Mathematics Basic Formulae Part - 2

         Mathematics Basic Formulae Part - 2

Previous Year Mathematics Question of Class X [Chapter - Quadratic Equations ]

 

Previous Year Mathematics Questions of Class 10 Chapter {Quadratic Equations } for Examination 2023

One mark Questions

1)  Value of the roots of the quadratic equation x² – x – 6 = 0 is ………………                        (CBSE 2020)

2)  The roots of the quadratic equation x² – 0.04 = 0 are …………..                                                 (CBSE 2020)

3)  Find the nature of the roots of the quadratic equation 4x² + 4√3 x + 3 = 0                                (CBSE 2019)

4)  Find the nature of the roots of the quadratic equation 2x² – 4x + 3 = 0                                     (CBSE 2019)

5)  If x = 3 is one root of the quadratic equation x² – 2kx – 6 = 0, then find the value of k.          (CBSE 2018)

6)  Find the values of k for which the quadratic equation 

    2x² – kx + k = 0 has two equal roots.            

Two mark Questions

1)  Determine the roots of the following quadratic equations 4√5x² – 17x + 3√5 = 0

2)  If roots of the quadratic equation (b – c)x² + (c – a)x + (a – b) = 0 are real and equal then prove that 2b = a + c

3)  Solve the quadratic equation √3x² + 10 x - 8√3 = 0.

4)  Solve the quadratic equation x² + x – (a + 1)(a + 2) = 0

5)  Find the value of a for which equation 5x² – kx + 1 = 0 has real roots (D ≥ 0).

6)  If the equation (1 + m²)x² + 2mcx + (c² – a²) = 0 has equal roots prove that c² = a²(1 + m²)

 

Three mark Questions

1)  Solve for x:1/x+4  -  1/x+7 = 11/30 ,  x ≠ - 4, 7                                                     (CBSE 2020)

2)  In a flight of 600 Km, an aircraft was slowed down due to bad weather. The average speed of the trip was reduced by 200 Km per hour and the time of flight increased by 30 minutes. Find the duration of flight.    (CBSE 2020)  

3)  A fast train takes 3 hours less than a slow train for a journey for 600 Km. If speed of the slow train is 10 Km per hour less than that of the fast train, find the speed of each train.                                     (CBSE 2020)

4)  Write all the values of p for which the quadratic equation x² + px + 16 = 0 has equal roots, find the roots of the equation so obtained.                                                                         (CBSE 2019)

5)  A plane left 13 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 Km from the usual speed. Find its usual speed.

6)  Find the roots of the quadratic equations 4x² + 4bx – (a² – b²) = 0 by the completing the square method.

7)  Using quadratic formula solve the following quadratic equation in x:  p²x² + (p² – q²)x – q² = 0

Four mark Questions

1)  If the price of book is reduced by Rs. 5, a person can buy 4 more books for Rs. 600. Find the original price of the book.  (CBSE 2020)

2)  Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46. Find the integers.                          (CBSE 2020)

3)  Solve the quadratic equation x² + 3x – (a² + a – 2) = 0      (CBSE 2019)

4)  In a class test the sum of Arun’s marks in Hindi and English is 30 had he got 2 marks more in Hindi and 3 marks less in English, the product of the marks would have been 210, find his marks in 2 subjects.          (CBSE 2019)

5)  A motor boat whose speed is 18 Km per hour in still water takes 1 hour more to go 24 km upstream then to return downstream to the same spot. Find the speed of the stream.                                        (CBSE 2018)

6)  A train travel at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km per hour more than its original speed, if it takes 3 hours to complete total journey. What is the original average speed?                              (CBSE 2018)

7) 2 water taps together can fill a tank in 31/10 hours the tap of smaller diameter take 2 hours more than the larger one to fill the tank separately. Find the time in which each tap can separately fill the tank.

8)  Find the positive value of k for which x² + kx + 64 = 0 and x² – 8x + k = 0 will have real roots.  

9)  Solve the quadratic equation by factorization method: 9x² – 9(a + b)x + (2a² + 5ab + 2b²) = 0

10)                    The denominator of a fraction is 1 more than twice the numerator if the sum of the fraction and its reciprocal is 16/11. Find the fraction.

11)        O Girl! Out of a group of swans 7/2 times the square root of the number are playing on the shore of a tank , the two remaining ones are playing with a morous fight in the water. What is the total number of swans?

12)                    If the price of a book is reduced by Rs. 5, a person can by 5 more books for Rs. 300. Find the original list price of the book.

 

Previous Year Mathematics Question of Class X [Chapter - Pair of Linear Equations of two Variables]

 

Previous Year Mathematics Questions of Class 10 Chapter {Pair of Linear Equations of two variables}        for Examination 2023

One mark Questions

1.       The value of K for which the system of linear equations x + 2y = 3, 5x + ky + 7 = 0 is inconsistent is …   {CBSE 2020}

2.       If x = a, y = b is the solution of the pair of linear equation x – y = 2 and x + y = , find the value of a and b.

