Page 1 CBSE IX | Mathematics Sample Paper 4 â€“ Solution CBSE Board Class IX Mathematics Sample Paper 4 â€“ Solution Time: 3 hrs Total Marks: 80 Section A 1. 56 0.056 1000 ? OR 4 4 3 4 3 3 3 3 3 2. Linear equation in two variables: An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero is called a linear equation in two variables x and y. 3. If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of the Parallelogram. ? Area of triangle = 1 2 ? Area of a Parallelogram ? Area of triangle Area of a Parallelogram = 1 2 Hence, the ratio is 1 : 2. 4. p(x) = x 3 + 10x 2 + px (x â€“ 1) is a factor of p(x). ? x â€“ 1 = 0 ? x = 1 Sunstituting x = 1 in p(x) = 0 ? 1 + 10 + p = 0 ? p = â€“11 OR g(x) = vx â€“ 3 = x 1/2 - 3. Here, one term of exponent is ½ which is not an integer. Hence, it is not a polynomial. Page 2 CBSE IX | Mathematics Sample Paper 4 â€“ Solution CBSE Board Class IX Mathematics Sample Paper 4 â€“ Solution Time: 3 hrs Total Marks: 80 Section A 1. 56 0.056 1000 ? OR 4 4 3 4 3 3 3 3 3 2. Linear equation in two variables: An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero is called a linear equation in two variables x and y. 3. If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of the Parallelogram. ? Area of triangle = 1 2 ? Area of a Parallelogram ? Area of triangle Area of a Parallelogram = 1 2 Hence, the ratio is 1 : 2. 4. p(x) = x 3 + 10x 2 + px (x â€“ 1) is a factor of p(x). ? x â€“ 1 = 0 ? x = 1 Sunstituting x = 1 in p(x) = 0 ? 1 + 10 + p = 0 ? p = â€“11 OR g(x) = vx â€“ 3 = x 1/2 - 3. Here, one term of exponent is ½ which is not an integer. Hence, it is not a polynomial. CBSE IX | Mathematics Sample Paper 4 â€“ Solution 5. Arranging the data in the ascending order: 35, 36, 39, 40, 41, 55, 61, 62, 65, 70, 71, ? Range = Maximum value â€“ Minimum value = 75 â€“ 35 = 40 6. Let the fourth angle of the quadrilateral be x. Sum of all angles of a quadrilateral = 360° ? 60° + 110° + 86° + x = 360° ? 256° + x = 360° ? x = 104° Section B 7. 1 2 1 2 12 27 ? ? ? ? 1 2 11 22 1 2 34 39 2 3 8. Let f(z) = 3z 3 + 8z 2 â€“ 1 The possible integral zeros of f(z) are â€“1 and 1. f(z) = 3z 3 + 8z 2 â€“ 1 f(â€“1) = 3(â€“1) 3 + 8(â€“1) 2 â€“ 1 0 â€“1 is not a zero of f(z) f(1) = 3(1) 3 + 8(1) 2 â€“ 1 0 1 is not a zero of f(z) Therefore, f(z) has no integral zero. 9. (i) Coordinates of A are (â€“7, 3) (ii) Abscissa of point D is 4. (iii) Point is B. (iv) Coordinates of C are (â€“3, â€“2) Page 3 CBSE IX | Mathematics Sample Paper 4 â€“ Solution CBSE Board Class IX Mathematics Sample Paper 4 â€“ Solution Time: 3 hrs Total Marks: 80 Section A 1. 56 0.056 1000 ? OR 4 4 3 4 3 3 3 3 3 2. Linear equation in two variables: An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero is called a linear equation in two variables x and y. 3. If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of the Parallelogram. ? Area of triangle = 1 2 ? Area of a Parallelogram ? Area of triangle Area of a Parallelogram = 1 2 Hence, the ratio is 1 : 2. 4. p(x) = x 3 + 10x 2 + px (x â€“ 1) is a factor of p(x). ? x â€“ 1 = 0 ? x = 1 Sunstituting x = 1 in p(x) = 0 ? 