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RD Sharma

A bucket is in the form of a frustum of a cone of height $30cm$ with radii of its lower and upper ends as $10cm$ and $20cm$ respectively. Find the capacity and surface area of the bucket. Also, find the cost of milk which can completely fill the container, at the rate of $Rs.25$ per litre.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

Let us assume R and r be the radii of the top and base of the bucket respectively, Let us assume h be its height of the bucket. Then, according to the question we have $R=20cm$, $r=10cm$, $h=30cm$...

A milk container of height $16cm$ is made of metal sheet in the form of frustum of a cone with radii of its lower and upper ends as $8cm$ and $20cm$ respectively. Find the cost of milk at the rate of $Rs.44$ per liter which the container can hold.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

As per the given information, A milk container in a form of frustum of a cone with, Radius of the lower end $\left( {{r}_{1}} \right)=8cm$ And radius of the upper end $\left( {{r}_{2}} \right)=20cm$...

A tent consists of a frustum of a cone capped by a cone. If radii of the ends of the frustum be $13m$ and $7m$ , the height of frustum be $8m$ and the slant height of the conical cap be $12m$ , find the canvas required for the tent.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

According to the given data in the question, Height of frustum (h) $=8m$ (given) Bigger and smaller radii of the frustum cone are $13cm$ and $7cm$. Therefore, ${{r}_{1}}=13cm$ and ${{r}_{2}}=7cm$...

The radii of circular bases of a frustum of a right circular cone are $12cm$ and $3cm$ and the height is $12cm$ . Find the total surface area and volume of frustum.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

The height of frustum cone $=12cm$ (given) Bigger and smaller radii of a frustum cone are $12cm$ and $3cm$ respectively. (given) Therefore , ${{r}_{1}}=12cm;{{r}_{2}}=3cm$ Let us assume that the...

If the radii of the circular ends of a bucket $24cm$ high are $5$ and $15cm$ respectively, find the surface area of the bucket.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

As per the given data in question, Height of the bucket (h) $=24cm$ Radius of the small and big circular ends of the bucket $5cm$ and $15cm$ respectively. So, ${{r}_{1}}=5cm,{{r}_{2}}=15cm$ Let us...

The height of a cone is $20cm$ . A small cone is cut off from the top by a plane parallel to the base. If its volume be $1/125$ of the volume of the original cone, determine at what height above the base the section is made.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

According to the given information, Let us asssume the radius of the small cone be r cm And, the radius of the big cone be R cm It is given, height of the big cone is $20cm$ Let us also assume the...

The height of a cone is $20cm$ . A small cone is cut off from the top by a plane parallel to the base. If its volume be $1/125$ of the volume of the original cone, determine at what height above the base the section is made.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

According to the given information, Let us asssume the radius of the small cone be r cm And, the radius of the big cone be R cm It is given, height of the big cone is $20cm$ Let us also assume the...

If the radii of the circular ends of a conical bucket which is $45cm$ high be $28cm$ and $7cm$ , find the capacity of the bucket.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

Given data as per the question, Height of the conical bucket asgiven in the question $=45cm$ Radii of the bigger and smaller circular ends of the conical bucket are $28cm$ and $7cm$ respectively....

The perimeters of the ends of a frustum of a right circular cone are $44cm$ and $33cm$ . If the height of the frustum be $16cm$ , find its volume, the slant surface and the total surface.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

As per the given data, Perimeter of the upper end of a frustum of a right circular cone $=44cm$ So, $2\pi {{r}_{1}}=44$ $2\left( 22/7 \right){{r}_{1}}=44$ (radius of upper end of a frustum of a...

A frustum of a right circular cone has a diameter of base $20cm$ , of top $12cm$ and height $3cm$ . Find the area of its whole surface and volume.

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

As per the given data, The base diameter of cone $\left( {{d}_{1}} \right)$ $=20cm$ So, the radius of the base of the cone $\left( r_{1}^{{}} \right)$ $=20/2cm=10cm$ The top diameter of...

A bucket has top and bottom diameters of $40cm$ and $20cm$ respectively. Find the volume of the bucket if its depth is $12cm$ . Also, find the cost of tin sheet used for making the bucket at the rate of $Rs120$ per $d{{m}^{2}}$

CBSE, Class 10, Exercise 16.3, Maths, RD Sharma, Surface Areas And Volumes

As per the given information, Diameter of the top of the bucket $=40cm$ So, the radius of the top of the bucket $\left( {{r}_{1}} \right)$ $=40/2=20cm$ Diameter of the bottom part of the bucket...

Find the ratio in which the point P(-1, y) lying on the line segment joining A $(-3,10)$ and B $(6,-8)$ divides it. Also find the value of y.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.3, Maths, RD Sharma

Let’s P divide A$(-3,10)$ and B$(6,-8)$ in the ratio of k:$1$ Given that the coordinates of P as ($-1$,y) Now, by using the section formula for x – coordinate we have $-1=6k–3/k+1$ $-(k+1)=6k–3$...

If the points A $(2,0)$ , B $(9,1)$ , C $(11,6)$ and D $(4,4)$ are the vertices of a quadrilateral ABCD. Then Determine whether ABCD is a rhombus or not.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.3, Maths, RD Sharma

Given that the points are A$(2,0)$, B$(9,1)$, C$(11,6)$ and D$(4,4)$. Now Coordinates of mid-point of AC are $(11+2/2,6+0/2)=(13/2,3)$ Coordinates of mid-point of BD are $(9+4/2,1+4/2)=(13/2,5/2)$...

At what ratio does the point $(-4,6)$ divide the line segment joining the points A $(-6,10)$ and B $(3,-8)$ ?

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.3, Maths, RD Sharma

Let’s the point $(-4,6)$ divide the line segment AB in the ratio k:$1$. Thus, by using the section formula, we have $(-4,6)=\left( \frac{3k-6}{k+1},\frac{-8k+10}{k+1} \right)$ $-4=\frac{3k-6}{k+1}$...

If we have the points $(-2,1)$ , $(1,0)$ , $(x,3)$ and $(1,y)$ form a parallelogram, then find the values of x and y.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.3, Maths, RD Sharma

Let’s A $(-2,1)$, B$(1,0)$, C$(x,3)$ and D$(1,y)$ be the given points of the parallelogram. As We know that the diagonals of a parallelogram bisect each other. Therefore, the coordinates of...

Find out the coordinates of a point A, where AB is the diameter of circle whose center is $(2,-3)$ and B is $(1,4)$ .

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.3, Maths, RD Sharma

Let’s the coordinates of point A be (x, y) If we have AB is the diameter, then the center in the mid-point of the diameter Thus , $(2,-3)=(x+1/2,y+4/2)$ $2=x+1/2$ and $-3=y+4/2$ $4=x+1$ and $-6=y+4$...

Find out the ratio in which the y-axis divides the line segment joining the points $(5,-6)$ and $(-1,-4)$ . Also find the coordinates of the point of division.

CBSE, Class 10, Co-ordinate Geometry, Exercise 14.3, Maths, RD Sharma

Let’s P$(5,-6)$ and Q$(-1,-4)$ be the given points. Let’s the y-axis divide the line segment PQ in the ratio k: $1$ Now, by using section formula for the x-coordinate (as it’s zero) Now we have...

On the set Z of integers a binary operation * is defined by a 8 b = ab + 1 for all a, b ∈ Z. Prove that * is not associative on Z.

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

\[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Z \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }a\text{ }*\text{ }\left( bc\text{ }+\text{ }1...

Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative?

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

\[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Z \\ a\text{ }*\text{ }b\text{ }=\text{ }3a\text{ }+\text{ }7b \\ b\text{ }*\text{ }a\text{ }=\text{ }3b\text{ }+\text{ }7a \\...

If the binary operation o is defined by a0b = a + b – ab on the set Q – {-1} of all rational numbers other than 1, show that o is commutative on Q – [1].

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

\[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q\text{ }\text{ }-\left\{ -1 \right\}. \\ Then\text{ }aob\text{ }=\text{ }a\text{ }+\text{ }b\text{ }-\text{ }ab \\ =\text{...

Check the commutativity and associativity of each of the following binary operations: (xv) ‘’ on Q defined by a b = gcd (a, b) for all a, b ∈ Q

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(xv) to check: commutativity of * \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }N,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }gcd\text{ }\left( a,\text{ }b \right) ...

Check the commutativity and associativity of each of the following binary operations: (xiii) ‘’ on Q defined by a b = (ab/4) for all a, b ∈ Q (xiv) ‘’ on Z defined by a b = a + b – ab for all a, b ∈ Z

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(xiii) to check :commutativity of * \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }\left( ab/4 \right) \\ =\text{ }\left(...

Check the commutativity and associativity of each of the following binary operations: (xi) ‘’ on N defined by a b = ab for all a, b ∈ N (xii) ‘’ on Z defined by a b = a – b for all a, b ∈ Z

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(xi) to check : commutativity of * \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }N,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }{{a}^{b}} \\ b\text{ }*\text{ }a\text{...

