Explanation

\(3x^{2} + 2y^{2} + xy + x - 7 = 0\)

We differentiate y implicitly with respect to x.

\(6x + 2(2y)\frac{\mathrm d y}{\mathrm d x} + y + x\frac{\mathrm d y}{\mathrm d x} + 1 = 0\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{- y - 1 - 6x}{4y + x}\)

= \(\frac{-(6x + y + 1)}{4y + x}\).

At the point (-2, 1),

\(\frac{\mathrm d y}{\mathrm d x} = \frac{-(-12 + 1 + 1)}{4(1) - 2}\)

= \(\frac{-(-10)}{2} = 5\)

Correct Answer: D. \(\frac{1}{36}\)

Explanation

P(c and e) = \(P(c) \times P(e)\)

P(3) = \(\frac{1}{6}\)

P(5) = \(\frac{1}{6}\)

P(3 and 5) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\)

Correct Answer: C. \(36^0\)

Explanation

t = ∠NTS (the angle between a tangent and a chord is equal to the angle in the alternate segment).

Therefore, t = 36°

Correct Answer: A. 13 cm

Correct Answer: A. 1/8

Explanation

(\(\frac{√3-3}{√3}\)) = \(\frac{√3}{√3}\)\

\(\frac{3-√3}{3}\)

= 1 - √3

20 = 15 + 10 - x

x = 25 - 20 = 5

P(both) = \(\frac{5}{20}\)

= \(\frac{1}{4}\) or 0.25

Correct Answer: A. 60\(^0\)

Explanation

\(PR^2 = PQ^2 + RQ^2 - 2(PQ)(RQ)cos Q\)

\(\implies cos Q = \frac{PQ^2 + RQ^2 - PR^2}{2(PQ)(RQ)}\)

\(\implies cos Q = \frac{8^2 + 5^2 - 7^2}{2\times8\times5}\)

\(\implies cos Q = \frac{64 + 25 - 49}{80}\)

\(\implies cos Q = \frac{40}{80} = 0.5\)

\(\implies Q = cos^{-1} (0.5) = 60^0\)

\(\therefore x = 60^0\)

Correct Answer: A. 35°

Explanation

In the diagram, QOR + 2m(vertically opposite angles)

So, m + 90° + 2m = 180°

(angles on str. line)

3m = 180° - 90°

3m = 90°

m = \(\frac{90^o}{3}\)

= 30°

substituting 30° for m in

2m + 4n = 200° gives

2 x 30° + 4n = 200°

60° + 4n = 200°

4n = 200° - 60°

= 140°

n = \(\frac{140°}{4}\)

= 35°

Explanation

(5, 2), (-4, k), (2, 1)

A straight line has constant gradient.

\(\frac{k - 2}{-4 - 5} = \frac{1 - k}{2 - (-4)}\)

\(\frac{k - 2}{-9} = \frac{1 - k}{6}\)

\(-9(1 - k) = 6(k - 2) \implies -9 + 9k = 6k - 12\)

\(9k - 6k = -12 + 9 \implies 3k = -3\)

\(k = -1\)

Correct Answer: D. -2x + 3y - 6 = 0

Explanation

Recall:

Where 'a' and 'b' are the x and y intercept respectively.

2x-3y = -6

2x - 3y + 6 = 0 -------(1)

multiply through by -1

-2x + 3y - 6 = 0