SAT (Stage-2) Mock Test - 24

For numbers to be closed under division,

a ÷ b = c (where, a, b and c are rational numbers)

a = 9, b = 0

9 ÷ 0 = 9/0= not defined

So, 9 ÷ 0 proves that rational numbers are not closed under division.

Sum of the roots will be (-3 + 4) = 1 and
Product of the roots = (-3)(4) = -12

For option 3, we have
Sum of the roots = 1
Product of the roots = -12

If 3 is the zero of the quadratic polynomial (k - 1) x² + kx + 1, then

9(k - 1) - 3k + 1 = 0

9k - 9 - 3k + 1 = 0

6k - 8 = 0

k = 8/6 = 4/3

a₁₈ - a₁₄ = 32
Thus,
a + 17d - (a + 13d) = 32 (a = first term, d = common difference)
4d = 32
d = 8

3 sinθ+ 2 cosθ= 2 ... (i)
Let 2 sinθ- 3 cosθ= x ... (ii)

Squaring both sides,
9 sin²θ + 4 cos²θ + 12 sinθ cosθ = 4
4 sin²θ + 9cos²θ - 12 sinθ cosθ = x²

Adding both the equations,
13 sin²θ + 13 cos²θ = 4 + x²
x² = 9
x = ±3

Hence, value of 2 sinθ- 3 cosθ= ±3

Lines make equal intercepts with axes.
Then, suppose equation of the line is:
x + y + a = 0 ... (1)
Point of intersection of the lines 3x + y + 1 = 0 and 2x - y + 3 = 0 is (– 4/5, 7/5).
Put the above value in equation (1):
– 4/5 + 7/5 + a = 0
⇒ a = – 3/5
Therefore, equation of the line is:
5x + 5y - 3 = 0

Favourable cases = 3, 6, 9, 12, 15, 18, 7, 14

Probability = 8/20 = 2/5