1 + 10 + p = 0 ? p = â€“11 OR g(x) = vx â€“ 3 = x 1/2 - 3. Here, one term of exponent is ½ which is not an integer. Hence, it is not a polynomial. CBSE IX | Mathematics Sample Paper 4 â€“ Solution 5. Arranging the data in the ascending order: 35, 36, 39, 40, 41, 55, 61, 62, 65, 70, 71, ? Range = Maximum value â€“ Minimum value = 75 â€“ 35 = 40 6. Let the fourth angle of the quadrilateral be x. Sum of all angles of a quadrilateral = 360° ? 60° + 110° + 86° + x = 360° ? 256° + x = 360° ? x = 104° Section B 7. 1 2 1 2 12 27 ? ? ? ? 1 2 11 22 1 2 34 39 2 3 8. Let f(z) = 3z 3 + 8z 2 â€“ 1 The possible integral zeros of f(z) are â€“1 and 1. f(z) = 3z 3 + 8z 2 â€“ 1 f(â€“1) = 3(â€“1) 3 + 8(â€“1) 2 â€“ 1 0 â€“1 is not a zero of f(z) f(1) = 3(1) 3 + 8(1) 2 â€“ 1 0 1 is not a zero of f(z) Therefore, f(z) has no integral zero. 9. (i) Coordinates of A are (â€“7, 3) (ii) Abscissa of point D is 4. (iii) Point is B. (iv) Coordinates of C are (â€“3, â€“2) CBSE IX | Mathematics Sample Paper 4 â€“ Solution 10. In parallelogram ABCD, CD = AB = 16 cm [Opposite sides of a parallelogram are equal] We know that, Area of parallelogram = Base × corresponding altitude Area of parallelogram ABCD = CD × AE = AD × CF 16 cm × 8 cm = AD × 10 cm Thus, the length of AD is 12.8 cm. OR Area of rhombus = 1 Product of diagonals 2 = 1 16 12 2 = 96 cm 2 11. Area of cross section of pipe = 5 cm 2 Speed of water flowing out of the pipe = 30 cm/sec Volume of water that flows out in 1 sec = 5 × 30 = 150 cm 3 Volume of water that flows out in 1 minute = 150 × 60 = 9000 cm 3 = 9 litres. OR l = 15 m, b = 12 m and h = 4.5 m Volume of cuboid = lbh = 15 × 12 × 4.5 = 810 m 3 Total surface area of cuboid = 2(lb + bh + lh) = 2(15 × 12 + 12 × 4.5 + 15 × 4.5) = 603 m 2 12. Let the measure of the smaller angle be x and that of the larger angle be y. The larger angle is 3° less than twice the measure of the smaller angle, so y = 2x - 3° â€¦.(1) Given, that the two angles are complementary, x + y = 90° ? x + (2x - 3°) = 90° ? x + 2x - 3° = 90° ? 3x = 93° ? x = 31° Substitute value of x in equation (1) y = 2(31) - 3° Page 4 CBSE IX | Mathematics Sample Paper 4 â€“ Solution CBSE Board Class IX Mathematics Sample Paper 4 â€“ Solution Time: 3 hrs Total Marks: 80 Section A 1. 56 0.056 1000 ? OR 4 4 3 4 3 3 3 3 3 2. Linear equation in two variables: An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero is called a linear equation in two variables x and y. 3. If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of the Parallelogram. ? Area of triangle = 1 2 ? Area of a Parallelogram ? Area of triangle Area of a Parallelogram = 1 2 Hence, the ratio is 1 : 2. 4. p(x) = x 3 + 10x 2 + px (x â€“ 1) is a factor of p(x). ? x â€“ 1 = 0 ? x = 1 Sunstituting x = 1 in p(x) = 0 ? 1 + 10 + p = 0 ? p = â€“11 OR g(x) = vx â€“ 3 = x 1/2 - 3. Here, one term of exponent is ½ which is not an integer. Hence, it is not a polynomial. CBSE IX | Mathematics Sample Paper 4 â€“ Solution 5. Arranging the data in the ascending order: 35, 36, 39, 40, 41, 55, 61, 62, 65, 70, 71, ? Range = Maximum value â€“ Minimum value = 75 â€“ 35 = 40 6. Let the fourth angle of the quadrilateral be x. Sum of all angles of a quadrilateral = 360° ? 60° + 110° + 86° + x = 360° ? 256° + x = 360° ? x = 104° Section B 7. 