Check the commutativity and associativity of each of the following binary operations: (ix) ‘’ on Q defined by a b = (a – b)2 for all a, b ∈ Q (x) ‘’ on Q defined by a b = a b + 1 for all a, b ∈ Q

CBSE, Class 12, Maths, RD Sharma

(ix) to check: commutativity of * \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }{{\left( a\text{ }-\text{ }b...

Check the commutativity and associativity of each of the following binary operations: (vii) ‘’ on Q defined by a b = a + a b for all a, b ∈ Q (viii) ‘’ on R defined by a b = a + b -7 for all a, b ∈ R

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(vii) to check : commutativity of * \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }+\text{ }ab \\ b\text{...

Check the commutativity and associativity of each of the following binary operations: (v) ‘o’ on Q defined by a o b = (ab/2) for all a, b ∈ Q (vi) ‘’ on Q defined by a b = ab2 for all a, b ∈ Q

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(v) to check: commutativity of o \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }o\text{ }b\text{ }=\text{ }\left( ab/2 \right) \\ =\text{ }\left(...

Check the commutativity and associativity of each of the following binary operations: (iii) ‘’ on Q defined by a b = a – b for all a, b ∈ Q (iv) ‘⊙’ on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(iii) to check: commutativity of * \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }-\text{ }b \\ b\text{...

Check the commutativity and associativity of each of the following binary operations: (i) ‘’ on Z defined by a b = a + b + a b for all a, b ∈ Z (ii) ‘’ on N defined by a b = 2ab for all a, b ∈ N

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(i) to check: commutativity of * \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Z \\ =>\text{ }a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }+\text{ }b\text{ }+\text{ }ab ...

Let A be any set containing more than one element. Let ‘’ be a binary operation on A defined by a b = b for all a, b ∈ A Is ‘*’ commutative or associative on A?

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

\[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }A \\ =>\text{ }a\text{ }*\text{ }b\text{ }=\text{ }b \\ b\text{ }*\text{ }a\text{ }=\text{ }a \\ Therefore\text{ }a\text{...

Determine which of the following binary operation is associative and which is commutative: (i) * on N defined by a * b = 1 for all a, b ∈ N (ii) * on Q defined by a * b = (a + b)/2 for all a, b ∈ Q

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(i) to prove: commutativity of * Let \[\begin{array}{*{35}{l}} a,\text{ }b\text{ }\in \text{ }N \\ a\text{ }*\text{ }b\text{ }=\text{ }1 \\ b\text{ }*\text{ }a\text{ }=\text{ }1 \\ =>a\text{...

Let ‘’ be a binary operation on N defined by a b = l.c.m. (a, b) for all a, b ∈ N (i) Find 2 * 4, 3 * 5, 1 * 6. (ii) Check the commutativity and associativity of ‘*’ on N.

Binary Operations, CBSE, Class 12, Exercise 3.2, Maths, RD Sharma

(i) Since, \[\begin{array}{*{35}{l}} a\text{ }*\text{ }b\text{ }=\text{ }1.c.m.\text{ }\left( a,\text{ }b \right) \\ 2\text{ }*\text{ }4\text{ }=\text{ }l.c.m.\text{ }\left( 2,\text{ }4 \right) \\...

Let S = {a, b, c}. Find the total number of binary operations on S.

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

Number of binary operations on a set with n elements is ${{n}^{{{n}^{2}}}}$ Here, S = {a, b, c} Number of elements in S = 3 Number of binary operations on a set with 3 elements is...

Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

LCM 1 2 3 4 5 1 1 2 3 4 5 2 2 2 6 4 10 3 3 5 3 12 15 4 4 4 12 4 20 5 5 10 15 20 5 Since,, all the elements are not in the set {1, 2, 3, 4, 5}. If we consider a = 2 and b = 3, a * b = LCM...

Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

\[\begin{array}{*{35}{l}} a~*~b~=\text{ }2a~+~b~-\text{ }3 \\ 3\text{ }*\text{ }4\text{ }=\text{ }2\text{ }\left( 3 \right)\text{ }+\text{ }4\text{ }-\text{ }3 \\ =\text{ }6\text{ }+\text{...

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.(iii) On R, define * by ab = ab2 (iv) On Z+ define by a * b = |a − b|

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

(iii) Since, on R, define by a*b = ab2 Let \[\begin{array}{*{35}{l}} a,\text{ }b\text{ }\in \text{ }R \\ \Rightarrow \text{ }a,\text{ }{{b}^{2}}~\in \text{ }R \\ \Rightarrow \text{ }a{{b}^{2}}~\in...

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this. (i) On Z+, defined * by a * b = a – b (ii) On Z+, define * by a*b = ab

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

(i)Since, On Z+, defined * by a * b = a – b If a = 1 and b = 2 in Z+, then \[\begin{array}{*{35}{l}} a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }-\text{ }b \\ =\text{ }1\text{ }-\text{ }2 \\...

Determine whether the following operation define a binary operation on the given set or not:(vii) ‘’ on Q defined by a b = (a – 1)/ (b + 1) for all a, b ∈ Q

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

(vii)Since, ‘*’ on Q defined by a * b = (a – 1)/ (b + 1) for all a, b ∈ Q If a = 2 and b = -1 in Q, \[\begin{array}{*{35}{l}} a\text{ }*\text{ }b\text{ }=\text{ }\left( a\text{ }-\text{ }1...

Determine whether the following operation define a binary operation on the given set or not: (v) ‘+6’ on S = {0, 1, 2, 3, 4, 5} defined by a +6 b
$\{_{a+b-6;ifa+b\ge 6}^{a+b;ifa+b<6}$
(vi) ‘⊙’ on N defined by a ⊙ b= ab + ba for all a, b ∈ N

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

(v) Given ‘+6’ on S = {0, 1, 2, 3, 4, 5} defined by a +6 b Consider the composition table, +6 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4 Here all...

Determine whether the following operation define a binary operation on the given set or not: (iii) ‘’ on N defined by a b = a + b – 2 for all a, b ∈ N (iv) ‘×6‘ on S = {1, 2, 3, 4, 5} defined by a ×6 b = Remainder when a b is divided by 6.

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

(iii) Given ‘*’ on N defined by a * b = a + b – 2 for all a, b ∈ N \[\begin{array}{*{35}{l}} If~a~=\text{ }1\text{ }and~b\text{ }=\text{ }1, \\ a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }+\text{...

A bag contains $3$ red balls, $5$ black balls and $4$ white balls. if A ball is drawn at random from the bag. Then What is the probability that the ball drawn is:(iii) black? (iv) not red

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(iii) Total number of black balls is $5$ As We know that, Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of getting a black ball $=5/12$ (iv) Total...

A bag contains $3$ red balls, $5$ black balls and $4$ white balls. if A ball is drawn at random from the bag. Then What is the probability that the ball drawn is: (i) white? (ii) red?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A bag contains $3$ red, $5$ black and $4$ white balls to find: Probability of getting a (i) White ball (ii) Red ball (iii) Black ball (iv) Not red ball So, Total number of balls...

A bag contains $10$ red and $8$ white balls. One ball is drawn at random. Find the probability that the ball drawn is white.

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A bag contains $10$ red and $8$ white balls To find: Probability that one ball is drawn at random and getting a white ball Now, Total number of balls $10+8=18$ Total number of white balls...

In a lottery of $50$ tickets numbered $1$ to $50$ , there’s one ticket is drawn. Find the probability that the drawn ticket bears a prime number.

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given: Tickets are marked numbers from $1$ to $50$. And, one ticket is drawn at random. to find: Probability of getting a prime number on the drawn ticket Total number of tickets are $50$. Tickets...

Two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than $10$ .

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that: A pair of dice is thrown To find: Probability that the total of numbers on the dice is greater than $10$ Let’s write the all possible events that can occur...

A and B throw a pair of dice. If A throws $9$ , find B’s chance of throwing a higher number

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given A pair of dice is thrown To find: Probability that the total of numbers on the dice is greater than $9$ Le t’s write the all possible events that can occur...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(xvii) a heart (xviii) a red card

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(xvii) Total number of heart cards is $13$ As We know that, Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of getting a heart card $=13/52=1/4$...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(xv) the ace of spades (xvi) a queen

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(xv) Total number of aces of spade is $1$ As We know that Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of getting an ace of spade $=1/52$ (xvi)...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(xiii) a seven of clubs (xiv) jack

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(xiii) Total number of $7$ of club is $1$ only. As We know that, Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of getting a $7$ of club $=1/52$...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(xi) a spade (xii) a black card

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(xi) Total number of spades is $13$ As We know that, Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of getting a spade $=13/54=1/4$ (xii) Total...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(ix) the seven of clubs (x) a ten

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(ix) Total number of card other than ace is $52–4=48$ As We know that, Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of getting other than ace...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(vii) neither an ace nor a king (viii) neither a red card nor a queen

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(vii) Total number of ace cards are $4$ and king are $4$ Total number of cards that are an ace or a king $=4+4=8$ Thus, the total number of cards that are neither an ace nor a king is $52–8=44$ As...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(v) neither a heart nor a king (vi) spade or an ace

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(v) Total number of heart cards are $13$ and king are $4$ in which king of heart is also included. Now, the total number of cards that are a heart and a king $=13+3=16$ Thus, the total number of...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is:(iii) black and a king (iv) a jack, queen or a king

CBSE, Class 10, Exercise 2.1, Maths, Polynomials, RD Sharma

(iii) Total number of cards which are black and a king card is $2$ As We know that, Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of getting a black...

