As per the given question,
How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
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Class 11
Let R = {(x, y): x, y ϵ Z and x2 + y2 ≤ 4}.
(i)Write R in roster form.
(ii)Find dom (R) and range (R).
Answer : Given: R = {(x, y): x, y ϵ Z and x2 + y2 ≤ 4} (i) R is Foster Form is, R = {(-2, 0), (-1, -1), (-1, 0), (-1, 1), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (1, -1), (1, 0), (1, 1), (2, 0)}...
Evaluate: When (i) n = 6, r = 2 (ii) n = 9, r = 5
, Solution: As per the given question,
Define a relation R from Z to Z, given by R = {(a, b): a, b ϵ Z and (a – b) is an integer. Find dom (R) and range (R).
Answer : Given: R = {(a, b): a, b ϵ Z and (a – b) is an integer The condition satisfies for all the values of a and b to be any integer. So, R = {(a, b): for all a, b ϵ (-∞, ∞)} Dom(R) = {-∞, ∞}...
Find x. If
Solution: As per the given question,
Let A = {1, 2, 3, 4, 6} and R = {(a, b) : a, b ϵ A, and a divides b}.
(i)Write R in roster form.
(ii)Find dom (R) and range (R).
Answer : Given: A = {1, 2, 3, 4, 6} (i) R = {(a, b) : a, b ϵ A, and a divides b} R is Foster Form is, R = {(1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}...
Compute
Solution: As per the given question,
Let R = {(x, x + 5): x ϵ {9, 1, 2, 3, 4, 5}}.
(i)Write R in roster form.
(ii)Find dom (R) and range (R).
Answer : Given: R = {(x, x + 5): x ϵ {9, 1, 2, 3, 4, 5}} (i) R is Foster Form is, R = {(9, 14), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} (ii) Dom(R) = {1, 2, 3, 4, 5, 9} Range(R) = {6, 7, 8, 9, 10,...
Is 3! + 4! = 7!?
Consider LHS 3! +4! Computing left hand side, we get $ \begin{array}{l} 3 !+4 !=(3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1) \\ =6+24 \\ =30 \end{array} $ Again consider RHS and computing we...
Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y): y = x + 1}.
(i)Write R in roster form.
(ii)Find dom (R) and range (R).
(iii) What is its co-domain?
Answer : Given: A = {1, 2, 3, 4, 5, 6} (i) R = {(x, y): y = x + 1} So, R is Roster Form is, R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} (ii) Dom(R) = {1, 2, 3, 4, 5} Range(R) = {2, 3, 4, 5, 6}...
Evaluate (i) 8! (ii) 4! – 3!
(i) Consider $8 !$ We know that $8 !=8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $=40320$ (ii) Consider 4!-3! $ 4 !-3 !=(4 \times 3 !)-3 ! $ Above equation can be written as $...
Let A = {(x, y): x + 3y = 12, x ϵ N and y ϵ N}.
(i)Write R in roster form.
(ii)Find dom (R) and range (R).
Answer : Given: A = {(x, y): x + 3y = 12, x ϵ N and y ϵ N} (i) So, R in Roster Form is, R = {(3, 3), (6, 2), (9, 1)} (ii) Dom(R) = {3, 6, 9} Range(R) = {1, 2, 3}
Let A = {1, 2, 3, 5} AND B = {4, 6, 9}. Let R = {(x, y): x ϵ A, y ϵ B and (x – y) is odd}. Write R in roster form.
Answer : Given: A = {1, 2, 3, 5} AND B = {4, 6, 9} R = {(x, y): x ϵ A, y ϵ B and (x – y) is odd} Therefore, R in Roster Form is, R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}
Let A = {2, 3, 4, 5} and B = {3, 6, 7, 10}. Let R = {(x, y): x ϵ A, y ϵ B and x is relatively prime to y}.
(i)Write R in roster form.
(ii)Find dom (R) and range (R).
Answer : Given: A = {2, 3, 4, 5} and B = {3, 6, 7, 10} (i) R = {(x, y), : x ϵ A, y ϵ B and x is relatively prime to y} So, R in Roster Form, R = {(2, 3), (2, 7), (3, 7), (3, 10), (4, 3), (4, 7), (5,...
Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
Given $5$ flags of different colours We know the signal requires $2$ flags. The number of flags possible for upper flag is $5.$ Now as one of the flag is taken, the number of flags remaining for...
Let A = {2, 4, 5, 7} and b = {1, 2, 3, 4, 5, 6, 7, 8}. Let R = {(x, y) x ϵ A, y ϵ B and x divides y}.(i) Write R in roster form. (ii) Find dom (R) and range (R).
