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ICSE Class X Sample / Model Paper 2026 : English Paper 1 (English Language)

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IGNITE PAPER 1 SECTION A (Attempt all the questions from this section) Question 1 [15] Choose the correct answer to the question from the given options. i. ii. iii. iv. v. vi. vii. viii. ix. x. The market price of micro oven is 10000. Dealer offer 20% discount on the market price. The selling price of micro oven: a. 10,000 c. 2000 b. 6000 d. 8000 2 The linear factors of the equation x + kx + 1 = 0 exists, if a. Both k 2 and k -2 c. k = 2 b. k 2 d. k - 2 3 2 When 6x + 2x x + 2 is divided by (x+2), then remainder is a. -36 c. 44 b. 56 d. 36 If both A + B and AB are defined, then which one of the following is true? a. A and B are square matrices of c. A and B are rectangular matrices of same order same order b. A and B are square matrices of d. A and B are rectangular matrices of different order different order 3 2 When ax + 6x + 4x + 5 is divided by (x + 3), the remainder is -7. The value of constant is a. 2 c. -3 b. -2 d. 3 Which of the following point is invariant with respect to the line y = -2? a. (2,3) c. (-2,3) b. (3,2) d. (3,-2) If the cost of an article is 25,000 and CGST paid by the owner is 2250, the rate of GST is a. 18% c. 10% b. 15% d. 9% A point M is reflected in X-axis to M (4,-5). M is the image of M, when reflected in the Y-axis. The coordinates of M when M is reflected in the origin, is a. (-4,-5) c. (4,5) b. (-4,5) d. (4,-5) If the ratio of the mode and median of a certain data is 6:5, then the ratio of its mean and median is a. 10:9 c. 10:8 b. 9:10 d. 8:10 The circumcentre of a triangle is the point which is: a. At equal distance from the three c. The point of intersection of the sides of the triangle three medians b. The point of intersection of the d. At equal distance from the three three altitudes of the triangle vertices of the triangle xi. xii. xiii. xiv. xv. If and are the roots of the equation x2 + x 6 = 0 such that > , then the product of the matrices +1 0 0 and is 5 4 5 4 a. c. 9 2 9 2 6 9 6 9 b. d. 13 6 13 6 Suppose there are fout points A(2,4), B(6,4), C(6,6) and D(2,6), which lie in the first quadrant. If we rotate only the axes at an angle of 900 in anti-clockwise direction, then what will be the new coordinates of the point C and what will be the name of the figure, when we join adjacent points. a. (6,-6); rectangle c. (-6,-6); square b. (6,4); square d. (2,-6); rectangle If P, Q, S and R are points on the circumference of a circle of radius r, such that PQR is an equilateral triangle and PS is a diameter of the circle. Then, the perimeter of the quadrilateral PQSR will be c. 2r a. 2( 3 + 1) d. 2 3 b. 2 3 + In a colony, the average age of the boys is 14 yr and the average age of the girls is 17yr. if the average age of the children in the colony is 15yr, then the ratio of the number of boys to that of girl is a. 1:1 c. 3:2 b. 1:2 d. 2:1 Assertion (A) : Three consecutive terms 2k + 1, 3k +3 and 5k 1 from an AP than K is equal to 6. Reason (R): In an AP a, a+d, a+2d, the sum of n terms of the AP be Sn = ( 2a +(n-1)d) a. Both A and R are true and R is the correct explanation of A b. Both A and R are true but R is not the correct explanation of A Question 2 i. ii. iii. ii. iii. [12] Rashmi has a 4 yr recurring deposit account in bank of Maharashtra and deposits 800 per month.[4] If she gets 48200 at the time of maturity, then find a. The rate of (simple) interest. b. The total interest earned by Rashmi Mr. Kamal reduces the number of workers of his factory in the ratio 9:7 and increases their wages in the ratio 13:20. In what ratio, the wages bill is increased or decreased? [4] Prove that: [4] (1+cot A cosec A)(1+tanA+secA) = 2 Question 3 i. c. A is true but R is false d. A is false but R is true [13] Circumference of the base of a cylinder, open at the top, is 132cm. the sum of radius and height is 41 cm. find the cost of polishing the outer surface area of cylinder at the rate of 10 per sq dm (decimetre). Take =22/7 [4] Find the equation of the line, which passes through the point(3,4) and sum of its intercepts on the axes is 14. [4] Using graph paper and taking 1cm = 1 unit along with X-axis and Y-axis [5] a. Plot the point A(-4,4) and B(2,2). b. Reflect A and B in the origin to get the images A and B , respectively. c. Write down the coordinates of A and B . d. Give the geometrical name for the figure ABA B . Section B (Attempt any 4 Questions) Question 4 i. ii. iii. [10] Find the value of p, if (p,-2), (-5,6) and (1,2) are collinear. If the 2nd term of an AP is 13 and 5th term is 25, then what is its 7th term? The centroid of a triangle is the point (6,-1). If two vertices are (3,4) and (-2,5), then find third vertex. [3] [3] [4] Question 5 i. ii. iii. [10] Two positive numbers are in the ratio 3:5 and the difference between their squares is 400. Find the number. [3] Draw a circle of radius 4cm. mark a point A outside the circle. Draw the tangents to the circle from point A, without using the centre of the circle. [3] An aeroplane at an altitude of 1500 metres finds that two ships are sailing towards it in the same direction. The angles of depression as observed from the aeroplane