Class 10 Mathematics - Chapter Polynomials NCERT Solutions | Find a cubic polynomial with the sum, su

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x² – 2x – 8 (ii) 4s² – 4s + 1 (iii) 6x² – 3 – 7x (iv) 4u² + 8u (v) t² – 15 (vi) 3x² – x – 4

The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12

(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

(iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :
(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2    (ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x      (iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²

On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 nd –2x + 4, respectively. Find g(x).

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

(i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0

Obtain all other zeroes of 3x⁴ + 6x³ – 2x² – 10x – 5, if two of its zeroes are

Use Euclid’s division algorithm to find the HCF of :
   (i) 135 and 225      (ii) 196 and 38220      (iii) 867 and 255

A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).

Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

Complete the following statements:
(i) Probability of an event E + Probability of the event ‘not E’ =                  .
(ii) The probability of an event that cannot happen is                  . Such an event is called                       .
(iii) The probability of an event that is certain to happen is                  . Such an event is called                  .
(iv) The sum of the probabilities of all the elementary events of an experiment is                   .

(v) The probability of an event is greater than or equal to                   and less than or equal to                  .

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

How many tangents can a circle have?

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

The coach of a cricket team buys 3 bats and 6 balls for ` 3900. Later, she buys another bat and 3 more balls of the same kind for ` 1300. Represent this situation algebraically and geometrically.

Which of the following experiments have equally likely outcomes? Explain.
(i) A driver attempts to start a car. The car starts or does not start.
(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
(iii) A trial is made to answer a true-false question. The answer is right or wrong.
(iv) A baby is born. It is a boy or a girl.

Suppose you drop a die at random on the rectangular region shown in Fig. 15.6. What is the probability that it will land inside the circle with diameter 1m?

Find the LCM and HCF of the following integers by applying the prime factorisation method.
   (i) 12, 15 and 21     (ii) 17, 23 and 29     (iii) 8, 9 and 25

Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?

Complete the following statements:
(i) Probability of an event E + Probability of the event ‘not E’ =                  .
(ii) The probability of an event that cannot happen is                  . Such an event is called                       .
(iii) The probability of an event that is certain to happen is                  . Such an event is called                  .
(iv) The sum of the probabilities of all the elementary events of an experiment is                   .

(v) The probability of an event is greater than or equal to                   and less than or equal to                  .

Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x² + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x² – 3x + 5 = 0  (iii) 2x²– 6x + 3 = 0

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
   (i) 26 and 91     (ii) 510 and 92     (iii) 336 and 54

Solve the following pair of linear equations by the substitution method.

Two water taps together can fill a tank in  hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Find the roots of the following quadratic equations by factorisation: