Solution: Given, \[\begin{array}{*{35}{l}} a\text{ }=\text{ }4\surd 6/\text{ }\left( \surd 2\text{ }+\text{ }\surd 3 \right) \\ a/2\surd 2\text{ }=\text{ }2\surd 3/\text{ }\left( \surd 2\text{...

### A point P (a, b) is reflected in the x-axis to P’ (2, -3). Write down the values of a and b. P” is the image of P, reflected in the y-axis. Write down the co-ordinates of P”. Find the co-ordinates of P”’, when P is reflected in the line, parallel to y-axis, such that x = 4.

A point \[P\text{ }\left( a,\text{ }b \right)\] is reflected in the \[x-hub\] to \[P'\text{ }\left( 2,\text{ }-\text{ }3 \right).\] We realize that, \[{{M}_{x}}~\left( x,\text{ }y \right)\text{...

### If x = 6ab/ (a + b), find the value of:

Solution: Given, \[\begin{array}{*{35}{l}} x\text{ }=\text{ }6ab/\text{ }\left( a\text{ }+\text{ }b \right) \\ \Rightarrow \text{ }x/3a\text{ }=\text{ }2b/\text{ }a\text{ }+\text{ }b \\...

### If (7a + 8b) (7c – 8d) = (7a – 8b) (7c + 8d); Prove that a: b = c: d

### If (fig 1) Then prove that x: y = u: v

SOLUTION: \[\begin{array}{*{35}{l}} 10x/\text{ }12y\text{ }=\text{ }10u/\text{ }12v \\ {} \\ x/y\text{ }=\text{ }u/v\text{ }\Rightarrow \text{ }x:\text{ }y\text{ }=\text{ }u:\text{ }v \\...

### A point P (-2, 3) is reflected in line x = 2 to point P’. Find the coordinates of P’.

The line \[x\text{ }=\text{ }2\]is a line corresponding to \[y-hub\]and a ways off of \[2\text{ }units\]from it. We should check the point \[P\text{ }\left( -\text{ }2,\text{ }3 \right).\] From...

### Given, a/b = c/d, prove that: (3a – 5b)/ (3a + 5b) = (3c – 5d)(3c + 5d)

### If a : b = c : d, prove that: (6a + 7b) (3c – 4d) = (6c + 7d) (3a – 4b).

Since, a/b = c/d \[\left( 6a\text{ }+\text{ }7b \right)\left( 3c\text{ }-\text{ }4d \right)\text{ }=\text{ }\left( 3a\text{ }-\text{ }4b \right)\left( 6c\text{ }+\text{ }7d \right)\]

### The points P (4, 1) and Q (-2, 4) are reflected in line y = 3. Find the co-ordinates of P’, the image of P and Q’, the image of Q.

The line \[y\text{ }=\text{ }3\] is a line corresponding to \[x-hub\]and a good ways off of \[3\] \[units\]from it. We should stamp the focuses \[P\text{ }\left( 4,\text{ }1 \right)\]and \[Q\text{...

### The point (-2, 0) on reflection in a line is mapped to (2, 0) and the point (5, -6) on reflection in the same line is mapped to (-5, -6). (i) State the name of the mirror line and write its equation. (ii) State the co-ordinates of the image of (-8, -5) in the mirror line.

(I) We realize that, impression of a point \[\left( x,\text{ }y \right)\]in \[y-hub\] is \[\left( -\text{ }x,\text{ }y \right).\] Thus, the point \[\left( -\text{ }2,\text{ }0 \right)\] when...

### Name a single transformation that maps P’ to P”.

Single change that maps \[P\text{ }to\text{ }P\]is the appearance in beginning.

### If a : b = c : d, prove that: (iii) xa + yb : xc + yd = b : d.

Since,a/b = c/d

### (i) Point P (a, b) is reflected in the x-axis to P’ (5, -2). Write down the values of a and b. (ii) P” is the image of P when reflected in the y-axis. Write down the co-ordinates of P”.

(I) As, \[{{M}_{x}}~\left( x,\text{ }y \right)\text{ }=\text{ }\left( x,\text{ }-y \right)\] \[P\text{ }\left( 5,\text{ }-2 \right)\text{ }=\text{ }reflection\text{ }of\text{ }P\text{ }\left(...

### If a : b = c : d, prove that: (i) 5a + 7b : 5a – 7b = 5c + 7d : 5c – 7d. (ii) (9a + 13b) (9c – 13d) = (9c + 13d) (9a – 13b).

(i) since, a/b = c/d (ii) since, a/b = c/d On cross-multiplication we have, \[\left( 9a\text{ }+\text{ }13b \right)\left( 9c\text{ }-\text{ }13d \right)\text{ }=\text{ }\left( 9c\text{ }+\text{ }13d...

### Points (3, 0) and (-1, 0) are invariant points under reflection in the line L1; points (0, -3) and (0, 1) are invariant points on reflection in line L2. (i)Write down the images of P and Q on reflection in L2. Name the images as P” and Q” respectively. (ii) State or describe a single transformation that maps P’ onto P”.

(i) \[P\text{ }=\text{ }Image\text{ }of\text{ }P\text{ }\left( 3,\text{ }4 \right)\text{ }in\text{ }{{L}_{2}}~=\text{ }\left( -3,\text{ }4 \right)\] Also, \[Q\text{ }=\text{ }Image\text{ }of\text{...

### What least number must be subtracted from each of the numbers 7, 17 and 47 so that the remainders are in continued proportion?

Let the number subtracted to be x. therefore, \[\begin{array}{*{35}{l}} \left( 7\text{ }-\text{ }x \right):\text{ }\left( 17\text{ }-\text{ }x \right)::\text{ }\left( 17\text{ }-\text{ }x...

### Points (3, 0) and (-1, 0) are invariant points under reflection in the line L1; points (0, -3) and (0, 1) are invariant points on reflection in line L2. (i) Name or write equations for the lines L1 and L2. (ii) Write down the images of the points P (3, 4) and Q (-5, -2) on reflection in line L1. Name the images as P’ and Q’ respectively.

(I) We realize that, each point in a line is invariant under the appearance in a similar line. As the focuses \[\left( 3,\text{ }0 \right)\]and \[\left( -\text{ }1,\text{ }0 \right)\]lie on the...

### If a/b = c/d, show that:

SOLUTION: Let a/b = c/d = k Therefore, a = bk and c = dk Taking L.H.S, Now, taking the R.H.S Thus, L.H.S = R.H.S

### If a, b, c are in continued proportion and a(b – c) = 2b, prove that: a – c = 2(a + b)/ a

a, b, c are in continued proportion. So, \[\begin{array}{*{35}{l}} a/b\text{ }=\text{ }b/c \\ \Rightarrow \text{ }{{b}^{2}}~=\text{ }ac \\ \end{array}\] And, \[\begin{array}{*{35}{l}} a\left(...

### If a, b, c are in continued proportion, show that

SOLUTION: a, b, c are in continued proportion. So, \[\begin{array}{*{35}{l}} a/b\text{ }=\text{ }b/c \\ \Rightarrow \text{ }{{b}^{2}}~=\text{ }ac \\ \end{array}\] \[\begin{array}{*{35}{l}} \left(...

### What least number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?

Let the number added to be x. therefore, \[\begin{array}{*{35}{l}} \left( 6\text{ }+\text{ }x \right):\text{ }\left( 15\text{ }+\text{ }x \right)\text{ }::\text{ }\left( 20\text{ }+\text{ }x...

### If x^2, 4 and 9 are in continued proportion, find x.

x2, 4 and 9 are in continued proportion So, we have \[\begin{array}{*{35}{l}} {{x}^{2}}/4\text{ }=\text{ }4/9 \\ {{x}^{2}}~=\text{ }16/9 \\ Thus,\text{ }x\text{ }=\text{ }4/3 \\...

### If x + 5 is the mean proportional between x + 2 and x + 9; find the value of x. Solution:

x + 5 is the mean proportional between x + 2 and x + 9. So, (x + 2), (x + 5) and (x + 9) are in continued proportion. \[\begin{array}{*{35}{l}} \left( x\text{ }+\text{ }2 \right):\text{ }\left(...

### Write down: (i) the image of A” of A, when A is reflected in the origin. (ii) the single transformation that maps A’ to A”.

(i) \[A\text{ }=\text{ }\left( -3,\text{ }-2 \right)\] (iv) Single change that maps \[A'\text{ }to\text{ }A''\]is the appearance in y-hub.

### Find the mean proportional between: (i) 6 + 3√3 and 8 – 4√3 (ii) a – b and a^3 – a^2b

(i) Let the mean proportional between \[6\text{ }+\text{ }3\surd 3~and\text{ }8\text{ }-\text{ }4\surd 3~\] be x. So, \[6\text{ }+\text{ }3\surd 3,\text{ }x\text{ }and\text{ }8\text{ }-\text{...

### Write down: (i) the geometrical name of the figure ABB’A’; (ii) the measure of angle ABB’;

According to the given question the solution is (I) The geometrical name From the diagram, it's obviously seen that \[ABBA\]is an isosceles trapezium. (ii) The measure of angle The proportion of...

### Find the third proportional to: (i) (ii) a – b and a^2 – b^2

(i) take the third proportional to and 4 be x. So, , 4, x are in continued proportion. \[\begin{array}{*{35}{l}} 8/3:\text{ }4\text{ }=\text{ }4:\text{ }x \\ \left( 8/3 \right)/\text{ }4\text{...

### Find the fourth proportional to: (i) 1.5, 4.5 and 3.5 (ii) 3a, 6a^2 and 2ab^2

(i) Let he fourth proportional to 1.5, 4.5 and 3.5 be x. \[\begin{array}{*{35}{l}} 1.5:\text{ }4.5\text{ }=\text{ }3.5:\text{ }x \\ 1.5~*~x\text{ }=\text{ }3.5~\times *4.5 \\ x\text{ }=\text{...

### Attempt this question on graph paper. (a) Plot A (3, 2) and B (5, 4) on graph paper. Take 2 cm = 1 unit on both the axes. (b) Reflect A and B in the x-axis to A’ and B’ respectively. Plot these points also on the same graph paper.

Solution: According to the question given the graph of first and second question is

### By increasing the cost of entry ticket to a fair in the ratio 10: 13, the number of visitors to the fair has decreased in the ratio 6: 5. In what ratio has the total collection increased or decreased?

Let take the cost of the entry ticket initially and at present to be 10x and 13x respectively. And let the number of visitors initially and at present be 6y and 5y respectively. Therefore,...

### The bus fare between two cities is increased in the ratio 7: 9. Find the increase in the fare, if: (i) the original fare is Rs 245; (ii) the increased fare is Rs 207.

From the question we have, Increased (new) bus fare = (9/7) x original bus fare (i) Increased (new) bus fare= \[=~9/7\text{ }*\text{ }Rs\text{ }245\text{ }=\text{ }Rs\text{ }315\] Thus, the increase...

### A point P is reflected in the origin. Co-ordinates of its image are (-2, 7). (i) Find the co-ordinates of P. (ii) Find the co-ordinates of the image of P under reflection in the x-axis.

(I) As, \[{{M}_{O}}~\left( 2,\text{ }-7 \right)\text{ }=\text{ }\left( -2,\text{ }7 \right)\] Thus, the co-ordinates of \[P\]are \[\left( 2,\text{ }-7 \right).\] (ii) Co-ordinates of the...

### A point P is reflected in the x-axis. Co-ordinates of its image are (-4, 5). (i) Find the co-ordinates of P. (ii) Find the co-ordinates of the image of P under reflection in the y-axis.

(I) As, \[{{M}_{x}}~\left( -4,\text{ }-5 \right)\text{ }=\text{ }\left( -4,\text{ }5 \right)\] Thus, the co-ordinates of \[P\]are \[\left( -4,\text{ }-5 \right).\] (ii) Co-ordinates of the...

### State the co-ordinates of the following points under reflection in the line y = 0: (-1, -3)

\[\left( -\text{ }1,\text{ }-\text{ }3 \right)\] The co-ordinate of the given point under appearance in the line \[y\text{ }=\text{ }0\] is \[\left( -\text{ }1,\text{ }3 \right).\]

### The work done by (x – 2) men in (4x + 1) days and the work done by (4x + 1) men in (2x – 3) days are in the ratio 3: 8. Find the value of x.

On assuming that the same amount of work is done one day by all the men and one day work of each man = 1 units, we get Amount of work done by (x – 2) men in (4x + 1) days = Amount of work done by (x...

### State the co-ordinates of the following points under reflection in the line y = 0: (i) (-3, 0) (ii) (8, -5)

(I) \[\left( -\text{ }3,\text{ }0 \right)\] The co-ordinate of the given point under appearance in the line \[y\text{ }=\text{ }0\] is \[~\left( -\text{ }3,\text{ }0 \right).\] (ii) \[\left(...

### State the co-ordinates of the following points under reflection in the line x = 0: (3, -4)

\[\left( 3,\text{ }-\text{ }4 \right)\] The co-ordinates of the given point under appearance in the line \[x\text{ }=\text{ }0\]are \[\left( -\text{ }3,\text{ }-\text{ }4 \right).\]

### The monthly pocket money of Ravi and Sanjeev are in the ratio 5: 7. Their expenditures are in the ratio 3: 5. If each saves Rs. 80 every month, find their monthly pocket money.

The pocket money of Ravi and Sanjeev are in the ratio 5: 7 Thus, we assume the pocket money of Ravi as 5k and that of Sanjeev as 7k. Also, The expenditure of Ravi and Snajeev are in the ratio 3: 5...

### State the co-ordinates of the following points under reflection in the line x = 0: (i) (-6, 4) (ii) (0, 5)

(I) \[\left( -\text{ }6,\text{ }4 \right)\] The co-ordinates of the given point under appearance in the line \[x\text{ }=\text{ }0\]are \[\left( 6,\text{ }4 \right).\] (ii) \[\left( 0,\text{ }5...

### A school has 630 students. The ratio of the number of boys to the number of girls is 3: 2. This ratio changes to 7: 5 after the admission of 90 new students. Find the number of newly admitted boys.

Let the number of boys be 3x. Then, the number of girls = 2x \[\begin{array}{*{35}{l}} \Rightarrow \text{ }3x\text{ }+\text{ }2x\text{ }=\text{ }630 \\ 5x\text{ }=\text{ }630 \\ x\text{ }=\text{...

### State the co-ordinates of the following points under reflection in origin: (0, 0)

\[\left( 0,\text{ }0 \right)\] The co-ordinates of the given point under appearance in beginning are \[\left( 0,\text{ }0 \right).\]

### State the co-ordinates of the following points under reflection in origin: (i) (-2, -4) (ii) (-2, 7)

(I) \[\left( -\text{ }2,\text{ }-\text{ }4 \right)\] The co-ordinates of the given point under appearance in beginning are \[\left( 2,\text{ }4 \right).\] (ii) \[\left( -\text{ }2,\text{ }7...

### State the co-ordinates of the following points under reflection in y-axis: (-8, -2)

\[\left( -\text{ }8,\text{ }-\text{ }2 \right)\] The co-ordinates of the given point under appearance in the y-pivot are \[\left( 8,\text{ }-\text{ }2 \right).\]

### Divide Rs 1290 into A, B and C such that A is 2/5 of B and B: C = 4: 3.

B: C = 4: 3 so, B/C = 4/3 ⇒ C = (3/4) B And, A = (2/5) B Since, \[\begin{array}{*{35}{l}} A\text{ }+\text{ }B\text{ }+\text{ }C\text{ }=\text{ }Rs\text{ }1290 \\ \left( 2/5 \right)\text{ } B\text{...

### State the co-ordinates of the following points under reflection in y-axis: (i) (6, -3) (ii) (-1, 0)

(I) \[\left( 6,\text{ }-\text{ }3 \right)\] The co-ordinates of the given point under appearance in the y-pivot are \[\left( -\text{ }6,\text{ }-\text{ }3 \right).\] (ii) \[\left( -\text{ }1,\text{...

### If the ratio between 8 and 11 is the same as the ratio of 2x – y to x + 2y, find the value of 7x/ 9y.

\[\left( 2x\text{ }-\text{ }y \right)/\text{ }\left( x\text{ }+\text{ }2y \right)\text{ }=\text{ }8/11\] On cross multiplying, we get \[\begin{array}{*{35}{l}} 11\left( 2x\text{ }-\text{ }y...

### Find x/y; when x^2 + 6y^2 = 5xy

Given, \[{{x}^{2}}~+\text{ }6{{y}^{2}}~=\text{ }5xy\] Dividing by y2 both side, we have Let \[\begin{array}{*{35}{l}} x/y\text{ }=\text{ }a \\ =>\text{ }{{a}^{2}}~\text{ }-5a\text{ }+\text{...

### SOLVE:

SOLUTION: Since, \[\begin{array}{*{35}{l}} 3\left( m\text{ }+\text{ }n \right)\text{ }=\text{ }2\left( m\text{ }+\text{ }3n \right) \\ 3m\text{ }+\text{ }3n\text{ }=\text{ }2m\text{ }+\text{ }6n ...

### What quantity must be subtracted from each term of the ratio 9: 17 to make it equal to 1: 3?

Let x be subtracted from each term of the ratio 9: 17. 27 – 3x = 17 – x 10 = 2x x = 5 Therefore, the required number that should be subtracted is 5.

### State the co-ordinates of the following points under reflection in x-axis: (0, 0)

\[\left( 0,\text{ }0 \right)\] The co-ordinates of the given point under appearance in the x-hub are \[\left( 0,\text{ }0 \right).\]

### State the co-ordinates of the following points under reflection in x-axis: (i) (3, 2) (ii) (-5, 4)

(I) \[\left( 3,\text{ }2 \right)\] The co-ordinates of the given point under appearance in the x-hub are \[\left( 3,\text{ }-2 \right).\] (ii) \[\left( -\text{ }5,\text{ }4 \right)\] The...

### A point P is its own image under the reflection in a line l. Describe the position of point the P with respect to the line l.

As, the picture of the point P is a similar point under the appearance in the line l we can say, point P is an invariant point. Subsequently, the situation of point P stays unaltered.

### Complete the following table:

Answer: As per question,

### The point P (5, -4) divides the line segment AB, as shown in the figure, in the ratio 2: 5. Find the co-ordinates of points A and B. Given AP is smaller than BP

Solution: From the graph given, It's clear that \[Point\text{ }A\] lies on\[x-pivot\]. So, its co-ordinates can be \[A\text{ }\left( x,\text{ }0 \right).\] \[Point\text{ }B\] lies on\[y-hub\]....

### Calculate the ratio in which the line joining A (6, 5) and B (4, -3) is divided by the line y = 2.

We realize that, The co-ordinates of each point on the line \[y\text{ }=\text{ }2\]will be of the sort \[\left( x,\text{ }2 \right).\] Thus, by area equation, we have \[2{{m}_{1}}~+\text{...

### Calculate the ratio in which the line joining the points (-3, -1) and (5, 7) is divided by the line x = 2. Also, find the co-ordinates of the point of intersection.

We realize that, The co-ordinates of each point on the line \[x\text{ }=\text{ }2\] will be of the kind \[\left( 2,\text{ }y \right).\] So from segment recipe, we have \[x\text{ }=\text{...

### P is a point on the line joining A (4, 3) and B (-2, 6) such that 5AP = 2BP. Find the co-ordinates of P.

\[5AP\text{ }=\text{ }2BP\] In this way, \[AP/BP\text{ }=\text{ }2/5\] Subsequently, the co-ordinates of the point \[P\]are \[\left( \left( 2x\left( -2 \right)\text{ }+\text{ }5\times 4...

### The line joining the points A (-3, -10) and B (-2, 6) is divided by the point P such that PB/AB = 1/5 Find the co-ordinates of P.

Let the directions of point \[P\]alone taken as \[\left( x,\text{ }y \right).\] Given, \[PB:\text{ }AB\text{ }=\text{ }1:\text{ }5\] In this way, \[PB:\text{ }PA\text{ }=\text{ }1:\text{ }4\]...

### Points A, B, C and D divide the line segment joining the point (5, -10) and the origin in five equal parts. Find the co-ordinates of A, B, C and D.

Answer: According to question,

### Find the ratio in which the join of (-4, 7) and (3, 0) is divided by the y-axis. Also, find the coordinates of the point of intersection.

We should accept \[S\text{ }\left( 0,\text{ }y \right)\]be the point on \[y-pivot\] what separates the line portion \[PQ\]in the proportion \[k:\text{ }1.\] Then, at that point, by segment equation,...

### In what ratio is the join of (4, 3) and (2, -6) divided by the x-axis. Also, find the co-ordinates of the point of intersection.

We should accept the point \[P\text{ }\left( x,\text{ }0 \right)\]on x-hub separates the line portion joining \[A\text{ }\left( 4,\text{ }3 \right)\text{ }and\text{ }B\text{ }\left( 2,\text{...

### In what ratio does the point (a, 6) divide the join of (-4, 3) and (2, 8)? Also, find the value of a.

We should accept the point \[P\text{ }\left( a,\text{ }6 \right)\]partitions the line portion joining \[A\text{ }\left( -\text{ }4,\text{ }3 \right)\text{ }and\text{ }B\text{ }\left( 2,\text{ }8...

### In what ratio does the point (1, a) divided the join of (-1, 4) and (4, -1)? Also, find the value of a.

How about we expect the point \[P\text{ }\left( 1,\text{ }a \right)\]partition the line portion AB in the proportion \[k:\text{ }1.\] Then, at that point, by segment recipe, we have \[1\text{...

### In what ratio is the line joining (2, -4) and (-3, 6) divided by the y-axis.

How about we expect the line joining focuses \[A\left( 2,\text{ }-\text{ }4 \right)\text{ }and\text{ }B\left( -\text{ }3,\text{ }6 \right)\]be isolated by point \[P\text{ }\left( 0,\text{ }y...

### In what ratio is the line joining (2, -3) and (5, 6) divided by the x-axis.

How about we accept the joining focuses as \[A\left( 2,\text{ }-\text{ }3 \right)\text{ }and\text{ }B\left( 5,\text{ }6 \right)\]be separated by point \[P\left( x\text{ },0 \right)\]in the...

### Calculate the co-ordinates of the point P which divides the line segment joining: (i) A (1, 3) and B (5, 9) in the ratio 1: 2. (ii) A (-4, 6) and B (3, -5) in the ratio 3: 2.

(I) Let's expect the co-ordinates of the point \[P\text{ }be\text{ }\left( x,\text{ }y \right)\] Then, at that point, by segment recipe, we have \[P\left( x,\text{ }y \right)\text{ }=\text{...

### In the given figure, ∆ ABC and ∆ AMP are right angled at B and M respectively. Given AC = 10 cm, AP = 15 cm and PM = 12 cm. (i) ∆ ABC ~ ∆ AMP. (ii) Find AB and BC.

(I) In\[\vartriangle \text{ }ABC\text{ }and\text{ }\vartriangle \text{ }AMP\], we have \[\angle BAC\text{ }=\angle PAM\text{ }\left[ Common \right]\] \[\angle ABC\text{ }=\text{ }\angle PMA\text{...

### D is a point on the side BC of triangle ABC such that angle ADC is equal to angle BAC. Prove that: CA^2 = CB x CD.

Answer: In \[\Delta \text{ }ADC\text{ }and\text{ }\Delta \text{ }BAC,\] \[\angle ADC\text{ }=\angle BAC\text{ }\left[ Given \right]\] \[\angle ACD\text{ }=\angle ACB\text{ }\left[ Common \right]\]...

### Given: ∠GHE = ∠DFE = 90o, DH = 8, DF = 12, DG = 3x – 1 and DE = 4x + 2. Find: the lengths of segments DG and DE.

Answer: In\[\Delta \text{ }DHG\text{ }and\text{ }\Delta \text{ }DFE\], \[\angle GHD\text{ }=\text{ }\angle DFE\text{ }=\text{ }{{90}^{o}}\] \[\angle D\text{ }=\angle D\text{ }\left[ Common \right]\]...

### State, true or false: The diagonals of a trapezium, divide each other into proportional segments.

True

### State, true or false: (i)All isosceles triangles are similar. (ii) Two isosceles-right triangles are similar.

(i) True (ii) True

### State, true or false: (ii) All equiangular triangles are similar. (ii) All isosceles triangles are similar.

(i) True (ii) False

### State, true or false: (i) Two similar polygons are necessarily congruent. (ii) Two congruent polygons are necessarily similar.

(i) False (ii) True

### Angle BAC of triangle ABC is obtuse and AB = AC. P is a point in BC such that PC = 12 cm. PQ and PR are perpendiculars to sides AB and AC respectively. If PQ = 15 cm and PR = 9 cm; find the length of PB.

In \[\Delta \text{ }ABC,\] \[AC\text{ }=\text{ }AB\text{ }\left[ Given \right]\] In this way, \[\angle ABC\text{ }=\angle ACB\][Angles inverse to rise to sides are equal.] In \[\Delta \text{...

### In the given figure, AB ‖ DC, BO = 6 cm and DQ = 8 cm; find: BP x DO.

Answer: In \[\Delta \text{ }DOQ\text{ }and\text{ }\Delta \text{ }BOP,\] \[\angle QDO\text{ }=\angle PBO\] \[\left[ As\text{ }AB\text{ }\left| \left| \text{ }DC\text{ }in\text{ }this\text{...

### In the given figure, AD = AE and AD^2 = BD x EC. Prove that: triangles ABD and CAE are similar.

Answer: In \[\Delta \text{ }ABD\text{ }and\text{ }\Delta \text{ }CAE,\] \[\angle ADE\text{ }=\angle AED\text{ }\left[ Angles\text{ }inverse\text{ }to\text{ }rise\text{ }to\text{ }sides\text{...

### In the given figure, DE ‖ BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm. (i) Write all possible pairs of similar triangles. (ii) Find the lengths of ME and DM.

Answer: (I) In\[\Delta \text{ }AME\text{ }and\text{ }\Delta \text{ }ANC\], \[\angle AME\text{ }=\angle ANC\] \[\left[ Since\text{ }DE\text{ }\left| \left| \text{ }BC\text{ }in\text{ }this\text{...

### In Δ ABC, BM ⊥ AC and CN ⊥ AB; show that:

Answer: In \[\Delta \text{ }ABM\text{ }and\text{ }\Delta \text{ }ACN,\] \[\angle AMB\text{ }=\text{ }\angle ANC\] [Since, BM ⊥ AC and CN ⊥ AB] \[\angle BAM\text{ }=\angle CAN\] [Common angle] Thus,...

### In Δ ABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that : (i) CB : BA = CP : PA (ii) AB x BC = BP x CA

(I) In\[\Delta \text{ }ABC\], we have \[\angle ABC\text{ }=\text{ }2\angle ACB\][Given] Presently, let \[\angle ACB\text{ }=\text{ }x\] In this way, \[\angle ABC\text{ }=\text{ }2x\] Likewise given,...

### In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that: (i) Δ AOB is similar to Δ COD. (ii) OA x OD = OB x OC.

(I) Given, \[AO\text{ }=\text{ }2CO\text{ }and\text{ }BO\text{ }=\text{ }2DO,\] \[AO/CO\text{ }=\text{ }2/1\text{ }=\text{ }BO/DO\] What's more, \[\angle AOB\text{ }=\angle DOC\] [Vertically inverse...

### P is a point on side BC of a parallelogram ABCD. If DP produced meets AB produced at point L, prove that: (i) DP : PL = DC : BL. (ii) DL : DP = AL : DC.

(I) As\[AD||BC\], we have \[AD||\text{ }BP\] too. Along these lines, by \[BPT\] \[DP/PL\text{ }=\text{ }AB/BL\] What's more, since \[ABCD\]is a parallelogram, \[AB\text{ }=\text{ }DC\] Consequently,...

### In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that: (i) Δ APB is similar to Δ CPD. (ii) PA x PD = PB x PC.

(I) In\[\vartriangle APB\text{ }and\text{ }\vartriangle CPD\], we have \[\angle APB\text{ }=\angle CPD\] [Vertically inverse angles] \[\angle ABP\text{ }=\angle CDP\] [Alternate points as, AB||DC]...

### In the figure, given below, straight lines AB and CD intersect at P; and AC || BD. Prove that: (i) ∆APC and ∆BPD are similar. (ii) If BD = 2.4 cm, AC = 3.6 cm, PD = 4.0 cm and PB = 3.2 cm; find the lengths of PA and PC.

Answer: (I) In \[\vartriangle \mathbf{APC}\text{ }\mathbf{and}\text{ }\vartriangle \mathbf{BPD}\]we have \[\angle APC\text{ }=\angle BPD\] [Vertically inverse angles] \[\angle ACP\text{ }=\angle...

### Does the line 3x – 5y = 6 bisect the join of (5, -2) and (-1, 2)?

It's realized that the given line will divide the join of \[A\text{ }\left( 5,\text{ }-\text{ }2 \right)\text{ }and\text{ }B\text{ }\left( -\text{ }1,\text{ }2 \right),\] if the co-ordinates of the...

### The line 3x/5 – 2y/3 + 1 = 0 contains the point (m, 2m – 1); calculate the value of m.

The condition of the given line is \[3x/5\text{ }-\text{ }2y/3\text{ }+\text{ }1\text{ }=\text{ }0\] On putting\[x\text{ }=\text{ }m,\text{ }y\text{ }=\text{ }2m\text{ }\text{ }1\], we have...

### For what value of k will the point (3, -k) lie on the line 9x + 4y = 3?

The given line condition is \[9x\text{ }+\text{ }4y\text{ }=\text{ }3.\] On putting\[x\text{ }=\text{ }3\text{ }and\text{ }y\text{ }=\text{ }-\text{ }k\], we have \[9\left( 3 \right)\text{ }+\text{...

### The line given by the equation 2x – y/3 = 7 passes through the point (k, 6); calculate the value of k.

Given line condition is \[2x\text{ }-\text{ }y/3\text{ }=\text{ }7\]goes through the \[point\text{ }\left( k,\text{ }6 \right).\] Thus, on subbing \[x\text{ }=\text{ }k\text{ }and\text{ }y\text{...

### State, true or false: if the point (2, a) lies on the line 2x – y = 3, then a = 5.

The \[point\text{ }\left( 2,\text{ }a \right)\] lies on the line \[2x\text{ }\text{ }y\text{ }=\text{ }3.\] Subbing \[x\text{ }=\text{ }2\text{ }and\text{ }y\text{ }=\text{ }a\] in the given...

### State, true or false: (i) the point (8, 7) lies on the line y – 7 = 0. (ii) the point (-3, 0) lies on the line x + 3 = 0.

(i) The given line is \[y\text{ }-\text{ }7\text{ }=\text{ }0\] Subbing \[y\text{ }=\text{ }7\]in the given condition, \[L.H.S\text{ }=\text{ }y\text{ }-\text{ }7\text{ }=\text{ }7\text{ }-\text{...

### State, true or false: (i) the line x/2 + y/3 = 0 passes through the point (2, 3). (ii) the line x/2 + y/3 = 0 passes through the point (4, -6).

(I) The given line is \[x/2\text{ }+\text{ }y/3\text{ }=\text{ }0\] Subbing \[x\text{ }=\text{ }2\text{ }and\text{ }y\text{ }=\text{ }3\]in the given condition, \[L.H.S\text{ }=\text{ }2/2\text{...

### Find, which of the following points lie on the line x – 2y + 5 = 0:

Given line condition is \[x\text{ }-\text{ }2y\text{ }+\text{ }5\text{ }=\text{ }0.\] (i) On subbing \[x\text{ }=\text{ }2\text{ }and\text{ }y\text{ }=\text{ }-\text{ }1.5\] in the given line...

### Find, which of the following points lie on the line x – 2y + 5 = 0: (i) (-5, 0) (ii) (5, 5)

Given line condition is \[x\text{ }\text{ }2y\text{ }+\text{ }5\text{ }=\text{ }0.\] (i) On subbing \[x\text{ }=\text{ }-\text{ }5\text{ }and\text{ }y\text{ }=\text{ }0\]in the given line condition,...

### Find, which of the following points lie on the line x – 2y + 5 = 0: (i) (1, 3) (ii) (0, 5)

Given line condition is \[x\text{ }\text{ }2y\text{ }+\text{ }5\text{ }=\text{ }0\] (I) On subbing \[x\text{ }=\text{ }1\text{ }and\text{ }y\text{ }=\text{ }3\]in the given line condition, we have...

### Find the nth term of the series: 1, 2, 4, 8, ……..

It's seen that, the initial term is \[\left( a \right)\text{ }=\text{ }1\] What's more, typical ratio \[\left( r \right)\text{ }=\text{ }2/\text{ }1\text{ }=\text{ }2\] We realize that, the overall...

### Find the 10th term of the G.P. :

Answer It can be written as \[12,\text{ }4,\text{ }4/3,\text{ }\ldots ..\] It's seen that, the initial term is \[\left( a \right)\text{ }=\text{ }12\] What's more, typical ratio \[\left( r...

### Find the 8th term of the sequence:

Answer It's seen that, the initial term is \[\left( a \right)\text{ }=\text{ }1\] What's more, typical ratio \[\left( r \right)\text{ }=\text{ }\surd 3/1\text{ }=\text{ }\surd 3\] We realize that,...

### Find the 9th term of the series: 1, 4, 16, 64, …..

It's seen that, the initial term is \[\left( a \right)\text{ }=\text{ }1\] What's more, typical ratio\[\left( r \right)\text{ }=\text{ }4/1\text{ }=\text{ }4\] We realize that, the overall term is...

### Find which of the following sequence form a G.P.: 9, 12, 16, 24, ………

Given arrangement: \[9,\text{ }12,\text{ }16,\text{ }24,\text{ }\ldots ...\] Since, \[12/9\text{ }=\text{ }4/3;\text{ }16/12\text{ }=\text{ }4/3;\text{ }24/16\text{ }=\text{ }3/2\] \[12/9\text{...

### Find which of the following sequence form a G.P.: (i) 8, 24, 72, 216, ……… (ii) 1/8, 1/24, 1/72, 1/216, ………

(i) Given arrangement:\[~8,\text{ }24,\text{ }72,\text{ }216,\text{ }\ldots \text{ }\ldots \] Since, \[24/8\text{ }=\text{ }3,\text{ }72/24\text{ }=\text{ }3,\text{ }216/72\text{ }=\text{ }3\]...

### How many terms are there in the series : 3/4, 1, 1 ¼, ……., 3?

Given series, \[3/4,\text{ }1,\text{ }1\text{ }{\scriptscriptstyle 1\!/\!{ }_4},\text{ }\ldots \text{ }.,\text{ }3\] Here, \[a\text{ }=\text{ }3/4~\] and \[d\text{ }=\text{ }1\text{ }-\text{...

### How many terms are there in the series : (i) 4, 7, 10, 13, …………, 148? (ii) 0.5, 0.53, 0.56, ……………, 1.1?

Given series, \[4,\text{ }7,\text{ }10,\text{ }13,\text{ }\ldots \text{ }\ldots \text{ },\text{ }148\] Here, \[a\text{ }=\text{ }4\text{ }and\text{ }d\text{ }=\text{ }7\text{ }\text{ }4\text{...

### Find the common difference and 99th term of the arithmetic progression:

Answer: \[i.e.,\text{ }31/4,\text{ }19/2,\text{ }45/4,\text{ }\ldots \ldots .\] So, \[a\text{ }=\text{ }31/4\] Common difference \[d\text{ }=\text{ }19/2\text{ }\text{ }31/4\text{ }=\text{ }\left(...

### Is 402 a term of the sequence: 8, 13, 18, 23,………….?

Given arrangement, \[8,\text{ }13,\text{ }18,\text{ }23,\ldots \ldots \ldots \ldots .\] \[d\text{ }=\text{ }13\text{ }\text{ }8\text{ }=\text{ }5\text{ }and\text{ }a\text{ }=\text{ }8\] General...

### Find the 50th term of the sequence: 1/n, (n+1)/n, (2n+1)/n, ……

Given arrangement, \[1/n,\text{ }\left( n+1 \right)/n,\text{ }\left( 2n+1 \right)/n,\text{ }\ldots \ldots \] Hence, \[a\text{ }=\text{ }1/n\] \[d\text{ }=\text{ }\left( n+1 \right)/n\text{ }\text{...

### Find the 100th term of the sequence: √3, 2√3, 3√3, ….

Given arrangement, \[\surd 3,\text{ }2\surd 3,\text{ }3\surd 3,\text{ }\ldots .\] Hence, \[a\text{ }=\text{ }\surd 3\] \[d\text{ }=\text{ }2\surd 3\text{ }\text{ }\surd 3\text{ }=\text{ }\surd 3\]...

### Find the 30th term of the sequence: 1/2, 1, 3/2, …….

Given arrangement, \[1/2,\text{ }1,\text{ }3/2,\text{ }\ldots \ldots .\] So, \[a\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\] \[d\text{ }=\text{ }1\text{ }-\text{ }{\scriptscriptstyle 1\!/\!{...

### Find the 24th term of the sequence: 12, 10, 8, 6,……

Given arrangement, \[12,\text{ }10,\text{ }8,~6,\ldots \ldots \] The normal distinction: \[10\text{ }-\text{ }12\text{ }=\text{ }-\text{ }2\] \[8\text{ }-\text{ }10\text{ }=\text{ }-\text{ }2\]...

### If the pth term of an A.P. is (2p + 3); find the A.P.

Given, \[{{p}^{th}}~\]term of an \[A.P.\text{ }=\text{ }\left( 2p\text{ }+\text{ }3 \right)\] In this way, on putting \[p\text{ }=\text{ }1,\text{ }2,\text{ }3,\text{ }\ldots \text{ },\]we have...

### The nth term of sequence is (2n – 3), find its fifteenth term.

Given, \[{{n}^{th}}\]term of arrangement is \[\left( 2n\text{ }\text{ }3 \right)\] Along these lines, the fifteenth term is when \[n\text{ }=\text{ }15\] \[{{t}_{15~}}=\text{ }2\left( 15...

### Which of the following sequences are in arithmetic progression? (i) 5, 9, 12, 18, …. (ii) 1/2, 1/3, 1/4, 1/5, ….

(i) \[5,\text{ }9,\text{ }12,\text{ }18,\text{ }\ldots .\] In the event that two networks are supposed to be equivalent, their comparing components are additionally equivalent. \[{{d}_{1}}~=\text{...

### Which of the following sequences are in arithmetic progression? (i) 2, 6, 10, 14, …. (ii) 15, 12, 9, 6, ….

(i) \[2,\text{ }6,\text{ }10,\text{ }14,\text{ }\ldots .\] In the event that two networks are supposed to be equivalent, their comparing components are additionally equivalent. \[{{d}_{1}}~=\text{...

### Find : (i) A + B – C (ii) A – B +C

If A= , B= , C= Answer (i) (ii)

### Find: (i) B + C (ii) A – C

If A= , B = , C= Answer (i) \[B\text{ }+\text{ }C\text{ }=~\] (ii) \[A\text{ }-\text{ }C\text{ }=\]

### If A = [8 -3] and B = [4 -5]; find: (i) A + B (ii) B – A

(i) \[A\text{ }+\text{ }B\text{ }=\text{ }\left[ 8\text{ }-3 \right]\text{ }+\text{ }\left[ 4\text{ }-5 \right]\] \[~=\text{ }\left[ 8+4\text{ }-3-5 \right]\text{ }=\text{ }\left[ 12\text{ }-8...

### Solve for a, b and c if

(i) (ii) Answer:- In the event that two networks are supposed to be equivalent, their comparing components are additionally equivalent. In this manner, (i) \[a\text{ }+\text{ }5\text{ }=\text{...

### In the given figure find x, y and z.

Answer Two matrices are said to be equal, when their corresponding elements are also equal. So, \[x\text{ }=\text{ }3,\] \[y\text{ }+\text{ }2\text{ }=\text{ }1~so,\text{ }y\text{ }=\text{ }-1\]...

### State, whether the following statements are true or false. If false, give a reason. A column matrix has many columns and one row.

False A section framework has just a single segment and many lines.

### State, whether the following statements are true or false. If false, give a reason. (i) Transpose of a 2 x 1 matrix is a 2 x 1 matrix. (ii) Transpose of a square matrix is a square matrix.

(I) False. The amount of grids\[A+B\] is conceivable just when the request for both the frameworks \[A\text{ }and\text{ }B\]are same. (ii) True

### State, whether the following statements are true or false. If false, give a reason. (i) If A and B are two matrices of orders 3 x 2 and 2 x 3 respectively; then their sum A + B is possible. (ii) The matrices A2 x 3 and B2 x 3 are conformable for subtraction.

(I) False. The amount of grids \[A\text{ }+\text{ }B\] is conceivable just when the request for both the frameworks \[A\text{ }and\text{ }B\]are same. (ii) True

### Find the number which bears the same ratio to 7/33 that 8/21 does to 4/9.

Let the required number to be x/y Since, Ratio of \[8/21\text{ }to\text{ }4/9\text{ }=\text{ }\left( 8/21 \right)/\text{ }\left( 4/9 \right)\text{ }=\text{ }\left( 8/21 \right)\text{ }x\text{...

### If (a – b): (a + b) = 1: 11, find the ratio (5a + 4b + 15): (5a – 4b + 3).

Since, \[\begin{array}{*{35}{l}} \left( a\text{ }\text{ }-b \right)/\text{ }\left( a\text{ }+\text{ }b \right)\text{ }=\text{ }1/\text{ }11 \\ 11a\text{ }\text{ }-11b\text{ }=\text{ }a\text{...

### If a: b = 3: 8, find the value of 4a + 3b/ 6a – b.

Since, a: b = 3: 8 Therefore, a/b = 3/8

### If x: y = 4: 7, find the value of (3x + 2y): (5x + y).

Since, x: y = 4: 7 Therefore, x/y = 4/7

### If a: b = 5: 3, find: 5a – 3b/ 5a + 3b

Since, a: b = 5: 3 Therefore, a/b = 5/3 Now,

### Solve: x4 – 10×2 + 9 = 0

Given condition, \[x4\text{ }\text{ }10x2\text{ }+\text{ }9\text{ }=\text{ }0\] \[x4\text{ }\text{ }x2\text{ }\text{ }9x2+\text{ }9\text{ }=\text{ }0\] \[x2\left( x2\text{ }\text{ }1 \right)\text{...

### Solve: x4 – 2×2 – 3 = 0

Given condition, \[x4\text{ }\text{ }2x2\text{ }\text{ }3\text{ }=\text{ }0\] \[x4\text{ }\text{ }3x2\text{ }+\text{ }x2\text{ }\text{ }3\text{ }=\text{ }0\] \[x2\left( x2\text{ }\text{ }3...

### Solve each of the following equation for x and give, in each case, your answer correct to 2 decimal places: (iii) 2×2 + 11x + 4 = 0

(iii) Given condition, \[2x2\text{ }+\text{ }11x\text{ }+\text{ }4\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }11\text{ }and\text{ }c\text{ }=\text{ }4\] Thusly,...

### Asharaf went to see a movie. He wanted to purchase a movie ticket for Rs. 80. As the ticket for Rs. 80 was not available, he purchased a ticket for Rs. 120 of upper class. How much extra GST did he pay for the ticket? (GST for a ticket below Rs. 100 is 18% and GST for a ticket above Rs. 100 is 28%)

From the inquiry, we have GST on ticket of \[Rs.\text{ }80\text{ }=\text{ }18percent\text{ }of\text{ }80\] \[=\text{ }18/100\text{ }x\text{ }80\text{ }=\text{ }Rs.\text{ }14.40\] GST on ticket of...