are 450 and 300 respectively. Find the distance between the two ships. [4] Question 6 i. ii. iii. [10] Solve the following inequation and graph the solution set on the number line x + 2/15 -8/15, x R. The following table shows the expenditure of 60 boys on books. Expenditure 20-25 25-30 30-35 35-40 40-45 (in ) No. of 4 7 23 18 6 students Find the mode of their expenditure. In the following figure, CM and RN are respectively the medians of ABC and PQR, if ABC ~ PQR, prove that A N [3] 45-50 2 [3] [4] p M B i. AMC ~ PNR ii. Question 7 i. ii. = C R [10] Let Vinod, Govind and Ankit be three dealers belonging to difference states. Dealer Vinod sells some products/services to dealer Govind for 1000 and dealer Govind sells the same products/services to dealer Ankit at a profit of 300. Calculate the tax liability of Govind, if the rate of GST in 12%. [3] 2 If the roots of the equation (b-c)x + (c-a)x + (a-b) = 0 are equal, then prove that 2b = a +c. [3] iii. The mean of the following distribution is 52 and the frequency of class-interval 30-40 is f. Find f.[4] Class10-20 20-30 30-40 40-50 50-60 60-70 70-80 interval Frequency 5 3 f 7 2 6 13 Find the value of f. Question 8 i. ii. [10] 2 + 1 5 = , then find the values of x, y, z and w. 2 3 + 0 13 In the given figure, O is the centre of circle and AOB = 1100 . calculate If [3] [3] c A iii. B a) ACO b) CAO If one zero of the polynomial 2x2-5x-(2k+1) is twice the other, then find both the zeroes of the polynomial and the value of k. Question 9 i. ii. iii. [10] A number is chosen from 1 to 100. Find the probability that it is a prime number. [3] The internal and external diameters of steel pipe of length 140 cm are 8 cm and 10 cm, respectively. Then, find the volume of steel. [3] 0 0 In the figure, AB is parallel to DC, BCE = 80 and BAC = 25 . find: [4] a) CAD b) CBD c) ADC A C Question 10 i. [4] 25 B 80 D E [10] If -5 is a root of the quadratic equation 2x2 + px -15 = 0 and the quadratic equation p(x2 + x)+k=0 has equal roots, then find value of k. [5] ii. Marks obtained by 200 students in an examination are given below: [5] Marks Number of students 0-10 5 10-20 11 20-30 10 30-40 20 40-50 28 50-60 37 60-70 40 70-80 29 80-90 14 90-100 6 Draw an ogive for the given distribution taking 2 cm = 10 marks at on one axis and 2cm = 20 students on the other axis. a) The median marks b) The number of students who failed, if minimum marks required to pass is 40. c) If scoring 85 and more marks is considered as grade one, find the number of students who secured grade one in the examination. IGNITE PAPER 2 Section - A (Answer all the question from this section) Question 1. Choose the correct answer from the option given below. [1 x 15] i. ii. iii. iv. v. vi. vii. viii. ix. x. The GST paid by a customer to the shopkeeper for an article which is priced at x is 30. The rate of GST charged is 6%, then x is equal to a) 500 c) 1500 b) 1000 d) 2500 If (-5) is the root of the equation 2x + px - 15 = 0 and the quadratic equation p(x + x) + k = 0 has equal roots, the value of k is a) 7/4 c) 7 b) 7/2 d) 3.5 Which of the following point is invariant with respect to the line x = -2 a) (3, 2) c) (2, 3) b) (3, -2) d) (-2, 3) For the given A.P. 2, 5, 8, 11...., the value of a - a is a) 4 c) 2 b) 3 d) 1 The locus of mid points of the radii of a circle is a) concentric circle c) diameter of the circle b) concentric circle with half the d) chord of the circle radius A vertical stick 12 cm long cast a shadow 8 cm long on the ground. At the same time a tower cast a shadow 40 m long. The height of the tower is a) 40 m c) 80 m b) 60 m d) 120 m If A = [3 x; 0 1], B = [9 16; 0 -y] and A = B, the value of x and y are a) x = 4, y = 1 c) x = -4, y = 1 b) x = 4, y = -1 d) x = -4, y = -1 The median of 3, 5, 0, 4, 9, 7, 6, 2, 8 is a) 0 c) 4 b) 2 d) 5 The surface area and volume of a sphere are numerically equal. Then the diameter of the sphere is a) 3 units c) 6 units b) 4 units d) 12 units Assertion (A): If (x - 1) is a factor of (ax - 1), then a = 1 Reason (R): If (x - a) is a factor of a polynomial p(x), then p(a) = 0 xi. xii. a) A is true, R is false c) both A and R are true b) A is false, R is true d) both A and R are false 8 + 4x 36 - 3x, then a) x < 4 c) x 4 b) x > 4 d) x 4 A man deposited 400 per month in a recurring deposit account for 18 months. The qualifying sum of money for calculation of interest is a) 3600 b) 7200 xiii. xiv. xv. c) 68400 d) 136800 (Sec A + tan A) (1 - Sin A) a) Sec A c) Cosec A b) Sin A d) Cos A The mean proportional between 16 and 128 is a) 64 c) 16 b) 32 d) 8 Assertion (A): If the probability of India winning a T-20 cricket match against Australia is 7/10, then the probability of India losing the match against Australia is 3/10 Reason (R): If E is the complementary event of event E, then P(E) + P(E') = 1 a) A is true, R is false b) A is false, R is true c) both A and R are true d) both A and R are false Question 2. i. A and B invest 12000 each in buying shares of two companies. A buys 15% 100 shares at a discount of 20, while B buys 25 shares at a premium of 20%. If both receive equal dividend at the end of the year, find rate percent of dividend declared by B's company. A shopkeeper bought an article at a discount of 25% from a wholesaler, the printed price of the