If A card is drawn at random from a pack of $52$ cards then Find the probability that the card drawn is: (i) a black king (ii) either a black card or a king

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A card is drawn at random from a pack of $52$ cards To Find: Probability of the following Total number of cards in a pack $=52$ (i) Number of cards which are black king $=2$ We know that...

If Three coins are tossed together then Find the probability of getting:(iii) at least one head and one tail (iv) no tails

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(iii) now For getting at least one head and one tail the cases are THT, TTH, THH, HTT, HHT, and HTH. Thus, the total number of favorable outcomes i.e. at least one tail and one head is $6$ We know...

If Three coins are tossed together then Find the probability of getting: (i) exactly two heads (ii) at most two heads

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given in the question Three coins are tossed simultaneously. When three coins are tossed then the outcome will be anyone of these combinations. TTT, THT, TTH, THH. HTT, HHT, HTH, HHH. Thus, the...

A die is thrown. Find the probability of getting:(v) a number greater than $5$ (vi) a number lying between $2$ and $6$

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(v) A number greater than $5$ is $6$ only. So, the number of favorable outcomes is $1$. As We know that, Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the...

A die is thrown. Find the probability of getting:(iii) a multiple of $2$ or $3$ (iv) an even prime number

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(iii)by Multiplying of $2$ are $3$ are $2,3,4$ and $6$. Thus, the number of favorable outcomes is $4$ As We know that, Probability = Number of favorable outcomes/ Total number of outcomes So, the...

A die is thrown. Find the probability of getting: (i) a prime number (ii) $2$ or $4$

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

We have to give that A dice is thrown once To find: (i) The Probability of getting a prime number (ii) The Probability of getting $2$ or $4$ (iii)The Probability of getting a multiple of $2$ or $3$....

There is the probability that it will rain tomorrow is $0.85$ . Then find the probability that it will not rain tomorrow?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given: Probability that it will rain tomorrow P(E) $=0.85$ We have to find that the Probability that it will not rain tomorrow P(E) As We know that sum of the probability of occurrence of an event...

A bag contains $6$ red, $8$ black and $4$ white balls. A ball is drawn at random. What is the probability that the ball drawn is not black?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A bag contains $6$ red, $8$ black and $4$ white balls and a ball is drawn at random to find: Probability that the ball drawn is not black so, Total number of balls $6+8+4=18$ therefore,...

What is the probability of a number selected from the numbers $1,2,3,…,15$ is a multiple of $4$ ?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given the Numbers are from $1$ to $15$. One number is selected to find: Probability that the selected number is a multiple of $4$ Thus, the Total number between from $1$ to $15$ to $15$ So, Numbers...

In a lottery there are $10$ prizes and $25$ blanks. Then What is the probability of getting a prize?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that in a lottery there are $10$ prizes and $25$ blanks. to find: Probability of winning a prize So, Total number of tickets is $10+25=35$ Thus, Total number of prizes carrying tickets is $10$...

Tickets numbered from $1$ to $20$ are mixed up and if a ticket is drawn at random. Then What is the probability that the ticket drawn has a number which is a multiple of $3$ or $7$ ?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that Tickets are marked from $1$ to $20$ are mixed up. One ticket is picked at random. to find: Probability that the ticket bears a multiple of $3$ or $7$ so, Total number of cards is $20$....

A bag contains $5$ white balls and $7$ red balls. If One ball is drawn at random. Then What is the probability that ball drawn is white?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A bag contains $7$ red and $5$ white balls and a ball is drawn at random to find: Probability that the ball drawn is white so Total number of balls $7+5=12$ therefore, Total number of...

If there’s the probability of winning a game is $0.3$ , then find that what is the probability of losing it?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that probability of winning a game P(E) $=0.3$ We have To Find that Probability of losing the game As We know that the sum of probability of occurrence of an event and probability of...

A bag contains $4$ red, $5$ black and $6$ white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(iii) Total number of black balls are $5$ We know that the Probability = Number of favorable outcomes/ Total number of outcomes Therefore, the probability of drawing black ball P(E) $=5/15=1/3$ As...

A bag contains $4$ red, $5$ black and $6$ white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is: (i) White (ii) Red

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A bag contains $4$ red, $5$ black and 6white balls and a ball is drawn at random to Find: Probability of getting a (i) white ball (ii) red ball (iii) not black ball (iv) red or white So,...

If $12$ defective pens are accidently mixed with $132$ good ones. Then It is not possible to just look at pen and tell whether or not it is defective. if One pen is taken out at random from this lot. Then Determine the probability that the pen taken out is good one.

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

We have, No. of good pens $=132$ No. of defective pens $=12$ Therefore, the total no. of pens $=132+12=144$ Then we have, the total no. of possible outcomes $=144$ Now, let E be the event of getting...

A box is given which contains $5$ red marbles, $8$ white marbles and $4$ green marbles. if One marble is taken out of the box at random. Then What is the probability that the marble taken out will be (i) red (ii) not green

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that, We have A box which containing 5 red, 8 white and 4 green marbles. Therefore, the total no. of possible outcomes $=17$ ($5$ red $+8$ white $+4$ green) (i) Let E be the Event of getting a...

A lot consists of $144$ ball pens of which $20$ are defective and others good. Then Nuri will buy a pen if it is good, but will not buy if it is defective. If The shopkeeper draws one pen at random and gives it to her. Then What is the probability that (i) She will buy it (ii) She will not buy it

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

We have, No. of good pens $=144–20=124$ No. of detective pens $=20$ Therefore, Total no. of possible outcomes $=144$ (total no. of pens) (i) So, for her to buy it the pen should be a good one. So,...

A bag contains $3$ red balls and $5$ black balls. if A ball is drawn at random from the bag. Then What is the probability that the ball drawn is (i) red (ii) not red

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that, A bag contains $3$ red and $5$ black balls. Therefore, the total no. of possible outcomes $=8$ ($3$ red $+5$ black) (i)Now Let E = event of getting red ball. So, No. of favorable...

Determine whether the following operation define a binary operation on the given set or not: (i) ‘’ on N defined by a b = ab for all a, b ∈ N. (ii) ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.

Binary Operations, CBSE, Class 12, Exercise 3.1, Maths, RD Sharma

(i) Given ‘*’ on N defined by a * b = ab for all a, b ∈ N. Let a, b ∈ N. Then, \[\begin{array}{*{35}{l}} {{a}^{b~}}\in ~N~~~~~~\left[ \because ~{{a}^{b}}\ne 0~and~a,\text{ }b~is~positive~integer...

A function $f: R \rightarrow R$ is defined as $f(x)=x^{3}+4 .$ Is it a bijection or not? In case it is a bijection, find $\mathbf{f}^{-1}$ (3).

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given that $f: R \rightarrow R$ is defined as $f(x)=x^{3}+4$ Injectivity of f: Let $x$ and $y$ be two elements of domain (R), Such that $f(x)=f(y)$ $\Rightarrow x^{3}+4=y^{3}+4$ $\Rightarrow...

If $f: R \rightarrow R$ be defined by $f(x)=x^{3}-3$ , then prove that $f^{-1}$ exists and find a formula for $f^{-1}$ . Hence, find $f^{-1}(24)$ and $f^{-1}(5)$ .

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R \rightarrow R$ be defined by $f(x)=x^{3}-3$ Now we have to prove that $\mathrm{f}^{-1}$ exists Injectivity of f: Let $x$ and $y$ be two elements in domain $(R)$, Such that,...

Consider f: $R^{+} \rightarrow[-5, \infty)$ given by $f(x)=9 x^{2}+6 x-5 .$ Show that $f$ is invertible with $f^{-1}(x)=(v(x+6)-1) / 3$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R^{+} \rightarrow[-5, \infty)$ given by $f(x)=9 x^{2}+6 x-5$ We have to show that $\mathrm{f}$ is invertible. \section{Injectivity of f:} Let $x$ and $y$ be two elements of domain...

If $f(x)=(4 x+3) /(6 x-4), x \neq(2 / 3)$ show that $f o f(x)=x$ , for all $x \neq(2 / 3)$ . What is the inverse of f?

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

It is given that $f(x)=(4 x+3) /(6 x-4), x \neq 2 / 3$ Now we have to show $\operatorname{fof}(x)=x$ $($ fof $)(x)=f(f(x))$ $=f((4 x+3) /(6 x-4))$ $=(4((4 x+3) /(6 x-4))+3) /(6((4 x+3) /(6 x-4))-4)$...

