Question Paper: Sections, Marks, and Instructions

• This question paper has 5 sections: A, B, C, D, and E.
• Section A has 20 MCQs carrying 1 mark each.
• Section B has 5 questions carrying 2 marks each.
• Section C has 6 questions carrying 3 marks each.
• Section D has 4 questions carrying 5 marks each.
• Section E has 3 case-based questions carrying 4 marks each, with sub-parts of 1, 1, and 2 marks respectively.
• All questions are compulsory.
• Neat figures should be drawn wherever required.

Section A

• Section A consists of 20 questions worth 1 mark each.
• If the sum of the zeroes of p(x) = (k² – 14)x² − 2x − 4 is 1, then the value of k is ±√18.
• If 3x + βy = 7, 12y + βx = 10 is a pair of inconsistent linear equations, then the possible value of β is ±6.
• In two concentric circles of radii p and q (p > q), the length of the chord of the larger circle which touches the smaller circle is 2√(p² - q²).
• If the first two terms of an arithmetic progression are −(2x + 3) and x, then the third term is 4x - 3.
• If four identical cubes of sides 3 cm each are joined together end to end to form a new block, then the volume of the combined block is 108 cm³.
• In two triangles ABC and PQR, if AB/QR = BC/PR = CA/PQ, then ΔQBA ~ ΔPQR.
• If the angle between two tangents drawn from an external point P to a circle of radius a and center O is 60°, then the length of OP is 2a.
• The least integral value of k for which the quadratic equation kx² – 5x + 1 = 0 does not have a real solution is 7.
• For a distribution of marks, the number of students who got less than 30 marks is 25, based on the provided data.
• In given figures, if ΔPQR ~ ΔXYZ, ∠RPQ = 65° and ∠PRQ = 40°, then the measure of ∠XYZ is 75°.
• If the equation x² + 4x + k = 0 has real and distinct roots, then the value of k is k < 4.
• If sec θ = 2/√3, then the value of θ is 30°.
• The volume of the largest sphere which can be carved out of the cube of side x cm is (1/6)πx³.
• If two dice are rolled, the probability of getting a sum of 6 is 5/36.
• The distance of (a, 4) from the origin is 5 units; then the value of a is ±3.
• If mode = 12.4 and mean = 10.5, then median is 11.13.
• The ratio in which the origin divides the line segment joining (-20, 30) and (2, -3) is 1:10.
• A letter is chosen at random from the word "ASSASSINATION"; the probability that it is a vowel is 6/13.
• Assertion (A): HCF (11, 17) is 1. Reason (R): If p and q are prime numbers, then HCF (p, q) = 1; both (A) and (R) are true, and (R) is the correct explanation of (A).
• Assertion (A): The minute hand of a clock is 84 cm long. The distance covered by the tip of the minute hand from 10:10 am to 10:25 am is 132 cm. Reason (R): The area of a sector = (θ/360°) × πr²; assertion (A) is true, but reason (R) is false.

Section B

• Section B consists of 5 questions of 2 marks each.
• A merchant has 120 litres of oil of one kind, 180 litres of another kind, and 240 litres of a third kind; the greatest capacity of tins to sell the three kinds of oil in equal capacity should be determined.
• Coloured pencils are available in packs of 12, and crayons in packs of 32; determine how many packs of each a girl needs to buy to have the same number of pencils and crayons.
• From a deck of 52 playing cards, the red face cards are removed, and 2 jokers are added; find the probability of drawing (i) a black jack and (ii) a diamond from the new deck.
• For a game consisting of tossing a coin 3 times; the probability of getting at most one tail, and at least two heads, needs to be found.
• The point on the y-axis equidistant from (2, -5) and (-2, 9) must be found.
• If tan θ = √2, determine the value of sin θ * cos θ.
• If P(9a - 2, -b) divides the line segment joining A(3a + 1, -3) and B(8a, 5) in the ratio 3:1, find the values of a and b.

Section C

• Section C consists of 6 questions of 3 marks each.
• Sides AB and BC and median AD of triangle ABC are proportional to sides PQ and QR and median PM of triangle PQR; it must be shown that ΔABC ~ ΔPQR.
• D and E are two points on side AB of triangle ABC such that AD = BE; if DP || BC and EQ || AC, prove that PQ || AB.
• A person on tour has ₹9600 for expenses; if the tour is extended by 4 days, daily expenses are cut down by ₹200; find the original duration of the tour.
• The zeroes of the quadratic polynomial 15y² + 14 = 41y need to be found, and the relationship between the zeroes and the coefficients of the polynomial must be verified.
• Evaluate the expression (sin60° × cot30° + tan45°) / (cos0° + cosec90° × sec45°).
• Find the area of the segment of a circle formed by a chord of length 5 cm subtending an angle of 90° at the center.
• Calculate the perimeter of an equilateral triangle if it inscribes a circle whose area is 154 cm². (Take √3 = 1.73.)
• Prove that √5 is an irrational number.

Section D

• Section D consists of 4 questions of 5 marks each.
• Draw the graphs of 2x + y = 6 and 2x − y + 2 = 0, shade the region bounded by these lines and the x-axis, and find the area of the shaded region.
• A train covered a certain distance at a uniform speed; if the train had been 10 km/h faster, it would have taken 2 hours less than the scheduled time. If it were slower by 10 km/h, it would have taken 3 hours more than the scheduled time; the distance covered by the train must be found.
• Prove that the lengths of tangents drawn from an external point to a circle are equal, and using this result, find the perimeter of triangle ABC, given AD = 15 cm, CF = 12 cm, and BE = 7 cm.
• Two poles of equal heights are standing opposite to each other on either side of the road, which is 80 m wide; from a point C between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively; find the height of each pole and the distances of the point C from the poles. (Take √3 = 1.73.)

Section E

• Section E consists of 3 case study-based questions of 4 marks each.

• Frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality; the mode and median of the data must be found.

• Marks obtained by 40 students of a class are shown; the mean of the distribution has been estimated as 35.75; calculate the values of f1 and f2, given that f1:f2 = 3:1.

• Nitesh wants to participate in a push-up challenge and can currently do 3000 push-ups in one hour, aiming to achieve 3900; each day, he practices and is able to do 5 more push-ups than the previous day; if on the first day of practice, he does 3000 push-ups, determine the following:

  • Form an A.P. representing the number of push-ups per day.
  • Find the minimum number of days he needs to practice before his goal is accomplished.
  • Find the total number of push-ups performed by Nitesh up to the day his goal is achieved.

• Find in how many days will he perform 30225 push-ups?

• Shweta went to a beach with her father; from where she was standing, a ship and a lighthouse are in a straight line; based on this:

  • The distance between Shweta and the ship is twice the height of the ship; determine the height of the ship.
  • Find the distance between Shweta and the top of the ship.
  • If the distance of Shweta from the lighthouse is twelve times the height of the ship, determine the ratio of the heights of the ship and the lighthouse.
  • What similarity criteria can be applied if the ship and lighthouse are considered straight lines? Also, prove DE × AC = AE × BC.

• Arpana is studying in X standard; after seeing a rolling pin made of wood with two small cylindrical handles with hemispherical ends:

  • Find the curved surface area of two identical cylindrical parts if the diameter and length of each cylindrical part are 2.5 cm and 5 cm, respectively.
  • Find the volume of the big cylindrical part whose diameter is 4.5 cm and height is 12 cm.
  • Find the difference of the volumes of the bigger cylindrical part and the total volume of the two small hemispherical ends.
  • Find the total volume of wood needed to make two small cylindrical handles with hemispherical ends or find the total volume of wood required to make two small cylindrical handles with hemispherical ends.