### Mr. Malik went on a tour to Goa. He took a room in a hotel for two days at the rate of Rs. 5000 per day. On the same day, his friend John also joined him. Hotel provided an extra bed charging Rs. 1000 per day for the bed. How much GST, at the rate of 28% is charged by the hotel in the bill to Mr. Malik for both the days?

From the inquiry, we have The amount of bill \[=\text{ }cost\text{ }of\text{ }hotel\text{ }room\text{ }x\text{ }no.\text{ }of\text{ }days\text{ }+\text{ }additional\text{ }bed\text{ }charges\text{...

### Solve each of the following equations for x and give, in each case, your answer correct to 3 decimal places: (i) 3×2 – 12x – 1 = 0 (ii) x2 – 16 x +6 = 0

(i) Given condition, \[3x2\text{ }\text{ }12x\text{ }\text{ }1\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }3,\text{ }b\text{ }=\text{ }-\text{ }12\text{ }and\text{ }c\text{ }=\text{ }-\text{ }1\]...

### Mr. Pankaj took Health Insurance Policy for his family and paid Rs. 900 as SGST. Find the total annual premium paid by him for this policy, rate of GST being 18%.

We should consider that the all out yearly premium paid by Mr. Pankaj be \[Rs.\text{ }X.\] Then, at that point, from the inquiry \[\begin{array}{*{35}{l}} 18percent\text{ }of\text{ }X\text{ }=\text{...

### The tax invoice of a telecom service in Meerut shows cost of services provided by it as Rs. 750. If the GST rate is 18%, find the amount of the bill.

From the inquiry, we have \[\begin{array}{*{35}{l}} GST\text{ }=\text{ }18percent\text{ }of\text{ }750 \\ =\text{ }18/100\text{ }x\text{ }750\text{ }=\text{ }Rs.\text{ }135 \\ \end{array}\] In...

### Solve each of the following equation for x and give, in each case, your answer correct to 2 decimal places: x2 – 5x – 10 = 0

Given condition, \[x2\text{ }\text{ }5x\text{ }\text{ }10\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-\text{ }5\text{ }and\text{ }c\text{ }=\text{ }-\text{ }10\] Thus,...

### For the following, find the amount of bill data:

SOLUTION:- In this way, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }GST\] \[\begin{array}{*{35}{l}} =\text{ }15,168\text{ }+\text{ }2634.24 \\ =\text{ }Rs.\text{ }17,802.24 ...

### Solve each of the following equation for x and give, in each case, your answer correct to 2 decimal places: x2 – 3x – 9 = 0

Given condition, \[x2\text{ }\text{ }3x\text{ }\text{ }9\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-\text{ }3\text{ }and\text{ }c\text{ }=\text{ }-\text{ }9\]...

### M/s Ram Traders, Delhi, provided the following services to M/s Geeta Trading Company in Agra (UP). Find the amount of bill:

SOLUTION:- In this manner, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }IGST\] \[\begin{array}{*{35}{l}} =\text{ }18,600\text{ }+\text{ }2007.2 \\ =\text{ }Rs.\text{...

### National Trading Company, Meerut (UP) made the supply of the following goods/services to Samarth Traders, Noida (UP). Find the total amount of bill if the rate of GST = 12%

SOLUTION:- In this manner, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }SGST\text{ }+\text{ }CGST\] \[=\text{ }17,220\text{ }+\text{ }2066.4\] \[=\text{ }Rs.\text{...

### Solve each of the following equation for x and give, in each case, your answer correct to 2 decimal places: 4×2 – 5x – 3 = 0

Given condition, \[4x2\text{ }\text{ }5x\text{ }\text{ }3\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }4,\text{ }b\text{ }=\text{ }-\text{ }5\text{ }and\text{ }c\text{ }=\text{ }-\text{ }3\] In...

### A dealer in Mumbai supplied some items at the following prices to a dealer in Delhi. Find the total amount of the bill.

SOLUTION:- Hence, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }IGST\] \[=\text{ }17,710\text{ }+\text{ }3187.8\] \[=\text{ }Rs.\text{ }20,897.80\]

### Solve each of the following equation for x and give, in each case, your answer correct to 2 decimal places : 4x + 6/x + 13 = 0

Given condition, \[4x\text{ }+\text{ }6/x\text{ }+\text{ }13\text{ }=\text{ }0\] Increasing by x the two sides, we get \[4x2\text{ }+\text{ }13x\text{ }+\text{ }6\text{ }=\text{ }0\] Here, \[a\text{...

### Solve each of the following equation for x and give, in each case, your answer correct to 2 decimal places: 2×2 – 10x + 5 = 0

Given condition, \[2x2\text{ }\text{ }10x\text{ }+\text{ }5\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }-\text{ }10\text{ }and\text{ }c\text{ }=\text{ }5\]...

### For the data given , find the amount of bill for the inter-state transaction.

SOLUTION:- Hence, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }IGST\] \[\begin{array}{*{35}{l}} =\text{ }37,850\text{ }+\text{ }5570 \\ =\text{ }Rs.\text{ }43,420 \\...

### Find the amount of bill for the following intra-state transaction of goods/services.

SOLUTION:- Hence, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }IGST\] \[=\text{ }37,850\text{ }+\text{ }5570\] \[=\text{ }Rs.\text{ }43,420\]

### Solve each of the following equations for x and give, in each case, your answer correct to one decimal place: (i) x2 – 8x +5 = 0 (ii) 5×2 + 10x – 3 = 0

(i) \[x2\text{ }\text{ }8x\text{ }+\text{ }5\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-\text{ }8\text{ }and\text{ }c\text{ }=\text{ }5\] By quadratic...

### Find the amount of bill for the following inter-state transaction of goods/services. The GST rate is 18%.

Solution:- Hence, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }IGST\] \[\begin{array}{*{35}{l}} =\text{ }44,210\text{ }+\text{ }3978.9\text{ }+\text{ }3978.9 \\ =\text{...

### Find the amount of bill for the following intra-state transaction of goods/services. The GST rate is 5%.

Solution:- Hence, the measure of bill \[=\text{ }Selling\text{ }cost\text{ }+\text{ }IGST\] \[\begin{array}{*{35}{l}} =\text{ }10,290\text{ }+\text{ }257.25\text{ }+\text{ }257.25 \\ =\text{...

### Solve equations using formula:

From the given condition, \[10x2\text{ }\text{ }60x\text{ }+\text{ }80\text{ }=\text{ }6x2\text{ }\text{ }30x\text{ }+\text{ }30\] \[4x2\text{ }\text{ }30x\text{ }+\text{ }50\text{ }=\text{ }0\]...

### A computer mechanic in Delhi charges repairing cost from five different persons A, B, C, D and E with certain discounts. The repairing costs and the corresponding discounts are as given below:

If the rate of GST is 18%, find the total money (including GST) received by the mechanic. Solution:- Hence, The all out cash (counting GST) got by the mechanic is \[18,820\text{ }+\text{...

### Solve equations using formula: 2x/ x – 4 + (2x – 5)/(x – 3) =

2x/ x – 4 + (2x – 5)/(x – 3) = Given condition, \[2x/x\text{ }\text{ }4\text{ }+\text{ }\left( 2x\text{ }\text{ }5 \right)/\left( x\text{ }\text{ }3 \right)\] \[25x2\text{ }\text{ }175x\text{...

### Solve equations using formula: √6×2 – 4x – 2√6 = 0

Given condition, \[\surd 6x2\text{ }\text{ }4x\text{ }\text{ }2\surd 6\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }\surd 6,\text{ }b\text{ }=\text{ }-\text{ }4\text{ }and\text{ }c\text{ }=\text{...

### For the following transaction from Delhi to Jaipur, fill in the blanks to find the amount of bill: MRP = Rs 50,000, Discount % = 20%, GST = 28% Discount = Selling price (discounted value) = CGST = SGST = IGST = Amount of Bill =

Given, MRP \[=\text{ }Rs\text{ }50,000\] Discount % \[=\text{ }20percent\] GST $=28percent$ Presently, \[Discount\text{ }=\text{ }20percent\text{ }of\text{ }50,000\text{ }\] \[=\text{ }\left( 20/100...

### For the following transaction within Delhi, fill in the blanks to find the amount of bill: MRP = Rs 12,000, Discount % = 30%, GST = 18% Discount = Selling price (discounted value) = CGST = SGST = IGST = Amount of Bill =

Given, MRP = Rs 12,000 Discount = 30% GST = 18% Presently, \[Discount\text{ }=\text{ }30%\text{ }of\text{ }12,000\] \[=\text{ }\left( 30/100 \right)\text{ }x\text{ }1200\text{ }=\text{ }Rs\text{...

### Solve equations using formula: 2x + 3/ x + 3 = x + 4/ x + 2

Given condition, \[2x\text{ }+\text{ }3/x\text{ }+\text{ }3\text{ }=\text{ }x\text{ }+\text{ }4/x\text{ }+\text{ }2\] On cross-duplicating, we have \[\left( 2x\text{ }+\text{ }3 \right)\text{...

