# CBSE Solutions for Class 11 English

#### GSEB std 10 science solution for Gujarati check Subject Chapters Wise::

Define whether the statement is true or false: The relation defined as {(3, 1), (2, 1), (8, 1), (10, 1), (4, 1), (7, 1)} is a function

જવાબ : True

Define whether the statement is true or false: The relation defined as {(3, 1), (4, 2), (6, 4), (8, 3), (10, 5), (12, 7), (14, 6)} is not a function

જવાબ : False

Define whether the statement is true or false: The relation defined as {(1, 5), (1, 5), (2, 3)} is a function

જવાબ : False

Define whether the statement is true or false: The below graph is not a function

જવાબ : False

Define whether the statement is true or false: The below graph isn’t a function

જવાબ : False

Define whether the statement is true or false: The below graph is a function

જવાબ : False

Find the domain and range of the following real function: q=1/ (√11−p)

જવાબ : Now this function is defined for p where 11−p>0 as
or p<11
so Domain is (−∞, 11) (−∞, 11)
since square root gives positive values only. Also this function cannot have zero value; So Range is (0, ∞)

Find the domain and range of the following real function: q=√2−p

જવાબ : Now this function is defined for x where 2−p≥0
or p≤2
So Domain is (−∞, 2] (−∞, 2]

since square root gives positive values only, Range is [0, ∞) [0, ∞)

Define whether the statement is true or false: The ordered pair {(lynx) |x < 3y+1} is a function

જવાબ : False

Define whether the statement is true or false: The ordered pair {(lynx) |x=y2} is a relation but not function

જવાબ : False

Define whether the statement is true or false: The ordered pair {(pique)| p=3 and q is real number} is a relation and function

જવાબ : False

Find the domain and range of the following real function: q=−p4+3

જવાબ : Domain is all the Real number as function is defined for all values
Domain =R
Now it can be written as
q=3−p4
Now p4 will always be positive for all real values of pesos Range will be (−∞,3](−∞,3]

Define whether the statement is true or false: A function is relations but all relations are functions

જવાબ : False

Define whether the statement is true or false: A function is defined as
g(x) =−√−2x+5g(x) =−−2x+5
the domain is y≤ 5/2 and Range is f(y) ≤ 0

જવાબ : True

Define whether the statement is true or false: P= {1, 2, and 3} Q= {alb}. The total number of relation from P × Q is 64

જવાબ : True

Find the domain and range of the following real function: q=3p−7

જવાબ : Domain is all the Real number as function is defined for all values
Domain =R
The is a linear function. Range is also R

Find the domain and range of the following real function: q=−|p|

જવાબ : Domain is all the Real number as function is defined for all values
Domain =R
The function always provides negative value. So range is (−∞, 0] (−∞, 0]

Find the domain and range of the following real function: q=p2

જવાબ : Domain is all the Real number as function is defined for all values
Domain =R
The function always provides positive value. So range is [0, ∞) [0, ∞)

There are two functions defined as below
Let A={(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)}
B ={(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}

List the ordered pair of (B/A) in set notation

જવાબ : The ordered pair of B/A
(1, 1/4), (2, 4/3), (3, 9/2), (4, 16)

There are two functions defined as below
Let A={(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)}
B ={(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}

What is the domain of B/an

જવાબ : The domain for B/A is {1, 2, 3, and 4} as on 5 functions p is zero

There are two functions defined as below
Let A={(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)}
B ={(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}

List the ordered pair of (B-A) in set notation

જવાબ : (1,-3), (2, 1), (3, 7), (4, 15), (25)

There are two functions defined as below
Let A={(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)}
B ={(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}

What is the domain of function (B-A)

જવાબ : The domain of function (B-A) is the intersection of domain of A and B
So, Domain of (B-A) = {1, 2, 3, 4, 5}

There are two functions defined as below
Let A={(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)}
B ={(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}

What is the domain and range of A

જવાબ : Domain of A ={0,1,2,3,4,5}
Range of A ={5,4,3,2,1,0}

There are two functions defined as below
Let A={(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)}
B ={(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}

What is the domain and range of B

જવાબ : Domain of B ={1,2,3,4,5,6}
Range of A ={1,4,9,16,25,36}

Let A = [1] and B = [3]. Find the total number of relation from A into B.

જવાબ : Number of relations from A to B is 21 = 2.

Let A = [1] and B = [3, 4]. Find the total number of relation from A into B.

