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Question Number 18384 Answers: 1 Comments: 0

Let a, b, c ∈ R, a ≠ 0, such that a and 4a + 3b + 2c have the same sign. Show that the equation ax^2 + bx + c = 0 can not have both roots in the interval (1, 2).

$Let a, b, c \in R, a \neq 0, such that a and$ $4 a + 3 b + 2 c have the same sign . Show$ $that the equation {a x}^{2} + b x + c = 0 can$ $not have both roots in the interval$ $(1, 2) .$

Question Number 19200 Answers: 0 Comments: 5

A river of width d is flowing with speed u as shown in the figure. John can swim with maximum speed v relative to the river and can cross it in shortest time T. John starts at A. B is the point directly opposite to A on the other bank of the river. If t be the time John takes to reach the opposite bank, match the situation in the column I to the possibilities in column II. Column I (A) John reaches to the left of B (B) John reaches to the right of B (C) John reaches the point B (D) John drifts along the bank while minimizing the time Column II (p) t = T (q) t > T (r) u < v (s) u > v

$A river of width d is flowing with speed$ $u as shown in the figure . John can swim$ $with maximum speed v relative to the$ $river and can cross it in shortest time$ $T . John starts at A . B is the point$ $directly opposite to A on the other$ $bank of the river . If t be the time John$ $takes to reach the opposite bank, match$ $the situation in the column I to the$ $possibilities in column II .$ $Column I$ $(A) John reaches to the left of B$ $(B) John reaches to the right of B$ $(C) John reaches the point B$ $(D) John drifts along the bank while$ $minimizing the time$ $Column II$ $(p) t = T$ $(q) t > T$ $(r) u < v$ $(s) u > v$

Question Number 18379 Answers: 1 Comments: 0

Question Number 18361 Answers: 0 Comments: 0

N propositions are judged by 2k−1 people. Each person assigns “true” to exactly M propositions and “false” to the other N−M (M ≤ N). To say a proposition is “approved” means it is true according to at least k judges. Find the minimum and maximum numbers of approved propositions given N, M and k.

$N propositions are judged by 2 k - 1 people .$ $Each person assigns ‘ ‘ {true}^{″} to$ $exactly M propositions and ‘ ‘ {false}^{″}$ $to the other N - M (M ⩽ N) .$ $To say a proposition is ‘ ‘ {approved}^{″} means$ $it is true according to at least k judges .$ $Find the minimum and maximum numbers$ $of approved propositions given N, M and k .$

Question Number 18377 Answers: 0 Comments: 0

Suppose one is given two vector field A and B in region of space such that, A(x,y,z) = 4xi + zj + y^2 z^2 k B(x,y,z) = yi +3j − yzk Find: C(x,y,z) if C = A ∧ B Also prove that, C(x,y,z) is perpendicular to A(x,y,z)

$Suppose one is given two vector field A and B in region of space such that,$ $A (x, y, z) = 4 xi + zj + y^{2} z^{2} k$ $B (x, y, z) = yi + 3 j - yzk$ $Find : C (x, y, z) if C = A \land B$ $Also prove that, C (x, y, z) is perpendicular to A (x, y, z)$

Question Number 18721 Answers: 1 Comments: 0

Let ABC and ABC′ be two non- congruent triangles with sides AB = 4, AC = AC′ = 2(√2) and angle B = 30°. The absolute value of the difference between the areas of these triangles is

$Let A B C and {A B C}^{'} be two non -$ $congruent triangles with sides A B = 4,$ $A C = {A C}^{'} = 2 \sqrt{2} and angle B = 30 ° .$ $The absolute value of the difference$ $between the areas of these triangles is$

Question Number 18357 Answers: 0 Comments: 0

From the topic transformer prove that: e = (√2) ε cos(ωt)

$From the topic transformer$ $prove that : e = \sqrt{2} ε \cos (ω t)$

Question Number 18355 Answers: 0 Comments: 0

Question Number 18349 Answers: 0 Comments: 0

Consider the iteration x_(k+1) =x_k −(([f(x)]^2 )/(f(x_k +f(x_k ))−f(x_k ))), k=0,1,2,... for the solution of f(x)=0. Explain the connection with Newton′s method, and show that (x_k ) converges quadratically if x_0 is sufficiently close to the solution.