1 2 1 2 12 27 ? ? ? ? 1 2 11 22 1 2 34 39 2 3 8. Let f(z) = 3z 3 + 8z 2 â€“ 1 The possible integral zeros of f(z) are â€“1 and 1. f(z) = 3z 3 + 8z 2 â€“ 1 f(â€“1) = 3(â€“1) 3 + 8(â€“1) 2 â€“ 1 0 â€“1 is not a zero of f(z) f(1) = 3(1) 3 + 8(1) 2 â€“ 1 0 1 is not a zero of f(z) Therefore, f(z) has no integral zero. 9. (i) Coordinates of A are (â€“7, 3) (ii) Abscissa of point D is 4. (iii) Point is B. (iv) Coordinates of C are (â€“3, â€“2) CBSE IX | Mathematics Sample Paper 4 â€“ Solution 10. In parallelogram ABCD, CD = AB = 16 cm [Opposite sides of a parallelogram are equal] We know that, Area of parallelogram = Base × corresponding altitude Area of parallelogram ABCD = CD × AE = AD × CF 16 cm × 8 cm = AD × 10 cm Thus, the length of AD is 12.8 cm. OR Area of rhombus = 1 Product of diagonals 2 = 1 16 12 2 = 96 cm 2 11. Area of cross section of pipe = 5 cm 2 Speed of water flowing out of the pipe = 30 cm/sec Volume of water that flows out in 1 sec = 5 × 30 = 150 cm 3 Volume of water that flows out in 1 minute = 150 × 60 = 9000 cm 3 = 9 litres. OR l = 15 m, b = 12 m and h = 4.5 m Volume of cuboid = lbh = 15 × 12 × 4.5 = 810 m 3 Total surface area of cuboid = 2(lb + bh + lh) = 2(15 × 12 + 12 × 4.5 + 15 × 4.5) = 603 m 2 12. Let the measure of the smaller angle be x and that of the larger angle be y. The larger angle is 3° less than twice the measure of the smaller angle, so y = 2x - 3° â€¦.(1) Given, that the two angles are complementary, x + y = 90° ? x + (2x - 3°) = 90° ? x + 2x - 3° = 90° ? 3x = 93° ? x = 31° Substitute value of x in equation (1) y = 2(31) - 3° CBSE IX | Mathematics Sample Paper 4 â€“ Solution ? y = 59° So, the measures of the two angles are 31° and 59°. Section C 13. Let x = 0.001 Then, x = 0.001001001â€¦â€¦â€¦.. (i) Therefore, 1000x = 1.001001001â€¦â€¦â€¦â€¦â€¦ (ii) Subtracting (i) from (ii), we get 999x = 1 ? x = 1 999 Hence, 0.001 = 1 999 OR 6 4 3 6 4 3 6 4 3 6 4 3 6 4 3 6 4 3 36 48 3 48 36 48 84 48 3 12 48 3 84 12 4 3 7 14. i. No. ? ?? ? ? ? 2 1 x 3x 2 x 3 2x x has negative power of ii. Yes ? ? ? ? 22 42 t t 3 t 3t iii. Yes ? ? 2 22 x 4x 5 x 4x 5 x 5 2x 2 2 2 2 2 2 ?? ? ? ? ? ? ? Page 5 CBSE IX | Mathematics Sample Paper 4 â€“ Solution CBSE Board Class IX Mathematics Sample Paper 4 â€“ Solution Time: 3 hrs Total Marks: 80 Section A 1. 56 0.056 1000 ? OR 4 4 3 4 3 3 3 3 3 2. Linear equation in two variables: An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero is called a linear equation in two variables x and y. 3. If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of the Parallelogram. ? Area of triangle = 1 2 ? Area of a Parallelogram ? Area of triangle Area of a Parallelogram = 1 2 Hence, the ratio is 1 : 2. 4. p(x) = x 3 + 10x 2 + px (x â€“ 1) is a factor of p(x). ? x â€“ 1 = 0 ? x = 1 Sunstituting x = 1 in p(x) = 0 ? 1 + 10 + p = 0 ? p = â€“11 OR g(x) = vx â€“ 3 = x 1/2 - 3. Here, one term of exponent is ½ which is not an integer. Hence, it is not a polynomial. CBSE IX | Mathematics Sample Paper 4 â€“ Solution 5. Arranging the data in the ascending order: 35, 36, 39, 40, 41, 55, 61, 62, 65, 70, 71, ? Range = Maximum value â€“ Minimum value = 75 â€“ 35 = 40 6. Let the fourth angle of the quadrilateral be x. Sum of all angles of a quadrilateral = 360° ? 