Answer : Given: A = {2, 4, 5, 7} and b = {1, 2, 3, 4, 5, 6, 7, 8} (i) R = {(x, y) x ϵ A, y ϵ B and x divides y} So, R in Roster Form, R = {(2, 2), (2, 4), (2, 6), (2, 8), (4, 4), (4, 8), (5, 5), (7,...
A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
Given $A$ coin is tossed $3$ times and the outcomes are recorded The possible outcomes after a coin toss are head and tail. The number of possible outcomes at each coin toss is $2.$ ∴The total...
Let A = {1, 3, 5, 7} and B = {2, 4, 6, 8}. Let R = {(x, y), : x ϵ A, y ϵ B and x > y}.
(i) Write R in roster form.
(ii)Find dom (R) and range (R).
Answer : Given: A = {1, 3, 5, 7} and B = {2, 4, 6, 8} (i) R = {(x, y), : x ϵ A, y ϵ B and x > y} So, R in Roster Form, R = {(3, 2), (5, 2), (5, ), (7, 2), (7, 4), (7, 6)} (ii) Dom(R) = {3, 5, 7}...
How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Let the five-digit number be $ABCDE.$ Given that first $2$ digits of each number is $ 67.$ Therefore, the number is $67CDE.$ As the repetition is not allowed and $6$ and $7$ are already taken, the...
Find the domain and range of each of the relations given below: (i) R = {(–1, 1), (1, 1), (–2, 4), (2, 4), (2, 4), (3, 9)}
(ii)R ={(x, y) : x + 2y = 8 and x, y ϵ N}
(iii) R = {(x, y), : y = |x – 1|, x ϵ Z and |x| ≤ 3}
Answer : (i) Given: R = {(–1, 1), (1, 1), (–2, 4), (2, 4), (2, 4), (3, 9)} Dom(R) = {x: (x, y) R} = {-2, -1, 1, 2, 3} Range(R) = {y: (x, y) R} = {1, 4, 9} (ii) Given: R = {(x, y): x +...
How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Suppose the $4$ digit code be $1234.$ Hence, the number of letters possible is $10.$ Let’s suppose any $1$ of the ten occupies place $1.$ So, as the repetition is not allowed, the number of letters...
How many 3-digits even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
Let the $3-digit$ number be $ABC,$ where $C$ is at the unit’s place, $B$ at the tens place and $A$ at the hundreds place. As the number has to even, the digits possible at $C$ are $2$ or $4$ or $6.$...
Let A and B be two nonempty sets.
(i) What do you mean by a relation from A to B?
(ii) What do you mean by the domain and range of a relation?
Answer : (i) If A and B are two nonempty sets, then any subset of the set (A × B) is said to a relation R from set A to set B. That means, if R be a relation from A to B then R ⊆ (A × B). Therefore,...
Let A = {a, b, c, d}, B = {c, d, e} and C = {d, e, f, g}. Then verify each of the following identities: (i) A × (B ∩ C) = (A × B) ∩ (A × C) (ii) A × (B – C) = (A × B) – (A × C) (iii) (A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B)
Answer : Given: A = {a, b, c, d,}, B = {c, d, e} and C = {d, e, f, g} Need to prove: A × (B ∩ C) = (A × B) ∩ (A × C) Left hand side, (B ∩ C) = {d, e} ⇒ A × (B ∩ C) = {(a, d), (a, e), (b, d), (b, e),...
How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) Repetition of the digits is allowed? (ii) Repetition of the digits is not allowed?
(i) Let the $3-digit$ number be $ABC,$ where $C$ is at the units place, $B$ at the tens place and $A$ at the hundreds place. Now when repetition is allowed, The number of digits possible at $C$ is...
Let A = {1, 2} and B = {2, 3}. Then, write down all possible subsets of A × B.
Answer : Given: A = {1, 2} and B = {2, 3} Need to write: All possible subsets of A × B A = {1, 2} and B = {2, 3} So, all the possible subsets of A × B are: (A × B) = {(x, y): x A and y B} =...
For any two sets A and B, show that A × B and B × A have an element in common if and only if A and B have an element in common.
Answer : We know, (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A) Here A and B have an element in common i.e., n(A ∩ B) = 1 = (B ∩ A) So, n((A × B) ∩ (B × A)) = n((A ∩ B) × (B ∩ A)) = n(A ∩ B) × n(B ∩ A) = 1...
If A and B be two sets such that n(A) = 3, n(B) = 4 and n(A ∩ B) = 2 then find.