Consider $f: R \rightarrow R^{+} \rightarrow[4, \infty)$ given by $f(x)=x^{2}+4 .$ Show that $f$ is invertible with inverse $\mathbf{f}^{-1}$ of f given by $f^{-1}(x)=V(x-4)$ where $R^{+}$ is the set of all non-negative real numbers.

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R \rightarrow R^{+} \rightarrow[4, \infty)$ given by $f(x)=x^{2}+4$ Now we have to show that $f$ is invertible, Consider injection of f: \section{RD Sharma Solutions for Class 12 Maths...

Consider $f: R \rightarrow R$ given by $f(x)=4 x+3 .$ Show that $f$ is invertible. Find the inverse of $\mathbf{f}$ .

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R \rightarrow R$ given by $f(x)=4 x+3$ Now we have to show that the given function is invertible. Consider injection of f: Let $x$ and $y$ be two elements of domain $(R)$, Such that...

Show that the function $f: Q \rightarrow Q$ , defined by $f(x)=3 x+5$ , is invertible. Also, find $f^{-1}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given function $f: Q \rightarrow Q$, defined by $f(x)=3 x+5$ Now we have to show that the given function is invertible. Injection of f: Let $x$ and $y$ be two elements of the domain (Q), Such that...

Let $A=\{1,2,3,4\} ; B=\{3,5,7,9\} ; C=\{7,23,47,79\}$ and f: $A \rightarrow B, g: B \rightarrow C$ be defined as $f(x)=2 x+1$ and $g(x)=x^{2}-2 .$ Express $(\text { gof })^{-1}$ and $f^{-1} o g^{-1}$ as the sets of ordered pairs and verify that (gof) $^{-1}=\mathbf{f}^{-1} \mathrm{og}^{-1}$ .

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

$\Rightarrow \mathrm{f}=\{(1,2(1)+1),(2,2(2)+1),(3,2(3)+1),(4,2(4)+1)\}$ $=\{(1,3),(2,5),(3,7),(4,9)\}$ Also given that $g(x)=x^{2}-2$ $\Rightarrow...

Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ and $g:\{a, b, c\} \rightarrow\{$ apple, ball, cat $\}$ defined as $f(1)$ $=a, f(2)=b, f(3)=c, g(a)=$ apple, $g(b)=$ ball and $g(c)=$ cat. Show that $f, g$ and gof are invertible. Find $\mathbf{f}^{-1}, \mathrm{~g}^{-1}$ and gof $^{-1}$ and show that $(\text { gof })^{-1}=f^{-1} 0 \mathrm{~g}^{-1}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f=\{(1, a),(2, b),(c, 3)\}$ and $g=\{(a$, apple), $(b, b a l l),(c$, cat $)\}$ Clearly, $f$ and $g$ are bijections. So, $f$ and $g$ are invertible. Now, $f^{-1}=\{(a, 1),(b, 2),(3, c)\}$ and...

Find $f^{-1}$ if it exists: f: A $\rightarrow$ , where (i) $A=\{0,-1,-3,2\} ; B=\{-9,-3,0,6\}$ and $f(x)=3 x$ (ii) $A=\{1,3,5,7,9\} ; B=\{0,1,9,25,49,81\}$ and $f(x)=x^{2}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

(i) Given $A=\{0,-1,-3,2\} ; B=\{-9,-3,0,6\}$ and $f(x)=3 x$. So, $f=\{(0,0),(-1,-3),(-3,-9),(2,6)\}$ \section{RD Sharma Solutions for Class 12 Maths Chapter 2 Function} Here, different elements of...

State with reason whether the following functions have inverse: (iii) $h:\{2,3,4,5\} \rightarrow\{7,9,11,13\}$ with $h=\{(2,7),(3,9),(4,11),(5,13)\}$

CBSE, Exercise 2.4, Function, Maths, RD Sharma

iii) Given $\mathrm{h}:\{2,3,4,5\} \rightarrow\{7,9,11,13\}$ with $\mathrm{h}=\{(2,7),(3,9),(4,11),(5,13)\}$ different elements of the domain have different images in the co-domain. $\Rightarrow...

State with reason whether the following functions have inverse: (i) $f:\{1,2,3,4\} \rightarrow\{10\}$ with $f=\{(1,10),(2,10),(3,10),(4,10)\}$ (ii) $g:\{5,6,7,8\} \rightarrow\{1,2,3,4\}$ with $g=\{(5,4),(6,3),(7,4),(8,2)\}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

(i) Given $\mathrm{f}:\{1,2,3,4\} \rightarrow\{10\}$ with $f=\{(1,10),(2,10),(3,10),(4,10)\}$ We have: $f(1)=f(2)=f(3)=f(4)=10$ $\Rightarrow \mathrm{f}$ is not one-one. $\Rightarrow \mathrm{f}$ is...

A bag contains $8$ red, $6$ white and $4$ black ball. if A ball is drawn at random from the bag. Find the probability that the drawn ball is(iii) Neither white nor black

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(iii) Let E be event of getting neither a white nor a black ball Therefore, No. of favorable outcomes $=18–6–4$ $=8$(Total balls – no. of white balls – no. of black balls) Thus, Probability, P(E) =...

A bag contains $8$ red, $6$ white and $4$ black ball. if A ball is drawn at random from the bag. Find the probability that the drawn ball is (i) Red or white (ii) Not black

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

As we know that total number of balls $=8+6+4=18$ So, Total no. of possible outcomes $=18$ (i) Let E = Event of getting red or white ball Now, No. of favorable outcomes $=14$($8$ red balls $+6$...

There are $30$ cards, of same size in a bag on which numbers $1$ to $30$ are written. If One card is taken out of the bag at random. Then Find the probability that the number on the selected card is not divisible by 3.

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that $30$ cards of same size in a bag on which numbers $1$ to $30$ are written. And, one card is taken out of the bag at random. to find: Probability that the number on the selected card is...

A bag contains $5$ red, $8$ white and $7$ black balls. If A ball is drawn at random from the bag. Then Find the probability that the drawn ball is(iii) neither white nor black.

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(iii) Let E be the Event of getting neither a white nor a black ball Therefore No. of favorable outcomes $=20–8–7=5$(total balls – no. of white balls – no. of black balls) Thus, Probability, P(E) =...

A bag contains $5$ red, $8$ white and $7$ black balls. If A ball is drawn at random from the bag. Then Find the probability that the drawn ball is (i) red or white (ii) not black

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

As we know that; Total number of possible outcomes $=20$ ($5$ red, $8$ white & $7$ black} (i) Let E = event of drawing a red or white ball So, No. of favorable outcomes $=13(5$ red $+8$ white)...

What is the probability of a number that selected at random from the number $1,2,2,3,3,3,4,4,4,4$ will be their average?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that the numbers are $1,2,2,3,3, 3,4,4,4,4$ So, Total number of possible outcomes $=10$ $Averageoftheno's=\frac{sumofnumbers}{tota\ln umber}$ $=\frac{1+2+2+3+3+3+4+4+4+4}{10}$ $=30/10$ $=3$...

In a class, there are $18$ girls and $16$ boys. The class teacher wants to choose one pupil for class monitor. Then What she does, she writes the name of each pupil on a card and puts them into a basket and mixes thoroughly. If A child is asked to pick one card from the basket. What is the probability that the name written on the card is: (i) The name of a girl (ii) The name of a boy?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that In a class there are $18$ girls and $16$ boys, the class teacher wants to choose one name. The class teacher writes all pupils’ name on a card and puts them in basket and mixes well...

Why is tossing a coin considered to be a fair way of deciding which team should choose ends in a game of cricket?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

So, No. of possible outcomes while tossing a coin $=2$ i.e., $1$ head or $1$ tail We know that Probability = Number of favorable outcomes/ Total number of outcomes P (getting head)$=1/2$ P (getting...

A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number, $1,2,3,….,12$ as shown in figure. What is the probability that it will point to:(iii) a number which is multiple of $3$ ? (iv) an even number?

CBSE, Class 10, Exercise 13.1, Maths, Maths, Probability, RD Sharma

(iii) So, Favorable outcomes i.e. to get a multiple of $3$ are $3,6,9,$ and $12$ Therefore, total number of favorable outcomes i.e. to get a multiple of $3$ is $4$ We know that the Probability =...

A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number, $1,2,3,….,12$ as shown in figure. What is the probability that it will point to: (i) $10$ ? (ii) an odd number?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A game of chance consists of spinning an arrow which is equally likely to come to rest pointing number $1,2,3…12$ to find: Probability of following So, Total numbers on the spin is 12 (i)...

Five cards are given– ten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random. Then (i) What is the probability that the card is a queen? (ii) If a king is drawn first and put aside, then what is the probability that the second card picked up is the (a) ace? (b) king?

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that Five cards-ten, jack, queen, king and Ace of diamond are shuffled face downwards. to find: Probability of following Total number of cards is $5$ (i) Now Total number of cards which is a...