60° + 110° + 86° + x = 360° ? 256° + x = 360° ? x = 104° Section B 7. 1 2 1 2 12 27 ? ? ? ? 1 2 11 22 1 2 34 39 2 3 8. Let f(z) = 3z 3 + 8z 2 â€“ 1 The possible integral zeros of f(z) are â€“1 and 1. f(z) = 3z 3 + 8z 2 â€“ 1 f(â€“1) = 3(â€“1) 3 + 8(â€“1) 2 â€“ 1 0 â€“1 is not a zero of f(z) f(1) = 3(1) 3 + 8(1) 2 â€“ 1 0 1 is not a zero of f(z) Therefore, f(z) has no integral zero. 9. (i) Coordinates of A are (â€“7, 3) (ii) Abscissa of point D is 4. (iii) Point is B. (iv) Coordinates of C are (â€“3, â€“2) CBSE IX | Mathematics Sample Paper 4 â€“ Solution 10. In parallelogram ABCD, CD = AB = 16 cm [Opposite sides of a parallelogram are equal] We know that, Area of parallelogram = Base × corresponding altitude Area of parallelogram ABCD = CD × AE = AD × CF 16 cm × 8 cm = AD × 10 cm Thus, the length of AD is 12.8 cm. OR Area of rhombus = 1 Product of diagonals 2 = 1 16 12 2 = 96 cm 2 11. Area of cross section of pipe = 5 cm 2 Speed of water flowing out of the pipe = 30 cm/sec Volume of water that flows out in 1 sec = 5 × 30 = 150 cm 3 Volume of water that flows out in 1 minute = 150 × 60 = 9000 cm 3 = 9 litres. OR l = 15 m, b = 12 m and h = 4.5 m Volume of cuboid = lbh = 15 × 12 × 4.5 = 810 m 3 Total surface area of cuboid = 2(lb + bh + lh) = 2(15 × 12 + 12 × 4.5 + 15 × 4.5) = 603 m 2 12. Let the measure of the smaller angle be x and that of the larger angle be y. The larger angle is 3° less than twice the measure of the smaller angle, so y = 2x - 3° â€¦.(1) Given, that the two angles are complementary, x + y = 90° ? x + (2x - 3°) = 90° ? x + 2x - 3° = 90° ? 3x = 93° ? x = 31° Substitute value of x in equation (1) y = 2(31) - 3° CBSE IX | Mathematics Sample Paper 4 â€“ Solution ? y = 59° So, the measures of the two angles are 31° and 59°. Section C 13. Let x = 0.001 Then, x = 0.001001001â€¦â€¦â€¦.. (i) Therefore, 1000x = 1.001001001â€¦â€¦â€¦â€¦â€¦ (ii) Subtracting (i) from (ii), we get 999x = 1 ? x = 1 999 Hence, 0.001 = 1 999 OR 6 4 3 6 4 3 6 4 3 6 4 3 6 4 3 6 4 3 36 48 3 48 36 48 84 48 3 12 48 3 84 12 4 3 7 14. i. No. ? ?? ? ? ? 2 1 x 3x 2 x 3 2x x has negative power of ii. Yes ? ? ? ? 22 42 t t 3 t 3t iii. Yes ? ? 2 22 x 4x 5 x 4x 5 x 5 2x 2 2 2 2 2 2 ?? ? ? ? ? ? ? CBSE IX | Mathematics Sample Paper 4 â€“ Solution iv. No ? 2 3x 6 x = ? ? ? 1/2 2 3x 6 x has fractional power of v. No ? 1 z z i.e. ? ? 1 zz has negative power of 15. When p(x) = ax 3 + 3x 2 â€“ 3 is divided by (x â€“ 4 ), the remainder is given by R1 = a(4) 3 + 3(4) 2 â€“ 3 = 64a + 45 When q(x) = 2x 3 â€“ 5x + a is divided by (x â€“ 4), the remainder is given by R2 = 2(4) 3 â€“ 5(4) + a = 108 + a Given: R1 + R2 = 0 ? 65a + 153 = 0 ? a = ?153 65 OR p = 2 â€“ a a + p â€“ 2 = 0 x + y + z = 0 where x = a, y = p and z = -2 x 3 + y 3 + z 3 = 3xyz ? x + y + z = 0 a 3 + p 3 + (-2) 3 = 3ap(-2) a 3 + p 3 â€“ 8 = -6ap a 3 + 6ap + p 3 â€“ 8 = 0 16. In ?DCB, ?DBC = ?DCB (given) DC = DB [Side opp. To equal ?â€™s are equal]â€¦..(i) In ?ABD and ?ACD AB = AC (given) BD = CD [from (i) ] AD = AD common ?ABD ? ? ACD [SSS Rule] ?BAD = ?CAD (CPCT) Hence, AD is bisector of ?BAC.Read More

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