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Three coins are tossed. Describe
(i) two events A and B which are not mutually exclusive.
(ii) two events A and B which are mutually exclusive but not exhaustive.

CBSE, Class 11, Exercise 33.2, Maths, Probability, RD Sharma

According to the question, three coins are tossed. So the sample space is, S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (i) The two events that are not mutually exclusive are: A: getting three heads...

Three coins are tossed. Describe
(i) two events A and B which are mutually exclusive.
(ii) three events A, B and C which are mutually exclusive and exhaustive.

CBSE, Class 11, Exercise 33.2, Maths, Probability, RD Sharma

According to the question, three coins are tossed. So the sample space is, S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (i) The two events which are mutually exclusive are when, A: getting no tails...

In a single throw of a die describe the following events:
(i) E = Getting an even number greater than 4.
(ii) F = Getting a number not less than 3.

CBSE, Class 11, Exercise 33.2, Maths, Probability, RD Sharma

According to the question, a dice is thrown once. Let us find the given events, and also find A ∪ B, A ∩ B, B ∩ C, E ∩ F, D ∩ F and $\bar{F}$ S = {1, 2, 3, 4, 5, 6} (i) E = Getting an even number...

In a single throw of a die describe the following events:
(i) C = Getting a multiple of 3
(ii) D = Getting a number less than 4

CBSE, Class 11, Exercise 33.2, Maths, Probability, RD Sharma

According to the question, a dice is thrown once. Let us find the given events, and also find A ∪ B, A ∩ B, B ∩ C, E ∩ F, D ∩ F and $\bar{F}$ S = {1, 2, 3, 4, 5, 6} (i) C = Getting multiple of 3 So,...

In a single throw of a die describe the following events:
(i) A = Getting a number less than 7
(ii) B = Getting a number greater than 7

CBSE, Class 11, Exercise 33.2, Maths, Probability, RD Sharma

According to the question, a dice is thrown once. Let us find the given events, and also find A ∪ B, A ∩ B, B ∩ C, E ∩ F, D ∩ F and $\bar{F}$ S = {1, 2, 3, 4, 5, 6} (i) A = getting a number below 7...

Three coins are tossed once. Describe the following events associated with this random experiment: A = Getting three heads, B = Getting two heads and one tail, C = Getting three tails, D = Getting a head on the first coin.
Which events are compound events?

CBSE, Class 11, Exercise 33.2, Maths, Probability, RD Sharma

According to the given quesion, there are three coins tossed once. When three coins are tossed, the sample spaces are: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} So, according to the question, A =...

List all events associated with the random experiment of tossing of two coins. How many of them are elementary events?

CBSE, Class 11, Exercise 33.2, Maths, Probability, RD Sharma

According to the question, two coins are tossed once. We know, when two coins are tossed then the total number of possible outcomes are will be $2^2=4$ So, the Sample space is {HH, HT, TT, TH} ∴...

Find the square root of the following complex numbers.
(i) -i

CBSE, Class 11, Complex Numbers, Exercise 13.3, Maths, RD Sharma

Solution: If $b>0, \sqrt{a+i b}=\pm\left[\left(\frac{a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}+i\left(\frac{-a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}\right]$ If $b<0, \sqrt{a+i...

Find the square root of the following complex numbers.
(i) 1 + 4√-3
(ii) 4i

CBSE, Class 11, Complex Numbers, Exercise 13.3, Maths, RD Sharma

Solution: If $b>0, \sqrt{a+i b}=\pm\left[\left(\frac{a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}+i\left(\frac{-a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}\right]$ If $b<0, \sqrt{a+i...

A coin is tossed repeatedly until a tail comes up for the first time. Write the sample space for this experiment.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, a coin is tossed and if the outcome is tail the experiment is over. If the outcome is Head, then the coin is tossed again. If the outcome is tail, then experiment is over,...

Find the square root of the following complex numbers.
(i) 8 – 15i
(ii) -11 – 60√-1

CBSE, Class 11, Complex Numbers, Exercise 13.3, Maths, RD Sharma

Solution: If $b>0, \sqrt{a+i b}=\pm\left[\left(\frac{a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}+i\left(\frac{-a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}\right]$ If $b<0, \sqrt{a+i...

Find the square root of the following complex numbers.
(i) 1 – i
(ii) – 8 – 6i

CBSE, Class 11, Complex Numbers, Exercise 13.3, Maths, RD Sharma

Solution: If $b>0, \sqrt{a+i b}=\pm\left[\left(\frac{a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}+i\left(\frac{-a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}\right]$ If $b<0, \sqrt{a+i...

A coin is tossed. If it shows tail, we draw a ball from a box which contains 2 red 3 black balls; it shows head, we throw a die. Find the sample space of this experiment.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, A coin is tossed and there is box which contains 2 red and 3 black balls. When coin is tossed, the outcomes will be {H, T} According to question, if tail is turned up,...

Find the square root of the following complex numbers.
(i) – 5 + 12i
(ii) -7 – 24i

CBSE, Class 11, Complex Numbers, Exercise 13.3, Maths, RD Sharma

Solution: If $b>0, \sqrt{a+i b}=\pm\left[\left(\frac{a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}+i\left(\frac{-a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}\right]$ If $b<0, \sqrt{a+i...

An experiment consists of tossing a coin and then tossing it second time if head occurs. If a tail occurs on the first toss, then a die is tossed once. Find the sample space.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, a coin is tossed and if the outcome is tail then, a die will be rolled. The possible outcome for coin is 2 that is {H, T} And, the possible outcome for die is 6 that is...

A coin is tossed twice. If the second throw results in a tail, a die is thrown. Describe the sample space for this experiment.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, A coin is tossed twice. If the second throw results in a tail, a die is thrown. When a coin tossed twice, then sample spaces will be, {HH, TT, HT, TH} Now, according to...

A coin is tossed and then a die is rolled only in case a head is shown on the coin. Describe the sample space for this experiment.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, a coin is tossed and the die is rolled. So, when coin is tossed there will be 2 events that is either Head or tail, According to question, If Head occurs on coin then Die...

Express the following complex numbers in the standard form a + ib:
(i) $\left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{3-4 i}{5+i}\right)$
(ii) $(5+\sqrt{2 i}) /(1-\sqrt{2 i})$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i)$\left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{3-4 i}{5+i}\right)$ Simplify and express in the standard form of $(a + ib),$ $\begin{aligned} \left(\frac{1}{1-4...

A coin is tossed and then a die is thrown. Describe the sample space for this experiment.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, a coin is tossed and then a die is thrown. So, when coin is tossed there will be 2 events, that is either Head or Tail. And, when die is thrown then there will be 6 faces...

What is the total number of elementary events associated to the random experiment of throwing three dice together?

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, three dice are thrown together. So there are 6 faces on die. As a result, the total numbers of elementary events on throwing three dice are $6^3=216$

Two dice are thrown. Describe the sample space of this experiment.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

As we know, there are 6 faces on a dice containing (1, 2, 3, 4, 5, 6). According to the question, two dice are thrown, so we have two faces of dice (one of each). As a result, the total sample space...

Write the sample space for the experiment of tossing a coin four times.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, a coin is tossed four time, so the no. of samples will be, $2^4=16$ So, S = {HHHH, TTTT, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, HTHT, THHT, THTH, TTHH, HTTT, THTT, TTHT,...

Express the following complex numbers in the standard form a + ib:
(i) $(1 + 2i)^{-3}$
(ii) (3 – 4i) / [(4 – 2i) (1 + i)]

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $(1+2 \mathrm{i})^{-3}$ Simplify and express in the standard form of $(a+i b)$, $\begin{array}{l} (1+2 i)^{-3}=1 /(1+2 i)^{3} \\ =1 /\left(1^{3}+3(1)^{2}(2 i)+2(1)(2 i)^{2}+(2...

If a coin is tossed three times (or three coins are tossed together), then describe the sample space for this experiment.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, a coin is tossed three times, so the no. of samples will be, $2^3=8$ So, S = {HHH, TTT, HHT, HTH, THH, HTT, THT, TTH} ∴ The sample space is {HHH, TTT, HHT, HTH, THH, HTT,...

A coin is tossed once. Write its sample space.

CBSE, Class 11, Exercise 33.1, Maths, Probability, RD Sharma

According to the question, a coin is tossed once. So, there are two possibilities, either Head (H) or Tail (T) will come. So, S = {H, T} ∴ The sample space is {H, T}

In a $\vartriangle ABC$ , D and E are points on AB and AC respectively, such that $DE||BC$ . If $AD=2.4cm$ , $AE=3.2cm$ , $DE=2cm$ and $BC=5cm$ . Find BD and CE.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

Given information: $\vartriangle ABC$ such that $AD=2.4cm$, $AE=3.2cm$, $DE=2cm$ and $BE=5cm$. Also $DE||BC$. Required to find: BD and CE. Proof: As $DE||BC$, as AB is transversal, $\angle...

In a $\vartriangle ABC$ , P and Q are the points on sides AB and AC respectively, such that $PQ||BC$ . If $AP=2.4cm$ , $AQ=2cm$ , $QC=3cm$ and $BC=6cm$ . Find AB and PQ.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

Given information: $\vartriangle ABC$, $AP=2.4cm$, $AQ=2cm$, $QC=3cm$, and $BC=6cm$. Also, $PQ||BC$. Required to find: AB and PQ. Proof: As it’s given that $PQ||BC$, By using Thales Theorem, we have...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively. For each of the following cases show that $DE||BC$ : iii) $AB=10.8cm$ , $BD=4.5cm$ , $AC=4.8cm$ , and $AE=2.8cm$ .iv) $AD=5.7cm$ , $BD=9.5cm$ , $AE=3.3cm$ , and $EC=5.5cm$ .

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

(iii) Given information: $=10.8 cm,$$BD=4.5cm$, $AC=4.8cm$, and $AE=2.8cm$. Required to prove: $DE||BC$. Proof: $AD=AB-DB=10.8-4.5=6.3$ And, $CE=AC-AE=4.8-2.8=2$ As ,we can see that...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively. For each of the following cases show that $DE||BC$ : (i) $AB=12cm$ , $AD=8cm$ , $AE=12cm$ , and $AC=18cm$ .ii) $AB=5.6cm$ , $AD=1.4cm$ , $AC=7.2cm$ , and $AE=1.8cm$ .

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

(i) Given information: $AB=12cm$, $AD=8cm$, $AE=12cm$, and $AC=18cm$ Required to prove: $DE||BC$. Proof: According to the given data, $BD=AB–AD=12–8=4cm$ And, $CE=AC–AE=18–12=6cm$ As we can see...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively such that $DE||BC$ .(xi) If $AD=4x-3$ , $AE=8x-7$ , $BD=3x-1$ , and $CE=5x-3$ , find the value of x.(xii) If $AD=2.5cm$ , $BD=3.0cm$ , and $AE=3.75cm$ , find the length of AC.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

(xi) Given information: $AD=4x–3$, $BD=3x–1$, $AE=8x–7$ and $EC=5x–3$ Required to find: x. As $DE||BC$, by using Thales Theorem, $AD/BD=AE/CE$ So, $(4x–3)/(3x-1)=(8x–7)/(5x–3)$...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively such that $DE||BC$ . (ix) If $AD=xcm$ , $DB=x-2cm$ , $AE=x+2cm$ , and $EC=x-1cm$ , find the value of x.(x) If $AD=8x-7cm$ , $DB=5x-3cm$ , $AE=4x-3cm$ , and $EC=(3x-1)cm$ , Find the value of x.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

(ix) Given information: $AD=x$, $DB=x–2$, $AE=x+2$ and $EC=x–1$ Required to find: the value of x. As $DE||BC$ given, By using Thales Theorem, $AD/BD=AE/CE$ So, $x/(x–2)=(x+2)/(x–1)$...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively such that $DE||BC$ .In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively such that $DE||BC$ .viii) If $AD/BD=4/5$ and $EC=2.5cm$ , Find AE.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

(vii) Given information: $AD=2cm$, $AB=6cm$ and $AC=9cm$ Required to find: AE Proof: $DB=AB–AD=6–2=4cm$ As $DE||BC$ given, by using Thales Theorem, $AD/BD=AE/CE$ $2/4=x/(9–x)$ $4x=18–2x$ $6x=18$...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively such that $DE||BC$ . (v) If $AD=8cm$ , $AB=12cm$ and $AE=12cm$ , find CE.(vi) If $AD=4cm$ , $DB=4.5cm$ and $AE=8cm$ , find AC.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

(v) Given information: $AD=8cm$, $AB=12cm$, and $AE=12cm$. Required to find: CE, Proof: As $DE||BC$ given, by using Thales Theorem, $AD/BD=AE/CE$ $8/4=12/CE$ $8\times CE=4\times 12cm$ $CE=\left(...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively such that $DE||BC$ .iii) If $AD/DB=2/3$ and $AC=18cm$ , Find AE iv) If $AD=4cm$ , $AE=8cm$ , $DB=x-4cm$ and $EC=3x-19$ , find x.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

iii) Given information: $AD/BD=2/3$ and $AC=18cm$ Required to find: AE. Proof: As $DE∥BC$, By using Thales Theorem, $AD/BD=AE/CE$ Let us assume, $AE=x$ and then, $CE=18–x$ $⇒23=x/(18–x)$ $3x=36–2x$...

In a $\vartriangle ABC$ , D and E are points on the sides AB and AC respectively such that $DE||BC$ . i) If $AD=6cm$ , $DB=9cm$ and $AE=8cm$ , Find AC.ii) If $AD/DB=3/4$ and $AC=15cm$ , Find AE.

CBSE, Class 10, Exercise 4.2, Maths, RD Sharma, Triangles

i) Given information : $\vartriangle ABC$, $DE||BC$, $AD=6cm$, $DB=9cm$ and $AE=8cm$. Required to find: AC. Proof: As $DE||BC$, by using Thales Theorem, $AD/BD=AE/CE$ Let us assume $CE=x$. So by...

Express the following complex numbers in the standard form a + ib:
(i) (2 + 3i) / (4 + 5i)
(ii) (1 – i)3 / (1 – i3)

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $(2+3 \mathrm{i}) /(4+5 \mathrm{i})$ Simplify and express in the standard form of $(a+i b)$, $(2+3 i) /(4+5 i)=$ [multiply and divide with (4-5i)] $\begin{array}{l} =(2+3 i) /(4+5 i)...

Find the number of terms of the A.P. $-12$ , $-9$ , $-6$ , . . . , $21$ . If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given data, First term of A.P, a $=-12$ Common difference of A.P., $d={{a}_{2}}-{{a}_{1}}=-9-\left( -12 \right)$ $d=–9+12=3$ And, we know that nth term $={{a}_{n}}=a+\left( n-1 \right)d$...

The first and the last term of an A.P are $17$ and $350$ respectively. If the common difference is $9$ , how many terms are there and what is their sum?

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given, In an A.P first term (a) $=17$ and the last term (l) of A.P. $=350$ And, the common difference (d) of A.P. $=9$ As we know that, ${{a}_{n}}=a+\left( n-1 \right)d$ so, by substitution,...

Find the equation of the circle with: (v) Centre (a, a) and radius √2 a.

CBSE, Class 11, Exercise 24.1, Maths, RD Sharma, The Circle

Centre (a, a) and radius √2 a. The radius is √2 a and the centre (a, a) By using the formula, The equation of the circle with centre (p, q) and radius ‘r’ is (x – p)2 + (y – q)2 = r2 Where, p = a,...

Find the equation of the circle with:(iii) Centre (0, – 1) and radius 1. (iv) Centre (a cos α, a sin α) and radius a.

CBSE, Circle, Class 11, Exercise 24.1, Maths, RD Sharma

(iii) Centre (0, -1) and radius 1. Given: The radius is 1 and the centre (0, -1) By using the formula, The equation of the circle with centre (p, q) and radius ‘r’ is (x – p)2 + (y – q)2 = r2 Where,...

In an A.P. , the first term is $2$ , the last term is $29$ and the sum of the terms is $155$ . Find the common difference of the A.P.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

According to the given information, The first term of the A.P. (a) $=2$ The last term of the A.P. (l) $=29$ And, sum of all the terms $\left( {{S}_{n}} \right)=155$ Let the common difference of the...

Find the equation of the circle with: (i) Centre (-2, 3) and radius 4. (ii) Centre (a, b) and radius $\sqrt{{{a}^{2}}+{{b}^{2}}}$

CBSE, Circle, Class 11, Exercise 24.1, Maths, RD Sharma

(i) Centre (-2, 3) and radius 4. Given: The radius is 4 and the centre (-2, 3) By using the formula, The equation of the circle with centre (p, q) and radius ‘r’ is (x – p)2 + (y – q)2 = r2 Where, p...

Express the following complex numbers in the standard form a + ib:
(i) $(2 + i)^3 / (2 + 3i)$
(ii) [(1 + i) (1 +√3i)] / (1 – i)

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i)$(2+i)^{3} /(2+3 i)$ Simplify and express in the standard form of (a +ib), $\begin{array}{l} (2+i)^{3} /(2+3 i)=\left(2^{3}+i^{3}+3(2)^{2}(i)+3(i)^{2}(2)\right) /(2+3 i) \\...

Express the following complex numbers in the standard form a + ib:
(i) $1/(2 + i)^2$
(ii) (1 – i) / (1 + i)

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $1 /(2+i)^{2}$ Simplify and express in the standard form of $(a+i b)$, $1 /(2+i)^{2}=1 /\left(2^{2}+i^{2}+2(2)(i)\right)$ $=1 /(4-1+4 i)\left[\right.$ since,$\left.i^{2}=-1\right]$ $=1...

Sum of the first 14 terms of an A.P. is $1505$ and its first term is $10$ . Find its 25th term.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given data in question, First term of the A.P (a) is $1505$ and ${{S}_{14}}=1505$ As we know that, the sum of first n terms of the A.P. ${{S}_{n}}=n/2\left( 2a+\left( n-1 \right)d \right)$ So, by...

Express the following complex numbers in the standard form a + ib:
(i) (1 + i) (1 + 2i)
(ii) (3 + 2i) / (-2 + i)

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $(1 + i) (1 + 2i)$ Simplify and express in the standard form of $(a + ib)$, $(1 + i) (1 + 2i) = (1+i)(1+2i)$ $= 1(1+2i)+i(1+2i)$ $= 1+2i+i+2i^2$ $= 1+3i+2(-1) [\text{since}\, i^2 =...

In an A.P. the sum of first ten terms is $-150$ and the sum of its next 10 term is $-550$ . Find the A.P.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let’s take a to be the first term of A.P.and d to be the common difference. And as we know that, sum of first n terms of an A.P. is given by, ${{S}_{n}}=n/2\left( 2a+\left( n-1 \right)d \right)$...

The nth term of an A.P is given by $(-4n+15)$ . Find the sum of first $20$ terms of this A.P.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

As per the given information, The nth term of the A.P $=(-4n+15)$ So, by putting the value of n as $1$ and $20$ we can find the first and 20th term of the A.P, $a=(-4(1)+15)=11$ And,...

Find the real values of $x$ and $y$ , if
(i) $\frac{(1+\mathbf{i}) \mathrm{x}-2 \mathrm{i}}{3+\mathbf{i}}+\frac{(2-3 \mathrm{i}) \mathbf{y}+\mathbf{i}}{3-\mathbf{i}}=\mathbf{i}$
(ii) $(1+i)(x+i y)=2-5 i$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $\frac{(1+\mathbf{i}) \mathbf{x}-2 \mathbf{i}}{\mathbf{3}+\mathbf{i}}+\frac{(2-\mathbf{3 i}) \mathbf{y}+\mathbf{i}}{\mathbf{3}-\mathbf{i}}=\mathbf{i}$ Given that $\begin{array}{l}...

Find the real values of $x$ and $y$ , if
(i) $(x+i y)(2-3 i)=4+i$
(ii) $(3 x-2 i y)(2+i)^{2}=10(1+i)$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $(x+i y)(2-3 i)=4+i$ On simplifying the expression we obtain, $\begin{array}{l} x(2-3 i)+i y(2-3 i)=4+i \\ 2 x-3 x i+2 y i-3 y i^{2}=4+i \\ 2 x+(-3 x+2 y) i-3 y(-1)=4+i\left[\text {...

Find the conjugates of the following complex numbers:
(i) [(1 + i) (2 + i)] / (3 + i)
(ii) [(3 – 2i) (2 + 3i)] / [(1 + 2i) (2 – i)]

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $[(1+i)(2+i)] /(3+i)$ As the given complex no. is not in the standard form of $(a+i b)$ Convert it to standard form, $\begin{aligned} \frac{(1+i)(2+i)}{3+i} =\frac{1(2+i)+i(2+i)}{3+i}...

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A coin is tossed once. Find the probability of: (i) getting a tail (ii) not getting a tail

CBSE, Class 10, Probability, Selina

Here, the sample space \[=\text{ }\left\{ H,\text{ }T \right\}\] \[i.e.\text{ }n\left( S \right)\text{ }=\text{ }2\] (i) If A = Event of getting a tail \[=\text{ }\left\{ T \right\}\] Then\[,\text{...

Find the conjugates of the following complex numbers:
(i) 4 – 5i
(ii) 1 / (3 + 5i)

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) 4 – 5i It is known that the conjugate of a complex number $(a + ib)$ is $(a – ib)$ $\therefore$ $(4 + 5i)$ is the conjugate of $(4 – 5i)$ (ii) 1 / (3 + 5i) As the given complex no. is...

The sum of first seven terms of an A.P. is $182$ . If its 4th and 17th terms are in ratio $1:5$ , find the A.P.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

It is given that, Sum of first seven term of A.P. that is ${{S}_{17}}=182$ And, as we know that the sum of first n terms of an A.P. is given by: ${{S}_{n}}=n/2\left( 2a+\left( n-1 \right)d \right)$...

Find the multiplicative inverse of the following complex numbers:
(i) 4 – 3i
(ii) √5 + 3i

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: (i) $4-3 \mathrm{i}$ Given that $4-3 i$ It is known that the multiplicative inverse of a complex number $(Z)$ is $Z^{-1}$ or $1 / Z$ Therefore, $\begin{array}{l} z=4-3 i \\...

If $x+i y=(a+i b) /(a-i b)$ , prove that $x^{2}+y^{2}=1$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $x+i y=(a+i b) /(a-i b)$ It is known that for a complex number $Z=(a+i b)$ it's magnitude is given by $\left.|z|=\sqrt{(} a^{2}+b^{2}\right)$ $\mathrm{So}$, $|\mathrm{a} /...

Find the modulus of [(1 + i)/(1 – i)] – [(1 – i)/(1 + i)]

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $[(1+i) /(1-i)]-[(1-i) /(1+i)]$ So, $Z=[(1+i) /(1-i)]-[(1-i) /(1+i)]$ $\begin{array}{l} =[(1+i)(1+i)-(1-i)(1-i)] /\left(1^{2}-i^{2}\right) \\...

If $z_{1}=(2-i), z_{2}=(-2+i)$ , find
(i) $\operatorname{Re}\left(\frac{\mathbf{z}_{1} \mathbf{z}_{2}}{\overline{\mathbf{z}_{1}}}\right)$
(ii) $\operatorname{Im}\left(\frac{1}{\mathbf{z}_{1} \overline{\mathbf{z}}_{1}}\right)$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $z_{1}=(2-i)$ and $z_{2}=(-2+i)$ (i) $\mathbf{R e}\left(\frac{\mathbf{z}_{1} \mathbf{z}_{2}}{\overline{\mathbf{z}_{1}}}\right)$ On rationalising the denominator, we get...

If $\mathbf{z}_{1}=\mathbf{2}-\mathbf{i}, \mathbf{z}_{2}=\mathbf{1}+\mathbf{i}$ , find $\left|\frac{\mathbf{z}_{1}+\mathbf{z}_{2}+1}{\mathbf{z}_{1}-\mathbf{z}_{2}+\mathbf{i}}\right|$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $\mathrm{z}_{1}=(2-\mathrm{i})$ and $\mathrm{z}_{2}=(1+\mathrm{i})$ It is known that, $|\mathrm{a} / \mathrm{b}|=|\mathrm{a}| /|\mathrm{b}|$ Therefore, $\begin{aligned}...

Find the least positive integral value of $n$ for which $[(1+i) /(1-i)]^{n}$ is real.

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $\begin{array}{l} {[(1+i) /(1-i)]^{n}} \\ Z=[(1+i) /(1-i)]^{n} \end{array}$ On multiplying and dividing by $(1+i)$, we get $\begin{array}{l} =\frac{1+i}{1-i} \times...

Find the real values of $\theta$ for which the complex number $(1+i \cos \theta) /(1-2 i \cos \theta)$ is purely real.

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $\begin{array}{l} (1+i \cos \theta) /(1-2 i \cos \theta) \\ Z=(1+i \cos \theta) /(1-2 i \cos \theta) \end{array}$ Multiply and divide by $(1+2 i \cos \theta)$ $\begin{array}{l}...

Find the smallest positive integer value of $n$ for which $(1+i)^{n} /(1-i)^{n-2}$ is a real number.

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $\begin{array}{l} (1+i)^{n} /(1-i)^{n-2} \\ Z=(1+i)^{n} /(1-i)^{n-2} \end{array}$ Multiply and divide by $(1-i)^{2}$ $\begin{array}{l} =\frac{(1+i)^{n}}{(1-i)^{n-2}} \times...

If $[(1+i) /(1-i)]^{3}-[(1-i) /(1+i)]^{3}=x+$ iy, find $(x, y)$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $[(1+i) /(1-i)]^{3}-[(1-i) /(1+i)]^{3}=x+i y$ On rationalizing the denominator, we obtain $\begin{array}{l} \left(\frac{1+i}{1-i} \times...

If $(1+i)^{2} /(2-i)=x+i y$ , find $x+y$

CBSE, Class 11, Complex Numbers, Exercise 13.2, Maths, RD Sharma

Solution: Given that $(1+i)^{2} /(2-i)=x+i y$ On expansion we obtain, $\begin{array}{l} \frac{1^{2}+i^{2}+2(1)(i)}{2-i}=x+i y \\ \frac{1+(-1)+2 i}{2-i}=x+i y \\ \frac{2 i}{2-i}=x+i y \end{array}$ On...

Find the values of the following expressions:
(i) ${(1 + i)}^6 + {(1 - i)}^3$

CBSE, Class 11, Complex Numbers, Exercise 13.1, Maths, RD Sharma

Solution: (i) ${(1 + i)}^6 + {(1 – i)}^3$ Let's simplify, ${(1 + i)}^6 + {(1 – i)}^3 = {(1 + i)^2 }^3 + (1 – i)^2 (1 – i)$ $= {\{1 + i^2 + 2i}\}^3 + (1 + i^2 – 2i)(1 – i)$ $= {\{1 – 1 + 2i}\}^3 + (1...

Find the values of the following expressions:
(i) $\frac{[i^{592} + i^{590} + i^{588} + i^{586} + i^{584}]} {[i^{582} + i^{580} + i^{578} + i^{576} + i^{574}]}$
(ii) $1 + i^2 + i^4 + i^6 + i^8 + \dots + i^{20}$

CBSE, Class 11, Complex Numbers, Exercise 13.1, Maths, RD Sharma

Solution: (i) $\frac{[i^{592} + i^{590} + i^{588} + i^{586} + i^{584}]} {[i^{582} + i^{580} + i^{578} + i^{576} + i^{574}]}$ Let us simplify we get, $\frac{[i^{592} + i^{590} + i^{588} + i^{586} +...

Find the values of the following expressions:
(i) $i + i^2 + i^3 + i^4$
(ii) $i^5 + i^{10} + i^{15}$

CBSE, Class 11, Complex Numbers, Exercise 13.1, Maths, RD Sharma

Solution: (i) $i + i^2 + i^3 + i^4$ Let's simplify, $i + i^2 + i^3 + i^4 = i + i^2 + i^2\times i + i^4$ $= i – 1 + (– 1) \times i + 1 [\text{since}\ i^4 = 1, i^2 = – 1]$ $= i – 1 – i + 1$ $= 0$...

Find the values of the following expressions:
(i) $i^{49} + i^{68} + i^{89} + i^{110}$
(ii) $i^{30} + i^{80} + i^{120}$

CBSE, Class 11, Complex Numbers, Exercise 13.1, Maths, RD Sharma

Solution: (i) $i^{49} + i^{68} + i^{89} + i^{110}$ Let's simplify, $i^{49} + i^{68} + i^{89} + i^{110} = i ^{(48 + 1)} + i^{68} + i^{(88 + 1)} + i^{(108 + 2)}$ $= {(i^4)}^{12} \times i + (i^4)^{17}...

Show that $1 + i^{10} + i^{20} + i^{30}$ is a real number?

CBSE, Class 11, Complex Numbers, Exercise 13.1, Maths, RD Sharma

Solution: Given that: $1 + i^{10} + i^{20} + i^{30} = 1 + i^{(8 + 2)} + i^{20} + i^{(28 + 2)}$ $= 1 + (i^4)^2 \times i^2 + (i^4)^5 + (i^4)^7 \times i^2$ $= 1 – 1 + 1 – 1 [\text{since}\, i^4 = 1, i^2...

Find the equation of the straight line drawn through the point of intersection of the lines x + y = 4 and 2x – 3y = 1 and perpendicular to the line cutting off intercepts 5, 6 on the axes.

CBSE, Class 11, Exercise 23.19, Maths, RD Sharma, Straight Lines

The lines x + y = 4 and 2x – 3y = 1 The equation of the straight line passing through the point of intersection of x + y = 4 and 2x − 3y = 1 is \[\begin{array}{*{35}{l}} x\text{ }+\text{...

Evaluate the following:
(i) $[i^{41} + {1/i}^ {257}]$
(ii) $(i^{77} + i^ {70} + i^ {87} + i^{ 414})^3$

CBSE, Class 11, Complex Numbers, Exercise 13.1, Maths, RD Sharma

Solution: (i) $\left[\mathrm{i}^{41}+1 / \mathrm{i}^{257}\right]$ Let's simplify, $\begin{array}{l} {[\mathrm{i} 41+1 / \mathrm{i} 257]=\left[\mathrm{i} 40+1+1 / \mathrm{i}^{256+1}\right]} \\...

Find the equation of the line passing through the point of intersection of 2x – 7y + 11 = 0 and x + 3y – 8 = 0 and is parallel to (i) x = axis (ii) y-axis.

CBSE, Class 11, Exercise 23.19, Maths, RD Sharma, Straight Lines

The equations, 2x – 7y + 11 = 0 and x + 3y – 8 = 0 The equation of the straight line passing through the points of intersection of 2x − 7y + 11 = 0 and x + 3y − 8 = 0 is given below:...

Find the equation of a straight line through the point of intersection of the lines 4x – 3y = 0 and 2x – 5y + 3 = 0 and parallel to 4x + 5 y + 6 = 0.

CBSE, Class 11, Exercise 23.19, Maths, RD Sharma, Straight Lines

Lines 4x – 3y = 0 and 2x – 5y + 3 = 0 and parallel to 4x + 5 y + 6 = 0 The equation of the straight line passing through the points of intersection of 4x − 3y = 0 and 2x − 5y + 3 = 0 is given below:...

Find the equations to the straight lines which pass through the point (h, k) and are inclined at angle tan-1 m to the straight line y = mx + c.

CBSE, Class 11, Exercise 23.18, Maths, RD Sharma, Straight Lines

The equation passes through (h, k) and make an angle of tan-1 m with the line y = mx + c Since, the equations of two lines passing through a point x1, y1 and making an angle α with the given line y...

Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75o to the straight line x + y + √3(y – x) = a.

CBSE, Class 11, Exercise 23.18, Maths, RD Sharma, Straight Lines

The equation passes through (0,0) and make an angle of 75° with the line x + y + √3(y – x) = a. Since, the equations of two lines passing through a point x1,y1 and making an angle α with the given...

Find the equation of the straight lines passing through the origin and making an angle of 45o with the straight line √3x + y = 11.

CBSE, Class 11, Exercise 23.17, Maths, RD Sharma, Straight Lines

Equation passes through (0, 0) and make an angle of 45° with the line √3x + y = 11. Since, the equations of two lines passing through a point x1,y1 and making an angle α with the given line y = mx +...

Prove that the area of the parallelogram formed by the lines a1x + b1y + c1 = 0, a1x + b1y + d1 = 0, a2x + b2y + c2 = 0, a2x + b2y + d2 = 0 is (FIG 1)sq. units. Deduce the condition for these lines to form a rhombus.

CBSE, Class 11, Exercise 23.17, Maths, RD Sharma, Straight Lines

FIG 1: SOLUTION: The given lines are \[\begin{array}{*{35}{l}} {{a}_{1}}x\text{ }+\text{ }{{b}_{1}}y\text{ }+\text{ }{{c}_{1}}~=\text{ }0\text{ }\ldots \text{ }\left( 1 \right) \\ {{a}_{1}}x\text{...

Show that the diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + n’ = 0, mx + ly + n = 0 and mx + ly + n’ = 0 include an angle π/2.

CBSE, Class 11, Exercise 23.17, Maths, RD Sharma, Straight Lines

The given lines are \[\begin{array}{*{35}{l}} lx\text{ }+\text{ }my\text{ }+\text{ }n\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right) \\ mx\text{ }+\text{ }ly\text{ }+\text{ }n\text{...

Prove that the lines 2x + 3y = 19 and 2x + 3y + 7 = 0 are equidistant from the line 2x + 3y = 6.

CBSE, Class 11, Exercise 23.16, Maths, RD Sharma, Straight Lines

The lines A, 2x + 3y = 19 and B, 2x + 3y + 7 = 0 also a line C, 2x + 3y = 6. Let d1 be the distance between lines 2x + 3y = 19 and 2x + 3y = 6, While d2 is the distance between lines 2x + 3y + 7 = 0...

Find the equation of two straight lines which are parallel to x + 7y + 2 = 0 and at unit distance from the point (1, -1).

CBSE, Class 11, Exercise 23.16, Maths, RD Sharma, Straight Lines

The equation is parallel to x + 7y + 2 = 0 and at unit distance from the point (1, -1) The equation of given line is x + 7y + 2 = 0 … (1) The equation of a line parallel to line x + 7y + 2 = 0 is...

The equations of two sides of a square are 5x – 12y – 65 = 0 and 5x – 12y + 26 = 0. Find the area of the square.

CBSE, Class 11, Exercise 23.16, Maths, RD Sharma, Straight Lines

Two side of square are 5x – 12y – 65 = 0 and 5x – 12y + 26 = 0 The sides of a square are 5x − 12y − 65 = 0 … (1) 5x − 12y + 26 = 0 … (2) Since, lines (1) and (2) are parallel. So, the distance...

Find the distance of the point of intersection of the lines 2x + 3y = 21 and 3x – 4y + 11 = 0 from the line 8x + 6y + 5 = 0.

CBSE, Class 11, Exercise 23.15, Maths, RD Sharma, Straight Lines

The lines 2x + 3y = 21 and 3x – 4y + 11 = 0 Solving the lines 2x + 3y = 21 and 3x − 4y + 11 = 0 we get: x = 3, y = 5 So, the point of intersection of 2x + 3y = 21 and 3x − 4y + 11 = 0 is (3, 5)....

Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are (a cos α, a sin α) and (a cos β, a sin β).

CBSE, Class 11, Exercise 23.15, Maths, RD Sharma, Straight Lines

Coordinates are (a cos α, a sin α) and (a cos β, a sin β). Equation of the line passing through (a cos α, a sin α) and (a cos β, a sin β) is

Find the perpendicular distance of the line joining the points (cos θ, sin θ) and (cos ϕ, sin ϕ) from the origin.

CBSE, Class 11, Exercise 23.15, Maths, RD Sharma, Straight Lines

The points (cos θ, sin θ) and (cos ϕ, sin ϕ) from the origin. The equation of the line joining the points (cos θ, sin θ) and (cos ϕ, sin ϕ) is given below:

Find the distance of the point (4, 5) from the straight line 3x – 5y + 7 = 0.

CBSE, Class 11, Exercise 23.15, Maths, RD Sharma, Straight Lines

The line: 3x – 5y + 7 = 0 Comparing ax + by + c = 0 and 3x − 5y + 7 = 0, we get: a = 3, b = − 5 and c = 7 So, the distance of the point (4, 5) from the straight line 3x − 5y + 7 = 0 is ∴ The...

Solve each of the following system of equations in R- $11-5 x>-4,4 x+13 \leq-11$

CBSE, Class 11, Exercise 15.2, Linear Inequations, Maths, RD Sharma

$11-5 x>-4$ and $4 x+13 \leq-11$ $11-5 x>-4$ $11-5 x-11>-4-11$ $-5 x>-15$ Dividing both the sides by 5 we get, $-5 x / 5>-15 / 5$ $-x>-3$ $x<3$ $\therefore \mathrm{X}...

Solve each of the following system of equations in R – 5x – 1 < 24, 5x + 1 > –24

CBSE, Class 11, Exercise 15.2, Linear Inequations, Maths, RD Sharma

$5 x-1<24$ and $5 x+1>-24$ $5 x-1<24$ $5 x-1+1<24+1$ $5 x<25$ Dividing both the sides by 5 we get, $5 x / 5<25 / 5$ $x<5$ $\therefore \mathrm{x} \in(-\infty, 5) \ldots(1)$ Now,...

Solve each of the following system of equations in R- 2x – 3 < 7, 2x > –4

CBSE, Class 11, Exercise 15.2, Linear Inequations, Maths, RD Sharma

$2 x-3<7$ and $2 x>-4$ $2 x-3<7$ $2 x-3+3<7+3$ $2 \mathrm{x}<10$ Dividing both the sides by 2 we get, $2 x / 2<10 / 2$ $x<5$ $\therefore \mathrm{x} \in(-\infty, 5) \ldots(1)$...

Solve each of the following system of equations in R- 2x + 6 ≥ 0, 4x – 7 < 0

CBSE, Class 11, Exercise 15.2, Linear Inequations, Maths, RD Sharma

$2 x+6 \geq 0$ and $4 x-7<0$ $2 x+6 \geq 0$ $2 x+6-6 \geq 0-6$ $2 x \geq-6$ Dividing both the sides by 2 we get, $2 x / 2 \geq-6 / 2$ $x \geq-3$ $\therefore \mathrm{x} \in[-3, \infty) \ldots(1)$...

Solve each of the following system of equations in R 2x – 7 > 5 – x, 11 – 5x ≤ 1

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

2x – 7 > 5 – x and 11 – 5x ≤ 1 Let us consider the first inequality. \[\begin{array}{*{35}{l}} 2x\text{ }-\text{ }7\text{ }>\text{ }5\text{ }-\text{ }x \\ 2x\text{ }-\text{ }7\text{ }+\text{...

Solve: (2x + 3)/4 – 3 < (x – 4)/3 – 2

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

\[\begin{array}{*{35}{l}} \left( 2x\text{ }+\text{ }3 \right)/4\text{ }-\text{ }3\text{ }<\text{ }\left( x\text{ }-\text{ }4 \right)/3\text{ }-\text{ }2 \\ \left( 2x\text{ }+\text{ }3...

Solve : 5x/2 + 3x/4 ≥ 39/4

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, Maths, RD Sharma, RD Sharma

\[\begin{array}{*{35}{l}} 5x/2\text{ }+\text{ }3x/4\text{ }\ge \text{ }39/4 \\ taking\text{ }LCM \\ \left[ 2\left( 5x \right)+3x \right]/4\text{ }\ge \text{ }39/4 \\ 13x/4\text{ }\ge \text{...

Solve : –(x – 3) + 4 < 5 – 2x

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

\[\begin{array}{*{35}{l}} - \left( x\text{ }-\text{ }3 \right)\text{ }+\text{ }4\text{ }<\text{ }5\text{ }-\text{ }2x \\ -x\text{ }+\text{ }3\text{ }+\text{ }4\text{ }<\text{ }5\text{...

Solve : (3x – 2)/5 ≤ (4x – 3)/2

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

(3x – 2)/5 ≤ (4x – 3)/2 Multiply both the sides by 5 we get, \[\begin{array}{*{35}{l}} \left( 3x\text{ }-\text{ }2 \right)/5\text{ }\times \text{ }5\text{ }\le \text{ }\left( 4x\text{ }-\text{ }3...

Solve : 3x + 9 ≥ –x + 19

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

\[\begin{array}{*{35}{l}} x\text{ }+\text{ }9\text{ }\ge -\text{ }x\text{ }+\text{ }19 \\ 3x\text{ }+\text{ }9\text{ }-\text{ }9\text{ }\ge \text{ }-x\text{ }+\text{ }19\text{ }-\text{ }9 \\...

Solve: 3x – 7 > x + 1

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

\[\begin{array}{*{35}{l}} 3x\text{ }-\text{ }7\text{ }>\text{ }x\text{ }+\text{ }1 \\ 3x\text{ }-\text{ }7\text{ }+\text{ }7\text{ }>\text{ }x\text{ }+\text{ }1\text{ }+\text{ }7 \\ 3x\text{...

Solve: 4x-2 < 8, when (i) x ∈ R (ii) x ∈ Z

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

\[\begin{array}{*{35}{l}} 4x\text{ }-\text{ }2\text{ }<\text{ }8 \\ 4x\text{ }-\text{ }2\text{ }+\text{ }2\text{ }<\text{ }8\text{ }+\text{ }2 \\ 4x\text{ }<\text{ }10 \\ \end{array}\]...

Solve: -4x > 30, when (i) x ∈ R (ii) x ∈ Z

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

-4x > 30 dividing by 4, we get \[\begin{array}{*{35}{l}} -4x/4\text{ }>\text{ }30/4 \\ -x\text{ }>\text{ }15/2 \\ x\text{ }<\text{ }\text{ }-15/2 \\ \end{array}\] (i) x ∈ R When x is...

Solve the following linear Inequations in R 2x < 50, when (i) x ∈ R (ii) x ∈ Z

CBSE, Class 11, Exercise 15.1, Linear Inequations, Maths, RD Sharma

12x < 50 dividing by 12, we get \[\begin{array}{*{35}{l}} 12x/\text{ }12\text{ }<\text{ }50/12 \\ x\text{ }<\text{ }25/6 \\ \end{array}\] (i) x ∈ R When x is a real number, the solution...

