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AlgebraQuestion and Answers: Page 336

Question Number 32977 Answers: 1 Comments: 0

Prove that ^n C_r +^n C_(r+1) =^(n+1) C_(r+1)

$${Prove}\:{that}\:\:^{{n}} {C}_{{r}} \:\:+\:^{{n}} {C}_{{r}+\mathrm{1}} \:=\:^{{n}+\mathrm{1}} {C}_{{r}+\mathrm{1}} \\ $$$$ \\ $$

Question Number 32889 Answers: 0 Comments: 1

Question Number 32868 Answers: 0 Comments: 0

Question Number 32832 Answers: 0 Comments: 0

Determine n such that 1001n+1 is perfect cube.

$$\mathrm{Determine}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{1001n}+\mathrm{1}\:\mathrm{is} \\ $$$$\mathrm{perfect}\:\mathrm{cube}. \\ $$

Question Number 32788 Answers: 0 Comments: 1

The least positive integral value of ′x′ satisfying : (e^x −2)(sin (x+(π/4)))(x−log_e 2_ )(sinx − cosx)<0

$$\boldsymbol{{T}}{he}\:{least}\:{positive}\:{integral}\:{value}\:{of} \\ $$$$'{x}'\:{satisfying}\:: \\ $$$$\left({e}^{{x}} −\mathrm{2}\right)\left(\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)\right)\left({x}−\mathrm{log}_{{e}} \:\underset{} {\mathrm{2}}\right)\left({sinx}\:−\:{cosx}\right)<\mathrm{0} \\ $$

Question Number 32780 Answers: 2 Comments: 0

10−8=(p/9)

$$\mathrm{10}−\mathrm{8}=\frac{{p}}{\mathrm{9}} \\ $$

Question Number 32768 Answers: 1 Comments: 0

Prove that a^2 +b^2 +c^2 ≥ab+bc+ca ∀ a,b,c∈R

$$\mathrm{Prove}\:\mathrm{that}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \geqslant{ab}+{bc}+{ca} \\ $$$$\forall\:{a},{b},{c}\in\mathbb{R} \\ $$

Question Number 32767 Answers: 0 Comments: 2

For a,b,c≥0 if a+b+c=n, determine minimum and maximum values of a^2 +b^2 +c^2 −ab−bc−ca.

$$\mathrm{For}\:{a},{b},{c}\geqslant\mathrm{0}\:\mathrm{if}\:{a}+{b}+{c}={n},\:\mathrm{determine} \\ $$$$\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{values}\:\mathrm{of} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}. \\ $$

Question Number 32682 Answers: 1 Comments: 0

If x_1 and x_(2 ) are roots of the equation acos 2x+bsin x = c and 2sin x_1 sinx_2 = sin x_1 +sinx_2 . Then the value of (b/(c−a)) is ?

$$\boldsymbol{{I}}{f}\:{x}_{\mathrm{1}} \:{and}\:{x}_{\mathrm{2}\:} \:{are}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${acos}\:\mathrm{2}{x}+{bsin}\:{x}\:=\:{c}\:{and}\: \\ $$$$\mathrm{2}{sin}\:{x}_{\mathrm{1}} {sinx}_{\mathrm{2}} =\:{sin}\:{x}_{\mathrm{1}} +{sinx}_{\mathrm{2}} .\:\boldsymbol{{T}}{hen}\: \\ $$$${the}\:{value}\:{of}\:\:\frac{{b}}{{c}−{a}}\:{is}\:? \\ $$

Question Number 32681 Answers: 0 Comments: 2

Total no. of polynomials of the form x^3 +ax^2 +bx+c that are divisible by x^2 +1, where a,b,c∈1,2,3,....,10 is 1) 10 2) 15 3) 5 4) 8

$$\boldsymbol{{T}}{otal}\:{no}.\:{of}\:{polynomials}\:{of}\:{the}\:{form} \\ $$$${x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c}\:\:{that}\:{are}\:{divisible}\:{by}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{1},\:{where}\:{a},{b},{c}\in\mathrm{1},\mathrm{2},\mathrm{3},....,\mathrm{10}\:{is}\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{10} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{15} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{5} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{8} \\ $$

Question Number 32650 Answers: 2 Comments: 0

f(x)=8x−34(√(25−4 (3/2)))

$${f}\left({x}\right)=\mathrm{8}{x}−\mathrm{34}\sqrt{\mathrm{25}−\mathrm{4}\:\frac{\mathrm{3}}{\mathrm{2}}} \\ $$