article being 32000. The shopkeeper sells it to a consumer at a discount of 10% on the printed price. If the sale is intra sale and rate of GST is 18%, find a) the price inclusive of GST at which the shopkeeper bought the article b) the price which the consumer pays for the article The mean of the following distribution is 50. If the sum of the frequencies is 120, find the values f and f ii. iii. Ci f 0-20 17 20-40 f 40-60 32 60-80 f 80-100 19 Question 3. i. If (x - 2) and (x + 1) are the factors of x + ax - bx - 6, find the values of a and b. With these values of a and b factorize the expression completely into linear factors. ii. A line AB meets x-axis at A and y-axis at B. Point P(4, -1) divides AB in the ratio 1:2, find n a) the co-ordinates of A and B, A b) the equation of line passing through P and perpendicular to AB diagram ala 1 B iii. Use graph paper to answer the following question. (take 2 cm = 1 unit on both the axes) a) Plot the point A(-4, 2) and B(2, 4) b) A' is the image of A when reflected in the y-axis and plot it on the graph and write down its coordinate c) B' is the image of B when reflected on the line AA' and write the co-ordinates B' d) Write the geometric name of ABA'B' Section B (Answer any four questions from this section) Question 4. i. A man deposits 1200 every month in a recurring deposit account for 2 years. If the rate of interest is 6% per annum, find the amount he will receive on maturity. ii. Solve the inequation: 2x - 5 5x + 4 < 11, x R and graph it on a number line iii. Solve the quadratic equation 7x2 + 2x 2 = 0. Give the answer correct to two places of decimal. Question 5. +3 4 5 4 = , find the value of x and y. 4 + 3 9 ii. By increasing the speed of a car by 10 km/h, the time of journey for a distance of 72 km is reduced by 36 minutes. Find the original speed of the car. i. If iii. Using componendo and dividendo, find the value of x, =3 Question 6. i. ii. iii. The sum of the 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms of the same A.P. is 34. Find the first three terms of the A.P. If P(3,4), Q(7,-2) and R(-2,-1) are the vertices of triangle PQR. Write down the equation of the median of the triangle through R. From the figure, ABC and CEF are two triangles where BA is parallel to CE and AF:AC = 5:8. A a) Prove that ADF ~ CEF b) Find AD if CE = 6cm c) If DF is parallel to BC, find area of ADF : area of ABC diagram C Find the sum of 10 terms of G.P, 1 + 3 + 3 + 3 3 + In the figure AB || CD and O is the center of the circle and ADC = 25 , find angle AEB. Give reasons in support of your answer. B A C iii. E B Question 7. i. ii. D D o E Calculate the mean of the given data by step-deviation method. Class Interval Frequency 10-20 20-30 30-40 40-50 50-60 60-70 70-80 5 3 4 7 2 6 13 Question 8. i. A cylindrical can of internal diameter 21 cm contains water. A solid sphere whose diameter is 10.5 cm is lowered into the cylindrical can. The sphere is completely immersed in water. Calculate rise in water level (no water overflows). ii. From the figure, PT is a tangent at R of a circle with center O. Diameter SQ when produced meet PT at P. If SPR = x and QRP = y , show that x + 2y = 90 . B S P T iii. Using ruler and compasses only, construct a quadrilateral ABCD in which AB = 6 cm, BC = 5 cm, B = 60 and AD = 5 cm and D is equidistant from AB and BC. Measure CD. Question 9 i. A box contain 15 cards numbered 1, 2, 3, 4, ..., 15 which are mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the card is a) divisible by 3 and 2 both. b) divisible by 3 or 2. ii. The daily wages of 160 workers in a building project are given below. Using a graph, draw an ogive to estimate a) the median wages of the workers. b) the upper quartile wages of workers. c) the percentage of workers who earn more than 45 per day. (Take 2 cm = 10 units on x-axis and 2cm = 20 units on y-axis). Wages in Rs Workers 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 12 20 30 38 24 16 12 8 Question 10 i. ii. Prove the identity: (1 + tan2 A) + (1 + 1/tan2A) = 1/ sin2A sin4A 2 1 Given, . = , write 3 4 a) the order of the matrix X. b) the matrix X. iii. From two points A and B on the same side of a building, the angle of elevation of the top of the building are 30 and 60 , respectively. If the height of the building is 10 m, then find the distance between A and B corrected to two decimal place. IGNITE PAPER 3 SECTION - A (40 Marks) (Attempt all the questions from this section) 1. Multiple Choice Question (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) The roots of the equation x2+1=0 a) 0 c) 1 b) -1 d) none of these. Which of the following equation represent a line equally inclined to the coordinate axes? a) x=5 c) 2x-2y=9 b) y=3x-2 d) none of these. The solution of the inequation 2x -5 5x+4<11,x E I is a) -3 ,- 1,0,1 c) -3 ,- 2 ,- 1,0,1 b) -4,3,0,1 d) -2 ,- 1,0,1 The sum of money required to buy 25, 40 shares at 38.50 is a) 961.50 c) 962.50 b) 962.25 d) 962.75 If an article with MRP 1200 is available at a discount of 20% and GST 18%, then CGST is a) 86.40 c) 172.80 b) 108.00 d) none of these. Krishna deposited 350 per month in a recurring deposit account for 15 months. The qualifying sum of money for the calculation of interest is a) 41000 c) 42000 b) 41500 d) 42500. AB is a chord of