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ICSE Class X Mid-term 2024 : Mathematics

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Dr.Vijayalakshmi Murali
Pearl Drop School, Pune

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PEARL DROP SCHOOL ICSE 2023-24 MIDTERM STD. X MATHEMATICS Date: 16th October 2023 Maximum Marks: 80 Time allowed: Two and half hours Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during first 15 minutes. This time is to be spent in reading the question paper. The time given at the head of this Paper is the time allowed for writing the answer Attempt all questions from Section A and any four questions from Section B. All working, including rough work, must be clearly shown, and must be done on the same sheet as the rest of the answer. Omission of essential working will result in loss of marks. The intended marks for questions or parts of questions are given in brackets [] Mathematical tables are provided. SECTION A (40 Marks) (Attempt all questions from this Section.) Question 1 Choose the correct answers to the questions from the given options (Do not copy the question, write the correct answers only) (i) The slope of the line perpendicular to the line 2x-3y=6 is: a) 3/2 b) 3/2 c) -2/3 d) 0 [10] (ii) A lighthouse is 80m high. The angle of elevation of its top, from a point 80m away from the foot along the same horizontal line is a) 300 b) 600 c) 450 d) 900 (iii) 8,8,8,8 . Is an a) Arithmetic Progression b) Geometric Progression c) Both Arithmetic and Geometric Progression d) Neither an Arithmetic progression nor Geometric Progression (iv) The table shows the values of x and y, where x is proportional to y. The values of P and Q are a) b) c) d) P=10 and Q=6 P=2 and Q=2 P=2 and Q=6 P=5 and Q=3 (v) A (1,4), B (x, 1) and C (7,4) are the vertices of triangle ABC. If the centroid of the triangle is G(4,3) then x is equal to a) 2 b) -4 c) 4 d) 1 (vi) If 2x+1 is a factor of 2 2 + 3 , then the value of a is a) 5/2 b) -5/2 c) 5 d) -5 (vii) The nth term of an Arithmetic Progression is 8n - 5. The 11th term is a) 83 b) -35 c) 93 d) -83 (viii) (sec - 1) (1 - cosec ) is equal to a) 1 b) -1 c) 0 d) 2 (ix) A shopkeeper buys goods worth 4000 and sells them adding a profit of 320. If the rate of GST is 5%, then the bill amount is: a) 4200 b) 4536 c) 4320 d) 3800 (x) Assertion (A) M is the mid-point of PQ. If P is (4, x), M is (y, -1) and Q is (-2,4) then x=1 and y=6 Reason (R): Mid-Point { a) b) c) d) (xi) 2 2 } Both A and R are and R is the correct explanation for A Both A and R are and R is not the correct explanation for A A is true but R is false A is false and R is true If the polynomial 22 11 + 1, is divided by x+1=0, then the remainder is a) b) c) d) (xii) 1 + 2 1 + 2 3 1 -1 -3 The seventh term of the G.P: 1, 3, 3, 3 3, a) 27 3 b) 27 c) 9 3 d) 18 3 (xiii) Points (3, 0) and (-1, 0) are invariant points under reflection in the line L1; Then line L1 is a) Y-axis b) X-axis c) X=3 d) X=-1 (xiv) If x+2 is the mean proportional between x+2 and x+9, then the value of x is; a) 3 b) 2 c) 7 d) 6 (xv) The coordinates of the point P( 3, -5 ) on reflecting on the y-axis are: a) (-3, 5) b) (-3, -5) c) (3, 5) d) (3, -5) Question 2 (i) (ii) (iii) Mrs. Gupta, a housewife buys the following items as listed in the table below Items Quantity Rate Discount Rice 3 kg 80/kg 5% Milk-food 2 kg 300/kg 10% She paid 1000 to the shopkeeper and if the GST for rice is 5% and Milk-food 12% respectively, find out how much the shopkeeper has to return rounded off to the nearest rupee? [5] Use the graph paper for this question. Take 2cm=1 unit on both x and y axis a) Plot the following points A (0,4), B (2, 3), C (1, 1) and D (2, 0) [5] b) Reflect B, C and D on the y-axis and write down their co-ordinates. Name the images as B , C and D respectively. c) Join the points A, B, C D, D C B and A in that order to get a closed figure. Write down the equation of the line about which if this closed figure obtained is folded, the two parts of the figure