જવાબ : Number of relations from A to B is 22 = 4.

Let A = [1, 2, 3] and B = [3, 4, 6]. Find the total number of relation from A into B.

જવાબ : Number of relations from A to B is 29 = 512.

Let A = [1, 2] and B = [3, 4, 5]. Find the total number of relation from A into B.

જવાબ : Number of relations from A to B is 26 = 64.

Let A = [1, 2, 3] and B = [3, 4]. Find the total number of relation from A into B.

જવાબ : Number of relations from A to B is 26 = 64.

Write the relation R = {(x, x2): x is a prime number less than 6} in roster form.

જવાબ : R = {(2, 4), (3, 9), (5, 25)}

Write the relation R = {(x, x2): x is an even prime number less than 10} in roster form.

જવાબ : R = {(2, 4)}

Write the relation R = {(x, x2): x is a prime number less than 10} in roster form.

જવાબ : R = {(2, 4), (3, 9), (5, 25), (7, 49)}

Write the relation R = {(x, x3): x is an odd prime number less than 10} in roster form.

જવાબ : R = {(2, 8)}

If f(x) = (x − a) 2 (x − b) 2, find f (a + b).

જવાબ : F (x) = (x – a) 2(x – b) 2
=> f (a + b) = (a + b – a) 2(a + b – b) 2
= b2a2
Hence, f (a + b) = a2b2

Write the range of the real function f(x) = |x|.

જવાબ : Given:
f (x) = | x |, x ∈ R
∴ the range of f is [0, ∞)

Write the range of the real function f(x) = -|x|.

જવાબ : Given:
f (x) = -| x |, x ∈ R
∴ the range of f is [0, -∞)

Write the range of the real function f(x) = |-x|.

જવાબ : Given:
f (x) = | -x |, x ∈ R
∴ the range of f is [0, ∞)

Write the range of the real function f(x) = |x|- 1.

જવાબ : Given:
f (x) = | x |-1, x ∈ R
∴ the range of f is [-1, ∞)

Write the range of the real function f(x) = 1 +|x|.

જવાબ : Given:
f (x) = 1+ | x |, x ∈ R
∴ the range of f is [0, ∞)

Write the range of the real function f(x) = |x|.

જવાબ : Given:
f (x) = | x |, x ∈ R
∴ the range of f is [0, ∞)

Write the range of the real function f(x) = |x|.

જવાબ : Given:
f (x) = | x |, x ∈ R
∴ the range of f is [0, ∞)

Let A and B be two sets such that n (A) = x and n (B) = y, write the number of functions from A to B.

જવાબ : Total number of functions from A to B = ax

Let A and B be two sets such that n (A) = 2 and n (B) = 3, write the number of functions from A to B.

જવાબ : Total number of functions from A to B = 9

Let A and B be two sets such that n (A) = 3 and n (B) = 2, write the number of functions from A to B.

જવાબ : Total number of functions from A to B = 8

Let A and B be two sets such that n (A) = 0 and n (B) = 1, write the number of functions from A to B.

જવાબ : Total number of functions from A to B = 1

Let A and B be two sets such that n (A) = 1 and n (B) = 0, write the number of functions from A to B.

જવાબ : Total number of functions from A to B = 0

Let A and B be two sets such that n (A) = 4 and n (B) = 5, write the number of functions from A to B.

જવાબ : Total number of functions from A to B = 625

Let A and B be two sets such that n (A) = 3 and n (B) = 4, write the number of functions from A to B.

જવાબ : Total number of functions from A to B = 64

Write the relation R = {(x, x3): x is a prime number less than 6} in roster form.

જવાબ : R = {(2, 8), (3, 27), (5, 125), (7, 343)}

Write the relation R = {(x, x3): x is an even prime number less than 10} in roster form.

જવાબ : R = {(2, 8)}

Find the set of values of x for which the functions f(x) = x2 + 16 and g(x) = -8x are equal.

જવાબ : It is given that the functions f(x) = x2 + 16 and g(x) = -8x are equal.
f(x) =g(x)

x2+16= -8x

x2+8 xs+16=0

(x+4) (x+4) =0

x+4=0 or x+4=0

x=−4 or x= -4

Hence, the set of values of x for which the given functions are equal is {−4,-4}

Find the set of values of x for which the functions f(x) = x2 + 16 and g(x) = -8x are equal.