If One card is drawn from a well shuffled deck of $52$ cards. Then Find the probability of getting:(v) A jack of hearts (vi) A spade

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(v) Total number of jack of hearts is $1$ We know that the Probability = Number of favorable outcomes/ Total number of outcomes Hence, the probability of getting a card which is a jack of hearts...

If One card is drawn from a well shuffled deck of $52$ cards. Then Find the probability of getting:(iii) A red face card (iv) A queen of black suit

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

(iii)Now, Total number of red face cards are $6$ So, Number of favorable outcomes i.e. total number of red face cards is $6$ We know that the Probability = Number of favorable outcomes/ Total number...

If One card is drawn from a well shuffled deck of $52$ cards. Then Find the probability of getting: (i) A king of red suit (ii) A face card

CBSE, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that One card is drawn from a well shuffled deck of $52$ playing cards to find: Probability of following we know that the Total number of cards are $52$ (i) Now, Total number of cards which...

A bag contains $3$ red balls and $5$ black balls. If A ball is drawn at random from the bag. Then What is the probability that the ball drawn is: (i) Red (ii) Back

CBSE, Chemistry, Class 10, Exercise 13.1, Maths, Probability, RD Sharma

Given that A bag contains $3$ red, and $5$ black balls. A ball is drawn at random to find: Probability of getting a (i) red ball (ii) white ball So, Total number of balls $3+5=8$ (i) we know that...

A bag contains $5$ black, $7$ red and $3$ white balls. If A ball is drawn from the bag at random. Then Find the probability that the ball drawn is:(iii) not black

CBSE, Class 10, Exercise 13.1, Probability, RD Sharma

(iii) Total number of black balls is $5$ We know that the Probability = Number of favourable outcomes/ Total number of outcomes Therefore, the probability of drawing black ball P(E)$=5/15=1/3$ But,...

A bag contains $5$ black, $7$ red and $3$ white balls. If A ball is drawn from the bag at random. Then Find the probability that the ball drawn is: (i) red (ii) black or white

CBSE, Class 10, Exercise 13.1, RD Sharma

Given that: A bag contains $7$ red, $5$ black and $3$ white balls and a ball is drawn at random to find: Probability of getting a (i) Red ball (ii) Black or white ball (iii) Not black ball So,Total...

A and B are two events such that P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35. Find
(i) P (A ∩ B′)
(ii) P (A′ ∩ B)

CBSE, Class 11, Exercise 33.4, Maths, Probability, RD Sharma

A and B are two events is given P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35 By definition of P (A or B) under axiomatic approach we can write, P (A ∪ B) = P (A) + P (B) – P (A ∩ B) (i) P (A ∩...

If A and B be mutually exclusive events associated with a random experiment such that P (A) = 0.4 and P (B) = 0.5, then find:
(i) P (A′ ∩ B)
(ii) P (A ∩ B′)

CBSE, Class 11, Exercise 33.4, Maths, Probability, RD Sharma

A and B are two mutually exclusive events is given to us. P (A) = 0.4 and P (B) = 0.5 By definition of mutually exclusive events we can write, P (A ∪ B) = P (A) + P (B) (i) P (A′ ∩ B) P (only B) = P...

A man saved ₹ 16500 in ten years. In each year after the first he saved ₹ 100/- more than he did in the preceding year. How much did he saved in the first year?

Arithmetic Progressions, CBSE, Class 11, Exercise 19.7, Maths, RD Sharma

Solution: As per the question: A man saved ₹$16500$ in ten years Let be his savings in the first year be ₹ $x$ Every year his savings increased by ₹ 100. Therefore, A.P will be $x$, $100 + x$, $200...

A chord AB of circle of radius $14cm$ makes an angle of ${{60}^{\circ }}$ at the center of a circle. Find the area of the minor segment of the circle.

As per the question it is given that Radius of the circle (r) $=14cm=OA=OB$ Angle subtended by the chord with the center of the circle, $\theta ={{60}^{\circ }}$ In triangle AOB, angle A $=$ angle B...

A chord $10cm$ long is drawn in a circle whose radius is $5\sqrt{2}cm$ . Find the area of both the segments.

As per the question it is given that, Radius of the circle, $r=5\sqrt{2}cm=OA=OB$ Length of the chord $AB=10cm$ In triangle OAB, $=O{{A}^{2}}+O{{B}^{2}}$ $={{\left( 5\sqrt{2} \right)}^{2}}+{{\left(...

A chord of a circle of radius $14cm$ makes a right angle at the center. Find the areas of the minor and major segments of the circle.

According to the question, it is given that, Radius (r) $=14cm$ Angle subtended by the chord with the center of the circle, $\theta ={{90}^{\circ }}$ Area of minor segment $=\theta /360\times \pi...

A chord PQ of length $12cm$ subtends an angle ${{120}^{\circ }}$ at the center of a circle. Find the area of the minor segment cut off by the chord PQ.

It is given in the question that $\angle POQ={{120}^{{}^\circ }}$ and $PQ=12cm$ Construct $OV\bot PQ$ $PV=PQ\times \left( 0.5 \right)=12\times 0.5=6cm$ As, $\angle POV={{120}^{\circ }}$ $\angle...

AB is a chord of a circle with center O and radius $4cm$ . AB is of length $4cm$ and divides the circle into two segments. Find the area of the minor segment.

As per the given data in the question, Radius of the circle with center ‘O’, $r=4cm=OA=OB$ Length of the chord $AB=4cm$ Thus, OAB is an equilateral triangle and angle $AOB={{60}^{\circ }}$ So, the...

The perimeter of a certain sector of a circle of radius is $5.6m$ and $27.2m$ . Find the area of the sector.

As per the given information, Radius of the circle $=5.6m=OA=OB$ (as shown in figure) Perimeter of the sector of the circle $=$ (AB arc length) $+OA+OB=27.2$ Let the angle subtended by an arc at the...

The perimeter of a sector of a circle of radius $5.7m$ is $27.2m$ . Find the area of the sector.

As per the given data, Radius of circle $=5.7cm=OA=OB$ [as shown in the figure above] Perimeter of the sector of circle $=27.2m$ Let the angle subtended by an arc at the centre of circle be $\theta...

In a circle of radius $35cm$ , an arc subtends an angle of ${{72}^{\circ }}$ at the center. Find the length of the arc and area of the sector.

As per the given data, Radius of circle $=35cm$ Angle subtended by an arc at the centre of the circle $={{72}^{\circ }}$ As we know that, Length of arc of a circle which subtends angle at the center...

Find the area of the sector of a circle of radius $5cm$ , if the corresponding arc length is $3.5cm$ .

As per the given information, Radius of the circle $=5cm$ Length of arc of the circle $=3.5cm$ Let us assume θ be the angle subtended by an arch at the centre of circle As we know that, Length of...

The area of a sector of a circle of radius $5cm$ is $5\pi c{{m}^{2}}$ . Find the angle contained by the sector.

As per the given information, Radius of circle $=5cm$ Angle subtended by arc at the center of circle ‘O’ Area of sector of circle $=5\pi c{{m}^{2}}$ As we know that, Area of the sector $=\theta...

The area of a sector of a circle of radius $2cm$ is $\pi c{{m}^{2}}$ . Find the angle contained by the sector.

As per the given information, Radius of circle $=2cm$ Angle subtended by arc at the centre ‘O’. Area of sector of circle with $2cm=\pi c{{m}^{2}}$ As we know the area of the sector of circle, Area...

A sector of a circle of radius $8cm$ contains an angle of ${{135}^{\circ }}$ . Find the area of sector.

As per given data, Radius of circle $=8cm$ Angle subtended by arc at the center of circle O $={{135}^{\circ }}$ As we know the formula of area of sector, Area of the sector $=\theta /360*\pi...

A sector of a circle of radius $4cm$ subtends an angle of ${{30}^{\circ }}$ . Find the area of the sector.

As per given information, Radius of the circle $=4cm$ Angle subtended by an arc at the center of circle O $={{30}^{\circ }}$ As we know the formula of area of sector, Area of the sector $=\theta...

Find the angle subtended at the centre of a circle of radius ‘a’ cm by an arc of length $\left( a\pi /4 \right)cm$ .

As per the given data, Radius of the circle $=$ a cm Length of arc as per given in the question $=a\pi /4cm$ $\theta =$ angle subtended by the arc at the center of circle As we know, Length of arc...

An arc of length $15cm$ subtends an angle of ${{45}^{\circ }}$ at the center of a circle. Find in terms of $\pi$ , the radius of the circle

As per the given data, Length of arc which subtends an angle of ${{45}^{\circ }}$ at the centre of a circle $=15cm$ Angle subtended at the centre of circle $\left( \theta \right)={{45}^{\circ }}$...

An arc of length $20\pi cm$ subtends an angle of ${{144}^{\circ }}$ at the center of a circle. Find the radius of the circle.

As per the given information, Length of arc $=20\pi cm$ And. Angle subtended by arc at the centre of circle $\left( \theta \right)={{144}^{\circ }}$ As we know that, Length of arc that subtends an...