the circle and AOC is its diameter such that / ACB = 50 . If AT is the tangent to the circle at the point A, then 2 BAT is equal to a) 65 c) 55 b) 60 d) 500 If 2,a,b,c,162 are the consecutive terms of a GP ., then the value of (a + b + c) is a) 68 c) 168 b) 78 d) 178 1 0 If A= then A5 is equal to 0 1 1 1 1 0 a) c) 0 1 0 1 1 0 1 1 b) d) 1 1 1 1 If the mean of the observations 5, 7, 12, 17, x - 4, x + 6, 42, 44, 46 and 65 is x, then the median of the observation is a) x+2 c) x+1 b) x d) x-1 For an event E, P(E)=1-a ; the value of P(not E) is a) a-1 c) 0 b) a d) 1 (xii) (xiii) (xiv) A solid cylinder is cut into two identical cylinders. Statement 1: The total volume of two new cylinders is equal to the volume of the original cylinder. Statement 2: The total surface area of two new cylinders together is equal to the surface area of the original cylinder. a) Both the statements are true. b) B)Both the statements are false c) C)Statement 1 is true and Statement 2 is false. d) D) Statement 2 is true and Statement 1 is false. Two men are born in the year 2020, what is the probability that they have the same birthday? a) 1/365 c) 2/365 b) 1/366 d) 1/183 Assertion (A) : The mean of the following data is 9. X f 4 6 5 10 Reason (R) : Mean = 9 10 10 7 15 8 (xv) a) Both (A) and (R) are true and (R) is the correct explanation of (A). b) Both (A) and (R) are true but (R) is not the correct explanation of (A). c) (A) is true but (R) is false. d) (A) is false but (R) is true. If tan A + sin A=m and tan A - sin A=n, then correct relation is c) m2+n2=4mn a) m2+n2=4 mn d) m2-n2=4mn b) m2-n2=4 mn Question 2 (i) When the polynomial x3 + 2x2 - 5ax - 7 is divided by (x - 1), the remainder is A and when the polynomial x3 + ax2 - 12x + 16 is divided by (x + 2), the remainder is B. Find the value of 'a' if 2A + B = 0. (ii) Determine whether the line through points (-2, 5) and (4, 3) is perpendicular to the line 3x = y + 1. Does the line 3x = y + 1 bisect the line segment joining the two given points? (iii)In the given figure, AC is the diameter of circle, centre O. CD and BE are parallel. AOB = 80 and ACE = 10 . Calculate: (a) BAC; (b) BEC; (c) BCD; (d) BED. C D 10 E Question 3 I B A (i) The product of 3rd and 8th terms of a G.P. is 243. If its 4th term is 3, find its 7th term. (ii) A solid, consisting of a right circular cone, standing on a hemisphere, is placed upright, in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left in the cylinder, having given that the radius of the cylinder is 3 cm and its height is 6 cm; the radius of the hemisphere is 2 cm and the height of the cone is 4 cm. Give your answer to the nearest cubic centimeter. (iii)Use graph paper for the following sum: Points (-5, 0) and (4, 0) are invariant points under reflection in the line L1; points (0, - 6) and (0, 5) are invariant on reflection in the line L2. a) Write equations for the lines Li and L2. b) Write down the images of P(2, 6) and Q (-8, - 3) on reflection in L1. Name the images as P' and Q' respectively. c) Write down the images of P and Q on reflection in L2. Name the images as P" and Q" respectively. d) State or describe a single transformation that maps Q' onto Q". SECTION B (40 MARKS) (Answer any Four) Question 4 (i) A man invested 45,000 in 15% 100shares quoted at 125. When the market value of these shares rose to 140, he sold some shares, just enough to raise 8,400. Calculate: (a) the number of shares he still holds; (b) the dividend due to him on these remaining shares. (ii) Solve the following inequation, write the solution set and represent it on the number line. -3 (x-7) 15-7x > x+ 1/3 , x = R. (iii)Prove that : (sin + cos )(cosec -sec ) = sec . cosec -2tan . Question 5 (i) In the given figure, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. a) Prove that PAD - PCB. b) Find the length of AB. T c) Find the length of tangent PT. B A C P D (ii) Shahrukh opened a Recurring Deposit Account in a bank and deposited 800 per month for 1 years. If he received 15,084 at the time of maturity, find the rate of interest per annum. (iii)Calculate the mean of the distribution, given below, using the step deviation method: Mark 11-20 21-30 31-40 41-50 51-60 61-70 No. of 2 6 10 12 9 7 students 71-80 4 Question 6 (i) Write down the equation of the line AB, through (3, 2) and perpendicular to the line 2y = 3x + 5. If line AB meets the x-axis at A and the y-axis at B, write down the co-ordinates of A and B. Calculate the area of triangle OAB, where O is the origin. (ii) In the given figure, tangent PT touches the circle with centre O at point R. Diameter SQ is produced to meet the tangent PT at P. Given and SPR = x and QRP = y Prove that : a) ORS = y b) Write an expression connecting x and y. S Q p 12 y T (iii)From the given table, calculate the total amount to be paid ( including GST ) for the bill : ABC Departmental Store S.no. Item (s) M.R.P. Discount Rate of GST A Cloths 1200 100 12% B Rice(5kg) 286 Nil 5% C Dry Fruits 500 10% 12% (500gm) Question 7 (i) From the top of a light house 100 m high, the angles of depression of two ships are observed as 480 and 360 respectively. Find the distance between the two ships(in the nearest metre) if: a) the ships are on the same side of the light house. b) the ships are on the opposite sides of the