Maths

If z is a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary, then the minimum value of $| z - (3 + 3 i) |$ is :

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Contributor-Level 10

Minimum distance Andhra Pradesh = OP – OA

$\begin{array}{l} = 3 \sqrt[]{2} - \sqrt[]{2} \\ = 2 \sqrt[]{2} \end{array}$

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3 months ago

Let f : R -> R be a continuous function. The $\underset{x \to \frac{π}{4}}{l i m} \frac{\frac{π}{4} \int_{2}^{s e c^{2} x} f (x) d x}{x^{2} - \frac{π^{2}}{16}}$ is equal to :

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Contributor-Level 10

$\underset{x \to \frac{π}{4}}{l i m} \frac{\frac{π}{4} . f (s e c^{2} x) 2 s e c^{2} x t a n x}{2 x}$ = 2f (2)

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3 months ago

Negation of the statement $(p \lor r) \Rightarrow (q \lor r)$ is :

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Contributor-Level 10

As $\sim (A \Rightarrow B) = A \land \sim B$

$\sim ((p v r) \Rightarrow (q v r))$

$= (p v r) \land (\sim r \land \sim q)$

$= (p \land \sim r) \lor (r \land \sim r) \land (\sim q)$

= $(p \land \sim r)$

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3 months ago

The area, enclosed by the curves y = sin x + cos x and y = |cos x – sin x| and the lines x = 0, $x = \frac{π}{2},$ is:

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Contributor-Level 10

Required area

$A = 2 \int_{}^{} ((s i n x + c o s x) - (c o s x - s i n x)) d x$

$= 2 \int_{0}^{π / 4} s i n x d x$

$= 2 \sqrt[]{2} (\sqrt[]{2} - 1)$

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3 months ago

Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g (1) is :

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Maths Application of Derivatives

Contributor-Level 10

Total ways = 6!

Ways satisfying g (3) = 2g (1) is 3

Number of onto function 3 * 4!

Probability = $\frac{1}{10}$

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3 months ago

An angle of intersection of the curves, $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 a n d x^{2} + y^{2} = a b, a > b$ is :

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Contributor-Level 10

$b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}$

$(b^{2} x^{2} + a^{2} (a b - x^{2})) = a^{2} b^{2}$

$x^{2} = \frac{b a^{2} (b - a)}{b^{2} - a^{2}}, y^{2} = \frac{a b^{2}}{a + b}$

points of intersection

$(a \sqrt[]{\frac{b}{a + b}}, b \sqrt[]{\frac{a}{a + b}})$

$t a n θ = | \frac{m_{1} - m_{2}}{1 + m_{1} . m_{2}} |$

$= | \frac{a - b}{\sqrt[]{a b}} |$

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3 months ago

Consider the system of linear equations

-x + y + 2z = 0

3x – ay + 5z = 1

2x – 2y – az = 7

Let S₁ be the set of all a $\in$ R for which the system is inconsistent and S₂ be the set of all $a \in R$ for which the system has infinitely many solutions. If $n (S_{1}) a n d n (S_{2})$ denote the number of elements in S₁ and S₂ respectively, then

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Contributor-Level 10

$Δ = 0 \Rightarrow a = 3, 4$

For a = 3, no solution

For a = 4, no solution

n (S₁) = 2, n (S₂) = 0,

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3 months ago

The distance of the point (-1, 2, -2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x – 2y + z = 0 is :

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Contributor-Level 10

Equation of intersection of line

$\frac{x - 0}{1} = y = \frac{z - 0}{- 1}$

Let $\vec{r} = \hat{i} - \hat{k}$

Direction ratio of $\vec{P Q}$

$\Rightarrow (λ + 1) (1) + (- 2) (0) + (2 - λ) (- 1) = 0$

$λ = \frac{1}{2} \Rightarrow Q (\frac{1}{2}, 0, \frac{- 1}{2})$

$P Q = \frac{\sqrt[]{34}}{2}$

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3 months ago

The range of the function $f (x) = l o g_{\sqrt[]{5}} (3 + c o s (\frac{3 π}{4} + x) + c o s (\frac{π}{4} + x) + c o s (\frac{π}{4} - x) - c o s (\frac{3 π}{4} - x)) i s :$

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Contributor-Level 10

$f (x) = l o g_{\sqrt[]{5}} (3 + \sqrt[]{2} (c o s x - s i n x))$

$- \sqrt[]{2} \leq c o s x - s i n x \leq \sqrt[]{2}$

$\Rightarrow 0 \leq f (x) \leq 2$

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3 months ago

Let A be the set of all points (α, β) such that the area of triangle formed by the points (5, 6), (3, 2) and (α, β) is 12 square units. Then the least possible length of a line segment joining the origin to a point in A, is :

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