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Question Number 19738 Answers: 1 Comments: 1

Locus of the point z satisfying the equation ∣iz − 1∣ + ∣z − i∣ = 2 is

$Locus of the point z satisfying the$ $equation ∣ i z - 1 ∣ + ∣ z - i ∣ = 2 is$

Question Number 19736 Answers: 1 Comments: 1

Question Number 19735 Answers: 1 Comments: 0

If z = λ + 3 + i(√(5 − λ^2 )), then the locus of z is a

$If z = λ + 3 + i \sqrt{5 - λ^{2}}, then the locus$ $of z is a$

Question Number 19733 Answers: 1 Comments: 0

If ∣z + 1∣ = (√2)∣z − 1∣, then the locus described by the point z in the argand diagram is a

$If ∣ z + 1 ∣ = \sqrt{2} ∣ z - 1 ∣, then the locus$ $described by the point z in the argand$ $diagram is a$

Question Number 19734 Answers: 1 Comments: 0

If the imaginary part of ((2z + 1)/(iz + 1)) is −2, then the locus of the point representing z in the complex plane is

$If the imaginary part of \frac{2 z + 1}{i z + 1} is - 2,$ $then the locus of the point representing$ $z in the complex plane is$

Question Number 19732 Answers: 1 Comments: 0

The locus of z given by ∣((z − 1)/(z − i))∣ = 1 is

$The locus of z given by ∣ \frac{z - 1}{z - i} ∣ = 1 is$

Question Number 19730 Answers: 1 Comments: 0

If z = x + iy and ∣z − 2i∣ = 1, then (1) z lies on x-axis (2) z lies on y-axis (3) z lies on a circle (4) None of these

$If z = x + i y and ∣ z - 2 i ∣ = 1, then$ $(1) z lies on x - axis$ $(2) z lies on y - axis$ $(3) z lies on a circle$ $(4) None of these$

Question Number 19729 Answers: 1 Comments: 0

2x + 9y^2 = 4 2x^2 − 45y^2 + xy = 0 Find the value of xy

$2 x + 9 y^{2} = 4$ $2 x^{2} - 45 y^{2} + x y = 0$ $Find the value of x y$

Question Number 19709 Answers: 1 Comments: 0

In the cyclic quadrilateral ABCD AB=7,BC=8,CD=8,DA=15. Calculate the angle ADC and the length ofAC.

$I n t h e c y c l i c q u a d r i l a t e r a l A B C D$ $A B = 7, B C = 8, C D = 8, D A = 15 .$ $C a l c u l a t e t h e a n g l e A D C a n d$ $t h e l e n g t h o f A C .$

Question Number 19704 Answers: 1 Comments: 0

What is the sum (in base 10) of all the natural numbers less than 64 which have exactly three ones in their base 2 representation?

$What is the sum (in base 10) of all the$ $natural numbers less than 64 which$ $have exactly three ones in their base 2$ $representation ?$

Question Number 19783 Answers: 1 Comments: 3

The sides of a triangle are of lengths (√((m^2 −n^2 ))) ,m^2 +n^2 , 2mn. Show that it is a right angle Δ.

$T h e s i d e s o f a t r i a n g l e a r e o f$ $l e n g t h s \sqrt{(m^{2} - n^{2})}, m^{2} + n^{2}, 2 m n .$ $S h o w t h a t i t i s a r i g h t a n g l e Δ .$

Question Number 19700 Answers: 1 Comments: 0

What is the maximum possible value of k for which 2013 can be written as a sum of k consecutive positive integers?

$What is the maximum possible value of$ $k for which 2013 can be written as a$ $sum of k consecutive positive integers ?$

Question Number 19699 Answers: 0 Comments: 0

Let S be a circle with centre O. A chord AB, not a diameter, divides S into two regions R_1 and R_2 such that O belongs to R_2 . Let S_1 be a circle with centre in R_1 , touching AB at X and S internally. Let S_2 be a circle with centre in R_2 , touching AB at Y, the circle S internally and passing through the centre of S. The point X lies on the diameter passing through the centre of S_2 and ∠YXO = 30°. If the radius of S_2 is 100 then what is the radius of S_1 ?

$Let S be a circle with centre O . A chord$ $A B, not a diameter, divides S into two$ $regions R_{1} and R_{2} such that O belongs$ $to R_{2} . Let S_{1} be a circle with centre in$ $R_{1}, touching A B at X and S internally .$ $Let S_{2} be a circle with centre in R_{2},$ $touching A B at Y, the circle S internally$ $and passing through the centre of S .$ $The point X lies on the diameter$ $passing through the centre of S_{2} and$ $∠ Y X O = 30 ° . If the radius of S_{2} is 100$ $then what is the radius of S_{1} ?$

Question Number 19698 Answers: 1 Comments: 0

Let f(x) = x^3 − 3x + b and g(x) = x^2 + bx − 3, where b is a real number. What is the sum of all possible values of b for which the equations f(x) = 0 and g(x) = 0 have a common root?

