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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Three

Construct a $3 \times 2$ matrix whose elements are given by $a_{i j} = e^{i x} \cdot s i n (j x)$ .

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Payal Gupta | Contributor-Level 10

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This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} L e t A = {[\begin{array}{l} a_{11} a_{12} \\ a_{21} a_{22} \\ a_{31} a_{32} \end{array}]}_{3 \times 2} \\ G i v e n t h a t a_{i j} = e^{i x} s i n j x \\ a_{11} = e^{x} s i n x a_{12} = e^{x} s i n 2 x \\ a_{21} = e^{2 x} s i n x a_{22} = e^{2 x} s i n 2 x \\ a_{31} = e^{3 x} s i n x a_{32} = e^{3 x} s i n 2 x \\ H e n c e, t h e m a t r i x A = [\begin{array}{l} e^{x} s i n x e^{x} s i n 2 x \\ e^{2 x} s i n x e^{2 x} s i n 2 x \\ e^{3 x} s i n x e^{3 x} s i n 2 x \end{array}] \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Three

Construct a_{2 × 2} matrix where

(i) $a_{i j} = \frac{{(i - 2 j)}^{2}}{2}$ .
(ii) $a_{i j} = ? - 2 i + 3 j ?$ .

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Payal Gupta | Contributor-Level 10

a year ago

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} L e t A = {[\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}]}_{2 \times 2} \\ (i) G i v e n t h a t a_{i j} = \frac{{(i - 2 j)}^{2}}{2} \\ a_{11} = \frac{{(1 - 2 \times 1)}^{2}}{2} = \frac{1}{2}; a_{12} = \frac{{(1 - 2 \times 2)}^{2}}{2} = \frac{9}{2} \\ a_{21} = \frac{{(2 - 2 \times 1)}^{2}}{2} = 0; a_{22} = \frac{{(2 - 2 \times 2)}^{2}}{2} = 2 \\ H e n c e, t h e m a t r i x A = [\begin{matrix} \frac{1}{2} & \frac{9}{2} \\ 0 & 2 \end{matrix}] \\ (i i) G i v e n t h a t a_{i j} = | - 2 i + 3 j | \\ a_{11} = | - 2 \times 1 + 3 \times 1 | = 1; a_{12} = | - 2 \times 1 + 3 \times 2 | = 4 \\ a_{21} = | - 2 \times 2 + 3 \times 1 | = - 1; a_{22} = | - 2 \times 2 + 3 \times 2 | = 2 \\ H e n c e, t h e m a t r i x A = [\begin{matrix} 1 & 4 \\ - 1 & 2 \end{matrix}] \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Three

In the matrix write:

(i) The order of the matrix .

(ii) The number of elements.

(iii) Write elements $a_{23}, a_{31}, a_{12}$ .

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Payal Gupta | Contributor-Level 10

a year ago

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} (i) T h e o r d e r o f t h e g i v e n m a t r i x A i s 3 \times 3 \\ (i i) T h e n u m b e r o f e l e m e n t s i n a m a t r i x A = 3 \times 3 = 9 \\ (i i i) a_{i j} = t h e e l e m e n t s o f i^{t h} r o w a n d j^{t h} c o l u m n . \\ S o, a_{23} = x^{2} - y, a_{31} = 0, a_{12} = 1 . \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Three

If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

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Payal Gupta | Contributor-Level 10

a year ago

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l} T h e p o s s i b l e o r d e r s t h a t a m a t r i x h a v i n g 28 e l e m e n t s a r e {28 \times 1, 1 \times 28, 2 \times 14, 14 \times 2, 4 \times 7, 7 \times 4} . \\ T h e p o s s i b l e o r d e r s o f a m a t r i x h a v i n g 13 e l e m e n t s a r e {1 \times 13, 13 \times 1} . \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Three

Express the matrix $[\begin{matrix} 2 & 3 & 1 \\ 1 & - 1 & 2 \\ 4 & 1 & 2 \end{matrix}]$ as the sum of a symmetric and a skew-symmetric matrix.

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Payal Gupta | Contributor-Level 10

