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
we know that
(i) Deg P(x) = deg g (x)
The degree of dividend or quotient can be equal, only if the divisor is a constant (degree 0)
Then, let p(x) = 3x2 – 6x + 5
Let g(x) = 3
Therefore, q(x) = x2 – 2x +1 and r(x) = 2
(ii) Deg q(x) = deg r(x)
Let p(x) = x2 + 1
Let g(x) = x
Therefore, q(x) = x + 1 and r(x) = 0
Here, we can see the degree of quotient is equal to the degree of remainder.
Hence, division algorithm is satisfied here.
(iii) deg r(x) = 0
The degree of remainder is zero, only if the remainder left after division algorithm is Constant.
Let p(x) = x2 + 1
Let g(x) = x
Therefore, q(x)= x and r(x) = 1
Here we can see the degree of remainder is zero.
Hence division algorithm is satisfied here.
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
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Find two consecutive positive integers, sum of whose squares is 365.
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.
Refer to Example 13. (i) Complete the following table:
(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability
Do you agree with this argument? Justify your answer.
Find two numbers whose sum is 27 and product is 182.
Prove that 3 + 2√5 is irrational.
Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
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
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to
(A) 60° (B) 70°
(C) 80° (D) 90°
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.
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 a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?
(ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective ?
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?
Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
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.
If P(E) = 0.05, what is the probability of ‘not E’?