Question Number 32648 Answers: 1 Comments: 0

Question Number 32647 Answers: 1 Comments: 0

Question Number 32632 Answers: 0 Comments: 2

Question Number 32629 Answers: 1 Comments: 0

If x^2 +y^2 =9 , 4a^2 +9b^2 =16, then maximum value of 4a^2 x^2 +9b^2 y^2 −12abxy is ?

$$\boldsymbol{{I}}{f}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{9}\:,\:\mathrm{4}{a}^{\mathrm{2}} +\mathrm{9}{b}^{\mathrm{2}} =\mathrm{16}, \\ $$$${then}\:\boldsymbol{{maximum}}\:{value}\:{of}\: \\ $$$$\mathrm{4}{a}^{\mathrm{2}} {x}^{\mathrm{2}} +\mathrm{9}{b}^{\mathrm{2}} {y}^{\mathrm{2}} −\mathrm{12}{abxy}\:{is}\:? \\ $$

Question Number 32563 Answers: 2 Comments: 2

The set of values of ′a′ for which all the solutions of the equation 4sin^4 x+asin^2 x+3=0 are real and distinct is ?

$$\boldsymbol{{T}}{he}\:{set}\:{of}\:{values}\:{of}\:'{a}'\:{for}\:{which}\: \\ $$$${all}\:{the}\:{solutions}\:{of}\:{the}\:{equation} \\ $$$$\mathrm{4sin}\:^{\mathrm{4}} {x}+{a}\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{3}=\mathrm{0}\:{are}\:{real}\:{and}\: \\ $$$${distinct}\:{is}\:? \\ $$

Question Number 32584 Answers: 1 Comments: 0

Question Number 32543 Answers: 1 Comments: 0

The coefficient of x^4 in the expansion of (1+5x+9x^2 +.....∞)(1+x^2 )^(11) is a) 171 b) 172 c) 173 d) 176

$$\boldsymbol{{T}}{he}\:{coefficient}\:{of}\:{x}^{\mathrm{4}} \:{in}\:{the}\:{expansion} \\ $$$${of}\:\left(\mathrm{1}+\mathrm{5}{x}+\mathrm{9}{x}^{\mathrm{2}} +.....\infty\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{11}} {is} \\ $$$$\left.{a}\right)\:\mathrm{171} \\ $$$$\left.{b}\right)\:\mathrm{172} \\ $$$$\left.{c}\right)\:\mathrm{173} \\ $$$$\left.{d}\right)\:\mathrm{176} \\ $$

Question Number 32541 Answers: 2 Comments: 0

Coefficient of x^5 in the expansion of (x^2 −x−2)^5 is

$${Coefficient}\:{of}\:{x}^{\mathrm{5}} \:{in}\:{the}\:{expansion} \\ $$$${of}\:\left({x}^{\mathrm{2}} −{x}−\mathrm{2}\right)^{\mathrm{5}} \:{is} \\ $$

Question Number 32532 Answers: 0 Comments: 3

If a,b,c are 3 positive numbers in an A.P and T= ((a+8b)/(2b−a))+((8b+c)/(2b−c)). Then the value of T^( 2 ) is ? Ans. given is 361.

$$\boldsymbol{{I}}{f}\:{a},{b},{c}\:{are}\:\mathrm{3}\:{positive}\:{numbers}\:{in}\:{an} \\ $$$$\boldsymbol{{A}}.\boldsymbol{{P}}\:{and}\: \\ $$$${T}=\:\frac{{a}+\mathrm{8}{b}}{\mathrm{2}{b}−{a}}+\frac{\mathrm{8}{b}+{c}}{\mathrm{2}{b}−{c}}. \\ $$$${Then}\:{the}\:{value}\:{of}\:{T}^{\:\:\mathrm{2}\:} \:{is}\:? \\ $$$${Ans}.\:{given}\:{is}\:\mathrm{361}. \\ $$

Question Number 32494 Answers: 0 Comments: 0

Question Number 32446 Answers: 1 Comments: 0

Question Number 32441 Answers: 1 Comments: 0

Question Number 32524 Answers: 0 Comments: 0

(1/2)x^2 +(√2)=

$$\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} +\sqrt{\mathrm{2}}= \\ $$

Question Number 32424 Answers: 1 Comments: 0

Question Number 32395 Answers: 0 Comments: 1

1+1

$$\mathrm{1}+\mathrm{1} \\ $$

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