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Range is the difference between highest and lowest values in a series. 

Cost of living index number/consumer price index number is the index number of the cost met by a specified class of consumers in buying a basket of goods and services. 

Uses:

• It is commonly used in fixation of salary, dearness allowances to the employees. 

• They are used by govt, in formulation of price policy, wage policy and general economic policies. 

• They are used in evaluation of purchasing power of money and deflating the money. 

• They are used for comparing cost of living of different classes of people.

(a) valuable 

Squares of S.D is known as variance 

(d) Spatial organization

(b) 2 × 2

If A is of order 2 × 3 and B is of order 3 × 2, then order of AB is 2 × 2.

Correct answer is

(b) \(\begin{bmatrix} 4 & 4 \\ 0 & 4 \\ \end{bmatrix}\)

[4, 4][0, 4]

Correct option: a) RS 25,600

(c) m = n

A = [a_ij]_m×n is a square matrix, if m = n.

Correct option: b) RS 19,200

We have matrix A =\(\begin{bmatrix} 2& -3 & 5 \\[0.3em] 3 & 2 & -4 \\[0.3em] 1 & 1 & -2 \end{bmatrix}\).

Now, determinant of matrix A is \(\begin{bmatrix} 2& -3 & 5 \\[0.3em] 3 & 2 & -4 \\[0.3em] 1 & 1 & -2 \end{bmatrix}\) = 2 \(\begin{vmatrix} 2& -4 \\[0.3em] 1 & -2 \end{vmatrix}\) - (-3) \(\begin{vmatrix} 3& -4 \\[0.3em] 1 & -2 \end{vmatrix}\) + 5\(\begin{vmatrix} 3& 2 \\[0.3em] 1 &1 \end{vmatrix}\)

= 2(−4 + 4) + 3(−6 + 4) + 5(3 − 2) = 2 × 0 + 3 × −2 + 5 × 1 = 0 − 6 + 5 = −1.

(By expanding determinant along row R₁ and \(\begin{vmatrix} a& b \\[0.3em] c &d \end{vmatrix}\) = ad - bc)

Hence, |A| = −1 ≠ 0. i.e., the inverse of matrix exists. 

We know that the cofactor of element a_ij of matrix A is given by A_ij= (−1)^i+j M_ij, where M_ij is minor of element a_ij of matrix A.

Now, the cofactor of element a₁₁ is A₁₁ = (−1)¹⁺¹ \(\begin{vmatrix} 2& -4 \\[0.3em] 1 &-2 \end{vmatrix}\) = (−4 + 4) = 0, 

The cofactor of element a₁₂ is A₁₂ = (−1)¹⁺² \(\begin{vmatrix} 3& -4 \\[0.3em] 1 &-2 \end{vmatrix}\) = −(−6 + 4) = −(−2) = 2, 

The cofactor of element a₁₃ is A₁₃ = (−1)¹⁺³ \(\begin{vmatrix} 3& 2 \\[0.3em] 1 &1 \end{vmatrix}\) = (3 − 2) = 1, 

The cofactor of element a₂₁ is A₂₁ = (−1) 2+1 \(\begin{vmatrix} -3& 5 \\[0.3em] 1 &-2 \end{vmatrix}\) = −(6 − 5) = −1, 

The cofactor of element a₂₂ is A₂₂ = (−1)²⁺² \(\begin{vmatrix} 2& 5 \\[0.3em] 1 &-2 \end{vmatrix}\) = (−4 − 5) = −9, 

The cofactor of element a₂₃ is A₂₃ = (−1)²⁺³ \(\begin{vmatrix} 2& -3 \\[0.3em] 1 &1 \end{vmatrix}\) = −(2 + 3) = −5, 

The cofactor of element a₃₁ is A₃₁ = (−1)³⁺¹ \(\begin{vmatrix} -3& 5 \\[0.3em] 2 &-4 \end{vmatrix}\) = (12 − 10) = 2,

The cofactor of element a₃₂ is A₃₂ = (−1)³⁺² \(\begin{vmatrix} 2& 5 \\[0.3em] 3 &-4 \end{vmatrix}\) = −(−8 − 15) = −(−23) = 23, 

The cofactor of element a₃₃ is A₃₃ = (−1)³⁺³ \(\begin{vmatrix}2& -3 \\[0.3em] 3 & 2 \end{vmatrix}\) = (4 + 9) = 13.

Therefore, the cofactor matrix of matrix A is \(\begin{bmatrix} 0& 2 & 1 \\[0.3em] -1 & -9 & -5 \\[0.3em] 2 &23 & 13 \end{bmatrix}\)

Now, the adjoint of matrix which is the transpose of cofactor matrix of matrix A

Therefore, a_dj A = \(\begin{bmatrix} 0& 2 & 1 \\[0.3em] -1 & -9 & -5 \\[0.3em] 2 &23 & 13 \end{bmatrix}'\) = \(\begin{bmatrix} 0& -1 & 2 \\[0.3em] 2 & -9 &23 \\[0.3em] 1 &-5 & 13 \end{bmatrix}\).

Therefore, the inverse matrix of matrix A is A⁻¹ \(\frac{adj A}{|A|}\) = \(\frac{1}{-1}\)\(\begin{bmatrix} 0& -1 & 2 \\[0.3em] 2 & -9 &23 \\[0.3em] 1 &-5 & 13 \end{bmatrix}\) = \(\begin{bmatrix} 0& 1 & -2 \\[0.3em] -2 & 9 &-23 \\[0.3em] -1 &5 & -13 \end{bmatrix}\)

Now, given system of equations are 

2x − 3y + 5z = 11 

3x + 2y − 4z = −5 

And x + y − 2z = −3.

The matrix form of given system of equations is AX = b, where A = \(\begin{bmatrix} 2& -3 & 5 \\[0.3em] 3 & 2 & -4 \\[0.3em] 1 & 1 & -2 \end{bmatrix}\), X = \(\begin{bmatrix} x \\[0.3em] y \\[0.3em] z \end{bmatrix}\)and b = \(\begin{bmatrix} 11 \\[0.3em] -5 \\[0.3em] -3 \end{bmatrix}.\)

Since, AX= b ⇒ X = A⁻¹ b = \(\begin{bmatrix} 0& 1 & -2 \\[0.3em] -2 & 9 &-23 \\[0.3em] -1 &5 & -13 \end{bmatrix}\)\(\begin{bmatrix} 11 \\[0.3em] -5 \\[0.3em] -3 \end{bmatrix}\) 

 \(\begin{bmatrix} 0& -5 +6 \\[0.3em] -22 & -45 +69 \\[0.3em] -11&-25+69 \end{bmatrix}\) = \(\begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\)

Hence, X = \(\begin{bmatrix} x \\[0.3em] y \\[0.3em] z \end{bmatrix}\) = \(\begin{bmatrix} 1 \\[0.3em] 2 \\[0.3em] 3 \end{bmatrix}\)

Hence, the solution of given system of equations is x =1, y = 2 and z = 3.

Option : (c)

\(\frac{x^2}{9}+\frac{y^2}{16}=1\;\Rightarrow\frac{2x}{9}+\frac{2y}{16}\frac{dy}{dx}=0\)

\(\Rightarrow\) slope of normal \(=\frac{-dx}{dy}=\frac{9y}{16x}\)

As curve’s tangent is parallel to y-axes

Therefore, the normal to the curve is parallel to x-axis

\(\Rightarrow\frac{9y}{16x}=0\Rightarrow y=0\;and\,x=\pm3\)

\(\therefore\,points=(\pm3,0)\)

Formal education curbs the potential of the child as it takes away from the child its freedom. The child is like a caged bird in school and just like the wings of the caged bird droop, the spirit of the child sags in school which instils in the child a sense of fear. Children, like plants, need to be nurtured and given care, if they are to ripen successfully and bear fruit later on. If children are not allowed to have a natural childhood, which is spent in playing freely among the natural things (hills, meadows, lambs, birds, etc.), then they will wither before they can blossom into mature adults. Spring stands for new growth and that is why the poet uses the metaphor of the spring to argue that a school does not allow spring to flourish.

