the total production in the first 7 years. (2 Marks) (OR) In which year will the production reach 1000 units? (2 Marks)

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38. Case Study 3 (Statistics)

The COVID-19 vaccination drive was conducted in a local municipal ward. The following distribution gives the ages of 80 individuals who received their booster shots on a particular day:

1. What is the modal class of the given distribution? (1 Mark)
2. Find the class mark and width of the modal class. (1 Mark)
3. Calculate the Mode of the given data. (2 Marks) (OR) Find the Mean age of the persons vaccinated using the Direct Method. (2 Marks)

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