a year ago

This is a long answer type question as classified in NCERT Exemplar

$\begin{array}{l} W e k n o w t h a t a n y s q u a r e m a t r i x c a n b e expresse d a s t h e s u m o f s y m m e t r i c a n d s k e w s y m m e t r i c m a t r i x i . e . \\ A = \frac{1}{2} [A + A^{'}] + \frac{1}{2} [A - A^{'}] \\ S o P = \frac{1}{2} [(\begin{matrix} 2 & 3 & 1 \\ 1 & - 1 & 2 \\ 4 & 1 & 2 \end{matrix}) + (\begin{matrix} 2 & 1 & 4 \\ 3 & - 1 & 1 \\ 1 & 2 & 2 \end{matrix})] \\ = \frac{1}{2} [\begin{matrix} 2 + 2 & 3 + 1 & 1 + 4 \\ 1 + 3 & - 1 - 1 & 2 + 1 \\ 4 + 1 & 1 + 2 & 2 + 2 \end{matrix}] = \frac{1}{2} [\begin{matrix} 4 & 4 & 5 \\ 4 & - 2 & 3 \\ 5 & 3 & 4 \end{matrix}] \\ = [\begin{matrix} 2 & 2 & \frac{5}{2} \\ 2 & - 1 & \frac{3}{2} \\ \frac{5}{2} & \frac{3}{2} & 2 \end{matrix}] \\ P^{'} = [\begin{matrix} 2 & 2 & \frac{5}{2} \\ 2 & - 1 & \frac{3}{2} \\ \frac{5}{2} & \frac{3}{2} & 2 \end{matrix}] = P \\ A s P^{'} = P ∴ P i s a s y m m e t r i c m a t r i x . \\ N o w, Q = \frac{1}{2} [A - A^{'}] \\ = \frac{1}{2} [(\begin{matrix} 2 & 3 & 1 \\ 1 & - 1 & 2 \\ 4 & 1 & 2 \end{matrix}) - (\begin{matrix} 2 & 1 & 4 \\ 3 & - 1 & 1 \\ 1 & 2 & 2 \end{matrix})] \\ = \frac{1}{2} [\begin{matrix} 2 - 2 & 3 - 1 & 1 - 4 \\ 1 - 3 & - 1 + 1 & 2 - 1 \\ 4 - 1 & 1 - 2 & 2 - 2 \end{matrix}] = \frac{1}{2} [\begin{matrix} 0 & 2 & - 3 \\ - 2 & 0 & 1 \\ 3 & - 1 & 0 \end{matrix}] \\ = [\begin{matrix} 0 & 1 & \frac{- 3}{2} \\ - 1 & 0 & \frac{1}{2} \\ \frac{3}{2} & \frac{- 1}{2} & 0 \end{matrix}] = - [\begin{matrix} 0 & - 1 & \frac{3}{2} \\ 1 & 0 & \frac{- 1}{2} \\ \frac{- 3}{2} & \frac{1}{2} & 0 \end{matrix}] = - Q \\ A s Q = - Q ∴ Q i s a s k e w s y m m e t r i c m a t r i x . \\ S o A = P + Q \\ A = [\begin{matrix} 2 & 2 & \frac{5}{2} \\ 2 & - 1 & \frac{3}{2} \\ \frac{5}{2} & \frac{3}{2} & 2 \end{matrix}] + [\begin{matrix} 0 & 1 & \frac{- 3}{2} \\ - 1 & 0 & \frac{1}{2} \\ \frac{3}{2} & \frac{- 1}{2} & 0 \end{matrix}] \\ = [\begin{matrix} 2 + 0 & 2 + 1 & \frac{5}{2} - \frac{3}{2} \\ 2 - 1 & - 1 + 0 & \frac{3}{2} + \frac{1}{2} \\ \frac{5}{2} + \frac{3}{2} & \frac{3}{2} &min \end{matrix} \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eight

The area of the region bounded by the curve x = 2y + 3 and the y lines. y = 1 and y = –1 is

(A) 4 sq units

(B) $\frac{3}{2}$ sq units

(C) 6 sq units

(D) 8 sq unit

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Payal Gupta | Contributor-Level 10

a year ago

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} G i v e n e q u a t i o n o f l i n e s a r e x = 2 y + 3, y = 1 a n d y = - 1 \\ ∴ R equired a r e a = \int_{- 1}^{1} (2 y + 3) d y \\ = 2 . \frac{1}{2} {[y^{2}]}_{- 1}^{1} + 3 {[y]}_{- 1}^{1} = (1 - 1) + 3 (1 + 1) = 6 s q . u n i t s \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eight

The area of the region bounded by the curve $y = x + 1$ and the lines $x = 2$ and $x = 3$ is:

(A) $\frac{7}{2}$   sq. units

(B) $\frac{9}{2}$   sq. units

(C) $\frac{11}{2}$   sq. units

(D) $\frac{13}{2}$   sq. units

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Answered by

Payal Gupta | Contributor-Level 10

a year ago

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} G i v e n e q u a t i o n o f l i n e s a r e y = x + 1, x = 2 a n d x = 3 \\ ∴ R equired a r e a = \int_{2}^{3} (x + 1) d x \\ = {[\frac{x^{2}}{2} + x]}_{2}^{3} = (\frac{9}{2} + 3) - (\frac{4}{2} + 2) \\ = \frac{15}{2} - 4 = \frac{7}{4} s q . u n i t s \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eight