i. - a) 

ii - b) 

iii - a) 

iv - b)

v - d)

i) (d) 

ii). (c) 

iii. (a) 

iv. (c) 

v. (b)

i. b 

ii. d

iii. c 

iv. b 

v. d

(i) - poem referred to the father swan who bit his three ducks because they quacked too much; allusion to Mr. Keesing 

- Poem written to turn the joke around on Mr. Keesing; metaphorically convey the value of talking 

- fortunate that Mr. Keesing got the joke, else, the disciplinary action could have been severe. 

(ii) - Poem discusses the way the world can end. 

- Fire stands for greed/lust/human desire; Ice stands for hatred/ coldness/human indifference; 

- world can end with either greed or indifference 

(iii) - It is justified as Rapunzel lived with a witch who would constantly keep a check on her and had several rules to be followed. 

- Amanda too, feels caged by the impositions 

- Amanda desires peace in isolation like Rapunzel had in her tower; clarifying she will not let her hair down for anyone

Competency applied for : 

content- 

• Stating Minimum 2 points + justification/ explanation 

• Depicting Support of / reference to textual evidence 

expression- 

• Applying logical progression, using appropriate linking devices (e.g. because/ besides, in order to, therefore etc.)- cohesion 

• Response relevance for overall sense according to the question attempted-coherence 

• Accuracy in use of grammatical structures, vocabulary and spellings 

NOTE— Writing in a paragraph recommended. Just listing of points shall result in deduction of marks allotted for cohesion & coherence.

(i) - she knew her son Lutkins was pretending to be Bill 

- the narrator’s predicament amused her 

(ii) -she was overfeeding Tricki; being overindulgent 

-she needed to cut down Tricki’s food intake to help him regain his health (be cruel to be kind) 

(iii) - Kind/encouraging/empathetic/assuring/any other relevant trait from the text. -Reason from text required

Competency applied for : 

conten t - 

• Stating Minimum 2 rationales 

• Depicting Support of / reference to textual evidence 

expression - 

• Applying logical progression, using appropriate linking devices (e.g. because/ besides, in order to, therefore etc.) - cohesion 

• Response relevance for overall sense according to the question attempted -

i. a) restless 

ii. d) abcd; abcb 

iii. c) freedom and captivity. 

iv. c) “Join me”. 

v. a) hopes to be free and in the wild, someday.

i) (c) 

ii) (b) 

iii). (b) 

iv). (c) 

v). (d) 

vi) (b) 

vii) (b) 

viii) (b)

Option : (D)

Z is maximum 180 at points C (15,15) and D(0, 20).

\(\Rightarrow\) Z is maximum at every point on the line segment CD.

Given polynomial p(x) = x²⁰¹⁰ + αx²⁰¹¹ + β 

Now, as the remainder is zero when p(x) is divided by (x- 1). So put x = 1. 

(1)²⁰¹⁰ + α(1)²⁰¹¹ + β = 0 

1 + α + β = 0

α + β = -1.... (1)

Similarly, put x = -1, In this case the remainder is 4

(-1)²⁰¹⁰ + α(-1)²⁰¹¹ + β = 4

1 -  α + β = 4

β - α = 3... (2) 

Adding (1) and (2) we get β = 1 and α = -2

c) Carbamazepine  

Root x \(= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

\(= \frac{-(-75) \pm \sqrt{75^2 - 4 \times 3 \times 250}}{2 \times 3}\)

\(= \frac{75 \pm \sqrt{5625 - 3000}}{6}\)

\(= \frac{75 \pm \sqrt{2625}}{6}\) \(= \frac{75 \pm 5\sqrt{105}}{6}\)

Given x² − x + 1 = 0

α + β = 1, αβ = 1

α² + β² = (α + β)² − 2αβ

= (1)² - 2 x 1

= 1 - 2

= -1

(a) Given hyperbola is 

9x² - 16y² - 144

⇒ \(\frac{x^2}{16}-\frac{y^2}9=1\)

