Correct Answer: B. -1

Explanation

P = x² + 4x - 2, Q = 2x - 1

Q - p = 2, (2x - 1) - (x² + 4x - 2) = 2

2x - 1 - x² - 4x + 2 = 2

-2x - x² + 1

-x² - 2x - 1 = 0

x² + 2x + 1 = 0

x² + x + x + 1 = 0

x(x + 1) + 1(x + 1) = 0

(x + 1)(x + 1) = 0

x + 1 = 0 or x + 1 = 0

x = -1 or x = -1

x = -1

Correct Answer: B. \(2x^{2} - 9x + 13 = 0\)

Explanation

Note: Given the sum of the roots and its product, we can get the equation using the formula:

\(x^{2} - (\alpha + \beta)x + (\alpha\beta) = 0\). This will be used later on in the course of our solution.

Given equation: \(2x^{2} - 5x + 6 = 0; a = 2, b = -5, c = 6\).

\(\alpha + \beta = \frac{-b}{a} = \frac{-(-5)}{2} = \frac{5}{2}\)

\(\alpha\beta = \frac{c}{a} = \frac{6}{2} = 3\)

Given the roots of the new equation as \((\alpha + 1)\) and \((\beta + 1)\), their sum and product will be

\((\alpha + 1) + (\beta + 1) = \alpha + \beta + 2 = \frac{5}{2} + 2 = \frac{9}{2} = \frac{-b}{a}\)

\((\alpha + 1)(\beta + 1) = \alpha\beta + \alpha + \beta + 1 = 3 + \frac{5}{2} + 1 = \frac{13}{2} = \frac{c}{a}\)

The new equation is given by: \(x^{2} - (\frac{-b}{a})x + (\frac{c}{a}) = 0\)

= \(x^{2} - (\frac{9}{2})x + \frac{13}{2} = 2x^{2} - 9x + 13 = 0\)

Correct Answer: B. 15/17

Correct Answer: B. 504

Correct Answer: A. 180°

Explanation

a + c + d - 180^o[sum of angles in triangle]

x = a[vertical opp. ?s]

y + x + d = 180^o[sum of ?s in a ?]

y + a + d = 180^o

y + a + d = a + c + d

y = c

but b + y = 180^o [sum of ?s on a straight line ]

b + c = 180^o; Ans = A = 180^o

Correct Answer: C. \(\frac{3}{5}\)

Correct Answer: A. 8π

Correct Answer: B. 12.6cm

Explanation

44 = 200/360 x 2 x 22/7 x r

r = 360 x 7 x 44/200 x 2 x 22

= 12.6cm

Correct Answer: C. r/8 cm

Correct Answer: B. 3

Explanation

When you have two lines, \(y_{1}, y_{2}\), perpendicular to each other, the product of their slopes = -1.

\(3x + 4y + 6 = 0 \implies 4y = -6 - 3x\)

\(\therefore y = \frac{-6}{4} - \frac{3}{4}x\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{-3}{4}\)

Also, \(4x - by + 3 = 0 \implies by = 4x + 3\)

\(y = \frac{4}{b}x + \frac{3}{b}\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{4}{b}\)

\(\frac{-3}{4} \times \frac{4}{b} = -1 \implies \frac{4}{b} = \frac{4}{3}\)

\(b = 3\)