Evaluate the following:
(i) $\frac{1} {i^{58}}$
(ii) $i^{37} + \frac{1}{i^{67}}$

CBSE, Class 11, Complex Numbers, Exercise 13.1, Maths, RD Sharma

Solution: (i) $\frac{1} {i^{58}}$ Let's simplify, $\frac{1} {i^{58}} = \frac{1} {i^ {56+2}}$ $= \frac{1}{ i^{56}} \times {i^{2}}$ $= \frac{1} {(i^4)^{14}} \times {i^{2}}$ $= \frac{1} {i^2} [\text...

The sum of first $7$ terms of an A.P. is $63$ and the sum of its next $7$ terms is $161$ . Find the 28th term of this A.P.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let’s take a to be the first term and d to be the common difference of the A.P And we know that, sum of first n terms of A.P. is ${{S}_{n}}=n/2\left( 2a+\left( n-1 \right)d \right)$ It is given that...

If the 10th term of an A.P. is $21$ and the sum of its first $10$ terms is $120$ , find its nth term.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let’s consider a to be the first term and d be the common difference of the AP And we know that, sum of first n terms of A.P is denoted by ${{S}_{n}}=n/2\left( 2a+\left( n-1 \right)d \right)$and...

The sum of first q terms of an A.P. is $162$ . The ratio of its 6th term to its 13th term is $1:2$ . Find the first and 15th term of the A.P.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let a be the first term and d be the common difference of the A.P. And as we know that, sum of first n terms of an A.P is: ${{S}_{n}}=n/2\left( 2a+\left( n-1 \right)d \right)$ Also, nth term of the...

The first and the last terms of an A.P are $5$ and $45$ respectively. If the sum of all its terms is $400$ , find its common difference.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given, First term $(a)=5$ and the last term $(l)=45$ Also, ${{S}_{n}}=400$ We know that, ${{a}_{n}}=a+\left( n-1 \right)d$ $⟹45=5+(n–1)d$ $⟹40=nd–d$ $⟹nd–d=40$ ….. (1) Next ${{S}_{n}}=n/2\left(...

The first and the last terms of an A.P. are $7$ and $49$ respectively. If sum of all its terms is $420$ , find the common difference.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given, First term $(a)=7$, last term $\left( {{a}_{n}} \right)=49$ and sum of n terms $\left( {{S}_{n}} \right)=420$ Now, we know that ${{a}_{n}}=a+\left( n-1 \right)d$ $⟹49=7+(n–1)d$ $⟹43=nd–d$...

In an A.P. the first term is $22$ , nth term is $-11$ and the sum of first n term is $66$ . Find n and the d, the common difference.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given, The first term of the A.P (a) $=22$ The nth term of the A.P (l) $=-11$ And, sum of all the terms ${{S}_{n}}=66$ Let the common difference of the A.P. be d. So, finding the number of terms by...

In an A.P. the first term is $8$ , nth term is $33$ and the sum of first n term is $123$ . Find n and the d, the common difference.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given, The first term of the A.P (a) $=8$ The nth term of the A.P (l) $=33$ And, the sum of all the terms ${{S}_{n}}=123$ Let the common difference of the A.P. be d. So, find the number of terms by...

The first term of an A.P. is $5$ , the last term is $45$ and the sum is $400$ . Find the number of terms and the common difference.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Sum of first n terms of an A.P is given by ${{S}_{n}}=n/2\left( 2a+\left( n-1 \right)d \right)$ Given, First term $(a)=5$, last term $\left( {{a}_{n}} \right)=45$ and sum of n terms $\left(...

If the sum of 7 terms of an A.P. is $49$ and that of $17$ terms is $289$ , find the sum of n terms.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given, Sum of 7 terms of an A.P. is $49$ $\Rightarrow {{S}_{7}}=49$ And, sum of 17 terms of an A.P. is $289$ $\Rightarrow {{S}_{17}}=289$ Let the first term of the A.P be a and common difference as...

Find the sum of first 51 terms of an A.P. whose second and third terms are $14$ and $18$ respectively.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let’s take the first term as a and the common difference as d. Given that, ${{a}_{2}}=14$and ${{a}_{3}}=18$ And, we know that ${{a}_{n}}=a+\left( n-1 \right)d$ So, ${{a}_{2}}=a+\left( 2-1 \right)d$...

In an A.P., if the 5th and 12th terms are $30$ and $65$ respectively, what is the sum of first 20 terms?

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let’s take the first term as a and the common difference to be d Given that, ${{a}_{5}}=30$and ${{a}_{12=65}}$ And, we know that ${{a}_{n}}=a+\left( n-1 \right)d$ So, ${{a}_{5}}=a+\left( 5-1...

In an A.P., if the first term is $22$ , the common difference is $-4$ and the sum to n terms is $64$ , find n.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given that, $a=22$, $d=–4$ and ${{S}_{n}}=64$ Let us consider the number of terms as n. For sum of terms in an A.P, we know that ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first...

Find the sum of n terms of the series $\left( 4-\frac{1}{n} \right)+\left( 4-\frac{2}{n} \right)+\left( 4-\frac{3}{n} \right)+....$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given in question, First term(a) $=4-1/n$ Common difference $={{a}_{n}}-{{a}_{n-1}}=\left( 4-2/n \right)-\left( 4-1/n \right)$ $=(4n-2-4n+1)/n=-1/n$ By using formula for sum of n terms of A.P, we...

If 12th term of an A.P. is $-13$ and the sum of the first four terms is $24$ , what is the sum of first 10 terms?

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let us assume the first term as a and the common difference as d. Given in question, ${{a}_{12}}=-13$ ${{S}_{4}}=24$ Also, as we know that ${{a}_{n}}=a+\left( n-1 \right)d$ So, for the 12th term...

The first term of an A.P. is $2$ and the last term is $50$ . The sum of all these terms is $442$ . Find the common difference.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given data according to question, The first term of the A.P is (a) $=2$ The last term of the A.P is (l) $=50$ Sum of all the terms ${{S}_{n}}=442$ So, let us assume the common difference of the...

The third term of an A.P. is $7$ and the seventh term exceeds three times the third term by $2$ . Find the first term, the common difference and the sum of first $20$ terms.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Let’s consider the first term of the A.P as a and the common difference as d. Given information in the question, ${{a}_{3}}=7$….(1) and, ${{a}_{7}}=3{{a}_{3}}+2$……(2) So, by using (1) in (2), we...

The first and the last terms of an A.P. are $17$ and $350$ respectively. If the common difference is $9$ , how many terms are there and what is their sum?

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given, the first term of the A.P (a) $=17$ The last term of the A.P (l) $=350$ The common difference (d) of the A.P. $=9$ Let the number of terms be n. And, as we know that; $l=a+(n–1)d$ So, by...

Find the sum: (vi) $34+32+30+...+10$ (vii) $25+28+31+...+100$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

We know that the sum of terms for an A.P is given by $S_{n}^{{}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first term for the given A.P. d = common difference of the given A.P. n = number...

Find the sum: (v) $7+10\frac{1}{2}+14+...+84$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma, RD Sharma

We know that the sum of terms for an A.P is given by $S_{n}^{{}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first term for the given A.P. d = common difference of the given A.P. n = number...

Find the sum: (iii) $(-5)+(-8)+(-11)+...+(-230)$ (iv) $1+3+5+7+...+199$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

We know that the sum of terms for an A.P is given by $S_{n}^{{}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first term for the given A.P. d = common difference of the given A.P. n = number...

Find the sum: (i) $2+4+6+..+200$ (ii) $3+11+19+...+803$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

We know that the sum of terms for an A.P is given by $S_{n}^{{}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first term for the given A.P. d = common difference of the given A.P. n = number...

Find the sum of (iii) all 3 – digit natural numbers which are divisible by 13. (iv) all 3 – digit natural numbers which are multiples of 11.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(iii) All 3 digit natural number which are divisible by $13$. So, we know that the first 3 digit multiple of $13$ is $104$ and the last $3$ digit multiple of $13$ is $988$. And, these terms...

Find the sum of (ii) the first $40$ positive integers divisible by (b) $5$ (c) $6$ .

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(b) First $40$ positive integers divisible by $5$ Hence, the first multiple of $5$ is $5$ and the 40th multiple is $200$. And, these terms will form an A.P. with the common difference of $5$. Here,...

Find the sum of (i) the first $15$ multiples of $8$ (ii) the first $40$ positive integers divisible by (a) $3$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

We know that the sum of n terms of an A.P is given by formula, ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first term for the A.P. d = common difference of the A.P. n = number...

Find the sum of the first (iii) 51 terms of the A.P. : whose second term is $2$ and fourth term is $8$ .

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(iii)Sum of $51$ terms of an AP whose second and fourth term is given by ${{a}_{2}}=2$ and ${{a}_{4}}=8$ As we know that, ${{a}_{2}}=a+d$ $2=a+d$ … let this be equation (1) Also, ${{a}_{4}}=a+3d$...

Find the sum of the first (i) 11 terms of the A.P. : $2,6,10,14$ , . . . (ii) 13 terms of the A.P. : $-6,0,6,12$ , . . .

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

As we know that the sum of terms for arithmetic progressions is given by ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first term for the A.P. d = common difference of the given...

The following distribution gives the daily income of $50$ workers of a factory: Convert the below distribution to a ‘less than’ type cumulative frequency distribution and draw its ogive.

CBSE, Class 10, Exercise 7.6, Maths, RD Sharma, Statistics

given data Daily income (in Rs): No of workers: $100–120$ $12$ $120–140$ $14$ $140–160$ $8$ $160–180$ $6$ $180–200$ $10$ Firstly, we prepare the cumulative frequency table by less than method as...

The monthly profits (in Rs) of $100$ shops are distributed as follows:

CBSE, Class 10, Exercise 7.6, Maths, RD Sharma, Statistics

Given data for frequency polygon is: Profit per shop No of shops: $0–50$ $12$ $50–100$ $18$ $100–150$ $27$ $150–200$ $20$ $200–250$ $17$ $250–300$ $6$ Solution: Doing for the less than method, we...

Draw an Ogive to represent the following frequency distribution:

CBSE, Class 10, Exercise 7.6, Maths, RD Sharma, Statistics

Given data for make an ogive is below: Class-interval $0–4$ $5–9$ $10–14$ $15–19$ $20–24$ No. of students $2$ $6$ $10$ $5$ $3$ Since the given frequency distribution is not continuous we will have...

The marks scored by $750$ students in an examination are given in the form of a frequency distribution table: Prepare a cumulative frequency distribution table by less than method and draw an ogive.