By comparing with \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

We get a² = 16, b² = 9

⇒ a = 4, b = 3

\(\therefore\) e = \(\sqrt{1+\frac{b^2}{a^2}}\) = \(\sqrt{1+\frac{3^2}{4^2}}\)

\(=\frac{4^2+3^2}{4^2}\) = \(\frac{5^2}{4^2}=\frac54\) 

\(\therefore\) coordinates of fozi are

(+ae, 0) = (\(\pm\)4 x 5/4,0) = (\(\pm5,0\))

Hence, (-5, 0) and (5, 0) are coordinates of foci or given hyperbola.

And latus rectum of hyperbola = \(\frac{2b^2}a=\frac{2\times9}4=\frac92\) unit

(b) Given ellipse is 3x² + 2y² = 18 

⇒ \(\frac{x^2}6+\frac{y^2}9=1\)

\(\therefore\) a² = 6 and b² = 9

⇒ a = √6 and b = 3

\(\because\) b > a

\(\therefore\) Focus of ellipse lies on y-axis.

Now, e = \(\sqrt{1-\frac{q^2}{b^2}}=\sqrt{1-\frac69}\) = \(\sqrt{\frac{9-6}9}\) = \(\frac{\sqrt3}3\) = \(\frac1{\sqrt3}\) 

\(\therefore\) Co-ordinates of foci are

(0, \(\pm\)be) = (0, \(\pm\)9 x \(\frac1{\sqrt3}\)) = (0, \(\pm\)\(\frac{3}{\sqrt3}\)) (0, \(\pm\sqrt3\))

Hence, (0, \(\sqrt3\)) and (0, \(-\sqrt3\)) are co-ordinates of foci or given ellipse.

And latus rectum of ellipse = \(\frac{2b^2}a=\frac{2\times3}{\sqrt6}\) = \(\frac{6}{\sqrt6}\) = \(\sqrt6\) unit

National income is a broader national level economic measure than is personal income. National income includes payments to individuals (income from wages and salaries, and other income), plus payments to government (taxes), plus retained income from the corporate sector (depreciation, undistributed profits), less adjustments (subsidies, government and consumer interest, and statistical discrepancy).

There are 8 persons (4 men and 4 women). A committee is to be formed consisting of 3 persons so that 1 particular man has to be in the committee. Excluding that particular man there are 7 persons ( 3 men and 4 women) from which 2 are to be chosen. And the choices are (i) 2 men and no woman (ii) 1 man and 1 woman and (iii) 2 women and no man. This leads to ³C₂ x ⁴C₀ +³C₁ x ⁴C₁ + ³C₀ x ⁴C₂ 

=3 x 1+3 x 4+1 x  6

=21

the, in, a, about.

Let (u,v) be the vertex of the parabola. 

The equation of required parabolas are of the form 

(x−u)²=4a(y−v) 

So there are two unknown constant.

Correct option is B) attention

Correct option is B) shouldn’t have used / should / have used

Correct option is C) could eat / has to be

Calculations :

Let the number be 'x'

Number after multiplying 'x' with 3003 = 3003x 

LCM of 7, 11 and 13 = 7 × 11 × 13 = 1001 

⇒ Result when 3003x is divided by 1001 and itself = 3003x ÷ 1001x 

⇒ 3 

∴ the result will be 3 

We know that whenever force is applied on a cross section or beam along its axis it either tries to compress it or elongate it. If applied force tries to compress the member force is known as compressive force as shown in fig a. If force applied on member tries to elongate it force is known as tensile force shown in figbIf a compressive force is applied on the member as in fig.a the member will always try to resist this force & a force equal in magnitude but opposite to direction of applied force will be induced in it as shown in figc Similarly induced force in member shown in figb will be as shown in fig.dFrom above we can conclude that if induced force in a member of loaded truss is like the figc we will say nature of applied force on the member is compressive. If Nature of induced force in a member of truss like shown in figd then we can say that Nature of force applied on the member is tensile.