3.       The area of the triangle formed by line  x/a  +  y/b  =  1 with the co ordinate axis is…………. ?

4.       The area of the triangle formed by the line x = 3, y = 4 and x = y is ………

5.       If am ≠ bl then the system of equations ax + by = c and    lx + my = n has a unique solution. ( T / F)

Two mark Questions

1.       Find the value of k so that the pair of equation x = 2y = 5 and 3x + ky + 15 = 0 has a unique solutions.    {CBSE 2019}

2.       The larger of two supplementary angles includes the smaller by 18⁰, Find the angles.            {CBSE 2019}

3.       Sumit is 3 times as old as his son. Five years later he shall be two and half times as old as his son. How old is Sumit at present.                                 {CBSE 2019}

4.       If the system of equations 6x + 2y = 3 and kx + y = 2 has a unique solution, find the value of K.

Three mark Questions

1.       Solve graphically 2x – 3y + 13 = 0,  3x – 2y + 12 = 0                      {CBSE 2020}

2.       If 2x + y = 23 and 4x – y = 19, find the value of (5y – 2x) and y/x - 2.                            {CBSE 2020}

3.       Solve using Cross Multiplication method : 2x + y = 5,             3x + 2y = 8

4.       For what value of m and n the following system of linear equations has infinitely many solutions: 3x + 4y = 12,              (m+n)x + 2(m – n)y = 5m – 1

5.       Draw the graph for x – y = -1 and 3x + 2y – 12 = 0. Also shade the region bounded by these lines and x axis.

6.       If 3 times the larger of the two numbers is divided by the smaller one, we get 4 as quotient and 3 as the remainder. Also if 7 times the smaller number is divided by the larger one, we get 5 as quotient and 1 as remainder. Find the numbers.

7.       If x + 1 is a factor of 2x³ + ax² + 2bx + 1, then find the value of a and b given that 2a – 3b = 4.

8.       A lending library has a fixed charge for the first three days and the additional charge for each day there after Sunita paid Rs. 27 for a book kept for seven days, while Raveena paid Rs. 21 for the book she kept for five days.  Find the fixed charge and the charge for add extra day.

Four mark Questions

1.       It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. How long would it take for each pipe to fill the pool separately?                {CBSE 2020}

2.       2 men and 7 boys can do a piece of work in 4 days. It is done by 4 men and 4 boys in 3 days. How long would it take for one man or one boy to do it?

3.       Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for ach wrong answer. Had 4 marks been awarded for each correct answer and 2 marks deducted for each wrong answer. Yash would have scored 50 marks. How many questions were there in test?

4.       Ravi divided to distribute Some amount to poor Students for their books. If there are 8 students less, every one will get Rs. 10 more. If there are 16 students more, every one will get Rs. 10 less. What is the number of students and how much does each get? What is the total amount distributed?

5.       Place A and B are 100 km apart on a highway. One car starts from A and another car starts From B at the same time. If the car travel in the same direction at different speeds they meet in 5 hours. If they travel towards each other they meet in 1 hour. What are the speeds of the two cars?

6.       The area of rectangle gets reduced by 9 square units if the length is reduced by 5 units and the breadth is increased by 3 units. If we increase the length by 3 units and breadth by 2 units the area is increased by 67 square units. Find the length and breadth of the rectangle.

7.       A boat covers 32 km upstream and 36 km downstream in 7 hours. Also it covers 40 km upstream and 48 km downstream in 9 hours find the speed of the boat in still water and that of the stream.

 

Previous Year Maths Question of Chapter Polynomials (Class X)

 

Previous Year Mathematics Questions of Class 10 Chapter {Polynomials} for Examination 2023

One mark Questions

1.       If α and β are the zeroes of the quadratic polynomial 2x² – 13x + 6, then α + β= ………………                (CBSE 2020)

2.       The maximum number of zeroes a cubic polynomial can have is ……………                                        (CBSE 2020)

3.       The Zeroes of the polynomial x² – 3x – m(m + 3) are ……………….                                                      (CBSE 2020)

4.       If one zero of the quadratic polynomial x² – 5x – 6 is 6 then find the other zero.

5.       Find the zeroes of polynomial √3 x² – 8x + 4√3

6.       If the graph of a polynomial intersects x – axis at exactly two points is it necessarily a quadratic polynomials.

7.       Find the quadratic polynomial whose zeroes are -3 and -5.

Two mark Questions

1.       A teacher asked 10 of his students to write a polynomial in one variable on a paper and then to handover the paper. The following were answers given by the students: 2x + 3,   3x² + 7x + 2 ,  4x³ + 3x² + 2 , 5x³ – 7x + 2,  2x² + 3 ,  5x – ½ ,  ax³ + bx² + cx + d ,   

Answer the following questions: 

(i) How many of the above ten, are not polynomials?                     (CBSE 2020)

(ii) How many of the above ten, are quadratic polynomials?