CBSE, Class 10, Exercise 7.6, Maths, RD Sharma, Statistics

Given data to make an ogive is Marks No. of Students $600–640$ $16$ $640–680$ $45$ $680–720$ $156$ $720–760$ $284$ $760–800$ $172$ $800–840$ $59$ $840–880$ $18$ Solution: Marks No. of Students Marks...

Draw an ogive by less than the method for the following data:

CBSE, Class 10, Exercise 7.6, Maths, RD Sharma, Statistics

Given data to make ogive is No. of rooms $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ No. of houses $4$ $9$ $22$ $28$ $24$ $12$ $8$ $6$ $5$ $2$ No. of rooms No. of houses Cumulative Frequency Less than...

Find the value of k, if points $A(7,-2)$ , $B(5,1)$ and $C(3,2k)$ are collinear.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Given, Points $A(7,-2)$, $B(5,1)$ and $C(3,2k)$ Given the area of$\vartriangle ABC$ is $=\frac{1}{2}\left\{ 7\left( 1-2k \right)+5\left( 2k+2 \right)+3\left( -2-1 \right) \right\}$...

Find the value of k if points $(k,3)$ , $(6,-2)$ and $(-3,4)$ are collinear.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume $A(k,3)$, $B(6,-2)$ and $C(-3,4)$ be the given points. Given area of $\vartriangle ABC$is $=\frac{1}{2}\left\{ k\left( -2-4 \right)+6\left( 4-3 \right)+\left( -3 \right)\left( 3+2 \right)...

If (x, y) be on the line joining the two points $(1,-3)$ and $(-4,2)$ . Prove that $x+y+2=0$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume $A(x,y)$, $B(1,-3)$ and $C(-4,2)$ be the given points. Given area of $\vartriangle ABC$ $=\frac{1}{2}\left\{ x\left( -3-2 \right)+1\left( 2-y \right)+\left( -4 \right)\left( y+3 \right)...

If the vertices of a triangle are $(1,-3)$ , $(4,p)$ and $(-9,7)$ and its area is $15$ sq. units, find the value (s) of p.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume $A(1,-3)$, $B(4,p)$ and $C(-9,7)$ be the vertices of $\vartriangle ABC$ Given, area of $\vartriangle ABC=15$ sq.units $15=\frac{1}{2}\left| 1\left( p-7 \right)+4\left( 7+3 \right)-9\left(...

Prove that the points $(a,b)$ , $\left( {{a}_{1}},{{b}_{1}} \right)$ and $\left( a-{{a}_{1}},b-{{b}_{1}} \right)$ are collinear if $a{{b}_{1}}={{a}_{1}}b$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume $A(a,b)$, $B\left( a_{1}^{{}},{{b}_{1}} \right)$and $C\left( a-{{a}_{1}},b-{{b}_{1}} \right)$ be the given points. So, given the area of $\vartriangle ABC$ $=\frac{1}{2}\left\{ a\left[...

For what value of a the points $(a,1)$ , $(1,-1)$ and $(11,4)$ are collinear?

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume, $A(a,1)$, $B(1,-1)$ and $C(11,4)$ be the given points Now the area of $\vartriangle ABC$is given by, $=\frac{1}{2}\left\{ a\left( -1-4 \right)+1\left( 4-1 \right)+11\left( 1+1 \right)...

If $A(-3,5)$ , $B(-2,-7)$ , $C(1,-8)$ and $D(6,3)$ are the vertices of a quadrilateral ABCD, find its area.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Now, join A and C. Then, we get $\vartriangle ABC$and $\vartriangle ADC$ Hence, The Area of quad. ABCD $=$ $ar\left( \vartriangle ABC \right)+ar\left( \vartriangle ADC \right)$ $=\frac{1}{2}\left|...

If $P(-5,-3)$ , $Q(-4,-6)$ , $R(2,-3)$ and $S(1,2)$ are the vertices of a quadrilateral PQRS, find its area.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

let’s join P and R. Now, $\vartriangle PSR$area is given by $=\frac{1}{2}\left| -5\left( 2+3 \right)+1\left( -3+3 \right)+2\left( -3-2 \right) \right|$ $=\frac{1}{2}\left| -5\times 5+1\times...

Find the area of the triangle PQR with $Q(3,2)$ and the mid-points of the sides through Q being $(2,-1)$ and $(1,2)$ .

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume the coordinates of P and R be $\left( {{x}_{1}},{{y}_{1}} \right)$and $\left( {{x}_{2}},{{y}_{2}} \right)$ respectively. Then, assume the points E and F be the centers of PQ and QR...

In $\vartriangle ABC$ , the coordinates of vertex $A(0,-1)$ and $D(1,0)$ and $E(0,1)$ respectively the mid-points of the sides AB and AC. If F is the mid-point of side BC, find the area of $\vartriangle DEF$ .

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume B(a, b) and C(p, q) be the other two vertices of the $\vartriangle ABC$ As, we know that D is the center of AB Then, coordinates of $D=(0+a/2,-1+b/2)$ $(1,0)=(a/2,b-1/2)$ $1=a/2$ and...

Find the area of a quadrilateral ABCD, the coordinates of whose vertices are $A(-3,2)$ , $B(5,4)$ , $C(7,6)$ and $D(-5,-4)$ .

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Join AC. So, we have formed two triangles Then, the $ar\left( ABCD \right)=ar\left( \vartriangle ABC \right)+ar\left( \vartriangle ACD \right)$ Area of $\vartriangle ABC$ is given by,...

Show that the following sets of points are collinear.(i) $(2,5)$ , $(4,6)$ and $(8,8)$ (ii) $(1,-1)$ , $(2,1)$ and $(4,5)$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Condition: For the 3 points to be collinear the area of the triangle formed with the 3 points has to be zero. (a) Assume $A(2,5)$, $B(4,6)$ and $C(8,8)$ be the given points Then, the area of...

The vertices of $\vartriangle ABC$ are $(-2,1)$ , $(5,4)$ and $(2,-3)$ respectively. Find the area of the triangle and the length of the altitude through A.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Let $A(-2,1)$, $B(5,4)$ and $C(2,-3)$ be the vertices of $\vartriangle ABC$ And assume AD be the altitude through A. Area of $\vartriangle ABC$ is given by $=1/2|(-2)(4+3)–5(-3–1)+2(1–4)|$...

The four vertices of a quadrilateral are $(1,2)$ , $(-5,6)$ , $(7,-4)$ and $(k,-2)$ taken in order. If the area of the quadrilateral is zero, find the value of k.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume $A(1,2)$, $B(-5,6)$, $C(7,-4)4$and $D(k,-2)$ be the given points Firstly, area of $\vartriangle ABC$ is given by $=1/2|(1)(6+4)-5(-4+2)+7(2-6)|$ $=1/2|10+30-28|$ $=1/2\times 12$ $=6$ Now, the...

Find the area of the quadrilaterals, the coordinates of whose vertices are(iii) $(-4,-2)$ , $(-3,-5)$ , $(3,-2)$ , $(2,3)$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Let $A(-4,2)$, $B(–3,–5)$, $C(3,-2)$ and $D(2,3)$ be the given points Firstly, area of $\vartriangle ABC$ is given by $=1/2|(-4)(-5+2)–3(-2+2)+3(-2+5)|$ $=1/2|(-4)(-3)–3(0)+3(3)|$ $=21/2$ then, the...

Find the area of the quadrilaterals, the coordinates of whose vertices are (i) $(-3,2)$ , $(5,4)$ , $(7,-6)$ and $(-5,-4)$ (ii) $(1,2)$ , $(6,2)$ , $(5,3)$ and $(3,4)$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

Assume $A(-3,2)$, $B(5,4)$, $C(7,-6)$ and $D(-5,– 4)$ be the given points. Given area of $\vartriangle ABC$ $=1/2[-3(4 +6)+5(-6–2)+7(2–4)]$ $=1/2[-3.1+5.(-8)+7(-2)]$ $=1/2[-30–40-14]$ $=– 42$ So,...

Find the area of a triangle whose vertices are (iii) $(a,c+a)$ , $(a,c)$ and $(-a,c-a)$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

(iii) Let $A=\left( {{x}_{1}}{{y}_{1}} \right)=\left( a,c+a \right),B=\left( {{x}_{2}},{{y}_{2}} \right)=\left( a,c \right)=C=\left( {{x}_{3}},{{y}_{3}} \right)=\left( -a,c-a \right)$ be the given...

Find the area of a triangle whose vertices are(i) $(6,3)$ , $(-3,5)$ and $(4,- 2)$ (ii) $\left[ \left( at_{1}^{2},a{{t}_{1}} \right),\left( at_{2}^{2},2at2 \right)\left( at_{3}^{2},2a{{t}_{3}} \right) \right]$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.5, Maths, RD Sharma

(i) Assume $A(6,3)$, $B(-3,5)$ and C(4,-2) be the given points As, we know that, area of a triangle is given by: \[1/2\left[ {{x}_{1}}\left( {{y}_{2}}-{{y}_{3}} \right)+{{x}_{2}}\left(...

(v) How many terms of the A.P. is $27$ , $24$ , $21$ . . . should be taken that their sum is zero?

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(v) Given A.P. in the question is $27$, $24$, $21$. . . As we know that, ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Here we have, the first term of A.P (a) $=27$ The sum of n terms of A.P...

(iii) How many terms of the A.P. $9$ , $17$ , $25$ , . . . must be taken so that their sum is $636$ ? (iv) How many terms of the A.P. $63$ , $60$ , $57$ , . . . must be taken so that their sum is $693$ ?

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(iii) Given AP in the question is $9$, $17$, $25$,… As we know that, ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Here we have, The first term of A.P as (a) = 9 and the sum of n terms of...

(i) How many terms of the sequence $18$ , $16$ , $14$ …. should be taken so that their sum is zero. (ii) How many terms are there in the A.P. whose first and fifth terms are $-14$ and $2$ respectively and the sum of the terms is $40$ ?

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(i) Given AP. in the question is $18$, $16$, $14$, … We know that, ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Here, The first term of A.P is(a) $=18$ The sum of n terms of A.P is $\left(...

If the sum of a certain number of terms starting from first term of an A.P. is $25$ , $22$ , $19$ , . . ., is $116$ . Find the last term.

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

According to the question, the sum of the certain number of terms of an A.P. $=116$ As we know that, ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right)d \right]$ Where; a = first term for the A.P. d =...

Find the sum of first 25 terms of an A.P whose nth term is given by ${{a}_{n}}=7-3n$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

According to the question, an A.P. where nth term is defined by ${{a}_{n}}=7-3n$ For calculating the sum of n term we define ${{S}_{n}}$ as, ${{S}_{n}}=n\left( a+1 \right)/2$ Where, a = the first...

