Correct Answer: A. \((2x - y)(2x + y)(4x^2 + y^2)\)

Explanation

16\(x^4 - y^4\)

= 2\(^4x^4 - y^4\)

= \((2x)^4 - y^4\)

= \(((2x)^2)^2 - (y^2)^2\)

Using a\(^2 - b^2\) = (a - b)(a + b) identity

= ((2x)\(^2 - y^2)((2x)^2 + y^2)\)

Using the identity one more time

= \((2x - y)(2x + y)((2x)^2 + y^2)\)

∴ \((2x - y)(2x + y)(4x^2 + y^2)\)

Correct Answer: D. pair of bisectors of the angles between the two lines

Explanation

The locus of a point equidistant from two intersecting lines is pair of bisectors of the angles between the two lines.

Correct Answer: D. \(\frac {8}{33}\)

Explanation

Let the number of white balls be n

The number of red balls = 8

Now, the probability of drawing a white ball = \(\frac {n}{n + 8}\)

The probability of drawing a red ball = \(\frac {8}{n + 8}\)

Since Pr(White ball) = \(\frac{1}{2}\) x Pr(Red ball)

\(\therefore \frac {n}{n + 8} = \frac {1}{2}\times \frac {8}{n + 8}\)

= \(\frac {n}{n + 8} = \frac {4}{n + 8}\)

\(\therefore n = 4\)

Number of white balls = 4

The possible number of outcomes = n + 8 = 4 + 8 = 12

Pr(Red ball and White ball) = Pr(Red ball) x Pr(White ball)

Pr(Red ball) = \(\frac {8}{12}\)

Pr(Red ball) = \(\frac {4}{11}\)

Pr(Red ball and white ball) = \(\frac {8}{12} \times \frac {4}{11}\)

\(\therefore\)Pr(Red ball and white ball) = \(\frac {8}{33}\)

Correct Answer: A. a 'b = b 'a

Explanation

A binary operation '*' defined on a set A is said to be commutative only if a*b=b*a, ∀a, b∈A.

If (a*b)*c=a*(b*c), then the operation is said to associative ∀ a, b∈ A.

If (b ο c)*a=(b*a) ο (c*a), then the operation is said to be distributive ∀ a, b, c ∈ A.

Correct Answer: A. dodecagon

Explanation

An interior angle of a regular polygon = \(\frac{(2n-4)\times 90}{n}\)

An exterior angle of a regular polygon = \(\frac{360}{n}\)

\(\frac{(2n-4)\times 90}{n}\) =5 \(\times\) \(\frac{360}{n}\) (Given)

= (2n-4) x 90 = 5 x 360

= 180n - 360 = 1800

= 180n = 1800 + 360

= 180n = 2160

= n = \(\frac{2160}{180}\) = 12

The polygon has 12 sides which is dodecagon

Correct Answer: A. 7.725

Explanation

\(\frac {N + 1}{2} = \frac {107 + 1}{2}\) th = 54th value i.e the median age is in the interval "7 - 8" (29 + 25 = 54)

Median = \(1_m + (\frac {^{\sum f} /_2 - f_b}{f_m})\)c

\(l_m\)=lower class boundary of the median age = 6.5

\(\sum f\) = 107

\(f_b\) = sum of all frequencies before the median age = 29

\(f_m\)=frequency of the median age = 40

c = class width = 8.5 - 6.5 = 2

Median = 6.5 + \((\frac {^{107}/_2 - 29}{40})\) x 2

= 6.5 + \((\frac {53.5 - 29}{40})\) x 2

= 6.5 + \((\frac {24.5}{40})\) x 2

= 6.5 + 1.225

\(\therefore\) median age = 7.725

Correct Answer: C. 13 cm

Explanation

Let the length of the longer side = \(x\) cm

∴ The length of the shorter side = (\(x\) - 6) cm

If we increase each side's length by 2 cm, it becomes

(\(x\) + 2) cm and (\(x\) - 4) cm respectively

Area of a rectangle = L x B

\(A_1 = x(x - 6) = x^2 - 6x\)

\(A_2 = (x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8\)

\(A_1 + 68 = A_2\) (Given)

⇒ \(x^2 - 6x + 68 = x^2 - 2x - 8\)

⇒ \(x^2 - x^2 - 6x + 2x\) = -8 - 68

⇒ -4\(x\) = -76

⇒ \(x\) = \(\frac{-76}{-4}\) = 19cm

∴ The length of the shorter side = \(x\) - 6 = 19 - 6 = 13 cm

Correct Answer: A. 2

Explanation

16\(^{0.16}\) × 16\(^{0.04}\) × 2\(^{0.2}\)

=2\(^{4(0.16)}\) × 2\(^{4(0.04)}\) × 2\(^{0.2}\)

=2\(^{0.64}\) × 2\(^{0.16}\) × 2\(^{0.2}\)

=2\(^{0.64\times0.16\times0.2}\)

=2\(^1\)

2

Correct Answer: A. 16

Explanation

Number of students who scored atleast 25% = 5 + 3 + 8 = 16

Correct Answer: C. 360

Explanation

Total number of students that scored at most 50 marks = 100 + 80 + 60 + 40 + 80 = 360