Find the angle subtended at the center of a circle of radius $5cm$ by an arc of length $5\pi /3cm$ .

As per the given information, Radius of the circle $=5cm$ Length of arc $=5\pi /3cm$ As we know that, Length of arc that subtends an angle in the center of circle $=\theta /360*2\pi rcm$ By using...

Find, in terms of $\pi$ , the length of the arc that subtends an angle of ${{30}^{\circ }}$ at the center of a circle of radius of $4cm$ .

As per the given information, Radius of the circle $=4cm$ Angle subtended by the arc at the centre ‘O’ $={{30}^{\circ }}$ As we know that, Length of arc that subtends an angle at center of circle...

Rain water, which falls on a flat rectangular surface of length $6m$ and breadth $4m$ is transferred into a cylindrical vessel of internal radius 20 cm .What will be the height of water in the cylindrical vessel if a rainfall of $1cm$ has fallen?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question, Length of the rectangular surface $=6m=600cm$ Breadth of the rectangular surface $=4m=400cm$ Height of the perceived rain $=1cm$ Then, Volume of the rectangular surface...

A cylindrical bucket, $32cm$ high and $18cm$ of radius of the base, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is $24cm$ , find the radius and slant height of the heap.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

It is given in the question that, Height of the cylindrical bucket $=32cm$ Radius of the cylindrical bucket $=18cm$ Height of conical heap $=24cm$ As we know that, Formula for volume of cylinder...

Find the volume largest right circular cone that can be cut out of a cube whose edge is $9cm$ .

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As per the question it is given that, The side of the cube $=9cm$ The largest cone that can be cut from cube will have the base diameter $=$ side of the cube $2r=9$ $r=9/2cm=4.5cm$ Now, Height of...

A well of diameter $3m$ is dug up to $14m$ deep. The earth taken out of it has been spread evenly all around it to a width of $4m$ to form an embankment. Find the height of the embankment.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As per the question it is given that, Diameter of the well $=3m$ Then, the radius of the well $=3/2m=1.5m$ Depth of the well (h) $=14m$ Width of the embankment (thickness) $=4m$ Therefore, the...

A well with inner radius $4m$ is dug up and $14m$ deep. Earth taken out of it has spread evenly all around a width of $3m$ it to form an embankment. Find the height of the embankment?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question it is given that, Inner radius of the well $=4m$ Depth of the well $=14m$ As we know that, Formula for Volume of the cylinder $=\pi {{r}^{2}}h$ $=\pi \times {{4}^{2}}\times...

A well of diameter $2m$ is dug $14m$ deep. The earth taken out of it is evenly spread all around it to form an embankment of height $40cm$ . Find the width of the embankment?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As per the question it is given that, Radius of the circular cylinder (r) $=2/2m=1m$ Height of the well (h) $=14m$ As we know that, Formula for volume of the solid circular cylinder $=\pi...

A $16m$ deep well with diameter $3.5m$ is dug up and the earth from it is spread evenly to form a platform $27.5m$ by $7m$ . Find the height of the platform?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

Consider the well to be a solid right circular cylinder Radius(r) of the cylinder $=3.5/2 m=1.75m$ Depth of the well or height of the cylinder (h) $=16m$ As we know that, Volume of the cylinder...

A path $2m$ wide surrounds a circular pond of diameter $40m$ . How many cubic meters of gravel are required to grave the path to a depth of $20cm$ ?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As per the question, Diameter of the circular pond $=40m$ So, the radius of the pond $=40/2=20m=r$ Thickness (width of the path) $=2m$ As the whole view of the pond looks like a hollow cylinder. And...

A spherical ball of radius $3cm$ is melted and recast into three spherical balls. The radii of two of the balls are $1.5cm$ and $2cm$ . Find the diameter of the third ball.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question it is given, Radius of the spherical ball $=3cm$ As we know that, The volume of the sphere $=4/3\pi {{r}^{2}}$ Now, it’s volume (V) $=4/3\pi {{r}^{3}}$ That the ball is...

A hollow sphere of internal and external radii $2cm$ and $4cm$ respectively is melted into a cone of base radius $4cm$ . Find the height and slant height of the cone.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As, per the question it is given The internal radius of hollow sphere $=2cm$ The external radius of hollow sphere $=4cm$ As we know that, Volume of the hollow sphere $4/3\pi \times \left(...

A hollow sphere of internal and external diameters $4cm$ and $8cm$ respectively is melted into a cone of base diameter 8 cm. Calculate the height of the cone?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question it is given that, Internal diameter of hollow sphere $=4cm$ So, the internal radius of hollow sphere $=2cm$ External diameter of hollow sphere $=8cm$ So, the external...

The diameters of the internal and external surfaces of a hollow spherical shell are $6cm$ and $10cm$ respectively. If it is melted and recast into a solid cylinder of diameter $14cm$ , find the height of the cylinder.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As per the question, Internal diameter of hollow spherical shell $=6cm$ Then, the internal radius of hollow spherical shell $=6/2=3cm=r$ External diameter of hollow spherical shell $=10cm$...

A solid cuboid of iron with dimensions $53cm\times 40cm\times 15cm$ is melted and recast into a cylindrical pipe. The outer and inner diameters of pipe are $8cm$ and $7cm$ respectively. Find the length of pipe.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

Assume the length of the pipe be h cm. Formula for volume of cuboid is $V=whl$ Now, Volume of cuboid $=\left( 53\times 40\times 15 \right)c{{m}^{3}}$ Internal radius of the pipe $=7/2cm=r$ External...

A solid metallic sphere of radius $5.6cm$ is melted and solid cones each of radius $2.8cm$ and height $3.2cm$ are made. Find the number of such cones formed.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

Assume the number of cones made be n It is given that, Radius of metallic sphere $=5.6cm$ Radius of the cone $=2.8cm$ Height of the cone $=3.2cm$ As we know that, Formula for volume of a sphere...

A cylindrical bucket, $32cm$ high and $18cm$ of radius of the base, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is $24cm$ , find the radius and slant height of the heap.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

It is given that, Height of the cylindrical bucket $=32cm$ Radius of the cylindrical bucket $=18cm$ Height of conical heap $=24cm$ As we know that, Volume of cylinder $=\pi \times {{r}^{2}}\times h$...

The surface area of a solid metallic sphere is $616c{{m}^{2}}$ . It is melted and recast into a cone of height $28cm$ . Find the diameter of the base of the cone so formed.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As per the question given, The height of the cone $=28cm$ Surface area of the solid metallic sphere $=616c{{m}^{3}}$ As we know that, Surface area of the sphere $=4\pi {{r}^{2}}$ Then, $4\pi...

How many coins $1.75cm$ in diameter and $2mm$ thick must be melted to form a cuboid $11cm\times 10cm\times 7cm$ ?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question, Diameter of the coin $=1.75cm$ Then, its radius $=1.74/2=0.875cm$ Thickness or the height $=2mm=0.2cm$ As we know that, Volume of the cylinder $\left( {{V}_{1}}...

The diameters of internal and external surfaces of a hollow spherical shell are $10cm$ and $6cm$ respectively. If it is melted and recast into a solid cylinder of length of $8/3$ , find the diameter of the cylinder?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As per the question given, Internal diameter of the hollow sphere $=6cm$ The internal radius of the hollow sphere $=6/2cm=3cm=r$ External diameter of the hollow sphere $=10cm$ Then, the external...

Find the equation of the circle passing through the origin and the points where the line 3x + 4y = 12 meets the axes of coordinates.

CBSE, Circle, Class 11, Exercise 24.3, Maths, RD Sharma

The line 3x + 4y = 12 The value of x is 0 on meeting the y – axis. So, \[\begin{array}{*{35}{l}} 3\left( 0 \right)\text{ }+\text{ }4y\text{ }=\text{ }12 \\ 4y\text{ }=\text{ }12 \\ y\text{...

The sides of a squares are x = 6, x = 9, y = 3 and y = 6. Find the equation of a circle drawn on the diagonal of the square as its diameter.

CBSE, Circle, Class 11, Exercise 24.3, Maths, RD Sharma

The sides of a squares are x = 6, x = 9, y = 3 and y = 6. assuming A, B, C, D be the vertices of the square. we get, the coordinates as: A = (6, 3) B = (9, 3) C = (9, 6) D = (6, 6) the equation of...

Find the equation of the circle, the end points of whose diameter are (2, -3) and (-2, 4). Find its centre and radius.

CBSE, Class 11, Exercise 24.3, Maths, RD Sharma

The diameters (2, -3) and (-2, 4). By using the formula, Centre = (-a, -b) \[\begin{array}{*{35}{l}} =\text{ }\left[ \left( 2-2 \right)/2,\text{ }\left( -3+4 \right)/2 \right] \\ =\text{ }\left(...