2.       Divide (2x² – x + 3) by (2 – x) and write the quotient and the remainder.                                           (CBSE 2020)

3.     If  α and β are the zeroes of the quadratic polynomial f(x) = 5x² - 7x + 1, find the value of  α/β + α/β  .             (CBSE 2020)

4.       If one of the zeroes of the quadratic polynomial p(x) = 4x² – 8kx – 9 is equal in magnitude but opposite in sign. Of the other, find the value of K.

5.       If one root of polynomial p(x) = 5x² + 13x + m is reciprocal of other, then find the value of m.

6.       For what value of h is the polynomial f(x) = 2x³ – hx² + 5x + 9, exactly divisibility by (x + 2).

Three mark Questions

1.       Find the quadratic polynomial, the sum of product of whose zeroes are -3 and 2 respectively. Hence find the zeroes.          (CBSE 2020)

2.       For what value of k is the polynomial f(x) = 3x⁴ – 9x³ + x² + 15x + k completely divisible by 3x² – 5?            (CBSE 2019)

3.       Find the zeroes of the quadratic polynomial 7y² – 11/3 y – 2/3 and verify th relationship between the zeroes and the coefficients.            (CBSE 2019)

4.       What must be added to the polynomial P(x) = x⁴ + 2x³ – 2x² + x – 1 so that the resulting polynomial is exactly divisible by x² + 2x – 3 ?

5.       If the polynomial 6x⁴ + 8x³ + 17x² + 21x + 7 is divided by another polynomial 3x² + 4x + 1, the remainder comes out to be ax + b, find a and b.

Four mark Questions

1.       Find the zeroes of the polynomial f(x) = x³ – 5x² – 2x + 24 if it is given that the product of its two zeroes is 12.

2.       If x² + x – 12 divides p(x) = x³ + ax² + bx – 84 exactly, find a and b.

 

Previous Year Maths Question of Chapter Real number (Class X)

 

Previous Year Mathematics Questions of Class 10 Chapter {Real Number} for Examination 2023

One mark Questions

     1.  The sum of exponents of prime factors in the prime factorization of 196 is             …......                                 (CBSE 2020)
2. Euclid’ division lemma states that for 2 positive integer a and b, there exists         unique integer q and r satisfying a = bq + r and ……………       (CBSE 2020)
3. The decimal expansion of 23/2⁵ x 5² will terminate after how many places of        decimal?                             (CBSE 2020)
4. If H.C.F of two numbers is 27 and their L.C.M is 162. If one of the number of         54 then find the other number. (CBSE 2020)
5.  2√3 is …………                              (CBSE 2020)
6. If H.C.F (336, 54) = 6, find L.C.M (336, 54).                   (CBSE 2019)
7.  What is the H.C.F of smallest prime number and the Smallest composite                 number.                                     (CBSE 2018)
8.  Explain why 7 x 11 x 13 + 13 is a composite number.
9.  Without actually performing the long division write 15/1600 in decimal form.
10.   Express 5050 as product of its prime factors, is it unique?
11.   Three measuring rods are 64 cm, 80 cm, and 96 cm in length, the least length       of cloth that can be measured an exact number of times, using any of the rods.

Two mark Questions

1.       Write the smallest number which is divisible by both 306 and 657.

2.       Given that √2 is an irrational. Prove that 5 + 3√2 is an irrational number.

3.       Prove that 1/√2 is an irrational number.

4.       Show that 5 - 3√2 is irrational.

5.       Show that any positive integer is of the form 3q or 3q + 1 or 3q + 2 for some         integer q.

7.       Using Euclid’s division algorithm, find the largest number that divides 1251,             9377 and 15628 leaving remainders 1, 2 and 3 respectively.

8.       Find the greatest number of 6 digit exactly divisible by 24, 15 and 36.

9.       Prove that √3 + √7 is irrational.

Three/ Four mark Questions

1.       Prove the √3 is an irrational number.                               (CBSE 2020)

2.       Prove the √5 is an irrational number.                               (CBSE 2020)

3.       Using Euclid algorithm, find the HCF of 271 and 1032. (CBSE 2020)

4.       Prove that 2 + 5√3 is an irrational numbers, given that √3 is an irrational                      number.                                                                                 (CBSE 2019)

5.       Using Euclid’s Algorithm, find the H.C.F of 2048 and 960.   (CBSE 2018)

6.       Find H.C.F and L.C.M of 404 and 96 and verify that H.C.F X L.C.M = Product             of the two given numbers.

7.     Prove that the square of any positive integer of the form 5q + 1 is of the same                 form.

8.      If d is the H.C.F of 56 and 72, find x, y satisfying d = 56x + 72y. Also show that x             and y are not Unique.

9.   In the morning walk three Persons step off together, their steps measure 80 cm, 85             cm and 90 cm respectively. What is the minimum distance each should walk, So             that he can cover the distance in complete step?