The document is an examination paper for Class IX Mathematics for the session 2023-2024, consisting of various sections including multiple-choice questions, problem-solving questions, and proofs. It covers topics such as geometry, algebra, trigonometry, and statistics. The exam includes a total of 80 marks with specific instructions for attempting questions.
The document is an examination paper for Class IX Mathematics for the session 2023-2024, consisting of various sections including multiple-choice questions, problem-solving questions, and proofs. It covers topics such as geometry, algebra, trigonometry, and statistics. The exam includes a total of 80 marks with specific instructions for attempting questions.
The document is an examination paper for Class IX Mathematics for the session 2023-2024, consisting of various sections including multiple-choice questions, problem-solving questions, and proofs. It covers topics such as geometry, algebra, trigonometry, and statistics. The exam includes a total of 80 marks with specific instructions for attempting questions.
The document is an examination paper for Class IX Mathematics for the session 2023-2024, consisting of various sections including multiple-choice questions, problem-solving questions, and proofs. It covers topics such as geometry, algebra, trigonometry, and statistics. The exam includes a total of 80 marks with specific instructions for attempting questions.
QUESTION 1: Choose the correct option: [15x1=15] i) The lengths of the diagonals of a rhombus are 6 cm and 8 cm . The length of the sides of rhombus is a) 9 cm b) 10 cm c) 5 cm d) 20 cm. ii) The compound interest on ₹1000 at 10% p.a. compounded annually for years is a) ₹ 155 b) ₹ 200 c) ₹ 210 d) ₹ 211 iii) The point which lies ony-axis at a distance of 5 units in the negative direction of y- axis is (a) (0, 5) (b) (5, 0) (c) (0, -5) (d) (-5, 0) iv) The value of 2 + log10 (0.01) is (a)4 (b)3 (c)1 (d)0 v) Two parallelograms are on equal bases and between the same parallels. The ratio of their areas is (a) 1 : 2 (b) 1 : 1 (c) 2 : 1 (d) 3 : 1 vi) If the diagonals of a quadrilateral PQRS bisect each other, then the quadrilateral PQRS must be a (a) parallelogram (b) rhombus (c) rectangle (d) square vii) If 2 sin 3θ = √3 then the value of is a) 300 b) 450 c) 200 d) 900 viii) If 8 = 64, what is the value of 3 2x +1? x+1
(a) 1 (b) 3 (c) 9 (d) 27
ix) In a ∆ABC, if AB = 6√3 cm, BC = 6 cm and AC = 12 cm, then ∠B is (a) 120° (b) 90° (c) 60° (d) 45° x) The distance between the points (4, p) and (1, 0) is 5 units, then the value of p is (a) 4 only (b) -4 only (c) ±4 (d) 0 xi) If the sum of cubes of two numbers is 72 and their sum is 6, then the product of these two numbers is (a) 24 (b) 8 (c) 6 (d) 3 xii) Factorise: 16-x4 (a) (4-x2)(2+x)(2-x) (b) (4+x2)(2+x)(2-x) (c) (2+x2)(2+x)(2-x) (d) (4+x)(2+x)(2-x) xiii) Find the value of m if x=3 and y=6 is a solution of the equation 6x+5y=m (a) 48 (b) 21 (c) 51 (d) -12 xiv) AD is a median of triangle ABC. Choose the correct statements. A. AB+AC B. AB+AC C. AB+AC D. AB+AC (a) B and C (b) A and B (c) A, B and C (d) A only. xv) Assertion (A): The area of a semi circle of radius 7 cm is 77 cm2 Reason ( R ) : The area of a circle is (a) A is true R is False (b) A is correct R is the correct explantion (c) A is False R is the coorect explanation (d) Both are correct but R is not the correct explantion of A. QUESTION 2:- a) Two equal sums were lent at 5% and 6% per annum compound interest for 2 years. If the difference in the compound interest was ₹422 find : (i) the equal sums (ii) compound interest for each sum. b) (i) Show that 2(Cos 4600+ Sin 4300)- (tan 2 600+Cot 2 450) +3 Sec 2 300= 1/ 4 (ii) Find A if (Sec A-2)(tan 3A-1)=0 c) A boat sails a distance of 44 km in 4 hours with the current. It takes 4 hours 48 minutes longer to cover the same distance against the current. Find the speed of the boat in still water and the speed of the current. [4+4+4=12] QUESTION 3: a) ABCD is a parallelogram. If the diagonal AC bisects ∠A, then prove that: (i) AC bisects ∠C (ii) ABCD is a rhombus (iii) AC ⊥ BD. b) In the figure (ii) given below, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)
c) Using a scale of 1 cm to 1 unit for both the axes, draw the graphs of the following equations : 6p = 5x + 10, y = 5x – 15. From the graph, find (i) the coordinates of the point where the two lines intersect. (ii) the area of the triangle between the lines and the x-axis. [4+4+5=13] SECTION – B [ATTEMPT ANY FOUR ] QUESTION 4 : a) Solve:
b) If p = xm+n . yt , q = xn+l. ym and r = xl+m . yn. Prove that pm-n. qn-l . rl-n = 1 c) The ends of a diagonal of a square have co-ordinates (-2, p) and (p, 2). Find p if the area of the square is 40 sq. units. [3+3+4=10] QUESTION 5: a) Solve: 97x + 53y = 177, 53x + 97y = 573 b) Given 4 sin θ =3 cos θ, find the values of : (i) sin θ (ii) cos θ (iii) cot2 θ – cosec2 θ. c) If , find the value of
[3+3+4=10] QUESTION 6: a) If tan (A + B) = and tan (A – B) = 1 and A, B (B < A) are acute angles, find the values of A and B. b) Draw a histogram showing marks obtained by the students of a school in a Mathematics paper carrying 60 marks :
c) In the figure (1) given below, ABCD is a parallelogram. P, Q are any two points on the sides AB and BC respectively. Prove that, area of ∆ CPD = area of ∆ AQD.
[3+3+4=10] QUESTION 7: a) Given log10 x= a, log10 y = b and log10 z =c, (i) write down 102a-3 in terms of x. (ii) write down 103b-1 in terms of y. (iii) if log10 P = 2a + b2– 3c, express P in terms of x, y and z. b) Factorise: (x2 – x) (4x2 – 4x – 5) – 6 c) Using ruler and compasses only, construct a regular hexagon having a length 5cm. [3+3+4=10] QUESTION 8: a) A line segment is of length 10 units and one of its end is (-2, 3). If the ordinate of the other end is 9, find the abscissa of the other end. b) In the given figure, AB=AC and AP=AQ. Prove that (i) ∆APC ≅ ∆AQB (ii) CP = BQ (iii) ∠APC = ∠AQB.
c) In the adjoining figure, ABCD is a rhombus and DCFE is a square. If ∠ABC = 56°, find (i) ∠DAG (ii) ∠FEG (iii) ∠GAC (iv) ∠AGC
[3+3+4=10] QUESTION 9: a) On what sum will the difference between the simple and compound interest for 3 years at 10% p.a. is ₹232·50 ? b) Show that
