#CBSE BOARD PAPER MARCH 2020 CLASS 10

10 - MATHS SAMPLE PAPERS 2019

GRADE 10 CBSE SAMPLE PAPER MATHEMATICS

LINEAR EQUATION IN TWO VARIABLES 4

1. Solve 11x + 1 5y + 23 = 0 , 7x – 2y – 20 = 0 by substitution method

2. Draw the graph of the equations 4x – 5 y + 16 = 0 and 2x + y – 6 = 0 and also determine the vertices of the triangle formed by these lines and the x axis.

3. Solve 2x – 5y + 8 = 0, x – 4y + 7 = 0 by substitution method

4. Two years ago Salim was thrice old as his daughter and six years later he will be four years older than twice her age. How old are they now.

5.  A fraction becomes 5/6 if 1 is added to each of the numerator and denominator. However if we subtract 5 from each, the fraction becomes 2/3  . Find the fraction.

NCERT SOLUTIONS 4. QUADRATIC EQUATIONS

 

1. Check whether the following are quadratic equations:

(i) (x + 1)² = 2(x – 3)

(ii) x² – 2x = (–2)(3 – x

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)

(iv) (x – 3)(2x + 1) = x(x + 5)

(v) (2x – 1) (x – 3) – (x + 5) (x – 1)

(vi) x² + 3x +1 = (x – 2)²

(vii) (x + 2)³ = 2x(x² – 1)

(viii) x³ – 4x² – x + 1 = (x – 2)³

Answer:

 (x + 1)² = 2(x – 3)

  (x + 1)² = 2 (x – 3)

x² + 2x + 1 = 2x – 6

 x² + 2x + 1 – 2x + 6 = 0

  x² + 7=0

This is of the form ax² + bx + c = 0

∴ (x + 1)² = 2(x – 3) is a quadratic equation.

         (ii) x²– 2x = (–2) (3 – x)

           x² – 2x = –6 + 2x

               x² – 2x – 2x + 6 = 0

               x² – 4x + 6 = 0

This is of the form ax² + bx + c = 0

∴ x² – 2x = (–2) (3 – x) is a quadratic equation.

         (iii) (x – 2) (x + 1) = (x – 1) (x + 3)

             x² – x – 2 = x² + 2x – 3

                 x² – x – 2 – x² – 2x + 3 = 0

                 –3x + 1 = 0

This is not the form of ax² + bx + c = 0

∴ (x – 2) (x + 1) = (x – 1) (x + 3) is not quadratic equation.

         (iv) (x – 3) (2x + 1) = x(x + 5)

          2x² + x – 6x – 3 = x² + 5x

          2x² – 5x – 3 – x² – 5x – 0

         x² + 10x – 3 = 0

                 This is of the form ax² + bx + c = 0

 ∴ (x – 3) (2x + 1) = x(x + 5) is a quadratic equation.

         (v) (2x – 1) (x – 3) = (x + 5) (x – 1)

              2x² – 6x – x + 3 = x² – x + 5x – 5

                2x² – x² – 6x – x + x – 5x + 3 + 5 = 0

                 x² – 11x + 8 = 0

                 This is of the form ax² + bx + c = 0

∴ (2x – 1) (x – 3) = (x + 5) (x – 1) is a quadratic equation.

         (vi) x² + 3x + 1 = (x – 2

             x² + 3x + 1 = (x – 2

                 x² + 3x + 1 = x² – 4x + 4

                x² + 3x + 1 – x² + 4x – 4 =0

                 7x – 3 = 0

This is not the form of ax² + bx + c = 0

 ∴ x² + 3x + 1 = (x – 2 is not a quadratic equation.

         (vii) (x + 2)³ = 2x(x2 – 1)

                  (a+b)³  = a3 + 3a2b + 3ab2 + b3

                  x³ + 3x²(2) + 3x(2)² + (2)³ = 2x³ – 2x

                 x³ + 6x² + 12x + 8 = 2x³ – 2x

                 x³ + 6x² + 12x + 8 – 2x³ + 2x = 0

                  –x³ + 6x² + 14x + 8 = 0

This is not the form of ax² + bx + c = 0

∴ (x + 2)³ = 2x(x² – 1) is not a quadratic equation.

         (viii) x³ – 4x² – x + 1 = (x – 2)³

                   We have:

                   x³ – 4x² – x + 1 = (x – 2)³

                  x³ – 4x² – x + 1 = x³ + 3x²(– 2) + 3x(– 2)² + (– 2)³

                  x³ – 4x² – x + 1 = x³ – 6x² + 12x – 8

                   x³ – 4x² – x – 1 – x³ + 6x² – 12x + 8 = 0

                   2x² – 13x + 9 = 0

This is of the form of ax² + bx + c = 0

∴ x³ – 4x² – x + 1 = (x – 2)³ is a quadratic equation.

2.   Represent the following situations in the form of quadratic equations:

   (i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

 (ii) The product of two consecutive positive integers is 306. We need to find the integers.

  (iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

 (iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Answer:

 (i) Let the breadth of the plot be x m.

Hence, the length of the plot is (2x + 1) m.

Area of a rectangle = Length × Breadth

                             (2x + 1) × x = 528

                          2x² + x = 528

                          2x² + x – 528 = 0

  Thus, the required quadratic equation is  2x² + x – 528 = 0

(ii) Let the consecutive integers be x and x + 1.

According to the question

 x (x + 1) = 306

    x² + x = 306

     x² + x – 306 = 0

iii) Let Rohan’s age be x.

Hence, his mother’s age = x + 26

3 years hence,

Rohan’s age = x + 3

Mother’s age = x + 26 + 3 = x + 29

According to the question

(x + 3) × (x + 29) = 360

                  x² + 29x + 3x + 87 = 360

                  x² + 29x + 3x + 87 – 360 = 0

                  x² + 32x – 273 = 0

(iv) Let the speed of train be x km/h.