$Let f (x) = x^{3} - 3 x + b and g (x) = x^{2} +$ $b x - 3, where b is a real number . What$ $is the sum of all possible values of b for$ $which the equations f (x) = 0 and g (x)$ $= 0 have a common root ?$

Question Number 22315 Answers: 0 Comments: 0

Prove that the greatest coefficient in the expansion of (x_1 +x_2 +x_3 +...+x_k )^n = ((n!)/((q!)^(k−r) [(q+1)!]^r )) , where n = qk + r, 0 ≤ r ≤ k − 1

$Prove that the greatest coefficient in$ $the expansion of {(x_{1} + x_{2} + x_{3} + . . . + x_{k})}^{n}$ $= \frac{n!}{{(q!)}^{k - r} {[(q + 1)!]}^{r}}, where n = q k + r,$ $0 ⩽ r ⩽ k - 1$

Question Number 19696 Answers: 1 Comments: 0

Let m be the smallest odd positive integer for which 1 + 2 + ... + m is a square of an integer and let n be the smallest even positive integer for which 1 + 2 + ... + n is a square of an integer. What is the value of m + n?

$Let m be the smallest odd positive$ $integer for which 1 + 2 + . . . + m is a$ $square of an integer and let n be the$ $smallest even positive integer for$ $which 1 + 2 + . . . + n is a square of an$ $integer . What is the value of m + n ?$

Question Number 22313 Answers: 1 Comments: 2

Question Number 19690 Answers: 1 Comments: 0

If ∣z − (4/z)∣ = 2, then find the maximum value of ∣z∣.

$If ∣ z - \frac{4}{z} ∣ = 2, then find the maximum$ $value of ∣ z ∣ .$

Question Number 19786 Answers: 1 Comments: 0

For natural numbers x and y, let (x, y) denote the greatest common divisor of x and y. How many pairs of natural numbers x and y with x ≤ y satisfy the equation xy = x + y + (x, y)?

$For natural numbers x and y, let (x, y)$ $denote the greatest common divisor of$ $x and y . How many pairs of natural$ $numbers x and y with x ⩽ y satisfy the$ $equation x y = x + y + (x, y) ?$

Question Number 19785 Answers: 1 Comments: 0

If x^((x^4 )) = 4, what is the value of x^((x^2 )) + x^((x^8 )) ?

$If x^{(x^{4})} = 4, what is the value of$ $x^{(x^{2})} + x^{(x^{8})} ?$

Question Number 19688 Answers: 1 Comments: 0

The vertices of a square are z_1 , z_2 , z_3 and z_4 taken in the anticlockwise order, then z_3 = (1) −iz_1 + (1 + i)z_2 (2) iz_1 + (1 + i)z_2 (3) z_1 + (1 + i)z_2 (4) (1 + i)z_1 + z_2

$The vertices of a square are z_{1}, z_{2}, z_{3}$ $and z_{4} taken in the anticlockwise order,$ $then z_{3} =$ $(1) - {i z}_{1} + (1 + i) z_{2}$ $(2) {i z}_{1} + (1 + i) z_{2}$ $(3) z_{1} + (1 + i) z_{2}$ $(4) (1 + i) z_{1} + z_{2}$

Question Number 19687 Answers: 1 Comments: 0

Let z_1 , z_2 , z_3 be three vertices of an equilateral triangle circumscribing the circle ∣z∣ = (1/2). If z_1 = (1/2) + (((√3)i)/2) and z_1 , z_2 , z_3 are in anticlockwise sense then z_2 is

$Let z_{1}, z_{2}, z_{3} be three vertices of an$ $equilateral triangle circumscribing the$ $circle ∣ z ∣ = \frac{1}{2} . If z_{1} = \frac{1}{2} + \frac{\sqrt{3} i}{2} and z_{1},$ $z_{2}, z_{3} are in anticlockwise sense then z_{2} is$

Question Number 19683 Answers: 1 Comments: 0

In an A.P; the common difference is −2 and the largest term exceeds the middle term by 58. Find the smallest term of the A.P.

$In an A . P; the common difference is - 2 and the largest term exceeds the middle term by 58 .$ $Find the smallest term of the A . P .$

Question Number 19679 Answers: 0 Comments: 1

Question Number 19675 Answers: 1 Comments: 1

Question Number 19668 Answers: 0 Comments: 2

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