CBSE Class 10 Mathematics Polynomials Worksheet Set 08

Short Answer Type Questions

Question. Divide \( (6 + 19x + x^2 - 6x^3) \) by \( (2 + 5x - 3x^2) \) and verify the division algorithm.
Answer: \( Q = 2x + 3, R = 0 \)

Question. If \( \alpha, \beta \) are zeroes of the polynomial \( x^2 - 2x - 8 \), then form a quadratic polynomial whose zeroes are \( 2\alpha \) and \( 2\beta \).
Answer: \( x^2 - 4x - 32 \)

Question. If \( \alpha, \beta \) are the zeroes of the polynomial \( x^2 - 2x - 8 \) then form a quadratic polynomial whose zeroes are \( 3\alpha \) and \( 3\beta \).
Answer: \( x^2 - 6x - 72 \)

Question. If one solution of the equation \( 3x^2 - 8x + 2k + 1 \) is seven times the other. Find the solutions and the value of k.
Answer: \( x = \frac{1}{3} \) and \( \frac{7}{3}, k = \frac{2}{3} \)

Long Answer Type Questions

Question. What must be added to the polynomial \( f(x) = x^4 + 2x^3 - 2x^2 + x - 1 \) so that the resulting polynomial is exactly divisible by \( x^2 + 2x - 3 \)?
Answer: \( -x + 2 \)

Question. Find the other zeroes of the polynomial \( 2x^4 - 3x^3 - 3x^2 + 6x - 2 \) if \( -\sqrt{2} \) and \( \sqrt{2} \) are the zeroes of the given polynomial.
Answer: Zeros are 1 and \( \frac{1}{2} \)

Question. If the remainder on division of \( x^3 + 2x^2 + kx + 3 \) by \( x - 3 \) is 21, find the quotient and the value of \( k \). Hence, find the zeroes of the cubic polynomial \( x^3 + 2x^2 + kx - 18 \).
Answer: \( k = -9 \), factors \( (x^2 + 5x + 6); -2, -3 \)

Question. Find all other zeroes of the polynomial \( p(x) = 2x^3 + 3x^2 - 11x - 6 \), if one of its zero is -3.
Answer: \( x = -\frac{1}{2}, x = 2 \)

Question. Divide \( 30x^4 + 11x^3 - 82x^2 - 12x + 48 \) by \( (3x^2 + 2x - 4) \) and verify the result by division algorithm.
Answer: \( Q = 10x^2 - 3x - 12, R = 0 \)

Question. Find all the zeroes of the polynomial \( x^4 - 5x^3 + 2x^2 + 10x - 8 \), if two of its zeroes are \( \sqrt{2}, -\sqrt{2} \).
Answer: zeroes are 1, 4, \( \sqrt{2}, -\sqrt{2} \)

Question. Find all the zeroes of the polynomial \( x^4 + x^3 - 9x^2 - 3x + 18 \), if two of its zeroes are \( \sqrt{3}, -\sqrt{3} \).
Answer: \( -\sqrt{3}, \sqrt{3}, -3, 2 \)

Question. Obtain all the zeroes of \( x^4 - 7x^3 + 17x^2 - 17x + 6 \) if two of its zeroes are 1 and 2.
Answer: zeroes are 1, 1, 2, 3

Question. Find all the zeroes of the polynomial \( 2x^4 - 10x^3 + 5x^2 + 15x - 12 \), if it is given that two of its zeroes are \( \sqrt{\frac{3}{2}} \) and \( -\sqrt{\frac{3}{2}} \).
Answer: zeroes are 4, 1, \( \sqrt{\frac{3}{2}}, -\sqrt{\frac{3}{2}} \)

Question. Find all zeroes of polynomial \( 4x^4 - 20x^3 + 23x^2 + 5x - 6 \), if two of its zeroes are 2 and 3.
Answer: 2, 3, \( \frac{1}{2}, -\frac{1}{2} \)

VALUE BASED QUESTIONS

Question. On the occasion of Diwali Rashmi want to donate blankets to poor people. She purchased two blankets more than the number of poor ones. If cost of each blanket is Rs. 2 less than the number of poor ones. Find the amount she paid for blankets and what values are reflected from the questions?
Answer: There are \( n \) poors. So number of blankets she purchased \( = n + 2 \).
Amount paid for one blanket \( (n - 2) \) so net amount she paid \( = (n + 2)(n - 2) = n^2 - 4 \).
Values:

• On such great occasions in place of buying crackers which pollute our atmosphere we should help poor persons.
• By donating blankets on Diwali we can save some lives from cold because immediately after Diwali winter season starts.
• We should respect poor persons.
• It is indeed our duty to help needy ones.