exactly coincide. If x = a cos + b sin and y = a sin - b cos , then find the value of (x + y ). [3] Question 3 (i) (ii) (iii) Find, by fractorisation method the values of a and b if (x -2) is a factor of ( ) = 3 + 2 + + 26 and f(3)=32 [4] From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be 45 and 60 respectively. If the height of the tower is 20 m. Find: (i) the height of the cliff (ii) the distance between the cliff and the tower. [4] The line segment joining the points A(3, -4) and B (-2, 1) is divided in the ratio 1: 3 at point P in it. Find the co-ordinates of P. Also, find the equation of the line through P and perpendicular to the line 5x 3y = 6 [4] SECTION B (40 Marks) (Attempt any four questions) Question 4 (i) (ii) (iii) Asharaf went to see a movie. He wanted to purchase a movie ticket for 80. As the ticket for 80 was not available, he purchased a ticket for 120 of upper class. How much extra GST did he pay for the ticket? (GST for a ticket below 100 is 18% and GST for a ticket above 100 is 28%). [4] In what ratio is the line joining (4,3) and (2,-6) is divided by the x-axis? Also find the co-ordinates of the point of intersection. [3] In an Arithmetic Progression (A.P.) the fourth and sixth terms are 8 and 14 respectively. Find the: (i) first term (ii) common difference (iii) sum of the first 20 terms. [3] Question 5 (i) (ii) (iii) M and N are two points on the X axis and Y axis respectively. P (3, 2) divides the line segment MN in the ratio 2 : 3. Find: (i) the coordinates of M and N (ii) slope of the line MN. [3] If sin + cos = 3 , then find the value of tan + cot . [3] The first term of a G.P. is -3 and the square of the 2nd term is equal to its 4th term. Find the 7th term of the G.P. [4] Question 6 (i) 1 2 Prove the identity: (1 ) = 2 3 +3 2 62 [3] (ii) If 3+3 2 = 63 find the ratio of p and q. [3] (iii) The first, last term and the common difference of an Arithmetic Progression are -5, 45 and 10 respectively. Find the following for the given AP. a) Number of terms 'n' and b) Sum of the 'n' terms. [4] Question 7 (i) (ii) (iii) Using the factor theorem, show that (x 2) is a factor of 3 + 2 4 4. Hence factorise the polynomial completely. [3] The following numbers, k + 3, k + 2, 3k 7 and 2k 3 are in proportion. Find the value of k. [3] A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7m. At a point in a plane the angle of elevation of the bottom and the top of the flagstaff are respectively 30 and 60 . Find the height of the tower [4] Question 8 (i) (ii) Point P (a, b) is reflected in the x-axis to P' (5, -2). Write down the values of a and b. (ii) P" is the image of P when reflected in the y-axis. Write down the co-ordinates of P". (iii) Name a single transformation that maps P' to P". [3] If 7x-15y=4x+y, find the value of x:y.Hence use componendo and dividendo 3 2 +2 2 (iii) to find the value 3 2 2 2 [3] Using remainder theorem, factorise completely 3x3 + 2x2 - 19x +6. [4] Question 9 (i) (ii) (iii) Find the G. P. whose 5th term is 45 and the 8th term is 384. 1 1 2 [3] Prove that 1+sin + 1 sin = 2 [3] Find: (a) Slope of CD (b) Equation of AB [4] Line AB is perpendicular to CD. Coordinate of B, C and D are respectively (4,0), (0, 1) and 4,3) Question 10 (i) (ii) Find a if A(2a+2,3), B(7,4) and C(2a+5, 2) are collinear. [3] The following bill shows the rate of GST and the marked price of the articles. Maharashtra Electronics S. No Item Marked Price Quantity Rate of GST 48000 1 LED TV Set 1 28% 10000 2 MP4 Player 1 18% (iii) Find the total amount to be paid (including GST) for the above bill. [3] Use graph paper for this question. The points A(2,3), B(4,5) and C(7,2) are the vertices of ABC. [4] a) Write down the co-ordinates of A , B and C if A B C is the image of ABC, when reflected about the origin. b) Write down the co-ordinates of A , B and C if A B C is the image of ABC, when reflected in the x-axis. c) Mention the special name of the quadrilateral BCC B and find its area.

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