The area of the region bounded by the curve <math> <mrow> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </math> and the lines <math> <mrow> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mrow> </math> and <math> <mrow> <mi>x</mi> <mo>=</mo> <mn>3</mn> </mrow> </math> is: (A) <math> <mrow> <mfrac> <mrow> <mn>7</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  sq. units (B) <math> <mrow> <mfrac> <mrow> <mn>9</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  sq. units (C) <math> <mrow> <mfrac> <mrow> <mn>1</mn> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  sq. units (D) <math> <mrow> <mfrac> <mrow> <mn>1</mn> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  sq. units

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eight

The area of the region bounded by the circle $x^{2} + y^{2} = 1$ is:

(A) $2 π$ sq. units

(B) $π$ sq. units

(C) $3 π$ sq. units

(D) $4 π$ sq. units

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Answered by

Payal Gupta | Contributor-Level 10

a year ago

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} G i v e n e q u a t i o n o f c i r c l e i s x^{2} + y^{2} = 1 \\ \Rightarrow y = \sqrt[]{1 - x^{2}} \\ ∴ Since t h e c i r c l e i s s y m m e t r i c a l a b o u t t h e a x e s . \\ ∴ R equired a r e a = 4 \times \int_{0}^{1} \sqrt[]{1 - x^{2}} d x \\ = 4 {[\frac{x}{2} \sqrt[]{1 - x^{2}} + \frac{1}{2} s i n^{- 1} x]}_{0}^{1} \\ = 4 [0 + \frac{1}{2} s i n^{- 1} (1) - 0 - 0] = 4 \times \frac{1}{2} \times \frac{π}{2} = π s q . u n i t s \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

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Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eight

The area of the region bounded by the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ is:

(A) $20 π$ sq. units

(B) $20 π^{2}$ sq. units

(C) $16 π^{2}$ sq. units

(D) $25 π$ sq. units

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Answered by

Payal Gupta | Contributor-Level 10

a year ago

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} G i v e n e q u a t i o n o f e l l i p s e i s \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 \\ \Rightarrow \frac{y^{2}}{16} = 1 - \frac{x^{2}}{25} \Rightarrow y^{2} = \frac{16}{25} (25 - x^{2}) \\ ∴ y = \frac{4}{5} \sqrt[]{25 - x^{2}} \\ ∴ Since t h e e l l i p s e i s s y m m e t r i c a l a b o u t t h e a x e s . \\ ∴ R equired a r e a = 4 \int_{0}^{5} \frac{4}{5} \sqrt[]{25 - x^{2}} d x = 4 \times \frac{4}{5} \int_{0}^{5} \sqrt[]{{(5)}^{2} - x^{2}} d x \\ = \frac{16}{5} {[\frac{x}{2} \sqrt[]{{(5)}^{2} - x^{2}} + \frac{25}{2} s i n^{- 1} \frac{x}{5}]}_{0}^{5} \\ = \frac{16}{5} [0 + \frac{25}{2} s i n^{- 1} (\frac{5}{5}) - 0 - 0] = \frac{16}{5} [\frac{25}{2} . s i n^{- 1} (1)] \\ = \frac{16}{5} [\frac{25}{2} . \frac{π}{2}] = 20 π s q . u n i t s \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

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Construct a $3 \times 2$ matrix whose elements are given by $a_{i j} = e^{i x} \cdot s i n (j x)$ .

Construct a_{2 × 2} matrix where

(i) $a_{i j} = \frac{{(i - 2 j)}^{2}}{2}$ .
(ii) $a_{i j} = ? - 2 i + 3 j ?$ .

In the matrix write:

(i) The order of the matrix .

(ii) The number of elements.

(iii) Write elements $a_{23}, a_{31}, a_{12}$ .

If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

Express the matrix $[\begin{matrix} 2 & 3 & 1 \\ 1 & - 1 & 2 \\ 4 & 1 & 2 \end{matrix}]$ as the sum of a symmetric and a skew-symmetric matrix.

The area of the region bounded by the curve x = 2y + 3 and the y lines. y = 1 and y = –1 is

(A) 4 sq units

(B) $\frac{3}{2}$ sq units

(C) 6 sq units

(D) 8 sq unit

The area of the region bounded by the curve $y = x + 1$ and the lines $x = 2$ and $x = 3$ is:

(A) $\frac{7}{2}$   sq. units

(B) $\frac{9}{2}$   sq. units

(C) $\frac{11}{2}$   sq. units

(D) $\frac{13}{2}$   sq. units

The area of the region bounded by the circle $x^{2} + y^{2} = 1$ is:

(A) $2 π$ sq. units

(B) $π$ sq. units

(C) $3 π$ sq. units

(D) $4 π$ sq. units

The area of the region bounded by the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ is:

(A) $20 π$ sq. units

(B) $20 π^{2}$ sq. units

(C) $16 π^{2}$ sq. units

(D) $25 π$ sq. units