(i)n(A × B)
(ii)n(B × A)
(iii) n(A × B) ∩ (B × A)
Answer : Given: n(A) = 3, n(B) = 4 and n(A ∩ B) = 2 n(A × B) = n(A) × n(B) ⇒ n(A × B) = 3 × 4 ⇒ n(A × B) = 12 n(B × A) = n(B) × n(A) ⇒ n(B × A) = 4 × 3 ⇒ n(B × A) = 12 (iii) n((A × B) ∩ (B × A)) =...
If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.
Answer : Given: A × B ⊆ C × D and A × B ≠ ϕ Need to prove: A ⊆ C and B ⊆ D Let us consider, (x, y) (A × B)---- (1) ⇒ (x, y) (C × D) [as A × B ⊆ C × D]---- (2) From (1) we can say that, x A...
(i) If A ⊆ B, prove that A × C ⊆ B × C for any set C.
(ii) If A ⊆ B and C ⊆ D then prove that A × C ⊆ B × D.
Answer : (i) Given: A ⊆ B Need to prove: A × C ⊆ B × C Let us consider, (x, y) (A × C) That means, x A and y C Here given, A ⊆ B That means, x will surely be in the set B as A is the subset of...
If A and B are nonempty sets, prove that A × B = B × A ⇔ A = B
Answer : Given: A = B, where A and B are nonempty sets. Need to prove: A × B = B × A Let us consider, (x, y) (A × B) That means, x A and y B As given in the problem A = B, we can write, ⇒...
For any sets A and B, prove that (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A).
Answer : Given: A and B two sets are given. Need to prove: (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A) Let us consider, (x, y) (A × B) ∩ (B × A) ⇒ (x, y) (A × B) and (x, y) (B × A) ⇒ (x A...
C. For any sets A, B and C prove that: A × (B – C) = (A × B) – (A × C)
Answer : Given: A, B and C three sets are given. Need to prove: A × (B – C) = (A × B) – (A × C) Let us consider, (x, y) A × (B – C) ⇒ x A and y (B – C ) ⇒ x A and (y B and y ∉ C) ⇒ (x A and y B)...
B. For any sets A, B and C prove that: A × (B ∩ C) = (A × B) ∩ (A × C)
Answer : Given: A, B and C three sets are given. Need to prove: A × (B ∩ C) = (A × B) ∩ (A × C) Let us consider, (x, y) A × (B ∩ C) ⇒ x A and y (B ∩ C) ⇒ x ⇒ (x A and (y...
A. For any sets A, B and C prove that: A × (B ???? C) = (A × B) ???? (A × C)
Answer : Given: A, B and C three sets are given. Need to prove: A × (B ???? C) = (A × B) ???? (A × C) Let us consider, (x, y) A × (B ???? C) ⇒ x A and y (B ???? C) ⇒ x A and (y B or...
Let A = {–3, –1}, B = {1, 3) and C = {3, 5). Find:
(iii)B × C
(iv)A × (B × C)
(iii) Given: B = {1, 3} and C = {3, 5} To find: B × C By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of...
Let A = {–3, –1}, B = {1, 3) and C = {3, 5). Find:
(i) A × B
(ii) (A × B) × C
Answer : (i) Given: A = {-3, -1} and B = {1, 3} To find: A × B By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered...
If A = {5, 7), find (i) A × A × A.
Answer : We have, A = {5, 7} So, By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q,...
Let A = {–2, 2} and B = (0, 3, 5). Find:
(iii)A × A
(iv) B × B
(iii) Given: A = {-2, 2} To find: A × A By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P...
Let A = {–2, 2} and B = (0, 3, 5). Find:
(i) A × B
(ii) B × A
Answer : (i) Given: A = {-2, 2} and B = {0, 3, 5} To find: A × B By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered...
Let A and B be two sets such that n(A) = 3 and n(B) = 2. If a ≠ b ≠ c and (a, 0), (b, 1), (c, 0) is in A × B, find A and B.
Answer : Since, (a, 0), (b, 1), (c, 0) are the elements of A × B. ∴ a, b, c Є A and 0, 1 Є B It is given that n(A) = 3 and n(B) = 2 ∴ a, b, c Є A and n(A) = 3 ⇒ A = {a, b, c} and 0, 1 Є B and n(B) =...
Let A × B = {(a, b): b = 3a – 2}. if (x, –5) and (2, y) belong to A × B, find the values of x and y.
Answer : Given: A × B = {(a, b): b = 3a – 2} and {(x, -5), (2, y)} Є A × B For (x, -5) Є A × B b = 3a – 2 ⇒ -5 = 3(x) – 2 ⇒ -5 + 2 = 3x ⇒ -3 = 3x ⇒ x = -1 For (2, y) Є A × B b = 3a – 2 ⇒ y = 3(2) –...