Find the sum of first 25 terms of an A.P whose nth term is given by ${{a}_{n}}=2-3n$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

According to the question, an A.P. where nth term is defined by ${{a}_{n}}=2-3n$ For calculating the sum of n term we define ${{S}_{n}}$ as, ${{S}_{n}}=n\left( a+1 \right)/2$ Where, a = the first...

Find the sum of first 20 terms the sequence whose nth term is ${{a}_{n}}=An+B$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

According to the question, an A.P. where nth term is defined by${{a}_{n}}=An+B$ According to the question we need to find the sum of first 20 terms. For calculating the sum of n term we define...

Solve the following quadratic equations: (xi) $x^{2}-(\sqrt{2}+i) x+\sqrt{2 i}=0$ (xii) $2 x^{2}-(3+7 i) x+(9 i-3)=0$

CBSE, Class 11, Exercise 14.2, Maths, Quadratic Equations, RD Sharma

(xi) $x^{2}-(\sqrt{2}+i) x+\sqrt{2 i}=0$ $x^{2}-(\sqrt{2 x}+i x)+\sqrt{2 i}=0$ $x^{2}-\sqrt{2 x}-i x+\sqrt{2 i}=0$ $x(x-\sqrt{2})-i(x-\sqrt{2})=0$ $(x-\sqrt{2})(x-i)=0$ $(x-\sqrt{2})=0$ or $(x-i)=0$...

Find the sum of first 15 terms of each of the following sequences having nth term as:(iii) ${{x}_{n}}=6-n$ (iv) ${{y}_{n}}=9-5n$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(iii) According to the question, an A.P. where nth term is defined by ${{x}_{n}}=6-n$ For calculating the sum of n term we define ${{S}_{n}}$as, ${{S}_{n}}=n\left( a+1 \right)/2$ Where, a = first...

Find the sum of first 15 terms of each of the following sequences having nth term as: (i) ${{a}_{n}}=3+4n$ (ii) ${{b}_{n}}=5+2n$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(i) According to the question, an A.P. where nth term is defined by ${{a}_{n}}=3+4n$ is given. For calculating the sum of n term we define ${{S}_{n}}$ as, ${{S}_{n}}=n\left( a+1 \right)/2$ Where, a...

Solve the following quadratic equations: (vii) $2 x^{2}+\sqrt{15 i x}-i=0$ (viii) $x^{2}-x+(1+i)=0$

CBSE, Class 11, Exercise 14.2, Maths, Quadratic Equations, RD Sharma

(vii) 2x2 + √15ix – i = 0 applying discriminant rule, x = (-b ±√(b2 – 4ac))/2a a = 2, b = √15i, c = -i =>15 – 8i = 16 – 1 – 8i 15 – 8i = 16 + (–1) – 8i = 16 + i2 – 8i [∵ i2 = –1] = 42 + (i)2 –...

Find the sum of last ten terms of the A.P. : $8$ , $10$ , $12$ , $14$ , .. , $126$

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given A.P.$=$ $8$, $10$, $12$, $14$, .. , $126$ Here, $a=8$ $d={{a}_{n}}-{{a}_{n-1}}=10-8=2$ As we know, ${{a}_{n}}=a+\left( n-1 \right)d$ So, to find the number of terms substituting the values in...

Solve the following quadratic equations: (iii) $(2+i) x^{2}-(5-i) x+2(1-i)=0$ (iv) $x^{2}-(2+i) x-(1-7 i)=0$

CBSE, Class 11, Exercise 14.2, Maths, Quadratic Equations, RD Sharma

(iii) (2 + i)x2 – (5- i)x + 2 (1 – i) = 0 applying discriminant rule, x = (-b ±√(b2 – 4ac))/2a a = (2+i), b = -(5-i), c = 2(1-i) since, i2 = –1 substituting –1 = i2 x = (1 – i) or 4/5 – 2i/5 ∴ The...

Find the sum of n terms of an A.P. whose the terms is given by ${{a}_{n}}=5-6n$ .

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

nth term of the A.P is given as ${{a}_{n}}=5-6n$. Let us put $n=1$ in nth term, we get ${{a}_{1}}=5-6.1=-1$ So, the first term (a) $=-1$ Last term $\left( {{a}_{n}} \right)=5-6n$ Then,...

Solving the following quadratic equations by factorization method: (iii) $x^{2}-(2 \sqrt{3}+3 i) x+6 \sqrt{3} i=0$ (iv) $6 x^{2}-17 i x-12=0$

CBSE, Class 11, Exercise 14.2, Maths, Quadratic Equations, RD Sharma

(iii) $x^{2}-(2 \sqrt{3}+3 i) x+6 \sqrt{3} i=0$ $x^{2}-(2 \sqrt{3} x+3 i x)+6 \sqrt{3} i=0$ $x^{2}-2 \sqrt{3} x-3 i x+6 \sqrt{3} i=0$ $x(x-2 \sqrt{3})-3 i(x-2 \sqrt{3})=0$ $(x-2 \sqrt{3})(x-3 i)=0$...

Find the sum to n terms of the A.P. $5$ , $2$ , $-1$ , $-4$ , $-7$ , …

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

Given AP is $5$, $2$, $-1$, -$4$, $-7$, …..,$n$. Here, $a=5$, $d={{a}_{n}}-{{a}_{n-1}}=2-5=-3$ As we know that, ${{S}_{n}}$ (sum of n terms) $=n/22a+(n-1)d$ $=n/22.5+(n-1)-3$ $=n/210-3(n-1)$...

Solve the following quadratic equations by factorization method only: $17 x^{2}-8 x+1=0$

CBSE, Class 11, Exercise 14.1, Maths, Quadratic Equations, RD Sharma

$17 x^{2}-8 x+1=0$ applying discriminant rule, $x=\left(-b \pm \sqrt{\left.\left(b^{2}-4 a c\right)\right) / 2 a}\right.$ $a=17, b=-8, c=1$ $\mathrm{x}=\left(-(-8) \pm...

Find the sum of the following arithmetic progressions:(vii) $\frac{x-y}{x+y},\frac{3x-2y}{x+y},\frac{5x-3y}{x+y}$ to n terms (viii) $-26$ , $-24$ , $-22$ , …. to $36$ terms

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(vii) First term (a) $=(x-y)/(x+y)$ Comman difference (d) $={{a}_{n}}-{{a}_{n-1}}=\left( 3x-2y \right)/\left( x+y \right)-\left( x-y \right)/\left( x+y \right)$ $=(3x-2y-x-y)/(x+y)$ $=(2x-y)/(x+y)$...

Find the sum of the following arithmetic progressions:(v) $a+b$ , $a-b$ , $a-3b$ , … to $22$ terms (vi) ${{\left( x-y \right)}^{2}},\left( {{x}^{2}}+{{y}^{2}} \right),{{\left( x+y \right)}^{2}}$ to $22$ terms

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(v) $a+b$, $a–b$, $a–3b$, ….. to $22$ terms First term (a) $=a+b$ Common difference (d) $={{a}_{n}}-{{a}_{n-1}}=-b-a-b=-2b$ Sum of n terms of A.P is ${{S}_{n}}=n/2\left\{ 2a\left( n-1 \right)d...

Solve the following quadratic equations by factorization method only: $\mathbf{x}^{2}-\mathbf{x}+1=0$

CBSE, Class 11, Exercise 14.1, Maths, Quadratic Equations, RD Sharma

$x^{2}-x+1=0$ $x^{2}-x+1 / 4+3 / 4=0$ $x^{2}-2(x)(1 / 2)+(1 / 2)^{2}+3 / 4=0$ $(x-1 / 2)^{2}+3 / 4=0\left[\right.$ Since, $\left.(a+b)^{2}=a^{2}+2 a b+b^{2}\right]$ $(x-1 / 2)^{2}+3 / 4 \times 1=0$...

Find the sum of the following arithmetic progressions: (iii) $3$ , $9/2$ , $6$ , $15/2$ , … to $25$ terms (iv) $41$ , $36$ , $31$ , … to $12$ terms

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

(iii) Given A.P $=3$, $9/2$, $6$, $15/2$ , … to $25$ terms First term (a) $=3$ Common difference (d) $={{a}_{n}}-{{a}_{n-1}}=9/2-3/2$ Sum of n terms ${{S}_{n}}$ and $n=25$ (given)...

Find the sum of the following arithmetic progressions: (i) $50$ , $46$ , $42$ , … to $10$ terms (ii) $1$ , $3$ , $5$ , $7$ , … to $12$ terms

Arithmetic Progressions, CBSE, Class 10, Exercise 9.6, Maths, RD Sharma

In an A.P if first term is given by $=a$, common difference $=d$, and if no. of terms are n. Therefore, sum of n terms of an A.P is given as: ${{S}_{n}}=n/2\left[ 2a+\left( n-1 \right) \right]$ (i)...

Solve the following quadratic equations by factorization method only: $5 x^{2}-6 x+2=0$

CBSE, Class 11, Exercise 14.1, Maths, Quadratic Equations, RD Sharma

$5 x^{2}-6 x+2=0$ applying discriminant rule, where, $x=\left(-b \pm \sqrt{\left.\left(b^{2}-4 a c\right)\right) / 2 a}\right.$ $a=5, b=-6, c=2$ => $\left.x=\left(-(-6) \pm...

Solve the following quadratic equations by factorization method only: $x^{2}-4 x+7=0$

CBSE, Class 11, Exercise 14.1, Maths, Quadratic Equations, RD Sharma

$x^{2}-4 x+7=0$ $x^{2}-4 x+4+3=0$ $x^{2}-2(x)(2)+2^{2}+3=0$ $(x-2)^{2}+3=0\left[\right.$ Since, $\left.(a-b)^{2}=a^{2}-2 a b+b^{2}\right]$ $(x-2)^{2}+3 \times 1=0$ since, $\mathrm{i}^{2}=-1...

Solve the following quadratic equations by factorization method only: $x^{2}+x+1=0$

CBSE, Class 11, Exercise 14.1, Maths, Quadratic Equations, RD Sharma

$\mathrm{x}^{2}+\mathrm{x}+1=0$ $x^{2}+x+1 / 4+3 / 4=0$ $x^{2}+2(x)(1 / 2)+(1 / 2)^{2}+3 / 4=0$ $(x+1 / 2)^{2}+3 / 4=0\left[\right.$ Since, $\left.(a+b)^{2}=a^{2}+2 a b+b^{2}\right]$ $(x+1 /...

