English Medium
Academic Year: 2024-2025
Date & Time: 10th March 2025, 10:30 am
Duration: 3h
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General Instructions:
Read the following instructions carefully and follow them:
- This question paper contains 38 questions. All questions are compulsory.
- This Question Paper is divided into FIVE Sections - Section A, B, C, D, and E.
- In Section - A question number 1 to 18 are Multiple Choice Questions (MCQs) and question number 19 & 20 are Assertion Reason based question of 1 mark each.
- In Section - B questions number 21 to 25 are Very Short-Answer (VSA) type questions of 2 marks each.
- In Section - C questions number 26 to 31 are Short Answer (SA) type questions carrying 3 marks each.
- In Section - D questions number 32 to 35 are Long Answer (LA) type questions carrying 5 marks each.
- In Section - E questions number 36 to 38 are Case Study/based integrated questions carrying 4 marks each. Internal choice is provided in 2 marks questions in each case study.
- There is no overall choice. However, an internal choice has been provided in 2 questions in Section - B, 2 questions in Section - C, 2 questions in Section - D, and 3 questions of 2 maarks in section E.
- Draw neat figures wherever required. Take π = `22/7` wherever required if not stated.
- Use of a calculator is NOT allowed.
Which of the following statements is true for a polynomial p(x) of degree 3?
p(x) has at most two distinct zeroes.
p(x) has at least two distinct zeroes.
p(x) has exactly three distinct zeroes.
p(x) has at most three distinct zeroes.
Chapter:
A pair of dice is thrown once. The probability that the sum of numbers appearing on top faces is at least 4 is ______.
`1/11`
`10/11`
`5/6`
`11/12`
Chapter:
If x = ab3 and y = a3b, where a and b are prime numbers, then [HCF (x, y) – LCM (x, y)] is equal to ______.
1 − a3b3
ab (1 − ab)
ab − a4b4
ab (1− ab) (1 + ab)
Chapter:
`(1 + sqrt3)^2 - (1 - sqrt3)^2` is ______.
a positive rational number
a negative integer
a positive irrational number
a negative irrational number
Chapter:
The value of ‘a’ for which ax2 + x + a = 0 has equal and positive roots is ______.
2
−2
`1/2`
`-1/2`
Chapter:
The distance of which of the following points from the origin is less than 5 units?
(3, 4)
(2, 6)
(−3, −4)
(1, 4)
Chapter:
The number of red balls in a bag is 10 more than the number of black balls. If the probability of drawing a red ball at random from this bag is `3/5`, then the total number of balls in the bag is ______.
50
60
80
40
Chapter:
The value of ‘p’ for which the equations px + 3y = p − 3, 12x + py = p has infinitely many solutions is ______.
−6 only
6 only
± 6
Any real number except ± 6.
Chapter:
ΔABC and ΔPQR are shown in the adjoining figures. The measure of ∠C is:
140°
80°
60°
40°
Chapter:
Which of the following is a trigonometric identity?
sin2 θ = 1 + cos2 θ
cosec2 θ + cot2 θ = 1
sec2 θ = 1 + tan2 θ
sin 2θ = 2 sin θ
Chapter:
Which of the following statement is true?
sin 20° > sin 70°
sin 20° > cos 20°
cos 20° > cos 70°
tan 20° > tan 70°
Chapter:
A 30 m long rope is tightly stretched and tied from the top of the pole to the ground. If the rope makes an angle of 60° with the ground, the height of the pole is ______.
`10sqrt3` m
`30sqrt3` m
15 m
`15sqrt3` m
Chapter:
If the mean and mode of the given set of observations are 10 and 13, respectively, then the value of the median is ______.
19
4
11
43
Chapter:
In the adjoining figure, AB is the chord of the larger circle which touches the smaller circle at P. If the length of AB = diameter of the inner circle = 2r, then the diameter of the larger circle is:
2r
4r
`2sqrt2r`
`sqrt2r`
Chapter:
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On the top face of the wooden cube of side 7 cm, hemispherical depressions of radius 0.35 cm are to be formed by taking out the wood. The maximum number of depressions that can be formed is ______.
400
100
20
10
Chapter:
The cumulative frequency for calculating median is obtained by adding the frequencies of all the ______.
classes up to the median class
classes following the median class
classes preceding the median class
all classes
Chapter:
A parallelogram having one of its sides 5 cm circumscribes a circle. The perimeter of a parallelogram is ______.
20 cm
less than 20 cm
more than 20 cm but less than 40 cm
40 cm
Chapter:
E and F are points on the sides AB and AC, respectively, of a ΔABC, such that `(AE)/(EB) = (AF)/(FC) = 1/2`. Which of the following relations is true?
EF = 2BC
BC = 2EF
EF = 3ВС
BC = 3EF
Chapter:
Assertion (A): A line drawn perpendicular to the tangent at the point of contact passes through the centre of the circle.
Reason (R): Lengths of tangents drawn from an external point to a circle are equal.
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
Chapter:
Assertion (A): 4n ends with the digit 0 for some natural number n.
Reason (R): For a number ‘x’ having 2 and 5 as its prime factors, xn always ends with the digit 0 for every natural number n.
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
Chapter:
Find the value of x for which (sin A + cosec A)2 + (cos A + sec A)2 = x + tan2 A + cot2 A.
Chapter:
Evaluate the following:
`(3 sin 30° - 4 sin^3 30°)/(2 sin^2 50° + 2 cos^2 50°)`
Chapter:
Saima and Arya were born in the month of June in the year 2012. Find the probability that:
- They have different dates of birth.
- They have the same date of birth.
Chapter:
Solve the following system of equations algebraically:
37x + 63y = 137
63x + 37y = 163
Chapter:
A 1.5 m tall boy is walking away from the base of a lamp post which is 12 m high, at the speed of 2.5 m/sec. Find the length of his shadow after 3 seconds.