light house. (ii) Use graph paper for the following questions (take 2 cm = 10 marks on one axis and 2 cm = 10 students on the other axis): The marks obtained by 100 students in a Mathematics test are given below: Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. of 3 7 12 17 23 14 9 6 5 4 students Draw an ogive for the given distribution and use the ogive to estimate: a) Median b) Number of students who obtained more than 85% marks in the test. c) Number of students failed, if the pass percentage was 35. Question 8 (i) A bag contains 3 red balls, 4 blue balls and 1 yellow ball, all the balls being identical in shape and size. If a ball is taken out of the bag without looking into it; find the probability that the ball is: (a) not yellow, (b) neither yellow nor red. (ii) Using properties of proportion solve for 'x': x4 + 1 / x2 = 17/4 (iii)The surface area of a solid sphere is increased by 21% without changing its shape. Find the percentage increase in its: (a) radius, (b) volume. Question 9 (i) The fourth term of an A.P. is 11 and the eighth term exceeds twice the fourth term by 5. Find the A.P and the sum of first 50 terms. (ii) The roots of the equation (q - r)x2 + (r - p)x + (p - q) = 0 are equal. Prove that: 2q = p + r. (iii)Use a graph paper for this question: The daily pocket expenses of 200 students in a school are given below : Pocket 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 Expenses ( ) No. of 10 14 28 42 50 30 14 12 students Draw a histogram representing the above distribution and estimate the mode from the graph. Question 10 2 a) If [x y] = [25] & [-x y] = [-2]; find x and y if (a) x and y W, (b) x and y Z . b) A trader buys x articles for a total cost of 600. a) Write down the cost of one article in terms of x. b) If the cost per article were 5 more, the number of articles that can be bought for 600 would be four less. Write down the equation in x for this situation and solve it for x. c) Use ruler and compasses for the following question taking a scale of 10m = 1cm. A park in a city is bounded by straight fences AB, BC, CD, & DA. Given that AB = 50m, BC=63m and ABC=750. D is a point equidistant from the fences AB & BC. If BAD=900, construct the outline of the park ABCD. Also locate a point P on the line BD for the flag post which is equidistant from the corners of the park A & B. IGNITE PAPER 4 Question 1 Choose the correct answers to the questions from the given options : i) If the equation x2 - kx + 9 = 0 does not possesses real roots, then: a) K <- 6 c) k>6 b) b)-6< k < 6 d) k = - 6,6 ii) Gogi purchased a washing machine, marked for Rs 25000. If the CGST charged is Rs 2250, then the rate of GST is: a) 9% c) 12% b) 6% d) 18% 3 2 iii) If on dividing 2x +3x - px +6 by x + 2 leaves a remainder 0, then the value of p is: a) p = 17 c) p = 1 b) p = - 1 d) p =- 17 4 3 1 iv) Given A= The order of matrix A is; 2 5 6 a) 1 x 2 c) 2 1 b) 2 2 d) 1 1 v) 20th term of an A.P, whose first three terms are -3, 4 and 11 is; a) 133 b) 16 c) 130 d) -130 vi) Assertion (A) : The locus of a point which is equidistant from two fixed points is the perpendicular bisector of the line segment joining the fixed points. Reason (R) : Any point on the bisector of an angle, is equidistant from the vertex of the angle. a) A is true, R is false. c) A is faise, R is true. b) Both A and R are true. d) Both A and R are false. vii) If in two triangles DEF and PQR, angle D = angle Q and angle E = angle R, then which of the following is not correct? a) = c) = b) = d) = viii) If x W, then the solution sot of the inequation 3x + 11 x + 8 is: a) {-1,0,1,2 ,..... } d) { x: x ER, x> ) b) {-2, - 1,0,1,2 ,..... } c) {0,1,2,3 ,..... } ix) Points (0,3) and (0 ,- 1) are invariant points under reflection in the line L1, Name the mirror line L1 a) Y-axis c) Origin b) X-axis d) line Y = 2 x) If a letter is chosen at random from the letters of English alphabet, then the probability that it is a letter of the word 'DELHI' is: a) c) b) d) xi) If a, b, c are in continued proportion ,then : a) a:b = a: c c) a: b2 = c : a2 b) b: c = a2 : b2 d) a : c = a2 : b2 xii) In a cylinder, if the radius is halved and the height Is doubled then the volume will be: a) same c) halved b) doubled d) four times xiii) In the adjoining figure, O is the centre of the circle. If the length of chord PQ is equal to the radius of the circle, then angle PRQ is: a) 60 c) 30 b) 45 d) 15 O R P xiv) If secA - tanA = k, then the value of secA Q + tanA is: c) 1 + a) 1 d) b) 1 xv) If P ( /3 , 4) is the mid point of the line segment joining the points Q(-6,5) and R (-2,3), then the value of a is: a) -4 c) 12 b) -6 d) -12 Question 2: a) A customer opens a recurring Deposit account and deposits Rs. 3000 every month. The bank offers an interest rate of 8% per annum. The customer achieves a maturity amount of Rs. 57420. Calculate the number of months required to reach the maturity amount. b) Given the polynomial P(x) = 2x3 - 7x2 + 7x - 2 and x =1 is the root of the polynomial, Use Factor Theorem to find the other roots of the polynomial. c) In the adjoining figure, CE is a tangent to the circle at point C. ABCD is a cyclic quadrilateral. If angle ABC =93 and angle DCE =35 . Find: i) ii) iii) iv) angle ADC angle CAD angle ACD. angle BAC + angle ACB D E A 930 B