10th Maths Quarterly Exam 2024 Question Paper & Answer Key | Ariyalur District

10th Maths Quarterly Exam 2024 Answer Key - Ariyalur District

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Solutions

PART-I (14x1=14)

Answer all the questions

1. Let n(A)=m and n(B)=n then the total number of non empty relations that can be defined from A to B is c) 2^mn-1
2. f(x)+f(1-x)=2 then f(1/2)=........ a) 1
3. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is d) 2520
4. Find the sum of n terms of the series is 2+2+2+... to n terms. b) 2n
5. If (x-6) is the HCF of pair of polynomials x²-2x-24 and x²-kx-6 then find the value of k. b) 5
6. The solution of (2x-1)²-9 is equal to a) -1, 2
7. 7^4k ≡ ____ (Mod 100) d) 1
8. If in triangle ABC and EDF, AB/DE = BC/FD then they will be similar when c) ∠B = ∠D
9. When proving that a quadrilateral is a trapezium, it is necessary to show a) Two sides are parallel (one pair of opposite sides)
10. If ΔABC is an isosceles triangle with ∠C = 90° and AC= 5 cm then AB is a) 5√2 cm
11. Find the slope of the equation is 2y= x+8 a) ½
12. The straight line given by the equation x=11 is d) parallel to the y axis
13. If sinθ = cosθ then 2tan²θ + sin²θ - 1 is equal to a) 3/2
14. If sinθ + cosθ = a and secθ + cosecθ = b then the value of b(a²-1) is equal to b) 2a

PART-II (10x2=20)

Answer any 10 questions. Question No. 28 is compulsory

15. A x B = {(3,2) (3, 4) (5,2) (5,4)}, then find A and B.
    Answer: A = {3, 5}, B = {2, 4}
16. Show that the function f: N → N defined by f(m)=m²+m+3 is one-one function.
    Answer: For f(m₁) = f(m₂), we get (m₁-m₂)(m₁+m₂+1) = 0. Since m₁, m₂ ∈ N, m₁+m₂+1 ≠ 0. Thus m₁=m₂. Hence, f is one-one.
17. Find the 3^rd term and 4^th term of a sequence if a_n = n² if n is odd; n²/2 if n is even.
    Answer: a₃ = 3² = 9; a₄ = 4²/2 = 8
18. Find the sum of the series. 1+4+9+16+......+2225
    Note: The question has a typo, as 2225 is not a perfect square. Assuming the series is 1² + 2² + ... + n², the last term must be a perfect square.
19. Simplify: (x²-16) / (x²+8x+16)
    Answer: (x-4)/(x+4)
20. Find the LCM of the expressions are (p²-3p+2), (p²-4).
    Answer: (p-1)(p-2)(p+2) or (p-1)(p²-4)
21. If the difference between a number and its reciprocal is 24/5, find the numbers.
    Answer: The numbers are 5 or -1/5.
22. The length of the tangent to a circle from a point p, which is 25cm away from the centre is 24 cm. what is the radius of the circle.
    Answer: 7 cm
23. In ΔABC, if DE || BC, AD=x, DB=x-2, AE=x+2 and EC=x-1 then find the length of the sides AB and AC.
    Answer: x=4. AB = 6, AC = 9.
24. Find the value of 'a' if the line through (-2, 3) and (8, 15) is perpendicular to y = ax+2.
    Answer: a = -5/6
25. Find the slope of a line joining the points are (14, 10) and (14, -6).
    Answer: Undefined
26. Prove that √(1+sinθ)/(1-sinθ) = secθ+tanθ.
    Answer: Proof involves multiplying numerator and denominator by (1+sinθ) inside the root.
27. Prove that secθ/sinθ - sinθ/cosθ = cotθ.
    Answer: Proof involves taking LCM and using the identity cos²θ = 1-sin²θ.
28. f(x)=3x-2, g(x)=2x+k and if fog=gof then find the value of k.
    Answer: k = -1

PART-III (10x5=50)

Answer any 10 questions. Question No. 42 is compulsory

29. A function f is defined by f(x) = 2x-3. Find i) (f(0)+f(1))/2 ii) x such that f(x)=0 iii) x such that f(x)=x iv) x such that f(x)=f(1-x).
    Answer: i) -2, ii) x=3/2, iii) x=3, iv) x=1/2
30. If f(x)=2x+3, g(x)=1-2x and h(x)=3x prove that fo(goh)=(fog)oh.
    Answer: Both LHS and RHS simplify to 5-12x. Hence proved.
31. Find the sum of all natural numbers between 100 and 1000 which are divisible by 11.
    Answer: 44550
32. Find the sum of series is 10³+11³+12³+...+20³.
    Answer: 42075
33. Find the GCD of the polynomials x³+x²-5x+3 and x³+3x²-x-3.
    Answer: (x-1)(x+3) or x²+2x-3
34. Find the square root of the expression 289x⁴ - 612x³ + 970x² - 684x + 361.
    Answer: |17x² - 18x + 19|
35. If vertices of a quadrilateral are at A(-4, -2), B(-3,k), C(3, -2) and D(2,3) and its area is 28 sq.units. Find the value of k.
    Answer: k = -5 or k = 11
36. Find the equation of the perpendicular bisector of the line joining the points A(-4, 2) and B(6,-4).
    Answer: 5x - 3y - 8 = 0
37. State and prove Thales theorem.
    Answer: Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. (Proof as per textbook).
38. An insect 8 m away initially from the foot of a lamp post which is 6m tall, crawls towards it moving through a distance. The distance from the top of the lamp post is equal to the distance it has moved. How far is the insect away from the foot of the lamp post?
    Answer: 1.75 m
39. Prove that √(1+cosθ)/(1-cosθ) + √(1-cosθ)/(1+cosθ) = 2cosecθ.
    Answer: Proof involves taking LCM and simplifying.
40. A straight line AB cuts the co-ordinate axes at A and B. If the mid-point of AB is (2,-3) find the equation of AB.
    Answer: 3x - 2y - 12 = 0
41. Let f: A→B be a function defined by f(x) = x/2 - 1 where A={2,4,6,10,12}, B={0,1,2,4,5,9} represented f by i) set of ordered pairs ii) a table iii) an arrow diagram iv) a graph.
    i) {(2,0), (4,1), (6,2), (10,4), (12,5)} ii) Table, arrow diagram, and graph to be drawn based on the ordered pairs.
42. The ratio of 6^th and 8^th term of an A.P is 7: 9. Find the ratio of 9^th term to 13^th term.
    Answer: 5:7

PART-IV (2x8=16)

Answer the following

43. a) Construct a triangle similar to a given triangle ABC with its sides equal to 6/5 of the corresponding sides of the triangle ABC (scale factor 6/5). (OR)
    b) Construct a triangle PQR such that QR=5 cm, ∠P=30° and the altitude from P to QR is of length 4.2 cm.
    Answer: Geometric constructions to be performed as per standard procedures.
44. a) Graph the quadratic equation x²-8x+16=0 and state the nature of their solution. (OR)
    b) A school announces that for a certain competition, the cash price will be distributed for all the participants equally as shown below.
    No of participants (x): 2, 4, 6, 8, 10
    Amount for each participant in (y): 180, 90, 60, 45, 36
    i) Find the constant of variation.
    ii) Graph the above data and hence, find how much will each participant get if the number of participants are 12.
    a) The graph is a parabola y=(x-4)² which touches the x-axis at (4,0). Nature of solution: Real and equal roots.
    b) i) It is an inverse variation. The constant of variation, k = xy = 360.
    ii) From the graph or by calculation (y=360/12), if there are 12 participants, each will get Rs. 30.