Let A = {2, 3} and B = {4, 5}. Find (A × B). How many subsets will (A × B) have?
Answer : Given: A = {2, 3} and B = {4, 5} To find: A × B By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs...
If A × B = {(–2, 3), (–2, 4), (0, 4), (3, 3), (3, 4), find A and B.
Answer : Here, A × B = {(–2, 3), (–2, 4), (0, 4), (3, 3), (3, 4)} To find: A and B Clearly, A is the set of all first entries in ordered pairs in A × B ∴ A = {-2, 0, 3} and B is the set of all...
Let A = {x ϵ W : x < 2}, B = {x ϵ N : 1 < x ≤ 4} and C = {3, 5}. Verify that: (i) A × (B ???? C) = (A × B) ???? (A × C) (ii) A × (B ∩ C) = (A × B) ∩ (A × C)
Answer : (i) Given: A = {x ϵ W : x < 2} Here, W denotes the set of whole numbers (non – negative integers). ∴ A = {0, 1} [∵ It is given that x < 2 and the whole numbers which are less than 2...
If A = {1, 3, 5) B = {3, 4} and C = {2, 3}, verify that:
(i) A × (B ???? C) = (A × B) ???? (A × C)
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
Answer : (i) Given: A = {1, 3, 5}, B = {3, 4} and C = {2, 3} H. S = A × (B ⋃ C) By the definition of the union of two sets, (B ⋃ C) = {2, 3, 4} = {1, 3, 5} × {2, 3, 4} Now, by the definition of the...
If A = {x ϵ N : x ≤ 3} and {x ϵ W : x < 2}, find (A × B) and (B × A). Is (A × B) = (B × A)?
Answer : Given: A = {x ϵ N: x ≤ 3} Here, N denotes the set of natural numbers. ∴ A = {1, 2, 3} [∵ It is given that the value of x is less than 3 and natural numbers which are less than 3 are 1 and...
If A = {2, 3, 5} and B = {5, 7}, find:
(iii)A × A
(iv)B × B
(iii) Given: A = {2, 3, 5} and B = {2, 3, 5} To find: A × A By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered...
If A = {2, 3, 5} and B = {5, 7}, find:
(i)A × B
(ii)B × A
Answer : (i) Given: A = {2, 3, 5} and B = {5, 7} To find: A × B By the definition of the Cartesian product, Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered...
If P = {a, b} and Q = {x, y, z}, show that P × Q ≠ Q × P.
Answer : Given: P = {a, b} and Q = {x, y, z} To show: P × Q ≠ Q × P Now, firstly we find the P × Q and Q × P By the definition of the Cartesian product, Given two non – empty sets P and Q. The...
If A = {9, 1} and B = {1, 2, 3}, show that A × B ≠ B × A.
Answer : Given: A = {9, 1} and B = {1, 2, 3} To show: A × B ≠ B × A Now, firstly we find the A × B and B × A By the definition of the Cartesian product, Given two non – empty sets P and Q. The...
Find the values of a and b, when:(a – 2, 2b + 1 = (b – 1, a + 2)
Since, the ordered pairs are equal, the corresponding elements are ∴, a – 2 = b – 1 …(i) & 2b + 1 = a + 2 …(ii) Solving eq. (i), we get a – 2 = b – 1 ⇒ a – b = -1 + 2 ⇒ a – b = 1 … (iii) Solving...
Find the values of a and b, when:
(i) (a + 3, b –2) = (5, 1)
(ii) (a + b, 2b – 3) = (4, –5)
Answer : Since, the ordered pairs are equal, the corresponding elements are equal. ∴, a + 3 = 5 …(i) and b – 2 = 1 …(ii) Solving eq. (i), we get a + 3 = 5 ⇒ a = 5 – 3 ⇒ a = 2 Solving eq. (ii), we...
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x: Sin (log sin x)
Differentiate the following functions with respect to x: . Sin (3x + 5)
Given Sin (3x + 5)
Differentiate the following functions from the first principles: 
In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (vii) 
(viii) 
(vii) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...
In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (v) 
(vi) 
(v) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...
Find the points of discontinuity, if any, of the following functions: (v) 
(vi) 
(v) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...
Find the points of discontinuity, if any, of the following functions: (iii) 
(i v) 
(iii) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...
Differentiate the following functions with respect to x: 
Answer: Consider y = $\begin{array}{l} \frac{{a{x^2} + bx + c}}{{p{x^2} + qx + r}} \end{array}$ y - fraction of two functions say u and v where, u = ax2 + bx + c v = px2 + qx + r ∴ y = u/v Using...