### Solve equations using formula: 4/x – 3 = 5/ (2x + 3)

Given condition, \[4/x\text{ }\text{ }3\text{ }=\text{ }5/\left( 2x\text{ }+\text{ }3 \right)\] \[\left( 4\text{ }\text{ }3x \right)/x\text{ }=\text{ }5/\left( 2x\text{ }+\text{ }3 \right)\] On...

### Solve equations using formula: x2 – 6 = 2 √2 x

Given condition, \[x2\text{ }\text{ }6\text{ }=\text{ }2\text{ }\surd 2\text{ }x\] \[x2\text{ }\text{ }2\surd 2\text{ }x\text{ }\text{ }6\text{ }=\text{ }0\] Here, \[a\text{ }=\text{...

### Solve equations using formula: 1/15 x2 + 5/3 = 2/3 x

Given condition, \[1/15\text{ }x2\text{ }+\text{ }5/3\text{ }=\text{ }2/3\text{ }x\] \[1/15\text{ }x2\text{ }\text{ }2/3\text{ }x\text{ }+\text{ }5/3\text{ }=\text{ }0\] Increasing by 15 on...

### Solve equations using formula: 2/3 x = -1/6 x2 – 1/3

Given condition, \[2/3\text{ }x\text{ }=\text{ }-\text{ }1/6\text{ }x2\text{ }\text{ }1/3\] \[1/6\text{ }x2\text{ }+\text{ }2/3\text{ }x\text{ }+\text{ }1/3\text{ }=\text{ }0\] Increasing by 6 on...

### Solve equations using formula: 2×2 + 7x + 5 = 0

Given condition, \[2x2\text{ }+\text{ }7x\text{ }+\text{ }5\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }7\text{ }and\text{ }c\text{ }=\text{ }5\] By quadratic equation,...

### Solve equations using formula: 3×2 + 2x – 1 = 0

Given condition, \[3x2\text{ }+\text{ }2x\text{ }\text{ }1\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }3,\text{ }b\text{ }=\text{ }2\text{ }and\text{ }c\text{ }=\text{ }-\text{ }1\] By quadratic...

### Solve equations using formula: x2 + 2x – 6 = 0

Given condition, \[x2\text{ }+\text{ }2x\text{ }\text{ }6\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }2\text{ }and\text{ }c\text{ }=\text{ }-\text{ }6\] By quadratic...

### Solve equations using formula: x2 + 6x – 10 = 0

Given condition, \[x2\text{ }+\text{ }6x\text{ }\text{ }10\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }6\text{ }and\text{ }c\text{ }=\text{ }-\text{ }10\] By quadratic...

### Solve using the formula: x2 – 10x + 21 = 0

Given condition, \[x2\text{ }\text{ }10x\text{ }+\text{ }21\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }1\] , \[b\text{ }=\text{ }-\text{ }10\] and \[c\text{ }=\text{ }21\] By quadratic equation,...

### Solve using the formula: x2 – 6x = 27

Given condition, \[~x2\text{ }\text{ }6x\text{ }=\text{ }27\] \[x2\text{ }\text{ }6x\text{ }\text{ }27\text{ }=\text{ }0\] Here, \[a\text{ }=\text{ }1\] , \[b\text{ }=\text{ }-\text{ }6\] and...

### Solve equations using factorization method:

\[2\left( 2x2\text{ }+\text{ }18 \right)\text{ }=\text{ }5\left( x2\text{ }\text{ }9 \right)\] \[4x2\text{ }+\text{ }36\text{ }=\text{ }5x2\text{ }\text{ }45\] \[x2\text{ }\text{ }81\text{ }=\text{...

### Solve equations using factorization method: 2×2 – 9x + 10 = 0, when: (i) x ∈ N (ii) x ∈ Q

Given condition, \[2x2\text{ }\text{ }9x\text{ }+\text{ }10\text{ }=\text{ }0\] \[2x2\text{ }\text{ }4x\text{ }\text{ }5x\text{ }+\text{ }10\text{ }=\text{ }0\] \[2x\left( x\text{ }\text{ }2...

### Solve equations using factorization method: 3x – 2/ 2x- 3 = 3x – 8/ x + 4

Given condition, \[3x\text{ }\text{ }2/2x-3\text{ }=\text{ }3x\text{ }\text{ }8/x\text{ }+\text{ }4\] On cross-duplicating we have, \[\left( 3x\text{ }\text{ }2 \right)\left( x\text{ }+\text{ }4...

### P is the solution set of 7x – 2 > 4x + 1 and Q is the solution set of 9x – 45 ≥ 5(x – 5); where x ∈ R. Represent: P ∩ Q’ on different number lines.

\[\begin{array}{*{35}{l}} P\text{ }=\text{ }\{x:\text{ }7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1,\text{ }x\in R\} \\ 7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1 ...

### Solve equations using factorization method: 4(2x – 3)2 – (2x – 3) – 14 = 0

Given condition, \[4\left( 2x\text{ }\text{ }3 \right)2\text{ }\text{ }\left( 2x\text{ }\text{ }3 \right)\text{ }\text{ }14\text{ }=\text{ }0\] Let substitute \[2x\text{ }\text{ }3\text{ }=\text{...

### P is the solution set of 7x – 2 > 4x + 1 and Q is the solution set of 9x – 45 ≥ 5(x – 5); where x ∈ R. Represent: (i) P ∩ Q (ii) P – Q

\[P\text{ }=\text{ }\{x:\text{ }7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1,\text{ }x\in R\}\] \[\begin{array}{*{35}{l}} 7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1 ...

### Given A = {x: -1 < x ≤ 5, x ∈ R} and B = {x: -4 ≤ x < 3, x ∈ R} Represent on different number lines: A – B

\[~A\text{ }\text{ }B\text{ }=\text{ }\left\{ x:\text{ }3\text{ }\le \text{ }x\text{ }\le \text{ }5,\text{ }x\in R \right\}\] Also, it very well may be addressed on a number line as:

### Solve equations using factorization method: (x + 3)2 – 4(x + 3) – 5 = 0

Given condition, \[\left( x\text{ }+\text{ }3 \right)2\text{ }\text{ }4\left( x\text{ }+\text{ }3 \right)\text{ }\text{ }5\text{ }=\text{ }0\] \[\left( x2\text{ }+\text{ }9\text{ }+\text{ }6x...

### Given A = {x: -1 < x ≤ 5, x ∈ R} and B = {x: -4 ≤ x < 3, x ∈ R} Represent on different number lines: (i) A ∩ B (ii) A’ ∩ B

(I) \[A\text{ }\cap \text{ }B\text{ }=\text{ }\{x:\text{ }-\text{ }1\text{ }<\text{ }x\text{ }<\text{ }3,\text{ }x\in R\}\] Also, it very well may be addressed on a number line as: (ii)...

### Illustrate the set {x: -3 ≤ x 2, x ∈ R} on the real number line.

We need to get that: Diagram of arrangement set of \[-\text{ }3\text{ }\le \text{ }x\text{ }<\text{ }0\text{ }or\text{ }x\text{ }>\text{ }2\text{ }=\]Graph of focuses which have a place with...

### Solve equations using factorization method: x2 – (a + b)x + ab = 0

Given condition, \[x2\text{ }\text{ }\left( a\text{ }+\text{ }b \right)x\text{ }+\text{ }ab\text{ }=\text{ }0\] \[x2\text{ }\text{ }ax\text{ }\text{ }bx\text{ }+\text{ }ab\text{ }=\text{ }0\]...

### Use real number line to find the range of values of x for which: -1 < x ≤ 6 and -2 ≤ x ≤ 3

\[-\text{ }1\text{ }<\text{ }x\text{ }\le \text{ }6\text{ }and\text{ }-\text{ }2\text{ }\le \text{ }x\text{ }\le \text{ }3\] Both the given inequations are valid in the reach where their charts...

### Use real number line to find the range of values of x for which: (i) x > 3 and 0 < x < 6 (ii) x < 0 and -3 ≤ x < 1

(I) \[x\text{ }>\text{ }3\text{ }and\text{ }0\text{ }<\text{ }x\text{ }<\text{ }6\] Both the given inequations are valid in the reach where their charts on the genuine number lines...

### Solve equations using factorization method: (x + 1) (2x + 8) = (x + 7) (x + 3)

Given condition, \[\left( x\text{ }+\text{ }1 \right)\text{ }\left( 2x\text{ }+\text{ }8 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }7 \right)\text{ }\left( x\text{ }+\text{ }3 \right)\]...

### Solve equations using factorization method: 2(x2 – 6) = 3(x – 4)

Given condition, \[2\left( x2\text{ }\text{ }6 \right)\text{ }=\text{ }3\left( x\text{ }\text{ }4 \right)\] \[2x2\text{ }\text{ }12\text{ }=\text{ }3x\text{ }\text{ }12\] \[2x2\text{ }=\text{ }3x\]...