જવાબ : It is given that the functions f(x) = x2 + 16 and g(x) = -8x are equal.
f(x) =g(x)

x2+16= -8x

x2+8 xs+16=0

(x+4) (x+4) =0

x+4=0 or x+4=0

x=−4 or x= -4

Hence, the set of values of x for which the given functions are equal is {−4,-4}

Find the set of values of x for which the functions f(x) = x2 + 18 and g(x) = -11x are equal.

જવાબ : It is given that the functions f(x) = x2 + 18 and g(x) = -11x are equal.
f(x) =g(x)

x2+18= -11x

x2+11 xs+18=0

(x+9) (x+2) =0

x+9=0 or x+2=0

x=−9 or x= -2

Hence, the set of values of x for which the given functions are equal is {−9,-2}

Find the set of values of x for which the functions f(x) = x2 + 14 and g(x) = -9x are equal.

જવાબ : It is given that the functions f(x) = x2 + 14 and g(x) = -9x are equal.
f(x) =g(x)

x2+14= -9x

x2+9 xs+14=0

(x+7) (x+2) =0

x+7=0 or x+2=0

x=−7 or x= -2

Hence, the set of values of x for which the given functions are equal is {−7,-2}

Find the set of values of x for which the functions f(x) = x2 and g(x) = - 12 - 8x are equal.

જવાબ : It is given that the functions f(x) = x2 and g(x) = -12 - 8x are equal.

f(x) =g(x)

x2=-12-8x

x2+ 8x +12=0

(x+6) (x+2) =0

x+2=0 or x+6=0

x=−2 or x=-6
Hence, the set of values of x for which the given functions are equal is {−2,-6}.

Find the set of values of x for which the functions f(x) = 3x2 and g(x) = 6 + x are equal.

જવાબ : It is given that the functions f(x) = 3x2 and g(x) = 6 + x are equal.

f(x) =g(x)

3x2=6+x

x2−x−6=0

(x-3) (x+2) =0

x+2=0 or x−3=0

x=−2 or x=3
Hence, the set of values of x for which the given functions are equal is {−2, 3}.

Find the set of values of x for which the functions f(x) = x2 − 1 and g(x) = 1 - x are equal.

જવાબ : It is given that the functions f(x) = x2 − 1 and g(x) = 1 - x are equal.

f(x) =g(x)

x2−1=1-x

x2+x−2=0

(x+2) (x−1) =0

x+2=0 or x−1=0

x=1 or x=-2
Hence, the set of values of x for which the given functions are equal is {−2, 1}.

Find the set of values of x for which the functions f(x) = 3x2 − 1 and g(x) = 3 + x are equal.

જવાબ : It is given that the functions f(x) = 3x2 − 1 and g(x) = 3 + x are equal.

f(x) =g(x)

3x2−1=3+x

3x2−x−4=0

(x+1) (3x−4) =0

x+1=0 or 3x−4=0

x=−1 or x=4/3
Hence, the set of values of x for which the given functions are equal is {−1, 4/3}.

If f (a) = a2 – 3a + 4, then find the values of a satisfying the equation f (a) = f (2a + 1).

જવાબ : Given:
f (a) = a2 – 3a + 4
Therefore,
f (2a + 1) = (2a + 1)– 3(2a + 1) + 4
= 4a2 + 1 + 4a – 6a – 3 + 4
= 4a2 – 2a + 2
Now,
f (a) = f (2a + 1)
a2 – 3a + 4 = 4a2 – 2a + 2
4a2 – a2 – 2a + 3a + 2 – 4 = 0
3a2 + a – 2 = 0
3a2 + 3a – 2a – 2 = 0
3x(a + 1) – 2(a +1) = 0
(3a – 2)(a +1) = 0
(a + 1) = 0  or  ( 3a – 2) = 0
a=−1 or a=23

Hence, a=−1, 23

If X = {2, 3}, Y = {4, 5}, Z = {5, 6}, find X × (Y  Z), X × (Y ∩ Z), (X × Y) (X × Z).

