### Dictionary Rank of a Word | Permutations & Combinations

PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

### Math Assignment Ch-2 Class X | Polynomials

Mathematics Assignment
Class 10, Chapter 2, Quadratic Polynomials
Important and useful questions on quadratic polynomial, Extra questions based on quadratic polynomial class 10 for the examination point of view. A complete and useful assignment on quadratic polynomial.

## LEVEL - 1

Q1- Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.

(a) 4x² - 4x + 1   [Ans; 1/2],

(b)  x² - 2$\sqrt{2}$x - 6     [Ans; 3$\sqrt{2}$, -$\sqrt{2}$],

(c)  x² - 8x + 12   [Ans; 6, 2]

(d) 2x² + 5x + 2    [Ans  -2, -1/2]

$e)\: \: \sqrt{3}x^{2}-8x+4\sqrt{3}\: \: \: \left [ Ans:\: \frac{2}{\sqrt{3}},\: 2\sqrt{3} \right ]$

$f)\: \: 4\sqrt{3}x^{2}+4\sqrt{3}x-3\sqrt{3}\: \: \: \left [ Ans:\: \frac{-3}{2},\: \frac{1}{2} \right ]$

(g)   x² - 8           [Ans; ± 2$\sqrt{2}$],

(h) 9y² - 6y + 1    [Ans; 1/3, 1/3],

(i)  3x² - 5x - 2      [Ans; 2, -1/3]

$j)\: \: 6x^{2}+17x+12\: \: \: \left [ Ans:\: \frac{-4}{3},\: \frac{-3}{2} \right ]$

$k)\: \: 24x^{2}-41x+12\: \: \: \left [ Ans:\: \frac{4}{3},\: \frac{3}{8} \right ]$
$l)\: \: 15x^{2}-x-28\: \: \: \left [ Ans:\: \frac{7}{5},\: \frac{-4}{3} \right ]$
$m)\: \: 5\sqrt{5}x^{2}+20x+3\sqrt{5}\: \: \: \left [ Ans:\: \frac{-3}{\sqrt{5}},\: \frac{-\sqrt{5}}{5} \right ]$

$n)\: \: 6\sqrt{3}x^{2}-47x+5\sqrt{3}\: \: \: \left [ Ans:\: \frac{5\sqrt{3}}{2},\: \frac{1}{3\sqrt{3}} \right ]$

Q2- Find sum and product of zeroes of

(a) 2x² + 2x + 3        [Ans; -1, 3/2],

(b) x² - 7x - 7            [Ans; 7, -7]

c)  $4\sqrt{3}x^{2}+7x-2\sqrt{3}\: \: \: \left [ Ans:\: \frac{-7}{4\sqrt{3}},\: \frac{-1}{2} \right ]$

Q3- Find a quadratic polynomial whose sum and product of zeros are

(a)    $\frac{1}{3}\: and\: \frac{1}{2}$       [Ans; 6x² - 2x + 3]

(b)  $0\: and\: \sqrt{3}\: \: \: \: \: [Ans:\: x^{2}+\sqrt{3}]$

(c)   5 ± $\sqrt{2}$         [x² - 10x + 23]

(d)  $-\sqrt{2}\: and\: \sqrt{3}$     [ Ans; x² + $\sqrt{2}$ x + $\sqrt{3}$ ]

(e)   $0\: and\: \frac{-\sqrt{7}}{2}$       $[Ans:\: 2x^{2}-\sqrt{7}]$

(f)     $\frac{5}{7}\: and\: \frac{10}{21}$        $[Ans:\: 21x^{2}-15x+10]$

Q4- Find quotient and remainder on dividing P(x) and g(x)

(a)  P(x) = x⁴ + x² + 1; g(x) = x² + x + 1

Ans; q(x) = x² - x + 1

(b)    P(y) = 4y⁴ - 10y³ - 10y² + 30y - 15; g(y) = 2y - 5

Ans; q(y) = 2x3 - 5x + 5/2

(c)   P(x) = 2x⁴ + 8x³ + 7x² + 4x + 5; g(x) = x + 3

Ans; q(x) = 2x³ + 2x² + x + 1

Q5- If x + a is the factor of 2x2 + 2ax + 5x + 10, then find a

[Ans a = 2]

Q6) For what value of k,  -4 is a zero of the polynomial  x2 – x - (2k + 2) ?

[Ans k = 9]

Q7) - Find a if 2x - 3 is a factor of 6x³ - x² - 10x + a.