### The diagram represents two inequations A and B on real number lines: (i) Write down A and B in set builder notation. (ii) Represent A ∩ B and A ∩ B’ on two different number lines.

SOLUTION:- (I) \[A\text{ }=\text{ }\{x\in R:\text{ }-\text{ }2\le x\text{ }<\text{ }5\}\] \[B\text{ }=\text{ }\left\{ x\in R:\text{ }-\text{ }4\le x\text{ }<\text{ }3 \right\}\] (ii) \[A\text{...

### Solve and graph the solution set of: (i) 3x – 2 > 19 or 3 – 2x ≥ -7, x ∈ R (ii) 5 > p – 1 > 2 or 7 ≤ 2p – ≤ 17, p ∈ R

(I) \[3x\text{ }\text{ }2\text{ }>\text{ }19\text{ }or\text{ }3\text{ }\text{ }2x\text{ }\ge \text{ }-\text{ }7\] \[\begin{array}{*{35}{l}} 3x\text{ }>\text{ }21\text{ }or\text{ }-\text{...

### Solve and graph the solution set of: x + 5 ≥ 4(x – 1) and 3 – 2x < -7, x ∈ R

\[\begin{align} & ~x\text{ }+\text{ }5\text{ }\ge \text{ }4\left( x\text{ }\text{ }1 \right)\text{ }and\text{ }3\text{ }\text{ }2x\text{ }<\text{ }-\text{ }7 \\ & \begin{array}{*{35}{l}}...

### Solve equations using factorization method: (2x – 3)2 = 49

Given condition, \[\left( 2x\text{ }\text{ }3 \right)2\text{ }=\text{ }49\] (2x – 3)2 = 49 Extending the L.H.S, we have \[4x2\text{ }\text{ }12x\text{ }+\text{ }9\text{ }=\text{ }49\] \[4x2\text{...

### Solve and graph the solution set of: (i) 2x – 9 25, x ∈ I

(I) \[2x\text{ }\text{ }9\text{ }<\text{ }7\text{ }and\text{ }3x\text{ }+\text{ }9\text{ }\le \text{ }25\] \[\begin{array}{*{35}{l}} 2x\text{ }<\text{ }16\text{ }and\text{ }3x\text{ }\le...

### Solve equations using factorization method: x + 1/x = 2.5

Given condition, \[x\text{ }+\text{ }1/x\text{ }=\text{ }2.5\] \[x\text{ }+\text{ }1/x\text{ }=\text{ }5/2\] Taking LCM on L.H.S, we have \[\left( x2\text{ }+\text{ }1 \right)/x\text{ }=\text{...

### Solve the following inequation and graph the solution set on the number line: 2x – 3 < x + 2 ≤ 3x + 5, x ∈ R.

Given inequation, \[\begin{align} & \begin{array}{*{35}{l}} 2x\text{ }\text{ }3\text{ }<\text{ }x\text{ }+\text{ }2\text{ }\le \text{ }3x\text{ }+\text{ }5 \\ 2x\text{ }\text{ }3\text{...

### If 5x – 3 ≤ 5 + 3x ≤ 4x + 2, express it as a ≤ x ≤ b and then state the values of a and b.

Given inequation, \[\begin{array}{*{35}{l}} 5x\text{ }\text{ }3\text{ }\le \text{ }5\text{ }+\text{ }3x\text{ }\le \text{ }4x\text{ }+\text{ }2 \\ 5x\text{ }\text{ }3\text{ }\le \text{ }5\text{...

### Given x ∈ {real numbers}, find the range of values of x for which -5 ≤ 2x – 3 < x + 2 and represent it on a real number line.

Given inequation, \[\begin{array}{*{35}{l}} -\text{ }5\text{ }\le \text{ }2x\text{ }\text{ }3\text{ }<\text{ }x\text{ }+\text{ }2 \\ -\text{ }5\text{ }\le \text{ }2x\text{ }\text{ }3\text{...

### Find the values of x, which satisfy the inequation:

Graph the solution on the number line. SOLUTION:- Given Inequation, Henceforth, the arrangement set is \[\{x\in N:\text{ }-\text{ }2\le x\le 3.75\}\] Also, as x ∈ N, the upsides of x are \[1,\text{...

### Solve equations using factorization method: x = (3x + 1)/ 4x

Given condition, \[x\text{ }=\text{ }\left( 3x\text{ }+\text{ }1 \right)/4x\] On increasing by 4x the two sides, we have \[4x\left( x \right)\text{ }=\text{ }3x\text{ }+\text{ }1\] \[4x2\text{...

### Find the range of values of x which satisfies

Graph these values of x on the number line. SOLUTION:- \[\Rightarrow -\text{ }3\le x\text{ }and\text{ }x\text{ }<\text{ }3\] Along these lines, \[3\text{ }\le \text{ }x\text{ }<\text{ }3\]...

### Solve equations using factorization method: 6/x = 1 + x

Given condition, \[6/x\text{ }=\text{ }1\text{ }+\text{ }x\] On increasing by x the two sides, we have \[6\text{ }=\text{ }x\left( 1\text{ }+\text{ }x \right)\] \[6\text{ }=\text{ }x\text{ }+\text{...

### List the elements of the solution set of the inequation -3 < x – 2 ≤ 9 – 2x; x ∈ N.

\[-\text{ }3\text{ }<\text{ }x\text{ }\text{ }2\text{ }\le \text{ }9\text{ }\text{ }2x\] \[\begin{array}{*{35}{l}} -\text{ }3\text{ }<\text{ }x\text{ }\text{ }2\text{ }and\text{ }x\text{...

### x ∈ {real numbers} and -1 < 3 – 2x ≤ 7, evaluate x and represent it on a number line.

\[\begin{array}{*{35}{l}} -\text{ }1\text{ }<\text{ }3\text{ }\text{ }2x\text{ }\le \text{ }7 \\ -\text{ }1\text{ }<\text{ }3\text{ }\text{ }2x\text{ }and\text{ }3\text{ }\text{ }2x\text{...

### Represent the solution of each of the following inequations on the real number line: (i)1 + x ≥ 5x – 11 (ii) (2x + 5)/3 > 3x – 3

(i) \[1\text{ }+\text{ }x\text{ }\ge \text{ }5x\text{ }\text{ }11\] \[\begin{array}{*{35}{l}} 12\text{ }\ge \text{ }4x \\ x\text{ }\le \text{ }3 \\ \end{array}\] The arrangement on number line is...

### Solve equations using factorization method: 9/2 x = 5 + x2

Given equation, \[9/2\text{ }x\text{ }=\text{ }5\text{ }+\text{ }x2\] On multiplying by 2 both sides, we have \[9x\text{ }=\text{ }2\left( 5\text{ }+\text{ }x2 \right)\] \[9x\text{ }=\text{...

### Represent the solution of each of the following inequations on the real number line: (i)x + 3 ≤ 2x + 9 (ii) 2 – 3x > 7 – 5x

(i) \[x\text{ }+\text{ }3\text{ }\le \text{ }2x\text{ }+\text{ }9\] \[\begin{array}{*{35}{l}} x\text{ }\text{ }2x\text{ }\le \text{ }-\text{ }3\text{ }+\text{ }9 \\ -\text{ }x\text{ }\le \text{ }6 ...

### Represent the solution of each of the following inequations on the real number line: (i) 4x – 1 > x + 11 (ii) 7 – x ≤ 2 – 6x

(I) \[4x\text{ }\text{ }1\text{ }>\text{ }x\text{ }+\text{ }11\] \[\begin{array}{*{35}{l}} 4x\text{ }\text{ }x\text{ }>\text{ }1\text{ }+\text{ }11 \\ 3x\text{ }>\text{ }12 \\ x\text{...

### Solve equations using factorization method: x(x – 5) = 24

Given condition, \[x\left( x\text{ }\text{ }5 \right)\text{ }=\text{ }24\] \[x2\text{ }\text{ }5x\text{ }=\text{ }24\] \[x2\text{ }\text{ }5x\text{ }\text{ }24\text{ }=\text{ }0\] \[x2\text{ }\text{...

### For the following inequation, graph the solution set on the real number line: (i) -4 ≤ 3x – 1 < 8 (ii) x -1 < 3- x ≤ 5

(I) \[-\text{ }4\text{ }\le \text{ }3x\text{ }\text{ }1\text{ }<\text{ }8\] \[\begin{array}{*{35}{l}} -\text{ }4\text{ }\le \text{ }3x\text{ }\text{ }1\text{ }and\text{ }3x\text{ }\text{ }1\text{...

### For each graph given alongside, write an inequation taking x as the variable:

(i) (ii) Solution:- (i) \[-4\text{ }\le \text{ }x\text{ }<\text{ }3,\text{ }x\in R\] (ii) \[-1\text{ }<\text{ }x\text{ }\le \text{ }5,\text{ }x\in R\]