Given:

A = {a, b, c, d, e, f}

B = {a, c, f, g} 

C = {a, d, e, g, h}

Concept used:

A ∩ B is the intersection of two sets that contain only the elements that are in both sets.

A ∪ B is the union of two sets that contain all the elements contained in either set (or both sets).

Calculation:

(A ∩ C) = {a, d, e}

⇒ (A ∩ C) ∪ B = {a, c, d, e, f, g}

Hence, option 4 is correct.

Given:

The given number = 438976

Concept:

Calculation skills trick

Calculation:

Step(1):

First, we will take the digit at the unit’s place.

At the unit place, we have 6.

Step(2):

Now consider the cube table, the cube of which number has 6 as at the unit digit place.

We see, 6 × 6 × 6 = 216.

Step(3):

That means the cube root of 438976 will have 6 at the unit place.

Step(4):

Now ignore the last 3 digits of 438976, i.e. 976

Step(5):

Taking 438 as a reference number, we can see from the cubes table 438 lies between cubes of 7 and 8.

Step(6): Here, we choose the lower number i.e. 7

∴ We get the two digit of the answer

∴ Cube root of 483976 = 76  

Given:

Expression is (2¹²⁰ – 2¹¹⁹ – 2¹¹⁸) (3¹¹⁰ – 3¹⁰⁹ – 3¹⁰⁸)

Concept used:

To find the number of zeroes at the end of the product, we need to calculate the number of 2’s and number 5’s or number of pairs of 2 and 5.

2 × 5 = 10 ⇒ Number of zeroes = 1 (number of pair = 1)

The number of pairs of 2 and 5 is same as the number of zeroes at the end of the product

Calculation:

 (2¹²⁰ – 2¹¹⁹ – 2¹¹⁸) (3¹¹⁰ – 3¹⁰⁹ – 3¹⁰⁸)

⇒ 2¹¹⁸ (2² – 2¹ – 2⁰) × 3¹⁰⁸ (3² – 3¹ – 3⁰)

⇒ 2¹¹⁸ (4 – 2 - 1) × 3¹⁰⁸ (9 – 3 - 1)

⇒ 2¹¹⁸ × 1 × 3¹⁰⁸ × 5

⇒ 2¹¹⁸ × 3¹⁰⁸ × 5

In the above expression, number of 5’s is 1, So only one pair of 2 and 5 is possible, number of zeroes would be 1

Given:

Expression is (4¹⁴⁴ – 4¹⁴³ – 4¹⁴¹) (5¹⁴⁰ – 5¹³⁹ – 5¹³⁸)

Concept used:

To find the number of zeroes at the end of the product, we need to calculate the number of 2’s and number 5’s or number of pairs of 2 and 5.

2 × 5 = 10 ⇒ Number of zeroes = 1 (number of pair = 1)

The number of pairs of 2 and 5 is same as the number of zeroes at the end of the product

Calculation:

(4¹⁴⁴ – 4¹⁴³ – 4¹⁴²) (5¹⁴⁰ – 5¹³⁹ – 5¹³⁸)

⇒ 4¹⁴² (4² – 4¹ – 4⁰) × 5¹³⁸ (5² – 5¹ – 5⁰)

⇒ 4¹⁴² (16 – 4 - 1) × 5¹³⁸ (25 – 5 - 1)

⇒ 4¹⁴² × 11 × 5¹³⁸ × 19

⇒ (2²)¹⁴² × 11 × 5¹³⁸ × 19

⇒ 2²⁸⁴ × 11 × 5¹³⁸ × 19

In the above expression, number of 5’s is less than the number of 2’s. So there are only 138 pairs of 2 and 5 are possible, number of zeroes would be 138

Four essential features of development communication

1.It is oriented to socio-economic development and happiness of the people and community at large. 

2.It aims at giving information and educating the community. 

3.It combines suitable mass media and interpersonal communication channels for greater impact. 

4.It is based on the audience characteristics and their environment

Four types of contrasting scheme -

OR 

Four basic shapes created by using different types of lines - 

1. Natural shapes 

2. Stylised shapes 

3. Geometric shapes 

4. Abstract shapes