[Ans; a = - 3]

Q8)  Find a and b so that x⁴ + x³ + 8x² + ax - b is divisible by x² + 1.

[Ans; a = 1, b= - 7]

Q9)  For what value of k is -3 a zero of x² + 11x + k.

Q10)  If 1 is the zero of ax² - 3(a - 1)x - 1 then find a.

Ans; a = 1

## LEVEL - 2

Q11)  Is (x + 2) a factor of 2x² + 3x + 1.

Q 12)  If α, β are the zeroes of P(x) = ax² + bx + c then find

a)   $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$        $\left [ Ans:\: \frac{b^{2}-2ac}{ac} \right ]$

b)  $\alpha ^{2}+\beta ^{2}$       $\left [ Ans:\: \frac{b^{2}-2ac}{a^{2}} \right ]$

c)   $\alpha ^{3}+\beta ^{3}$       $\left [ Ans:\: \frac{3abc-b^{3}}{a^{3}} \right ]$

d)   $\frac{1}{\alpha ^{3} }+\frac{1 }{\beta ^{3} }$      $\left [ Ans:\: \frac{3abc-b^{3}}{c^{3}} \right ]$

e)   $\frac{\alpha ^{2}}{\beta }+\frac{\beta ^{2}}{\alpha }$      $\left [ Ans:\: \frac{3abc-b^{3}}{a^{2}c} \right ]$

Q13- Verify that 1, 1/2, -2 are the zeroes of P(x) = 2x³ + x² - 5x + 2.

Q14- If the zeroes of x2 – Kx + 6 are in the ratio 3 : 2, find K

Q15. What is the degree of the polynomial (x + 1)(x2 – x - x4 + 1)

Q16- Find all zeroes of 2x³ + x² - 6x - 3 if its 2 zeroes are  ± $\sqrt{3}$

Q17- Find all the zeroes of  2x³ + x² - 5x + 2 if one of zero is 1/2.

Ans; 1, - 2

Q18- If -1 is one of the zero of P(x) = 3x³ - 5x² - 11x - 3. Find other zeroes and verify the relation between  zeroes and coefficients.

[Ans; All zeroes are  -1,  3, -1/3]

Q19) Obtain all zeroes of f(x) = x3 + 13x2 + 32x + 20, if one zero is -2

[Ans: All zeroes are  -10,  -1,  - 2]

Q20)  Find all the zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5 if its two zeroes are  $\pm \sqrt{\frac{5}{3}}$

[Ans:  - 1, - 1]

Q21) Find all zeroes of 2x⁴ - 3x³ - 3x² + 6x - 2 if its two zeroes are   $\pm \sqrt{2}$

[Ans: 1]

Q22) Obtain all zeroes of f(x) = 2x4 - 2x3 - 7x2 + 3x + 6, if its two zeroes are  $\pm \sqrt{\frac{3}{2}}$     [Ans:  2, -1]

Q23) Obtain all zeroes of p(x) =  2x4 + x3 – 14x2 – 19x – 6, if two of zeroes are -2 and -1

[Ans: -1/2, 3,  -2, -1]

Q24) Find all zeroes of P(x) = x⁴ + x³ - 23x² - 3x + 60 if its 2 zeroes are   $\pm \sqrt{3}$

Q25)  Find all zeroes of P(x) = x⁴ + x³ - 34x² - 4x + 120 if its two zeroes are  ± 2,     [Ans; 5, - 6]

Q26)  P(x) = x⁴ - 5x + 6,   g(x) = 2 - x² find q(x) and r(x) if P(x) is divided by g(x).

[Ans: q(x) = - x² - 1,  r(x) = - 5x + 10]

Q27- On dividing 10x⁴ - 6x³ - 40x² + 41x - 5 by g(x) the quotient is 5x - 3 and remainder is 2x + 4, Find g(x).

[Ans: 2x³ - 8x + 3]

Q28- If one zero of the P(x) = 5x² + 13x + k is the reciprocal of the other then the value of k is?

[Ans; 5]

Q29- What must be subtracted from 8x⁴ + 14x³ - 2x² + 7x - 8 so that the resulting polynomial is divisible by  4x² + 3x - 2.

[Ans; 14x - 10]

Q30) What must be subtracted from z⁵ - 9z + 6 so that it is exactly divisible by z² + 3.

[ Ans; 6]

Q31)  What must be added to the polynomial p(x) = x4 + 2x3 - 13x2 - 12x + 21 so that the resulting polynomial is exactly divisible by x2 - 4x + 3

[Ans:   - 2x + 3]

Q32- What must be added to x⁴ + 2x³ - 2x² - x - 2 so that the resulting polynomial is  divisible by  x² + 2x - 3.