$C o n s i d e r t h e i t e r a t i o n$ $x_{k + 1} = x_{k} - \frac{{[f (x)]}^{2}}{f (x_{k} + f (x_{k})) - f (x_{k})}, k = 0, 1, 2, . . .$ $f o r t h e s o l u t i o n o f f (x) = 0 . E x p l a i n t h e$ $c o n n e c t i o n w i t h {N e w t o n}^{'} s m e t h o d, a n d s h o w$ $t h a t (x_{k}) c o n v e r g e s q u a d r a t i c a l l y i f x_{0} i s$ $s u f f i c i e n t l y c l o s e t o t h e s o l u t i o n .$

Question Number 18342 Answers: 2 Comments: 0

Question Number 19199 Answers: 1 Comments: 0

y=tan x^(tan x^(tan x) )

$y = \tan x^{\tan x^{\tan x}}$

Question Number 18369 Answers: 1 Comments: 0

Prove that a^4 + b^4 + c^4 ≥ abc(a + b + c)

$Prove that a^{4} + b^{4} + c^{4} ⩾ a b c (a + b + c)$

Question Number 18366 Answers: 1 Comments: 0

Question Number 18365 Answers: 1 Comments: 0

Question Number 18364 Answers: 0 Comments: 0

Question Number 18363 Answers: 0 Comments: 0

Question Number 18362 Answers: 0 Comments: 0

Question Number 18333 Answers: 0 Comments: 1

Question Number 18469 Answers: 1 Comments: 0

Find Z_x and Z_y for each of the functions below (a) Z = 8x^2 y + 14xy^2 + 5y^2 x^3 (b) Z = 4x^3 y^2 + 2x^2 y^3 − 7xy^5

$Find Z_{x} and Z_{y} for each of the functions below$ $(a) Z = {8 x}^{2} y + {14 xy}^{2} + {5 y}^{2} x^{3}$ $(b) Z = {4 x}^{3} y^{2} + {2 x}^{2} y^{3} - {7 xy}^{5}$

Question Number 18327 Answers: 1 Comments: 1

Question Number 18323 Answers: 0 Comments: 0

Σ((cos 2rθ)/(sin^2 2rθ−sin^2 θ))

$Σ \frac{\cos 2 r θ}{\sin^{2} 2 r θ - \sin^{2} θ}$

Question Number 18322 Answers: 1 Comments: 1

The pulley arrangements are identical. The mass of the rope is negligible. In (a), the mass m is lifted up by attaching a mass (2m) to the other end of the rope. In (b), m is lifted up by pulling the other end of the rope with a constant downward force F = 2mg. In which case, the acceleration of m is more?

$The pulley arrangements are identical .$ $The mass of the rope is negligible . In$ $(a), the mass m is lifted up by attaching$ $a mass (2 m) to the other end of the rope .$ $In (b), m is lifted up by pulling the$ $other end of the rope with a constant$ $downward force F = 2 m g . In which$ $case, the acceleration of m is more ?$

Question Number 18318 Answers: 0 Comments: 3

∫ ((x + sinx)/(cosx)) dx

$\int \frac{x + sinx}{cosx} dx$

Question Number 18307 Answers: 1 Comments: 0

Question Number 18306 Answers: 1 Comments: 0

Question Number 18320 Answers: 1 Comments: 0

In a triangle ABC with fixed base BC, the vertex A moves such that cos B + cos C = 4 sin^2 (A/2) . If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C respectively, then (1) b + c = 4a (2) b + c = 2a (3) Locus of point A is an ellipse (4) Locus of point A is a pair of straight lines

$In a triangle A B C with fixed base B C,$ $the vertex A moves such that \cos B +$ $\cos C = 4 \sin^{2} \frac{A}{2} . If a, b and c denote$ $the lengths of the sides of the triangle$ $opposite to the angles A, B and C$ $respectively, then$ $(1) b + c = 4 a$ $(2) b + c = 2 a$ $(3) Locus of point A is an ellipse$ $(4) Locus of point A is a pair of straight$ $lines$

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