Differentiate the following functions with respect to x: 
Answer: Consider, y = $\begin{array}{l} \frac{{{e^x} - \tan x}}{{\cot x - {x^n}}}\\ \end{array}$ y - fraction of two functions say u and v where, u = ex – tan x v = cot x – xn ∴ y = u/v Using...
Differentiate the following functions with respect to x: 
Answer: Consider, y = $\begin{array}{l} \frac{{x + {e^x}}}{{1 + \log x}}\\ \end{array}$ y - fraction of two functions say u and v where, u = x + ex v = 1 + log x ∴ y = u/v Using quotient rule of...
Differentiate the following functions with respect to x: 
Answer: Consider, y = $\begin{array}{l} \frac{{2x - 1}}{{{x^2} + 1}}\\ \end{array}$ y - fraction of two functions say u and v where, u = 2x – 1 v = x2 + 1 ∴ y = u/v Using quotient rule of...
Differentiate the following functions with respect to x: 
Answer: Consider, y = $\begin{array}{l} \frac{{{x^2} + 1}}{{x + 1}}\\ \end{array}$ y - is a fraction of two functions say u and v where, u = x2 + 1 v = x + 1 ∴ y = u/v Using quotient rule of...
Differentiate the following functions with respect to x: 
Answer: Consider, y = xn loga x y - product of two functions say u and v where, u = xn v = loga x ∴ y = uv Using product rule of differentiation,
Differentiate the following functions with respect to x: 
Answer: Consider, y = xn tan x y - product of two functions say u and v where, u = xn v = tan x ∴ y = uv Using product rule of differentiation,
Differentiate the following functions with respect to x: 
Answer: Consider, y = x2 ex log x y - product of two functions say u and v where, u = x2 v = ex ∴ y = uv Using product rule of differentiation.
Differentiate the following functions with respect to x: 
Answer: Consider, y = x3 ex y - product of two functions say u and v where, u = x3 v = ex ∴ y = uv Using product rule of differentiation.
Differentiate the following functions with respect to x: 
Answer: Consider, y = x3 sin x y - product of two functions say u and v where, u = x3 v = sin x ∴ y = uv Using product rule of differentiation.
Differentiate the following with respect to x: 
Answer: f (x) = (2x2 + 1) (3x + 2) f (x) = 6x3 + 4x2 + 3x + 2 Differentiate on both the sides with respect to x,
Differentiate the following with respect to x: 
Answer: f (x) = ex log a + ea log x + ea log a elog f(x) = f(x) f(x) = ax + xa + aa Differentiate on both the sides with respect to x,
Differentiate the following with respect to x: 
Answer: Differentiate on both the sides with respect to x,
Differentiate the following with respect to x: 
Answer: f (x) = 3x + x3 + 33 Differentiate on both the sides with respect to x,
Differentiate the following with respect to x: 
Answer: f (x) = x4 – 2sin x + 3 cos x Differentiate on both the sides with respect to x,
Differentiate the following from first principles: (i) sin √2x (ii) cos √x
Answers: (i) f (x) = sin √2x f (x + h) = sin √2(x+h) (ii) f (x) = cos √x f (x + h) = cos √(x+h)
Differentiate the following from first principles: 
Answers: (i) f (x) = tan2 x (ii) f (x) = tan (2x + 1)
Differentiate each of the following from first principles: (i) √(sin 2x) (ii) sin x/x
Answers: (i) f (x) = √(sin 2x) (ii) f (x) = sin x/x
Differentiate each of the following from first principles: 
Answer: f (x) = eax+b
Differentiate each of the following from first principles: 
Answers: (i) f (x) = e-x (ii) f (x) = e3x
Differentiate each of the following from first principles: ![\begin{array}{l} [{x^2} - 1]/x \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cbad99f6b77a0956f6abbc11e4384a23_l3.png "Rendered by QuickLaTeX.com")
Answer: f (x) = [x2 – 1]/ x
Differentiate each of the following from first principles: ![\begin{array}{l} (i)1/{x^3}\\ (ii)[{x^2} + 1]/x\\ \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fc3e197c0c7f12ff44e3389048b6803c_l3.png "Rendered by QuickLaTeX.com")
Answers: (i) f (x) = 1/x3 ∴ Derivative of f(x) = 1/x3 is -3x-4 (ii) f (x) = [x2 + 1]/ x = 1 – 1/x2 ∴ Derivative of f(x) = 1 – 1/x2
Differentiate each of the following from first principles: (i) 2/x (ii) 1/√x
Answers: (i) f (x) = 2/x ∴ Derivative of f(x) = 2/x is -2x-2 (ii) f (x) = 1/√x ∴ Derivative of f(x) = 1/√x is -1/2 x-3/2
Find the derivatives of the following functions at the indicated points: (i) 2 cos x at x = π/2 (ii) sin 2xat x = π/2
Answers: (i) Using derivative formula, (ii) Using derivative formula, Multiplying numerator and denominator by 2,
Find the derivatives of the following functions at the indicated points: (i) sin x at x = π/2 (ii) x at x = 1
Answers: (i) Using derivative formula, 1 – cos x = 2 sin2(x/2) (ii) Using derivative formula,
Find the derivative of f(x) = tan x at x = 0
Answer: Using derivative formula,
Find the derivative of f(x) = cos x at x = 0
Answer: Using derivative formula,
Find the derivative of f(x) = x at x = 1
Answer: Using derivative formula,
Find the derivative of f(x) = 99x at x = 100.