 

In the second case, speed = x – 8 km/h

According to the question,

T2  - T1 = 3

480x – 480x + 3840 = 3x² – 24x

3x2 – 24x -3840 = 0

x2 – 8x – 1280 = 0

EXERCISE 4.2

1.  Find the roots of the following quadratic equations by factorisation:

         (i) x² – 3x – 10 = 0

        (ii) 2x² + x – 6 = 0

         (iii) X² +7X + 5  = 0

         (iv) 2x² – x + 8 = 0

         (v) 100x² – 20x + 1 = 0

Answer:

 (i) x² – 3x – 10 = 0

           x² – 3x – 10 = 0

              x² – 5x + 2x – 10 = 0

              x (x – 5) + 2(x – 5) = 0

              (x – 5) (x + 2) = 0

              x – 5 = 0 ⇒ x = 5

             or x + 2 = 0 ⇒ x = –2

             Thus, the required roots are x = 5 and x = –2.

         (ii) 2x² + x – 6 = 0

                We have:

                2x² + x – 6 = 0

                2x² + 4x – 3x – 6 = 0

                2x(x + 2) – 3 (x + 2) = 0

                (x + 2) (2x – 3) = 0

                 x + 2 = 0 ⇒ x = –2

                or 2x – 3 = 0 ⇒ x = 3/2

                Thus, the required roots are x = –2 and 3/2

WORKSHEET STATISTICS GRADE 10

SET - 1

1. Calculate the mean for the following distribution:

x	5	6	7	8	9
f	4	8	14	11	3

2.  Find the mean of the following frequency distribution:

Class	6	6 -12	12-18	18-24	24-30
Frequency	6	8	10	9	7

 3.  Find the mean of the following frequency distribution:

Class	50 - 70	70 - 90	90 - 110	110 - 130	130 - 150	150- 170
Frequency	18	12	13	27	8	22

4. Find the mean of the following frequency distribution:

Class	8	8 - 16	16 -24	24 - 32	32 – 40
Frequency	6	7	10	8	9

  5. Find the mean of the following frequency distribution:

Class	25-29	30-34	35-39	40-44	45-49	50-54	55-59
Frequency	14	22	16	6	5	3	4

6. If the mean of the following distribution is 27, find the value of p.

Classes	0 - 10	10 - 20	20-30	30-40	40 - 50
Frequency	8	p	12	13	10

 SET - 2

1. The following is the distribution of height of students of a certain class in certain city:

Height in cm	160 -162	163 - 165	166 - 168	169 - 171	171 - 174
No. of students	15	118	142	127	18

2. Calculate the missing frequency from the following distribution, it being gives that the median of the

     distribution is 24.

Classes	0 - 10	10 - 20	20-30	30-40	40 - 50
Frequency	5	25	x	18	7

3. An in complete distribution is given below:

Variable	10 - 20	20 – 30	30 -40	40- 50	50 - 60	60 - 70	70 - 80
Frequency	12	30	x	65	y	25	18

you are given that the median value is 46 and the total number of items is 230.

   (i) Using the median formula fill up missing frequencies.

    (ii) Calculate the mean the completed distribution.

4. A survey regarding the height(in cm) of 51 girls of class X of a school was conducted and the following data was obtained.

Height in cm	Number of girls
Less than 140 Less than 145 Less than 150 Less than 155 Less than 160 Less than 165	4 11 29 40 46 51

     Find the median of height.

5. The distribution below gives the weight of 30 students in a class. Find the median weight of students.

Weight in kg	40 -45	45-50	50-55	55 - 60	60 - 65	65 - 70	70 - 75
No. of students	2	3	8	6	6	3	2

  SET - 3

1. Find the mode of the following:

Class	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	5	8	7	12	28	20	10	10

2. The following is the height of students of a certain class in a certain city: find the mode.

Height in(cm)	160 - 162	163 - 165	166 - 168	169 - 171	171 - 174
No. students	15	118	142	127	18

3. The following table shows the ages of the patients admitted in a hospital during a year.

Age in years	5 - 15	15 - 25	25 - 35	35 - 45	45 – 55	55 - 65
Frequency	6	11	21	23	14	5

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.4. Compare the modal ages of two groups of students appearing for an entrance test:

Age in (years)	16 -18	18 - 20	20 - 22	22 -24	24 -26
Group A	50	78	46	28	23
Group B	54	89	40	25	17

5. Calculate the value of mode for the following frequency distribution:

Class	Frequency
1 – 4 5 – 8 9 -12 13 – 16 17 – 20 21 – 24 25 -28 29 – 32 33 – 36 37 -40	2 5 8 9 12 14 14 15 11 13

 SET - 4

1. Draw an ogive to represent the following frequency distribution:

Class	0 - 4	5 - 9	10 - 14	15 -19	20 - 24
No. students	2	6	10	5	3

 2. The following table gives the height of trees. Draw less than ogive and more than ogive.

Height	No. of trees
Less than 7 Less than 14 Less than 21 Less than 28 Less than 35 Less than 42 Less than 49 Less than 56	26 57 92 134 216 287 341 360

3.  The following distribution gives the daily income of 50 workers of a factory:

Class	100 - 120	120 - 140	140 - 160	160-180	180-200
No. students	12	14	8	6	10

Convert the above distribution  to a less than type cumulative frequency distribution and

 draw its ogive and hence find the median.

4.  Draw the both ogives in the same graph paper.

Classes	0 - 10	10 - 20	20-30	30-40	40 - 50
Frequency	5	25	15	18	7