F Question 3: a) The adjoining figure represents a solid consisting of a cylinder surmounted by a cone at one end and a hemisphere at the other end. Given that common radius = 3.5 cm, the height of the cylinder = 6.5 cm and the total height = 12.8cm, calculate the volume of the solid, correct to the nearest cm3. asem iasim b) The first four terms of an A.P. are 1,9,p and q. Find the i) Values of p and q. ii) 16th term of an A.P. iii) The sum of first 44 terms. c) Use graph paper for this question Points A and B, co-ordinates (-2,4) and (4.1) respectively. Find: i) The co-ordinates of A', the image of A in the line x=0. ii) The co-ordinates of B', the image of B in Y-axis. iii) The co-ordinates of A", the image of A in the line BB'. iv) Assign a special name to the figure B'ABA". v) Find the perimeter of the closed figure. SECTION -B (40 MARKS) (Attempt any four questions) Question 4 a) Solve the inequation and represents the solution set on the real number line. 3 + 8 14 +2 + 2 , 3 3 b) Use 2cm = 10 units on X-axis and 2cm = 5units on y-axis. Estimate the mode for the following frequency distribution of marks obtained in a Geometry test. Marks No. of students 0-10 7 10-20 13 20-30 15 30-40 12 40-50 3 c) If (a3 +3ab2) : (x3 + 3xy2) = (3a2b + b3) : (3x2y + y3) ; then using the properties of proportion prove that x : a = y : b. Question 5 a) Comment upon the nature of roots of the quadratic equation, if possible find x correct to 3 significant figures. 2 2 - 4x -3 =0 b) The first term of a G.P. is 2. If the sum of its third and fifth terms is 180, find the common ratio and G.P. c) Mrs. Verma bought the following articles for her family: S. no. 1 2 3 Article Almonds Hair oil Apple juice Kamad Departmental Store: Bill Price Quantity 800 2kg 350 2 400 5 Discount 10 per kg Nil 5 each Rate of GST 18% 5% 12% Calculate: i) ii) total GST paid. total amount of bill including GST. Question 6 a) A man invests Rs 4500 in shares of a company which is paying 7.5% dividend. If Rs 100 shares are available at a discount of 10%. Find: i) number of shares he purchases. ii) his annual income b) In the given figure AB = 7cm and BC = 9cm. i) Prove that AACD ~ ADCB ii) Find the length of CD. A B c D c) In the given figure, line APB meets the X-axis in A and Y-axis in B. P is a point (4,2) and AP: PB = 1:2. i) Find the Co-ordinates of A and B. ii) Equation of a line parallel to the line AB through point (1,1). Y B Pla 2 A Question 7 a) The daily wages of 160 workers in a building project are given below: Wages 130-140 140-150 150-160 160-170 170-180 No. of workers 48 34 26 32 20 On a graph sheet draw a cumulative frequency curve. Tako 2cm = Rs. 10 along one axis and 2cm = 20 workers along the other axis. Use it to estimate; i. ii. iii. The median wage Inter-quartile range Number of workers who earn between Rs145 to Rs 165 b) The horizontal distance between two A towers is 120m. The angle of elevation of the top and angle of depression of bottom of the first tower as observed from the second tower are 30 and 24 respectively. Find the height of two E towers. Give your answer correct one decimal place. A Question 8 E C B D a) Using ruler and compass only, construct triangle ABC such that BC= 5cm and AB =6.5cm and __ ABC = 120 . Construct: i. A locus of all the points equidistant from AB and BC. ii. A locus of all the points equidistant from BC and AC. Hence construct a circle touching all three sides of a triangle internally. b) i. ii. Find the value of ' p' if the lines 5x - 3y + 2 =0 and 6x - py + 7 = 0 are perpendicular to each other. Also find the equation of a line passing through (-2 ,- 1) and making an angle of 450 in the positive direction of x-axis. c) If A= 1 3 3 2 1 B= and A2 5B2 = 5C , find the matrix where C is 2 X 2 matrix. 4 3 2 Question 9 a) Rs 7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received Rs 100 more. Find the original number of children. b) In the given figure, angle QPS = angle RPT and angle PRQ = angle PTS. i) Prove that triangles PQR and PST are similar. ii) If = , then find QR:PR iii) Find area ( PQR) : area ( PST) l P T S R c) Prove that : cos2 / 1- tan + sin3 / sin - cos = 1+sin cos Question 10 a) Find the number of coins, 2.4 cm in diameter and 2mm thick, to be melted to form a Right circular cylinder of height 12cm and diameter 6cm. b) Find the probability that a number selected at random from the numbers 1,2,3 ,.........., 34,35 is: i) divisible by 3 and 5 ii) multiple of 3 or 5 iii) two digit prime numbers whose sum of digits is odd. c) The weights of 50 apples were recorded as given below. Calculate the mean weight, to the nearest gram by Step deviation Method. Weight in Grams No. of apples 80-85 85-90 90-95 95-100 100-105 105-110 110-115 5 8 10 12 8 4 3 IGNITE PAPER 5 SECTION A [40 marks] (Attempt all questions from this Section) Question 1 Choose the correct answers to the questions from the given options. i) A polynomial in x is x + ax - 5x - 6. Which of the following is a factor of the given polynomial so that the value of a is 2? a) (x-1) c) (x-2) b) (x-3) d) (x-4) ii) Prakhar deposited 1000 per month in a recurring deposit account for 1 year. The qualifying sum of money for the calculation of interest is: a) 120 c) 12065 b) 12000 d) 78000 iii) In the given diagram, B = F