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AP SSC · Class 10

Mathematics Important Questions

High-frequency problems from BSE AP Class 10 Maths public exams — algebra, geometry, trigonometry, and statistics questions from all 14 NCERT chapters.

Ch. 1 — Real Numbers

  1. 1Find the HCF and LCM of 306 and 657 using prime factorisation. Verify that HCF × LCM = product of the numbers.
  2. 2Prove that √5 is irrational.
  3. 3Check whether 6ⁿ can end with the digit 0 for any natural number n.
  4. 4Use Euclid's division algorithm to find the HCF of 867 and 255.

Ch. 2 — Polynomials

  1. 1If one zero of the polynomial 3x² − kx − 2 is 2, find the value of k and the other zero.
  2. 2Find a quadratic polynomial whose sum of zeros is 4 and product of zeros is 1.
  3. 3Find the zeros of the polynomial x² − 2x − 8 and verify the relationship between the zeros and coefficients.
  4. 4Divide the polynomial p(x) = x³ − 3x² + 5x − 3 by g(x) = x² − 2 and find the quotient and remainder.

Ch. 3 — Pair of Linear Equations in Two Variables

  1. 1Solve: 2x + 3y = 11 and 2x − 4y = −24. Hence find the value of m if 2x + my = 10 has the same solution.
  2. 2A fraction becomes 9/11 if 2 is added to both numerator and denominator; if 3 is added to both it becomes 5/6. Find the fraction.
  3. 3Solve graphically: x + y = 5 and 2x − y = 4. Find the coordinates of the point of intersection.
  4. 4For what value of k will the pair kx + 3y = k − 3 and 12x + ky = k have no solution?

Ch. 4 — Quadratic Equations

  1. 1Find the roots of 3x² − 5x + 2 = 0 by factorisation.
  2. 2The product of two consecutive positive integers is 306. Find the integers using a quadratic equation.
  3. 3Find the discriminant of 2x² − 3x + 5 = 0 and x² − 4x + 4 = 0, and state the nature of the roots.
  4. 4Solve 2x² − 7x + 3 = 0 by the method of completing the square.

Ch. 5 — Arithmetic Progressions

  1. 1Find the 10th term and the sum of the first 20 terms of the AP: 3, 7, 11, …
  2. 2The sum of n terms of an AP is 5n² − 3n. Find the AP and its 20th term.
  3. 3How many terms of the AP 3, 5, 7, … must be taken so that their sum is 120?
  4. 4Which term of the AP 21, 18, 15, … is −81? Is any term of this AP equal to zero?

Ch. 6 — Triangles

  1. 1Prove the Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.
  2. 2State and prove that the ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides.
  3. 3In ΔABC, D is the midpoint of BC and AD ⊥ BC. Prove that AB² − AC² = 2BC × BD.
  4. 4Sides AB and BC and median AD of ΔABC are proportional to sides PQ, QR and median PM of ΔPQR. Show that ΔABC ~ ΔPQR.

Ch. 7 — Coordinate Geometry

  1. 1Find the area of the triangle whose vertices are A(2, 3), B(−1, 0), C(2, −4).
  2. 2Find the coordinates of the point which divides the line joining (4, −3) and (8, 5) in the ratio 3 : 1 internally.
  3. 3Find the distance between the points (−5, 7) and (−1, 3). Show whether the points (1,5), (2,3), (−2,−11) are collinear.
  4. 4Find the coordinates of the midpoint of the line segment joining (3, 4) and (−1, 2).

Ch. 8 — Introduction to Trigonometry

  1. 1Prove that sin²θ + cos²θ = 1. Hence derive 1 + tan²θ = sec²θ.
  2. 2If tan θ = 4/3, evaluate (sin θ + cos θ). Verify sin²θ + cos²θ = 1.
  3. 3Evaluate: (sin 30° + cos 60°) / (tan 45°) without using tables.
  4. 4Prove the identity: (1 + cosθ)/(1 − cosθ) = (cosecθ + cotθ)².

Ch. 9 — Some Applications of Trigonometry

  1. 1From the top of a 75 m high tower, the angles of depression of two cars on either side are 30° and 45°. Find the distance between the cars.
  2. 2A ladder leaning against a wall makes an angle of 60° with the ground. If the foot of the ladder is 2.5 m from the wall, find the length of the ladder.
  3. 3The angle of elevation of the top of a building from a point 30 m away is 30°. Find the height of the building.
  4. 4An observer 1.5 m tall is 28.5 m from a tower. The angle of elevation of the top of the tower is 45°. Find the height of the tower.

Ch. 10 — Tangents and Secants to a Circle

  1. 1Prove that the lengths of tangents drawn from an external point to a circle are equal.
  2. 2Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
  3. 3A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the length PQ.
  4. 4Prove that the parallelogram circumscribing a circle is a rhombus.

Ch. 11 — Areas Related to Circles

  1. 1A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment.
  2. 2Find the area of a sector of a circle with radius 6 cm if the angle of the sector is 60°.
  3. 3The length of the minute hand of a clock is 14 cm. Find the area swept by it in 5 minutes.
  4. 4Find the area of the shaded region if a square of side 14 cm has a quadrant of a circle removed at each corner.

Ch. 12 — Surface Areas and Volumes

  1. 1A solid is in the form of a cone mounted on a hemisphere with the same base radius 7 cm. If the height of the cone is 14 cm, find the total surface area and volume.
  2. 2Two spheres of radii 8 cm and 6 cm are melted and recast into a cone with base radius 10 cm. Find the height of the cone.
  3. 3A cylindrical vessel of radius 7 cm and height 10 cm is full of water. Find the volume of water (use π = 22/7).
  4. 4A frustum of a cone has radii 4 cm and 10 cm and height 8 cm. Find its volume and curved surface area.

Ch. 13 — Statistics

  1. 1Find the mean of a given grouped data using the step-deviation method.
  2. 2Find the median for a given frequency distribution with classes 10–20, 20–30, etc.
  3. 3Find the mode from a frequency distribution table and identify the modal class.
  4. 4The mean of a frequency distribution is 18. Find the missing frequency f given the other class frequencies.

Ch. 14 — Probability

  1. 1Two dice are thrown simultaneously. Find the probability that: (i) the sum is 8, (ii) the sum is even, (iii) both show the same number.
  2. 2A bag has 5 red, 4 blue and 3 green balls. Find the probability of drawing a ball that is NOT green.
  3. 3One card is drawn from a well-shuffled deck of 52 cards. Find the probability that it is (i) a king, (ii) a red face card.
  4. 4A box contains 90 discs numbered 1 to 90. If one disc is drawn at random, find the probability that it bears a perfect square number.

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