Solve the following quadratic equations by factorization method only: $x^{2}+2 x+5=0$

CBSE, Class 11, Exercise 14.1, Maths, Quadratic Equations, RD Sharma

$x^{2}+2 x+5=0$ $x^{2}+2 x+1+4=0$ $x^{2}+2(x)(1)+1^{2}+4=0$ $(x+1)^{2}+4=0\left[\right.$ since, $\left.(a+b)^{2}=a^{2}+2 a b+b^{2}\right]$ $(x+1)^{2}+4 \times 1=0$ Since, $i^{2}=-1 \Rightarrow...

If $(-2,3)$ , $(4,-3)$ and $(4,5)$ are the mid-points of the sides of a triangle, find the coordinates of its centroid.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.4, Maths, RD Sharma

The directions of the centroid are just the normal of the directions of the vertices. So to track down the x facilitate of the orthocenter, include the three vertex x organizes and gap by three....

$A(3,2)$ and $B(-2,1)$ are two vertices of a triangle ABC whose centroid G has the coordinates $(5/3,-1/3)$ . Find the coordinates of the third vertex C of the triangle.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.4, Maths, RD Sharma

The directions of the centroid are just the normal of the directions of the vertices. So to track down the x facilitate of the orthocenter, include the three vertex x organizes and gap by three....

For any two sets A and B, prove that: A‘ – B‘ = B – A

CBSE, Class 11, Exercise 1.7, Exercise 1.7, Maths, Maths, RD Sharma, RD Sharma, Sets

Answer: To show, A’ – B’ ⊆ B – A Consider, x ∈ A’ – B’ x ∈ A’ and x ∉ B’ [A ∩ A’ = ϕ] x ∉ A and x ∈ B x ∈ B – A x ∈ A’ – B’ ∴ A’ – B’ = B – A Thus,...

Find the third vertex of a triangle, if two of its vertices are at $(-3,1)$ and $(0,-2)$ and the centroid is at the origin.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.4, Maths, RD Sharma

The centroid is the middle place of the item. The point in which the three medians of the triangle converge is known as the centroid of a triangle. It is likewise characterized as the mark of...

Two vertices of a triangle are $(1,2)$ , $(3,5)$ and its centroid is at the origin. Find the coordinates of the third vertex.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.4, Maths, RD Sharma

The directions of the centroid are just the normal of the directions of the vertices. So to track down the x facilitate of the orthocenter, include the three vertex x organizes and gap by three....

Find the centroid of the triangle whose vertices are: (i) $(1,4)$ , $(-1,-1)$ and $(3,-2)$ (ii) $(-2,3)$ , $(2,-1)$ and $(4,0)$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.4, Maths, RD Sharma

As we know that the coordinates of the centroid of a triangle whose vertices are $\left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right),\left( {{x}_{3}},{{y}_{3}} \right)$Are $\left(...

On which axis do the following points lie? (iii) $R(-4,0)$ (iv) $S(0,5)$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.1, Maths, RD Sharma

A diagram comprises of two tomahawks called the x (even) and y (vertical) tomahawks. These tomahawks compare to the factors we are relating. In financial aspects we will generally give the tomahawks...

On which axis do the following points lie? (i) $P(5,0)$ (ii) $Q(0,-2)$

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.1, Maths, RD Sharma

A diagram comprises of two tomahawks called the x (even) and y (vertical) tomahawks. These tomahawks compare to the factors we are relating. In financial aspects we will generally give the tomahawks...

A target is shown in fig. below consists of three concentric circles of radii, $3cm$ , $7cm$ and $9cm$ respectively. A dart is thrown and lands on the target. What is the probability that the dart will land on the shaded region?

CBSE, Class 10, Exercise 13.2, Maths, Probability, RD Sharma

Given in the question, 1st circle – with radius $3$ 2nd circle – with radius $7$ 3rd circle – with radius $9$ So, their areas would be Area of 1st circle $=\pi {{\left( 3 \right)}^{2}}=9\pi $ Area...

In the accompanying diagram, a fair spinner is placed at the center O of the circle. Diameter AOB and radius OC divide the circle into three regions labeled X, Y and Z.? If $\angle B0C={{45}^{\circ }}$ . What is the probability that the spinner will land in the region X?

CBSE, Class 10, Exercise 13.2, Maths, Probability, RD Sharma

Given in the question, $\angle BOC={{45}^{\circ }}$ $\angle AOC=180-45={{135}^{\circ }}$[ linear pair] Area of circle $=\pi {{r}^{2}}$ Area of region $x=\theta /360\times \pi {{r}^{2}}$...

Suppose you drop a tie at random on the rectangular region shown in fig. below. What is the probability that it will land inside the circle with diameter $1m$ ?

CBSE, Class 10, Exercise 13.2, Maths, Probability, RD Sharma

Formula of getting the probability is probability= number of favorable outcomes/ total number of outcomes. So, area of a circle with radius $0.5m$ A circle $={{\left( 0.5 \right)}^{2}}=0.25\pi...

For any two sets, prove that: (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) We know that, A ∪ (A ∩ B) [A ∪ A = A] (A ∪ A) ∩ (A ∪ B) ∴ A ∩ (A ∪ B) = A (ii) A ∩ (A ∪ B) = A We know that, (A ∩ A) ∪ (A ∩ B) [A ∩ A = A] ∴ A ∪ (A ∩ B) =...

For three sets A, B, and C, show that (i) A ∩ B = A ∩ C need not imply B = C. (ii) A ⊂ B ⇒ C – B ⊂ C – A

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) Consider, A = {1, 2} B = {2, 3} C = {2, 4} A ∩ B = {2} A ∩ C = {2} Thus, A ∩ B = A ∩ C and B is not equal to C. (ii) A ⊂ B C–B ⊂ C–A Consider, x ∈ C– B x ∈ C and x ∉ B x ∈ C and x ∉ A...

For any two sets A and B, show that the following statements are equivalent: (i) A ∪ B = B (ii) A ∩ B = A

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) A ∪ B = B Proving, (iii)=(iv) Let us take, A ∪ B = B A ∩ B = A. A ⊂ B and A ∩ B = A Thus, (iii)=(iv) is proved. (ii) A ∩ B = A Proving, (iv)=(i) Let us take, A ∩ B = A A ⊂ B A ∩ B = A...

For any two sets A and B, show that the following statements are equivalent: (i) A ⊂ B (ii) A – B = ϕ

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) A ⊂ B Proving, (i)=(ii) ( A ⊂ B) A–B = {x ∈ A: x ∉ B} All element of A is also an element of B ∴ A–B = ϕ Thus, (i)=(ii) Proved. (ii) A – B = ϕ Proving, (ii)=(iii) Let us take, A–B = ϕ...

For any two sets A and B, prove that A ⊂ B ⇒ A ∩ B = A

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answer: A ⊂ B ⇒ A ∩ B = A Consider, p ∈ A ⊂ B x ∈ B Let, p ∈ A ∩ B x ∈ A and x ∈ B x ∈ A and x ∈ A ∴ (A ∩ B) = A

For any two sets A and B, prove that (i) B ⊂ A ∪ B (ii) A ∩ B ⊂ A

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) Consider, p ∈ B p ∈ B ∪ A ∴ B ⊂ A ∪ B (ii) Consider, p ∈ A ∩ B p ∈ A and p ∈ B ∴ A ∩ B ⊂ A

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that: (i) (A ∪ B)’ = A’ ∩ B’ (ii) (A ∩ B)’ = A’ ∪ B’

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) LHS, A ∪ B = {x: x ∈ A or x ∈ B} A ∪ B = {2, 3, 5, 7, 9} (A∪B)’ = Complement of (A∪B) with U. (A∪B)’ = U – (A∪B)’ U – (A∪B)’ = {x ∈ U: x ∉ (A∪B)’} U = {2, 3, 5, 7, 9} (A∪B)’ = {2, 3,...

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A – (B ∩ C) = (A – B) ∪ (A – C) (ii) A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) LHS, (B ∩ C) = {x: x ∈ B and x ∈ C} (B ∩ C) = {5, 6} A – (B ∩ C) = {x ∈ A: x ∉ (B ∩ C)} A = {1, 2, 4, 5} (B ∩ C) = {5, 6} (A – (B ∩ C)) = {1, 2, 4} RHS, A – B = {x ∈ A: x ∉ B} A = {1,...

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A ∩ (B – C) = (A ∩ B) – (A ∩ C) (ii) A – (B ∪ C) = (A – B) ∩ (A – C)

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) LHS, B–C = {x ∈ B: x ∉ C} B = {2, 3, 5, 6} C = {4, 5, 6, 7} B–C = {2, 3} (A ∩ (B – C)) = {x: x ∈ A and x ∈ (B – C)} (A ∩ (B – C)) = {2} RHS, (A ∩ B) = {x: x ∈ A and x ∈ B} (A ∩ B) =...

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answers: (i) LHS, (B ∩ C) = {x: x ∈ B and x ∈ C} (B ∩ C) = {5, 6} A ∪ (B ∩ C) = {x: x ∈ A or x ∈ (B ∩ C)} A ∪ (B ∩ C) = {1, 2, 4, 5, 6} RHS, (A ∪ B) = {x: x ∈ A or x ∈ B} (A ∪ B) = {1, 2, 4, 5, 6}....

Find the smallest set A such that A ∪ {1, 2} = {1, 2, 3, 5, 9}.

CBSE, Class 11, Exercise 1.6, Maths, RD Sharma, Sets

Answer: A ∪ {1, 2} = {1, 2, 3, 5, 9} The smallest set of A, A = {1, 2, 3, 5, 9} – {1, 2} ∴ A = {3, 5, 9}

If A and B are two sets such that A ⊂ B, then Find: (i) A ⋂ B (ii) A ⋃ B

CBSE, Class 11, Exercise 1.5, Maths, RD Sharma, Sets

Answers: (i) A ∩ B - A intersection B (Same elements of A and B). A ⊂ B denotes that both A and B have the same elements. ∴ A ∩ B = A (ii) A ∪ B - A union B (Elements of either A or B or in both A...

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) A ∪ B (ii) A ∪ C

CBSE, Class 11, Exercise 1.5, Maths, RD Sharma, Sets

Answers: (i) A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8} A ∪ B = {x: x ∈ A or x ∈ B} ∴ A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8} (ii) A = {1, 2, 3, 4, 5} C = {7, 8, 9, 10, 11} A ∪ C = {x: x ∈ A or x ∈ C} ∴ A ∪ C...

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) B ∪ C (ii) B ∪ D

CBSE, Class 11, Exercise 1.5, Maths, RD Sharma, Sets

Answers: (i) B = {4, 5, 6, 7, 8} C = {7, 8, 9, 10, 11} B ∪ C = {x: x ∈ B or x ∈ C} ∴ B ∪ C = {4, 5, 6, 7, 8, 9, 10, 11} (ii) B = {4, 5, 6, 7, 8} D = {10, 11, 12, 13, 14} B ∪ D = {x: x ∈ B or x ∈ D}...