and = . F CE B A D Which of the following options is correct? d) The similarity of given triangles cannot a) ABC DFE be determined. b) ABC DEF c) ABC EFD iv) In the given figure, PA and PB are tangents at points A and B respectively to a circle with centre O. If APB=400, then the measure of reflex AOB is: A P 740 0 B a) 1400 c) 2200 b) 2800 d) 3200 v) If the angle of depression of an object from a 75 m high vertical tower is 600, then the distance of the object from the tower is: c) 75 m a) 25 3 m d) 150 m b) 75 3 m vi) Assume that the volumes of a solid cylinder and a solid cone of same radii are same. Statement 1: The height of the solid cylinder is three times the height of the solid cone. Statement 2: The ratio between the heights of the solid cylinder and the solid cone is 3: 1. Which of the following options is valid? a) Both the statements are true. d) Statement 1 is false and Statement 2 is b) Both the statements are false. true. c) Statement 1 is true and Statement 2 is false. vii) What is the mean of the numbers in progression 2, 4, 6, 8, ........., 40? a) 20 c) 21 b) 41 d) 42 viii) Ten cards (identical in all respects) are numbered 1 to 10 as shown. 1 6 2 7 3 4 5 8 9 10 A card is selected from these cards at random. The probability that the selected card bears a composite number is: a) c) b) d) 8 5 2 and B= 3 7 0 Assertion (A): Product BA of the two given matrices is possible. Reason (R): Product of the two matrices is possible if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. a) A is true, R is false. d) Both A and R are true, and R is the b) A is false, R is true. incorrect reason for A. c) Both A and R are true, and R is the correct reason for A. x) The table given below shows the values of x and y, where x is proportional (directly proportional) to y. ix) Given that A= x y The values of a and b are: a 12 24 b 15 20 a) a=16 and b=18 b) a=9 and b=32 c) a=32 and b=9 d) a=18 and b=16 xi) A retailer purchases an article for 8000 from a wholesaler and sells it to a consumer at 15% profit. The sales are intra-state and the rate of GST is 10%. The amount of tax (under GST) paid by the consumer for the purchase is: a) 120 c) 460 b) 680 d) 920 xii) The point of intersection of the lines (x+y=8) and (x-y=0) lies on the line (mx-2y=0). The value of m is: a) 4 c) 3 b) 2 d) 1 2 xiii) If (x=-2) is a root of the equation (x +3k-x=0), the value of k is: a) -2 c) b) d) 2 xiv) Mr Gupta invests in 100, 18% shares of Company A available at 40 each. Mr Agarwal invests in 50, 24% shares of Company B available at 40 each. Use this information to state which of the following statements is true. a) The rate of return for Mr Gupta is 18%. b) The rate of return for Mr Agarwal is 24%. c) Both Mr Gupta and Mr Agarwal have the same rate of return of 30%. d) The given information is insufficient to compute the rate of return for Mr Gupta and Mr Agarwal. xv) The real number lines for two linear inequations A and B are as given below: 5th oct 7757 4 5 footstffffeffett 000 I 2 A B can be represented by: (a) 4 577700 2 (b) 000 5 4 3 2 (c) faa fftt ioo.ge (d) stffette es 1 2 Question 2 (i) (ii) Nidhi factorised the polynomial 2x + 5x - 11x - 14 and found the result as (x + 1)(x - 2)(2x + 3). Using remainder and factor theorem, verify whether her result is correct or not. If not, write the correct factorisation of the polynomial, showing essential working. In the given figure, M is the mid-point of the line segment joining the points A(0, 4) and B(6, 0). M also divides the line segment OP in the ratio 1 : 3. Find: Y 10,4 A x 0 P m B 6,0 a) 1 the coordinates of M. b) the coordinates of P. c) the equation of the line passing through M and perpendicular to OP. (iii) In the given figure, I is the incentre of ABC. BI when produced meets the circumcircle of ABC at D. A D I 781 C B Given that BAC = 60 and ACB = 70 , find: (a) (b) (c) (d) DCA DAC DCI AIC Question 3 (i) (ii) The 4th term of an A.P. is equal to 3 times its first term. The 7th term of the A.P. exceeds twice its 3rd term by 1. Find the first term, the common difference and the sum of its first 10 terms. Ria has a rectangular piece of paper of dimensions 33 cm by 24 cm. Find the volume of the largest cylinder that Ria can form by rolling the paper along its longer side as shown in the given figure. (take = ) 24cm 33cm (iii) Study the given graph and answer the questions that follow: y ia X P 0 2 2 3 4 5 is p y (a) Write the coordinates of points P and S. (b) Given that point Q is the image of point P under reflection in a line, write the equation of the line of reflection. (c) Write the coordinates of the image of S when reflected in the line found in (b). (d) Write the coordinates of a point which is invariant under reflection in the y-axis. (e) State the geometrical name of the closed figure obtained on joining the points P, Q, R and S in order. SECTION B [40 marks] (Attempt any four questions from this Section) Question 4 (i) Solve the following inequation and represent the solution set on the number line: < , where x I. (ii) Prove the following trigonometric identity: (sec A - tan A) (1 + sin A) = 1 - sin A (iii) Suresh invested 60000 in 100 shares, paying 10% dividend quoted at 120. After a year, when the price of the shares rose to 200 per share, Suresh