Chapter:
In the parallelogram ABCD, side AD is produced at point E, and BE intersects CD at point F. Prove that ΔABE ∼ ΔCFB.
Chapter:
Find the coordinates of the point C, which lies on the line AB produced such that AC = 2BС, where the coordinates of points A and B are (–1, 7) and (4, –3), respectively.
Chapter:
α and β are zeroes of a quadratic polynomial x2 − ax − b. Obtain a quadratic polynomial whose zeroes are 3α + 1 and 3β + 1.
Chapter:
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Rectangle ABCD circumscribes the circle of radius 10 cm. Prove that ABCD is a square. Hence, find the perimeter of ABCD.
Chapter:
Let x and y be two distinct prime numbers, and p = x2y3, q = xy4, r = x5y2. Find the HCF and LCM of p, q, and r. Further check if HCF (p, q, r) × LCM (p, q, r) = p × q× r or not.
Chapter:
The two angles of a right angled triangle other than 90° are in the ratio 2 : 3. Express the given situation algebraically as a system of linear equations in two variables and hence solve it.
Chapter:
Р(x, y), Q(–2, –3), and R(2, 3) are the vertices of a right triangle PQR right-angled at P. Find the relationship between x and y. Hence, find all possible values of x for which y = 2.
Chapter:
Prove that `(cosA + sinA - 1)/(cosA - sinA + 1)` = cosec A − cot A.
Chapter:
If cot θ + cos θ = p and cot θ − cos θ = q, prove that `p^2 - q^2 = 4sqrt(pq)`.
Chapter:
The following table shows the number of traffic challans issued in the month of April by the traffic police:
| Number of Challans | Number of Days |
| 0 − 10 | 3 |
| 10 − 20 | 5 |
| 20 − 30 | 10 |
| 30 − 40 | 9 |
| 40 − 50 | 2 |
| 50 − 60 | 1 |
| Total | 30 |
Find the ‘mean’ and ‘mode’ of the above data.
Chapter:
The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given triangle.
Chapter:
Express the equation `(x - 2)/(x - 3) + (x - 4)/(x - 5) = 10/3`; (x ≠ 3, 5) as a quadratic equation in standard form. Hence, find the roots of the equation so formed.
Chapter:
The corresponding sides of ΔABC and ΔPQR are in the ratio 3 : 5. AD ⊥ BC and PS ⊥ QR, as shown in the following figures:
- Prove that ΔADC ~ ΔPSR.
- If AD = 4 cm, find the length of PS.
- Using (ii) find ar(ΔABC) : ar(ΔPQR).
Chapter:
State the basic proportionality theorem. Use it to prove the following:
If three parallel lines l, m, n are intersected by transversals q and s as shown in the adjoining figure, then `(AB)/(BC) = (DE)/(EF)`.
Chapter:
In order to provide shelter to flood victims, a shed was constructed using tin sheets, which is in the form of a cuboid surmounted by a half cylinder as shown below:
The length, breadth and height of the cuboidal portion are 10 m, 7 m and 3 m, respectively. The diameter of the cylindrical portion is 7 m. Find the cost of tin sheets required to make the shed at the rate of ₹ 70 per square metre, given that the shed is open from the front side and closed from the back side.
Chapter:
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Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of the cable car ride from the base point to the top-most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which the car moves. The distance of the first pole from the base point is 200 m and subsequent poles are installed at an equal interval of 150 m. Further, the distance of the last pole from the top is 300 m.
|
Based on the above information, answer the following questions using arithmetic progression:
- Find the distance of the 10th pole from the base. (1)
- Find the distance between the 15th pole and the 25th pole. (1)
-
- Find the time taken by the cable car to reach the 15th pole from the top if it is moving at the speed of 5 m/sec and coming from the top. (2)
OR - Find the total number of poles installed along the entire journey. (2)
- Find the time taken by the cable car to reach the 15th pole from the top if it is moving at the speed of 5 m/sec and coming from the top. (2)
Chapter:
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A drone was used to facilitate the movement of an ambulance on a straight highway to a point P on the ground where there was an accident. The ambulance was travelling at a speed of 60 km/h. The drone stopped at a point Q, 100 m vertically above the point P. The angle of depression of the ambulance was found to be 30° at a particular instant.
|
Based on the above information, answer the following questions:
- Represent the above situation with the help of a diagram. (1)
- Find the distance between the ambulance and the site of the accident (P) at the particular instant. (Use `sqrt3` = 1.73) (1)
-
- Find the time (in seconds) in which the angle of depression changes from 30° to 45°. (2)
OR - How long (in seconds) will the ambulance take to reach point P from a point T on the highway, such that the angle of depression of the ambulance at T is 60° from the drone? (2)
- Find the time (in seconds) in which the angle of depression changes from 30° to 45°. (2)
Chapter:
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The Olympic symbol, comprising five interlocking rings, represents the union of the five continents of the world and the meeting of athletes from all over the world at the Olympic Games. In order to spread awareness about the Olympic Games, students of Class X took part in various activities organised by the school. One such group of students made 5 circular rings in the school lawn with the help of ropes. Each circular ring required 44 m of rope. Also, in the shaded regions, as shown in the figure, students made rangoli showcasing various sports and games. It is given that ΔOAВ is an equilateral triangle, and all unshaded regions are congruent.
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Based on the above information, answer the following questions:
- Find the radius of each circular ring. (1)
- What is the measure of ∠AOB? (1)
-
- Find the area of the shaded region R1. (2)
OR - Find the length of rope around the unshaded regions. (2)
- Find the area of the shaded region R1. (2)
Chapter:
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