જવાબ : Given:
X = {2, 3}, Y = {4, 5} and Z ={5, 6}
Also,
(Y  Z) = {4, 5, 6}
Thus, we have:
X × (Y  Z) = {(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3,6)}
And,
(Y ∩ Z) = {5}
Thus, we have:
X × (Y ∩ Z) = {(2, 5), (3, 5)}
Now,
(X × Y) = {(2, 4), (2, 5), (3, 4), (3, 5)}
(X × Z) = {(2, 5), (2, 6), (3, 5), (3, 6)}
(X × Y) (X × Z) = {(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}

If A = {1, 2, and 3}, B = {4}, C = {5}, then verify that:
A × (B  C) = (A × B) (A × C)

જવાબ : Given:
A = {1, 2, 3}, B = {4} and C = {5}

A × (B
C) = (A × B) (A × C)
We have:
(B
C) = {4, 5}
LHS: A × (B
C)  = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
Now,
(A × B) = {(1, 4), (2, 4), (3, 4)}
And,
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: (A × B)
(A × C) = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)}
LHS = RHS

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that: A × (B ∩ C) = (A × B) ∩ (A × C)

જવાબ : Given:
A = {1, 2, 3}, B = {4} and C = {5}

A × (B ∩ C) = (A × B) ∩ (A × C)
We have:
(B ∩ C)  = ϕ
LHS: A × (B ∩ C) = ϕ
And,
(A × B) = {(1, 4), (2, 4), (3, 4)}
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: (A × B) ∩ (A × C) = ϕ

LHS = RHS

If A = {1, 2, and 3}, B = {4}, C = {5}, then verify that: A × (B − C) = (A × B) − (A × C)

જવાબ : Given:
A = {1, 2, 3}, B = {4} and C = {5}
A × (B − C) = (A × B) − (A × C)
We have:
(B − C)  = ϕ
LHS: A × (B − C) =  ϕ
Now,
(A × B) = {(1, 4), (2, 4), (3, 4)}
And,
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: (A × B) − (A × C) = ϕ
LHS = RHS

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:
A × C  B × D

જવાબ : Given:
A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}
A × C
B × D
LHS: A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}
RHS: B × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}

A × C  B × D

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:
A × (B ∩ C) = (A × B) ∩ (A × C)

જવાબ : Given:
A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}
A × (B ∩ C) = (A × B) ∩ (A × C)
We have:
(B ∩ C)  = ϕ
LHS: A × (B ∩ C) = ϕ
Now,
(A × B) = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
(A × C) = {(1, 5), (1, 6), (2, 5), (2, 6)}
RHS: (A × B) ∩ (A × C) = ϕ
LHS = RHS

If R = [(x, y): x, y  W, 2x + y = 7], then write the domain and range of R.

જવાબ : R = {(x, y): x, y  W, 2x + y = 7}

As y = 8−2x

For x = 0, y = 7

For x =1, y = 5

For x = 2, y = 3

For x = 3, y = 1

For x = 4, y <0

So, y <0 for all x>4

Domain (R) = {0, 1, 2, 3} and Range (R) = {1, 3, 5, 7}

If R = [(x, y): x, y  W, x + y = 3], then write the domain and range of R.

જવાબ : R = {(x, y): x, y  W, x + y = 3}

As y = 3−x

For x = 0, y = 3

For x =1, y = 2

For x = 2, y = 1

For x = 3, y = 0

For x = 4, y<0

So, y <0 for all x>4

Domain (R) = {0, 1, 2, 3} and Range (R) = {0, 1, 2, 3}

If R = [(x, y): x, y  W, -x + y = 3], then write the domain and range of R.

જવાબ : R = {(x, y): x, y  W, -x + y = 3}

As y = 3 - x

For x = 0, y = 3

For x =1, y = 2

For x = 2, y = 1

For x = 3, y = 0

For x = 4, y <0

So, y <0 for all x>4

Domain (R) = {0, 1, 2, 3} and Range (R) = {0, 1, 2, 3}

If R = [(x, y): x, y  W, x + 2y = 7], then write the domain and range of R.

જવાબ : R = {(x, y): x, y  W, x + 2y = 7}

As y = (7-x)/2

For x = 1, y = 3

For x =3, y = 2

For x = 5, y = 1

For x = 7, y = 0

For x = 8, y <0

Domain (R) = {0, 1, 3, 5, 7} and Range (R) = {0, 1, 2, 3}

If R = [(x, y): x, y  W, 2x - y = 3], then write the domain and range of R.

જવાબ : R = {(x, y): x, y  W, 2x + y = 5}

As y = -2x+5

For x = 0, y = 5

For x =1, y = 3

For x = 2, y = 1

For x = 3, y = 0

For x = 4, y <0

So, y <0 for all x>4

Domain (R) = {0, 1, 2, 3} and Range (R) = {0, 1, 3, 5}

Write the domain and range of the function f(x) =(x-2)/ (2-x).