sold all the shares and invested the proceeds in 12% 400 shares at 500. Calculate: a) the sale proceeds b) the number of 500 shares he bought. c) the change in his annual income from dividend. Question 5 (i) The scale of map is 1 : 50000. In the map, a triangular plot PQR of land has the dimensions as shown in the following figure. P 3cm (Figure not drawn to scale) Q 4cm R Calculate: (ii) (a) the actual length of side QR, in km, of the land. (b) the actual area of the plot in sq. km. Dheeraj deposited 600 per month in a recurring deposit account in a bank. The bank pays interest at the rate of 12% per annum. If Dheeraj gets 1026 as interest at the time of maturity, find: (a) the number of monthly instalments deposited by Dheeraj. (iii) (b) the total amount of money deposited by him in the account. The following table shows the ages of the students of a secondary school noted for official record: Age(Years) 6-8 8-10 10-12 12-14 14-16 No. of students 3 7 20 17 3 (a) Calculate the mean by short-cut method. Express your answer correct to one decimal place. (b) Find the median class of the distribution by calculating cumulative frequencies. Question 6 (i) (ii) Find the equation of a straight-line AB perpendicular to the line PQ whose equation is x - 2y - 6 = 0. The line AB cuts an intercept of 2 units from the positive y-axis. Hence, find the point of intersection of the two lines AB and PQ. To celebrate New Year Party in a school, the Principal asked the Students Council to buy necessary items for decoration from the market and to mark the bill of payment to the Accounts Section. The following bill shows the GST rate and the marked price of items along with the discount. Prakash Departmental Store S. No. Item MRP 1. Artificial flowers 360 (1 packet) 2. Balloons (1 packet) 200 3. Buntings (1 400 packet) Find the total bill amount including GST. (iii) Discount 5% Rate of GST 18% 10% 20 5% 5% In the given figure, AB and CD are two chords of a circle intersecting at the centre O. Given that C ABC=350, find: (a) CAB (b) DAO (c) BOD Show your steps and give valid reasons. 351 B A Question 7 (i) The weights of 50 workers are given below: D Weight (in 50-60 60-70 70-80 80-90 90-100 100-110 110-120 kg) No. of 4 7 11 14 6 5 3 workers Draw an ogive of the given distribution using a graph sheet. Take 2cm=10 kg on one axis and \(2~cm=5\) workers on the other axis. Use the graph to estimate the following: (a) the upper and lower quartiles. (b) If weighing 95 kg and above is considered overweight, find the number of workers who are overweight. (ii) In the given figure, the horizontal distance between two vertical towers AB and CD on the same level ground is 150 m. The angle of elevation of the top and the angle of depression of the bottom of the tower AB as observed from the top of the tower CD are 30 and 24 respectively. Find the height of the two towers. Give your answers correct to 3 significant figures. (Use Mathematical Tables for this question.) A C Question 8 D B (i) A bag contains 12 yellow cards, 12 blue cards and 12 orange cards. Each set of cards are numbered 1 to 12. The cards are well-shuffled and then a card is drawn at random from the bag. Find the probability that the card drawn is: a. a yellow card b. a card with a prime number c. a blue or an orange with a number which is a multiple of 5. (ii) Using ruler and compasses only, construct a triangle PQR where PQ=3cm, QR=4 cm and PQR=900. Hence, construct a circumcircle circumscribing triangle PQR. Measure and write down the radius of the circumcircle. (iii) If (8x-13y=5x+3y) use properties of proportion to find the value of 9 + 5 9 5 Question 9 (i) (ii) (iii) Solve the following equation for x and give your answer correct to two decimal places: 3x2 + 5x 1 = 0 cos 45 sin 30 tan 45 cos 90 Given A= and B= sin 0 cot 45 2 cos 0 sin 0 (a) Evaluate AB by showing the steps of multiplication of two matrices. (b) What special name can be assigned to matrix B? (c) What inference can be drawn from the result obtained in part (a)? Use ruler and compasses only for this question. (a) Construct ABC, where AB = 3.5 cm, BC = 6 cm and ABC = 60 . (b) Construct the locus of points inside the triangle which are equidistant from BA and BC. (c) Construct the locus of points inside the triangle which are equidistant from B and C. (d) Mark the point P which is equidistant from AB, BC and also equidistant from B and C. Measure and record the length of PB. Question 10 (i) (ii) The 4th, 6th and the last terms of a geometric progression are 6, 24 and 384 respectively. If the common ratio is positive, find the first term, common ratio and the number of terms in the progression. The surface area of a solid metallic sphere is 1386cm2 It is melted and recast into solid right circular cones of radius 3.5 cm and height 3 cm. Calculate: a. The radius of the sphere b. The number of cones recast . Take = The histogram given below represents the heights of the pupils of a school. Study the graph and answer the following questions. b. Write the frequency for each class interval. c. State the modal class. d. Estimate the modal height. y 28 24 20 Number of Pupils (iii) 8 4 O M 120.5 1305 140.5 150.5 160.5 170.5 180.5 Height (in cm)

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