If A = 1 2 4 2 then show that | 2 A | = 4 | A | .

Given A = 1 2 4 2 ⇒ A = 1 × 2 - 4 × 2 = - 6 ⇒ 4 A = 4 - 6 = - 24 Now 2 A = 2 1 2 4 2 = 2 4 8 4 2 A = 2 × 4 - 8 × 4 = - 24 Hence proved.

If Δ 1 = x s i n θ c o s θ - s i n θ - x 1 c o s θ 1 x and Δ 2 = x s i n 2 θ c o s 2 θ - s i n 2 θ - x 1 c o s 2 θ 1 x , x ≠ 0 ; then for all θ ∈ 0 , π 2 :

Δ 1 + Δ 2 = - 2 ( x 3 + x - 1 )

Δ 1 - Δ 2 = x ( c o s 2 θ - c o s 4 θ )

Show that the lines x + 3 - 3 = y - 1 1 = z - 5 5 and x + 1 - 1 = y - 2 2 = z - 5 5 are coplanar.

Consider any two points A ( x 1 , y 1 , z 1 ) and B ( x 2 , y 2 , z 2 ) and consider two vectors a → and b → having direction ratios as ( a 1 , b 1 , c 1 ) and ( a 2 , b 2 , c 2 ) respectively. Then the condition for them to be coplanar is A B → . ( a → × b → ) = 0 The two lines given in the question are x + 3 - 3 = y - 1 1 = z - 5 5 and x + 1 - 1 = y - 2 2 = z - 5 5 . These are in the form of x - x 1 a 1 = y - y 1 b 1 = z - z 1 c 1 . So, on comparing all the three lines, we get The points for the first line, x 1 = - 3 , y 1 = 1 , z 1 = 5 , a 1 = - 3 , b 1 = 1 , c 1 = 5 and Similarly, for the second line, x 2 = - 1 , y 2 = 2 , z 2 = 5 , a 2 = - 1 , b 2 = 2 , c 2 = 5 . Now the direction ratios of the vector A B → = ( - 1 - - 3 ) , ( 2 - 1 ) , ( 5 - 5 ) = 2 , 1 , 0 . Now applying the condition for coplanarity, we get x 2 - x 1 y 2 - y 1 z 2 - z 1 a 1 b 1 c 1 a 2 b 2 c 2 = 2 1 0 - 3 1 5 - 1 2 5 = 2 5 - 10 - 1 - 15 + 5 + 0 = 2 × - 5 - 1 × - 10 = 2 × - 5 - 1 × - 10 = - 10 + 10 = 0 Since the condition is satisfied, the two lines are coplanar.

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Who is known as the father of macroeconomics?

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Important Questions on Evolution

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Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

How is the detritus food chain connected with the grazing food chain in a natural ecosystem?

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Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

A = [\begin{array}{l} 1 & 2 \\ 4 & 2 \end{array}]

then show that

| 2 A | = 4 | A |

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

If 20J of energy is trapped at producer level, then how much energy will be available to peacock as food in the following chain?
plant

\to

mice

\to

snake

\to

peacock

HARD

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Let

A = [\begin{matrix} 2 & b & 1 \\ b & b^{2} + 1 & b \\ 1 & b & 2 \end{matrix}],

where

b > 0

. Then the minimum value of

\frac{\det (A)}{b}

is:

MEDIUM

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Δ_{1} = |\begin{matrix} x & s i n θ & c o s θ \\ - s i n θ & - x & 1 \\ c o s θ & 1 & x \end{matrix}|

and

Δ_{2} = |\begin{matrix} x & s i n 2 θ & c o s 2 θ \\ - s i n 2 θ & - x & 1 \\ c o s 2 θ & 1 & x \end{matrix}|,

x \neq 0;

then for all

θ \in (0, \frac{π}{2})

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Explain the role of the ‘first trophic level’ in an ecosystem.

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Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Show that the lines

\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}

and

\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}

are coplanar.

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Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Give a diagrammatic representation of energy flow through different trophic levels.

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

The sum of the real roots of the equation

|\begin{matrix} x & - 6 & - 1 \\ 2 & - 3 x & x - 3 \\ - 3 & 2 x & x + 2 \end{matrix}| = 0,

is equal to:

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

There are two values of

a

which makes determinant,

Δ = |\begin{matrix} 1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2 a \end{matrix}| = 86

, then sum of these number is

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Let

d \in R,

and

A = [\begin{matrix} - 2 & 4 + d & (\sin θ) - 2 \\ 1 & (\sin θ) + 2 & d \\ 5 & (2 \sin θ) - d & (- \sin θ) + 2 + 2 d \end{matrix}],

θ \in [0, 2 π] .

If the minimum value of

d e t (A)

8,

then a value of

d

is:

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Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

S = \{x \in [0, 2 π] : |\begin{matrix} 0 & \cos x & - \sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{matrix}| = 0\},

then

\sum_{x \in S} \tan (\frac{π}{3} + x)

is equal to:

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

What is a trophic level in an ecosystem ? What is ‘standing crop’ with reference to it?

HARD

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

A = [\begin{matrix} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{matrix}]

, then for all

θ \in (\frac{3 π}{4}, \frac{5 π}{4})

\det (A)

lies in the interval :

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Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Describe the DDT biomagnification occurring in an aquatic food chain. State the negative effects the process has on the organisms at the last trophic level of the food chain.

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Write the names of the following:

A 2 mya primate that lived in East African grasslands.

MEDIUM

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Explain energy flow in ecosystem.

EASY

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

The inverse of matrix is

[\begin{matrix} 2 & 5 & 0 \\ 0 & 1 & 1 \\ - 1 & 0 & 3 \end{matrix}]

MEDIUM

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

Let the numbers

2, b, c

be in an A.P. and

A = [\begin{matrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2} \end{matrix}]

. If

d e t (A) \in [2,16],

then

c

lies in the interval:

MEDIUM

Biology>Genetics and Evolution>Evolution>A Brief Account of Evolution- Geological Time Scale

|\begin{matrix} x^{2} + x & x + 1 & x - 2 \\ 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 \\ x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix}| = a x - 12

, then

a

is equal to:

Who is known as the father of macroeconomics?

Important Questions on Evolution

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