Answer: Using derivative formula,
Find the derivative of f(x) = x2 – 2 at x = 10
Answer: Using derivative formula, = 0 + 20 = 20 Derivative of f(x) = x2 – 2 at x = 10 is 20
Find the derivative of f(x) = 3x at x = 2
Answer: Using derivative formula,
Evaluate the following limit: 
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x + a\sin bx} \right)^{1/x}} \end{array}$ ...
Evaluate the following limit: 
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x + \sin x} \right)^{1/x}}\\ \end{array}$ f (x) = cos x + sin x – 1 g (x) =...
Evaluate the following limit: 
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x} \right)^{1/\sin x}}\\ \end{array}$
Evaluate the following limit: 
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to {0^ + }} {\left\{ {1 + {{\tan }^{\sqrt x }}} \right\}^{1/2x}}\\ \end{array}$
Evaluate the following limit: 
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to \pi } {\left( {1 - \frac{x}{\pi }} \right)^2}\\ \end{array}$
Prove that: sin2 π/18 + sin2 π/9 + sin2 7π/18 + sin2 4π/9 = 2
Let us consider LHS: \[si{{n}^{2}}~\pi /18\text{ }+\text{ }si{{n}^{2}}~\pi /9\] \[+\text{ }si{{n}^{2}}~7\pi /18\text{ }+\text{ }si{{n}^{2}}~4\pi /9\] Or, \[si{{n}^{2}}~\pi /18\text{ }+\text{...
Prove that:
Solution: As per the given question, \[1\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved.
Prove that:
(i) (ii) Solution: (i) \[1\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved. (ii) \[\left\{ 1+cotx\left-( -cosecx \right) \right\}\left\{ 1+cotx+\left( -cosecx \right)...
Prove that:
(i) (ii) Solution: (i) \[1\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved. (ii) \[1\text{ }+\text{ }1\] \[2\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{...
Prove that: 3 sin π/6 sec π/3 – 4 sin 5π/6 cot π/4 = 1
(i) \[3\text{ }sin\text{ }\pi /6\text{ }sec\text{ }\pi /3\text{ }-\text{ }4\text{ }sin\text{ }5\pi /6\text{ }cot\text{ }\pi /4\text{ }=\text{ }1\] Let us consider LHS: \[3\text{ }sin\text{ }\pi...
Prove that: (i) cos 570o sin 510o + sin (-330o) cos (-390o) = 0 (ii) tan 11π/3 – 2 sin 4π/6 – 3/4 cosec2 π/4 + 4 cos2 17π/6 = (3 – 4√3)/2
(i) \[cos\text{ }{{570}^{o}}~sin\text{ }{{510}^{o}}~+\text{ }sin\text{ }(-{{330}^{o}})\text{ }cos\text{ }(-{{390}^{o}})\] \[=\text{ }0\] Let us consider LHS: \[cos\text{ }{{570}^{o}}~sin\text{...
Prove that: (i) cos 24o + cos 55o + cos 125o + cos 204o + cos 300o = 1/2 (ii) tan (-125o) cot (-405o) – tan (-765o) cot (675o) = 0
(i) \[cos\text{ }{{24}^{o}}~+\text{ }cos\text{ }{{55}^{o}}~+\text{ }cos\text{ }{{125}^{o}}~+\text{ }cos\text{ }{{204}^{o}}~\] \[~+\text{ }cos\text{ }{{300}^{o}}~=\text{ }1/2\] Let us consider LHS:...