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 – sin α + i cos α
(ii) (1 – i) / (cos π/3 + i sin π/3)

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: (i) $1-\sin \alpha+i \cos \alpha$ Given that $Z=1-\sin \alpha+i \cos a$ Using the formulas, $\operatorname{Sin}^{2} \theta+\cos ^{2} \theta=1$ $\operatorname{Sin} 2 \theta=2 \sin \theta...

The letters of the word ‘CLIFTON’ are placed at random in a row. What is the chance that two vowels come together?

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

Given is The word ‘CLIFTON’. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes In the random arrangement of the alphabets of word “CLIFTON” we have...

A box contains 100 bulbs, 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that:
(i) all 10 are defective
(ii) all 10 are good

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

Given is a box contains 100 bulbs, 20 of which are defective. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes Ten bulbs are drawn at random for...

The face cards are removed from a full pack. Out of the remaining 40 cards, 4 are drawn at random. What is the probability that they belong to different suits?

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, the face cards are removed from a full pack of 52. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes Four cards are drawn...

A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that:
one is red

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

Given is a bag containing 6 red, 4 white and 8 blue balls. By using the formula of probability we get, P (E) = favourable outcomes / total possible outcomes Two balls are drawn at random, so the...

A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that:
(i) one is red and two are white
(ii) two are blue and one is red

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

Given is a bag containing 6 red, 4 white and 8 blue balls. By using the formula of probability we get, P (E) = favourable outcomes / total possible outcomes Two balls are drawn at random, so the...

A bag contains 7 white, 5 black and 4 red balls. If two balls are drawn at random, find the probability that:
both the balls are of the same colour

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

Given is a bag containing 7 white, 5 black and 4 red balls. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes Two balls are drawn at random, so total...

A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that one is red, one is white and one is blue

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

Given is a bag containing 6 red, 4 white and 8 blue balls. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes As three balls are drawn, se we need to...

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 + i tan α
(ii) tan α – i

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: (i) $1+\mathrm{i} \tan \alpha$ Given that $Z=1+\mathrm{i}$ tan $\alpha$ It is known to us that the polar form of a complex number $Z=$ In which, $\begin{array}{l} \left.|Z|=\text { modulus...

A bag contains
$\mathbf{3}$
white,
$\mathbf{5}$
black and
$\mathbf{2}$
red balls, all of the same shape and size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is:
$\left( \mathbf{i} \right)$
a black ball.
$\left( \mathbf{ii} \right)$
a red ball.

CBSE, Class 10, Probability, Selina

Total number of balls \[=\text{ }3\text{ }+\text{ }5\text{ }+\text{ }2\text{ }=\text{ }10\] So, the total number of possible outcomes \[=\text{ }10\] \[\left( i \right)~\] There are \[5\] black...

In shutting a pack of 52 playing cards, four are accidently dropped; find the chance that the missing cards should be one from each suit

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, a pack of 52 cards is given from which 4 are dropped. By using the formula of probability, we have, P (E) = favourable outcomes / total possible outcomes Now find the...

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:
(i) not a diamond card
(ii) a black card

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

A Pack of 52 cards is given By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes We know, a card is drawn from a pack of 52 cards, so number of...

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:
(i) neither an ace nor a king
(ii) a diamond card

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

A Pack of 52 cards is given By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes We know, a card is drawn from a pack of 52 cards, so number of...

Find the equation of the circle which circumscribes the triangle formed by the lines: (i) x + y + 3 = 0, x – y + 1 = 0 and x = 3 (ii) 2x + y – 3 = 0, x + y – 1 = 0 and 3x + 2y – 5 = 0

CBSE, Class 11, Exercise 24.2, Maths, RD Sharma, The Circle

(i) The lines \[\begin{array}{*{35}{l}} x\text{ }+\text{ }y\text{ }+\text{ }3\text{ }=\text{ }0 \\ x\text{ }-\text{ }y\text{ }+\text{ }1\text{ }=\text{ }0 \\ x\text{ }=\text{ }3 \\ \end{array}\]...

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:
(i) a black king
(ii) either a black card or a king

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

A Pack of 52 cards is given By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes We know, a card is drawn from a pack of 52 cards, so number of...

In a single throw of three dice, find the probability of getting the same number on all the three dice

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, three dice are rolled over. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes So, we have to find the probability of...

A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that:
(i) All the three balls are white
(ii) All the three balls are red

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, a bag contains 8 red and 5 white balls. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes Total number of ways of drawing...

Write $\left(\mathrm{i}^{25}\right)^{3}$ in polar form.

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: Given that $Z=\left(i^{25}\right)^{3}$ $\begin{array}{l} =\dot{i}^{75} \\ =\mathrm{i}^{74} \cdot \mathrm{i} \\ =\left(\mathrm{i}^{2}\right)^{37} \cdot \mathrm{i} \\ =(-1)^{37} \cdot...

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) $sin 120^o - i cos 120^o$
(ii) -16 / (1 + i√3)

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: It is known to us that the polar form of a complex number $Z=x+i$ iy is given by $Z=|Z|(\cos \theta+i \sin \theta)$ In which, $\begin{array}{l} \left.|Z|=\text { modulus of complex number...

Three coins are tossed together. Find the probability of getting:
(i) exactly two heads
(ii) at least two heads

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, three coins are tossed together. By using the formula of probability, we get, P (E) = favourable outcomes / total possible outcomes Total number of possible outcomes is...

In a simultaneous throw of a pair of dice, find the probability of getting:
(i) a total of 9 or 11
(ii) a total greater than 8

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, a pair of dice has been thrown So the number of elementary events in sample space will be $6^2=36$ n (S) = 36 By using the formula of probability, we get, P (E) =...

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1/(1 + i)
(ii) (1 + 2i) / (1 – 3i)

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

In a simultaneous throw of a pair of dice, find the probability of getting:
(i) a sum more than 7
(ii) neither a doublet nor a total of 10

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, a pair of dice has been thrown So the number of elementary events in sample space will be $6^2=36$ n (S) = 36 By using the formula of probability, we get, P (E) =...

In a simultaneous throw of a pair of dice, find the probability of getting:
(i) an even number on one and a multiple of 3 on the other
(ii) neither 9 nor 11 as the sum of the numbers on the faces

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, a pair of dice has been thrown So the number of elementary events in sample space will be $6^2=36$ n (S) = 36 By using the formula of probability, we get, P (E) =...

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1 – i
(ii) (1 – i) / (1 + i)

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1 + i
(ii) √3 + i

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

In a simultaneous throw of a pair of dice, find the probability of getting:
(i) 8 as the sum
(ii) a doublet

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, a pair of dice has been thrown So the number of elementary events in sample space will be $6^2=36$ n (S) = 36 By using the formula of probability, we get, P (E) =...

A die is thrown. Find the probability of getting: A multiple of 2 or 3

CBSE, Class 11, Exercise 33.3, Maths, Probability, RD Sharma

According to the question, a die is thrown. The total number of outcomes will be, n (S) = 6 By using the formula we get, P (E) = favourable outcomes / total possible outcomes Let E be the event of...

Show that the points (3, -2), (1, 0), (-1, -2) and (1, -4) are con – cyclic.

CBSE, Class 11, Exercise 24.2, Maths, RD Sharma, The Circle

The points (3, -2), (1, 0), (-1, -2) and (1, -4) the circle passes through the points A, B, C. therefore, the equation of the circle: x2 + y2 + 2ax + 2by + c = 0….. (1) Substituting the points A (3,...

Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on line x – 4y = 1.

CBSE, Circle, Class 11, Exercise 24.2, Maths, RD Sharma

The points (3, 7), (5, 5) The line x – 4y = 1…. (1) the equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (2) substituting the centre (-a, -b) in equation (1) we get,...

If $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$ are two complex number such that $\left|\mathbf{z}_{1}\right|=\left|\mathbf{z}_{2}\right|$ and $\arg \left(\mathbf{z}_{1}\right)+\arg \left(\mathbf{z}_{2}\right)=\pi$ , then show that $\mathbf{z}_{1}=-\overline{\mathbf{z}_{2}}$

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: Given that $\left|z_{1}\right|=\left|z_{2}\right|$ and $\arg \left(z_{1}\right)+\arg \left(z_{2}\right)=\pi$ Let's assume $\arg \left(z_{1}\right)=\theta$ $\arg...

Find the equation of the circle which passes through (3, – 2), (- 2, 0) and has its centre on the line 2x – y = 3.

CBSE, Circle, Class 11, Exercise 24.2, Maths, RD Sharma

The line 2x – y = 3 … (1) The points (3, -2), (-2, 0) The equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (2) substituting the centre (-a, -b) in equation (1) we get,...

If $z_{1}, z_{2}$ and $z_{3}, z_{4}$ are two pairs of conjugate complex numbers, prove that arg $\left(z_{1} / z_{4}\right)+\arg \left(z_{2} / z_{3}\right)=0$

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: Given that $\begin{array}{l} z_{1}=\bar{z}_{2} \\ z_{3}=\bar{z}_{4} \end{array}$ It is known that $\arg \left(\mathrm{z}_{1} / \mathrm{z}_{2}\right)=\arg \left(\mathrm{z}_{1}\right)-\arg...