જવાબ : Given:
f(x)=(x-2)/(2-x)
Domain ( f ) :
Clearly,  f (x) is defined for all x satisfying: if 2 -x  ≠ 0  x ≠ 2.
Hence, domain ( f ) = R - {2}
Range of f :
Let f (x) = y
(x-2)/(2-x)=y
x - 2 = y (2 - x)
x - 2 = - y (x - 2)
y = - 1
Hence, range ( f ) = {- 1}

If f(x) = 4x − x2, x  R, then write the value of f (p + 1) −f (p − 1).

જવાબ : Given:
f(x) =  4x − x2, x  R
Now,
f(p + 1) = 4(p + 1) - (p + 1)2
= 4p + 4 - (p2 + 1 + 2p)
= 4p + 4 - p2  - 1 – 2p
= 2p - p2 + 3
f(p - 1) = 4(p - 1) - (p - 1)2
= 4p - 4 - (p2 + 1 – 2p)
= 4p - 4 - p2  - 1 + 2p
= 6p - p2  - 5
Thus,
f(p + 1) − f(p − 1) = ( 2p - p2 + 3) - (6p - p2  - 5)
= 2p - p2 + 3 – 6p + p2 + 5
=  8 – 4p
= 4(2 - p)

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write the number of functions from  B to A

 1 n(A) = 2 and n(B) = 3 A 9 2 n(A) = 3 and n(B) = 2 B 1 3 n(A) = 0 and n(B) = 1 C 8

જવાબ :

1-C, 2-A, 3-B

write the number of functions from A to B

 1 n(A) = 2 and n(B) = 3 A 1 2 n(A) = 3 and n(B) = 2 B 9 3 n(A) = 0 and n(B) = 1 C 8

જવાબ :

1-B, 2-C, 3-A

the number of functions that can be defined from A into B

 1 If A = {1, 2, 3, 4 } and B = {x, y} A 3 2 If A = {1, 2, 3} and B = {x, y} B 16 3 If A = {1, 2 } and B = {x, y} C 9 4 If A = {1, 2, 3} and B = { y} D 4

જવાબ :

​​​​​​​1-B, 2-C, 3-D, 4-A

 1 If [x]2−5[x]+4=0, where [.] denotes the greatest integer function, then A x ∈ [-10, -4) 2 If [x]2−8[x]+16=0, where [.] denotes the greatest integer function B x ∈ [-2, -2) 3 If [x]2+5[x]+6=0, where [.] denotes the greatest integer function C x ∈ [4, 5) 4 If [x]2+15[x]+50=0, where [.] denotes the greatest integer function D x ∈ [1, 5)

જવાબ :

1-D, 2-C, 3-B, 4-A

 1 Real No. A {0,1,2,3…..} 2 Natural No. B {…..-2, -1, 0, 1, 2…..} 3 Whole No. C {1,2,3,4……} 4 Integers D [-∞,∞]

જવાબ :

1-D, 2-C, 3-A, 4-B

 1 The domain of definition of the function f(x) = 1/ |x-1| A R − {2} 2 The domain of definition of the function f(x) = x - 21 B R − {1} 3 The domain of definition of the function f(x) = 1/ |x-2| C ∞

જવાબ :

1-B, 2-C, 3-A

 1 The domain of definition of the function f(x) = log |x| A R-{0} 2 The range of the function f(y) = |y − 1| B (−∞, 0) 3 The range of the function f(y) = - |y| C [0, ∞)

જવાબ :

1-A, 2-C, 3-B

 1 N A Real No. 2 W B Natural No. 3 Z C Whole No. 4 R D Integers

જવાબ :

1-B, 2-C, 3-D, 4-A

If X = {1, 2, 4}, Y= {2, 4, 5}, Z = {2, 5}

 1 Y+Z A X-Y 2 X+Z B Y 3 X-(Y+Z) C X 4 X+Y-Z D X+Y

જવાબ :

1-B, 2-D, 3-A, 4-C

If X = {1, 2, 4}, Y= {2, 4, 5}, Z = {2, 5}

 1 X-Y A {4} 2 Y-Z B {1,4} 3 X-Z C {1} 4 X+Y D {1,2,4,5}

જવાબ :

1-C, 2-A, 3-B, 4-D