[Ans; 3x - 1]

Q33 If α, β are the roots of P(x) = x² - 6x + k then what is the value of k if 3α + 2β = 20.

[Ans; k = - 16]

Q34- If α, β are the zeroes of P(y) = 2y² + 7y + 5 then find α + β +  αβ.

Q35. A polynomial P(x) is divisible by  (x - 4) and 2 is the zero of P(x), then write the corresponding polynomial.

Q36)  If α, β are the zeroes of P(x) = x- 5x + 4, then find the value of

a)   $\frac{1}{\alpha }+\frac{1}{\beta }$     $\left [Ans:\: \frac{-27}{4} \right ]$

b)   $\alpha ^{4}\beta ^{3}+\alpha ^{3}\beta ^{4}$     [Ans:  320]

Q37)  If α, β  are the zeroes of P(x) = x2 - 1, then find a quadratic polynomial whose zeroes are $\frac{2\alpha }{\beta }\: and\: \frac{2\beta }{\alpha }$

Ans[x2 + 4x + 4]

Q38.  If α, β are the zeros of the polynomial 25p2 – 15p + 2, then find a quadratic polynomial whose zeroes are   $\frac{1 }{2\alpha }+ \frac{1 }{2\beta }$

Q39) If α, β are the zeroes of  4x2 + 4x + 1, then form a quadratic polynomial whose zeroes are  2α, 2β

Q40) Find k so that x2 + 2x + k is a factor of  2x4 + x3 – 14x2 + 5x + 6

Ans[k = - 3, - 1]

Q41) 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

Ans [a = 1 & b = 2]

Q42) Using division algorithm show that 6y5 + 15y4 + 16y3 + 4y2 + 10y – 35  is divisible by  3y2 + 5

Q43) On dividing the polynomial 9x4 – 4x2 + 5 by 3x2 + x - 1   remainder is ax - b. Find a and b

Q 44) If one zero of  (a2 + 9) x2 + 13x + 6a  is the reciprocal of the other then find the value of a

Q45) If sum of zeroes of ky2 + 2y – 3k  is equal to the twice their product , find the value of k

Q 46) If  α, β  are the zeroes of  P(x) = x2 – x – k  such that α – β  = 9, find the value of k.

## LEVEL - 3

Q 47)  If p and q are the zeroes of 2x2 – 7x + 3 then find the value of p2 + q2

Q 48) A quadratic polynomial 2x2 - 3x + 1 has zeroes as . Now form the quadratic polynomial whose zeroes are  3α and  3β

Q 49) Polynomial x4 + 7x3 + 7x2 + px + q  is exactly divisible by x2 + 7x + 12, then find the value of p and q .

Q 50) If α, β are the zeroes of  x2 + 4x + 3, find the polynomial whose zeroes are  $1+\frac{\beta }{\alpha }$  and  $1+\frac{\alpha }{\beta }$

Q 51) If x - $\sqrt{5}$   is the factor of x3 - 3 $\sqrt{5}$ x2 - 5x + 15$\sqrt{5}$  then find all zeroes of this polynomial.

Q 52) If x3 + 8x2 + kx + 18 is divisible by x2 + 6x + 9 then find the value of k.

Q 53) If 3x4 – 9x3 + x2 + 15x + k is divisible by 3x2 - 5 , find the value of k and other two zeroes  of the polynomial .

Q 54) On dividing x3 - 8x2 + 20x - 10 by g(x)  the quotient and remainder were x - 4 and 6 respectively . Find g(x)

Q 56) Find all zeroes of x4 - 17x2 - 36x - 20, if two of its zeroes are  5 and  -2

Q 57)  Find all zeroes of  x4 – 3  x3 + 3x2 + 3  x - 4 if two of its zeroes are $\sqrt{2}$ and 2$\sqrt{2}$

Q 58) What must be added or subtracted to 8x4 + 14x3 - 2x2 + 8x - 12 so that 4x2 + 3x - 2 is a factor of p(x).

Q 59) For what value of p, - 4 is the zero of P(x) = x2 - 2x - (7p + 3)

Q 60) If x4 + 2x3 + 8x2 + 12x + 18 is divided by another polynomial x2 + 5, the remainder is px + q . Find the value of p and q .

Q 61)  Find a quadratic polynomial whose zeroes are   $\frac{3\pm \sqrt{5}}{5}$

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