Question. Aayush went to school by cycle at the speed of x km/hr and cover the distance in (x - 2) hrs. His brother Anuj goes on foot with the speed of (x - 2) km/hr and takes x hrs to reach school. Represent the distance covered by both of them in the form of polynomials. Express their values to reach to school.
Answer: Distance covered by both of them is a polynomial \( x \times (x - 2) \) or \( (x - 2) \times x \)

\( \implies \) \( x^2 - 2x \).
Values:

• We should use cycle or go on foot for good health.
• By avoiding scooters etc. we save energy and save environment from pollution.

Question. Number of members in 3 distinct families is (x + 2), (x + 3) and (x - 2). Find the number of member in a joint family having 8 members more then product of members in above three families. Values which reflect by the problem.
Answer: Number of members in joint family \( = (x + 2) \times (x + 3) \times (x - 2) + 8 = x^3 + 3x^2 - 4x - 4 \)
Values:

• Living in joint family is the rich culture of our country.
• In joint family we feel secure, more caring about each other and live without fear.

Question. Product of timings of 3 major activities for one student out of 24 hrs is \( x^3 + 12x^2 + 44x + 48 \) and product of two of its activities is \( x^2 + 6x + 8 \). If quotient of these two is the time of study. Find number of hours for a student to study. What values that reflect, if least time is given to activity playing?
Answer: Quotient \( = \frac{x^3 + 12x^2 + 44x + 48}{x^2 + 6x + 8} = (x + 6) \)
Values:

• Management of time is very necessary at every stage.
• We have got timings for activity playing is \( (x + 2) \) because \( x^3 + 12x^2 + 44x + 48 = (x + 2)(x + 4)(x + 6) \) hence to achieve academic target we should play only when it is necessary.

Question. If zeroes of polynomial: \( ax^2 + bx + c \) are reciprocal of each other then \( a = c \) [T/F]
Answer: T

Question. If zeroes of polynomial: \( x^2 - kx + 2 \) are differ by 1 then value of \( k \) is ....... (fill in blank)
Answer: \( k = \pm 3 \)

Question. If sum of zeroes of polynomial: \( ax^2 + bx + c \) is equal to product of zeroes than value of \( b + c \) is .........
Answer: zero

Question. Value of 'b' for which polynomial: \( ax^2 + bx + c \) has its zeroes equal in magnitude but opposite in sign is...........
Answer: \( b = 0 \)

Question. If one of the zeroes of polynomial: \( ax^2 + bx + c \) is zero then value of \( c \) is .........
Answer: \( c = 0 \)

Question. Sum of zeroes of polynomial \( x^2 - 2x + 1 \) is equal to sum of zeroes of polynomial \( x^3 - 2x^2 + x \), then find the product of all 3 zeroes of second polynomial.
Answer: zero

Question. If zeroes of polynomial: \( x^3 - 3x^2 + x + 1 \) are \( a - b, a, a + b \), then find value of 'a'.
Answer: \( a = 1 \)

Question. Find a quadratic polynomial whose zeroes are 0 and 1.
Answer: \( x^2 - x \)

Question. If zeroes of polynomial: \( \frac{1}{3}x^2 + (k - 2) + 2x \) are reciprocal of each other. Find value of \( k \).
Answer: \( k = \frac{7}{3} \)

Question. If one of the zeroes of polynomial \( x^2 - kx + 16 \) is cube of other. Find value of \( k \).
Answer: \( k = 10 \)

Question. \( \alpha, \beta, \gamma \) are zeroes of polynomial: \( 2x^3 + 3x^2 + 4x + 8 \) then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} \).
Answer: \( -\frac{1}{2} \)

Question. If the polynomial: \( x^4 + 2x^3 + 8x^2 + 12x + 8 \) is divided by another polynomial \( (x^2 + 5) \), the remainder is \( (ax + b) \). Find \( a - b \).
Answer: \( a = 2, b = -7, a - b = 9 \)

Question. If two zeroes of polynomial: \( x^4 - 6x^3 - 26x^2 + 138x - 35 \) are \( (2 + \sqrt{3}) \) and \( (2 - \sqrt{3}) \) find other zeroes.
Answer: 7 and -5

Question. Find all zeroes of polynomial \( p(x) = x^3 - 5x^2 - 2x + 24 \), if it is given that the product of its two zeroes is 12.
Answer: 3, 4 and -2

Question. If \( \alpha \) and \( \beta \) are zeroes of polynomial \( p(x) = 3x^2 - 4x + 1 \), find a cubic polynomial whose zeroes are \( 0, \frac{\alpha^2}{\beta} \) and \( \frac{\beta^2}{\alpha} \).
Answer: \( x^3 - \frac{28}{9}x^2 + \frac{x}{3} \)