Prove that: (i) tan 225o cot 405o + tan 765o cot 675o = 0 (ii) sin 8π/3 cos 23π/6 + cos 13π/3 sin 35π/6 = 1/2
(i) \[tan\text{ }{{225}^{o}}~cot\text{ }{{405}^{o}}~+\text{ }tan\text{ }{{765}^{o}}~cot\text{ }{{675}^{o}}~=\text{ }0\] Let us consider LHS: \[tan\text{ }225{}^\circ ~\text{ }cot\text{ }405{}^\circ...
Find the values of the following trigonometric ratios: (i) cos 39π/4 (ii) sin 151π/6
(i) \[cos\text{ }39\pi /4\] \[cos\text{ }39\pi /4\text{ }=\text{ }cos\text{ }{{1755}^{o}}\] \[=\text{ }cos\text{ }{{\left( 90\times 19\text{ }+\text{ }45 \right)}^{o}}\] Since,\[{{1755}^{o}}\] lies...
Find the values of the following trigonometric ratios: (i) cos 19π/4 (ii) sin 41π/4
(i) \[cos\text{ }19\pi /4\] \[cos\text{ }19\pi /4\text{ }=\text{ }cos\text{ }{{855}^{o}}\] \[=\text{ }cos\text{ }{{\left( 90\times 9\text{ }+\text{ }45 \right)}^{o}}\] Since,\[{{855}^{o}}\] lies in...
Find the values of the following trigonometric ratios: (i) cosec (-20π/3) (ii) tan (-13π/4)
(i) \[cosec\text{ }\left( -20\pi /3 \right)\] \[cosec\text{ }\left( -20\pi /3 \right)\] \[=\text{ }cosec\text{ }{{\left( -1200 \right)}^{o}}\] Or, \[=\text{ }-\text{ }cosec\text{ }{{\left( 1200...
Find the values of the following trigonometric ratios: (i) cos 19π/6 (ii) sin (-11π/6)
(i) \[cos\text{ }19\pi /6\] \[cos\text{ }19\pi /6\text{ }=\text{ }cos\text{ }{{570}^{o}}\] \[=\text{ }cos\text{ }{{\left( 90\times 6\text{ }+\text{ }30 \right)}^{o}}\] Since,\[{{570}^{o}}\] lies in...
Find the values of the following trigonometric ratios: (i) tan 7π/4 (ii) sin 17π/6
(i) \[tan\text{ }7\pi /4\] \[tan\text{ }7\pi /4\text{ }=\text{ }tan\text{ }{{315}^{o}}\] \[=\text{ }tan\text{ }{{\left( 90\times 3\text{ }+\text{ }45 \right)}^{o}}\] Since,\[{{315}^{o}}\] lies in...
Find the values of the following trigonometric ratios: (i) tan 11π/6 (ii) cos (-25π/4)
(i) \[tan\text{ }11\pi /6\] \[tan\text{ }11\pi /6\text{ }=\text{ }{{\left( 11/6\text{ }\times \text{ }180 \right)}^{o}}\] \[=\text{ }{{330}^{o}}\] Since,\[{{330}^{o}}\] lies in the \[IV\text{...
Find the values of the following trigonometric ratios: (i) sin 5π/3 (ii) sin 17π
(i) \[sin\text{ }5\pi /3\] \[5\pi /3\text{ }=\text{ }{{\left( 5\pi /3\text{ }\times \text{ }180 \right)}^{o}}\] \[=\text{ }{{300}^{o}}\] Or, \[=\text{ }{{\left( 90\times 3\text{ }+\text{ }30...
If cos x = -3/5 and π
Solution: According to the given question: \[cos\text{ }x=\text{ }-3/5\text{ }and\text{ }\pi \text{ }<x\text{ }<\text{ }3\pi /2\] We know that in the third quadrant, \[tan\text{ }x\text{...
If sin x + cos x = 0 and x lies in the fourth quadrant, find sin x and cos x.
According to the given question: \[Sin\text{ }x\text{ }+\text{ }cos\text{ }x\text{ }=\text{ }0\text{ }and\text{ }x\]lies in fourth quadrant. \[Sin\text{ }x~=\text{ }-cos\text{ }x\] \[Sin\text{...
If sin x = 3/5, tan y = 1/2 and π/2 < x< π< y< 3π/2 find the value of 8 tan x -√5 sec y.
According to the given question: \[sin\text{ }x\text{ }=\text{ }3/5,\text{ }tan\text{ }y\text{ }=\text{ }1/2\] And \[\pi /2\text{ }<\text{ }x<\text{ }\pi <\text{ }y<\text{ }3\pi /2~\] We...
If sin x = 12/13 and lies in the second quadrant, find the value of sec x + tan x.
