CBSE Solutions for Class 11 English

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

Sets Relation and Function Trignometric Functions Mathematical Induction Complex Numbers and Quardratic Equations Linear Equations Permutation and Combination Binomial Theorem Sequence and Series Straight Lines Conic Section Introduction to 3D Geometry Limits and Derivatives Mathematical Reasoning Statistics Probability
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

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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

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જવાબ : False


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

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જવાબ : False


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

 

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જવાબ : False


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

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જવાબ : False


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

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જવાબ : False


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

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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

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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

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જવાબ : False


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

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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
 

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જવાબ : False


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

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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

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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

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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

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જવાબ : True


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

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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|

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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

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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

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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

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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

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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)

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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

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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

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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.

 

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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.

 

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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.

 

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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.

 

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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.

 

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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.

 

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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.

 

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જવાબ : R = {(2, 4)}


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

 

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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.

 

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જવાબ : R = {(2, 8)}


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

 

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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|.

 

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


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

 

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


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

 

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


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

 

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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|.

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


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

 

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


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

 

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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.

 

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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.

 

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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.

 

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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.

 

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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.

 

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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.

 

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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.

 

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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.

 

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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જવાબ : Given:
f (a) = a2 – 3a + 4
Therefore,
f (2a + 1) = (2a + 1)2 – 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).

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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)

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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)

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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)

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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

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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)

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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.

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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.

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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.

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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.

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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.

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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).

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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).

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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
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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
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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
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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)
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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

[-∞,∞]
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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

∞
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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, ∞)
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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
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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
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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}
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જવાબ :

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

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