Express $\sin \pi / 5+i(1-\cos \pi / 5)$ in polar form.

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: Given that $Z=\sin \pi / 5+i(1-\cos \pi / 5)$ Using the formula, $\begin{array}{l} \sin 2 \theta=2 \sin \theta \cos \theta \\ 1-\cos 2 \theta=2 \sin ^{2} \theta \end{array}$ Therefore,...

A copper rod of diameter $1cm$ and length $8cm$ is drawn into a wire of length $18m$ of uniform thickness. Find the thickness of the wire?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

As, per the question, Diameter of the copper wire $=1cm$ Radius of the copper wire $=1/2cm=0.5cm$ Length of the copper rod $=8cm$ As we know that, Formula for volume of the cylinder $=\pi...

Find the equation of the circle passing through the points : (i) (5, 7), (8, 1) and (1, 3) (ii) (1, 2), (3, – 4) and (5, – 6)

CBSE, Circle, Class 11, Exercise 24.2, Maths, RD Sharma

(i) (5, 7), (8, 1) and (1, 3) By using the standard form of the equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (1) Substitute the given point (5, 7) in equation (1), we get...

A copper sphere of radius $3cm$ is melted and recast into a right circular cone of height $3cm$ . Find the radius of the base of the cone?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question it is given that, Radius of the copper sphere $=3cm$ As we know that, Volume of the sphere $=4/3\pi {{r}^{3}}$ $=4/3\pi \times {{3}^{3}}$ ….. (i) The copper sphere is...

An iron spherical ball has been melted and recast into smaller balls of equal size. If the radius of each of the smaller balls is $1/4$ of the radius of the original ball, how many such balls are made? Compare the surface area, of all the smaller balls combined together with that of the original ball.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

Assume the radius of the big ball be $xcm$ The radius of the small ball $=x/4cm$ Let the number of balls $=n$ Then according to the question, we have Volume of n small balls $=$ Volume of the big...

The diameter of a metallic sphere is equal to $9cm$ . It is melted and drawn into a long wire of diameter $2mm$ havinThe diameter of a metallic sphere is equal to $9cm$ . It is melted and drawn into a long wire of diameter $2mm$ having uniform cross-section. Find the length of the wire.g uniform cross-section. Find the length of the wire.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question it is given that, Radius of the sphere $=9/2cm$ Its volume will be $=4/3\pi {{r}^{3}}=4/3\pi {{\left( 9/2 \right)}^{3}}$ Then, the radius of the wire $=2mm=0.2cm$ Assume...

A solid metallic sphere of radius $10.5cm$ is melted and recast into a number of smaller cones, each of radius $3.5cm$ and height $3cm$ . Find the number of cones so formed.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

It is given that, Radius of metallic sphere $=R=10.5cm$ So, its volume $=4/3\pi {{R}^{3}}=4/3\pi {{\left( 10.5 \right)}^{3}}$ We also have, Radius of each cone $=r=3.5cm$ Height of each cone...

Three cubes of a metal whose edges are in the ratio $3:4:5$ are melted and converted into a single cube whose diagonal is $12\sqrt{3}cm$ . Find the edges of the three cubes.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

Assume the edges of three cubes (in cm) be $3x$, $4x$ and $5x$ respectively. Then, the volume of the cube after melting will be $={{\left( 3x \right)}^{3}}+{{\left( 4x \right)}^{3}}+{{\left( 5x...

How many spherical lead shots of diameter $4cm$ can be made out of a solid cube of lead whose edge measures $44cm$ .

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question, The radius of each spherical lead shot $=r=4/2=2cm$ Volume of each spherical lead shot $=4/3\pi {{r}^{3}}=4/3\pi {{2}^{3}}c{{m}^{3}}$ Edge of the cube $=44cm$ Volume of...

Find the coordinates of the centre radius of each of the following circle: (iii) $\frac{1}{2}({{x}^{2}}+{{y}^{2}})+x\cos \theta +y\sin \theta -4=0$ (iv) ${{x}^{2}}+{{y}^{2}}-ax-by=0$

CBSE, Circle, Class 11, Exercise 24.2, Maths, RD Sharma

The equation of the circle is (Multiply by 2 we get) \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }2x\text{ }cos\text{ }\theta \text{ }+\text{ }2y\text{ }sin\text{ }\theta \text{...

A circle whose centre is the point of intersection of the lines 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 passes through the origin. Find its equation.

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

Since, circle has the centre at the intersection point of the lines 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 and passes through the origin Since, the equation of the circle with centre (p, q) and having...

Find the equation of the circle which has its centre at the point (3, 4) and touches the straight line 5x + 12y – 1 = 0.

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

It is given that we need to find the equation of the circle with centre (3, 4) and touches the straight line 5x + 12y – 1 = 0. Since, circle with centre (3, 4) and having a radius 62/13. As, the...

Find the equation of the circle (iii) Which touches both the axes and passes through the point (2, 1). (iv) Passing through the origin, radius 17 and ordinate of the centre is – 15.

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

(iii) Let the circle touches the x-axis at the point (a, 0) and y-axis at the point (0, a). Then the centre of the circle is (a, a) and radius is a. => (x – p)2 + (y – q)2 = r2 substituting the...

How many spherical lead shots each of diameter $4.2cm$ can be obtained from a solid rectangular lead piece with dimensions $66cm\times 42cm\times 21cm$ .

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question Radius of each spherical lead shot $=r=4.2/2=2.1cm$ The dimensions of the rectangular lead piece $=66cm\times 42cm\times 21cm$ So, the volume of a spherical lead shot...

Find the number of metallic circular discs with $1.5cm$ base diameter and of height $0.2cm$ to be melted to form a right circular cylinder of height 10 cm and diameter $4.5cm$ .

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

It is given in the question that, Radius of each circular disc $=r =1.5/2=0.75cm$ Height of each circular disc $=h=0.2cm$ Radius of cylinder $=R=4.5/2=2.25cm$ Height of cylinder $=H=10cm$ So, the...

25 circular plates, each of radius $10.5cm$ and thickness $1.6cm$ , are placed one above the other to form a solid circular cylinder. Find the curved surface area and the volume of the cylinder so formed.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

Given, 250 circular plates each with radius $10.5cm$ and thickness of $1.6cm$. As the plates are placed one above the other, the total height becomes $=1.6\times 25=40cm$ As we know that, Curved...

Find the equation of the circle (i) which touches both the axes at a distance of 6 units from the origin. (ii) Which touches x – axis at a distance of 5 from the origin and radius 6 units.

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

(i) . A circle touches the axes at the points (±6, 0) and (0, ±6). So, a circle has a centre (±6, ±6) and passes through the point (0, 6). Since, the radius of the circle is the distance between the...

Find the equation of the circle whose centre lies on the positive direction of y – axis at a distance 6 from the origin and whose radius is 4.

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

It is given that the centre lies on the positive y – axis at a distance of 6 from the origin, we get the centre (0, 6). As circle with centre (0, 6) and having radius 4. Since, the equation of the...

50 circular plates each of diameter $14cm$ and thickness $0.5cm$ are placed one above the other to form a right circular cylinder. Find its total surface area.

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question, 50 circular plates each with diameter $14cm$ Radius of circular plates $=7cm$ Thickness of plates $=0.5cm$ We have to find the total surface area As these plates is one...

A cylindrical vessel having diameter equal to its height is full of water which is poured into two identical cylindrical vessels with diameter $42cm$ and height $21cm$ which are filled completely. Find the diameter of the cylindrical vessel?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

It is given that, The diameter of the cylinder $=$ the height of the cylinder $⇒h=2r$, where h – height of the cylinder and r – radius of the cylinder As we know that, Volume of a cylinder $=\pi...

Find the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6).

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

Centre is (1, 2) and which passes through the point (4, 6). Where, p = 1, q = 2 By using the formula, \[\begin{array}{*{35}{l}} {{\left( x\text{ }-\text{ }p \right)}^{2}}~+\text{ }{{\left(...

What length of a solid cylinder $2cm$ in diameter must be taken to recast into a hollow cylinder of length $16cm$ , external diameter $20cm$ and thickness $2.5mm$ ?

CBSE, Class 10, Exercise 16.1, Maths, RD Sharma, Surface Areas And Volumes

According to the question, Diameter of the solid cylinder $=2cm$ Length of hollow cylinder $=16cm$ The solid cylinder is recast into a hollow cylinder of length $16cm$, with external diameter of...

Find the centre and radius of each of the following circles: (i) (x – 1)2 + y2 = 4 (ii) (x + 5)2 + (y + 1)2 = 9

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

\[~{{\left( x\text{ }-\text{ }1 \right)}^{2}}~+\text{ }{{y}^{2}}~=\text{ }4\] using the standard equation formula, \[{{\left( x\text{ }-\text{ }a \right)}^{2}}~+\text{ }{{\left( y\text{ }-\text{ }b...