According to the given question: \[Sin\text{ }x\text{ }=\text{ }12/13\text{ }and\text{ }x\]lies in the second quadrant. We know, in second quadrant, \[sin\text{ }x\text{ }and\text{ }cosec\text{...
Find the values of the other five trigonometric functions in each of the following: (i) tan x = 3/4, x in quadrant III (ii) sin x = 3/5, x in quadrant I
(i) \[tan\text{ }x\text{ }=\text{ }3/4,\text{ }x\text{ }in\text{ }quadrant\text{ }III\] In third quadrant, \[tan\text{ }x\text{ }and\text{ }cot\text{ }x\] are positive. \[sin\text{ }x,\text{...
Find the values of the other five trigonometric functions in each of the following: (i) cot x = 12/5, x in quadrant III (ii) cos x = -1/2, x in quadrant II
(i) \[cot\text{ }x\text{ }=\text{ }12/5,\text{ }x\text{ }in\text{ }quadrant\text{ }III\] In third quadrant, \[tan\text{ }x\text{ }and\text{ }cot\text{ }x\]are positive. \[sin\text{ }x,\text{...
A(1, 2, 3), B(0, 4, 1), C(-1, -1, -3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle ∠BAC meets BC.
Given: The vertices of a triangle are \[A\text{ }\left( 1,\text{ }2,\text{ }3 \right),\text{ }B\text{ }\left( 0,\text{ }4,\text{ }1 \right),\text{ }C\text{ }\left( -1,\text{ }-1,\text{ }-3 \right)\]...
The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C.
Given: The mid-points of the sides of a triangle \[ABC\] is given as \[\left( -2,\text{ }3,\text{ }5 \right),\text{ }\left( 4,\text{ }-1,\text{ }7 \right)\text{ }and\text{ }\left( 6,\text{ }5,\text{...
If the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divided AB.
Given: The points \[A\text{ }\left( 3,\text{ }2,\text{ }-4 \right),\text{ }B\text{ }\left( 9,\text{ }8,\text{ }-10 \right)\]and \[C\text{ }\left( 5,\text{ }4,\text{ }-6 \right)\] By using the...
Find the ratio in which the line segment joining the points (2, -1, 3) and (-1, 2, 1) is divided by the plane x + y + z = 5.
Given: The points \[\left( 2,\text{ }-1,\text{ }3 \right)\text{ }and\text{ }\left( -1,\text{ }2,\text{ }1 \right)\] By using the section formula, Let \[C\left( x,\text{ }y,\text{ }z \right)\]be any...
Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane.
Given: The points \[\left( 2,\text{ }4,\text{ }5 \right)\text{ }and\text{ }\left( 3,\text{ }5,\text{ }4 \right)\] By using the section formula, We know X coordinate is always 0 on yz-plane So, let...
Show that the three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB.
Given: The points \[A\text{ }\left( 2,\text{ }3,\text{ }4 \right),\text{ }B\text{ }\left( -1,\text{ }2,\text{ }-3 \right)\text{ }and\text{ }C\text{ }\left( -4,\text{ }1,\text{ }-10 \right)\] By...
A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find the coordinates.
Given: The points \[A\text{ }\left( 2,\text{ }-3,\text{ }4 \right)\text{ }and\text{ }B\text{ }\left( 8,\text{ }0,\text{ }10 \right)\] By using the section formula, Let Point \[C\left( x,\text{...
The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the length AD.
Given: The vertices of the triangle are \[A\text{ }\left( 5,\text{ }4,\text{ }6 \right),\text{ }B\text{ }\left( 1,\text{ }-1,\text{ }3 \right)\text{ }and\text{ }C\text{ }\left( 4,\text{ }3,\text{ }2...
Show that the points A(1, 3, 4), B(-1, 6, 10), C(-7, 4, 7) and D(-5, 1, 1) are the vertices of a rhombus.
Given: The points \[A\text{ }\left( 1,\text{ }3,\text{ }4 \right),\text{ }B\text{ }\left( -1,\text{ }6,\text{ }10 \right),\text{ }C\text{ }\left( -7,\text{ }4,\text{ }7 \right)\] and \[D\text{...
Prove that the point A(1, 3, 0), B(-5, 5, 2), C(-9, -1, 2) and D(-3, -3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
Given: The points \[A\text{ }\left( 1,\text{ }3,\text{ }0 \right),\text{ }B\text{ }\left( -5,\text{ }5,\text{ }2 \right),\text{ }C\text{ }\left( -9,\text{ }-1,\text{ }2 \right)\] and \[D\text{...
Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of squares.
Given: The points \[A\left( 3,3,3 \right),B\left( 0,6,3 \right),C\left( 1,7,7 \right)\text{ }and\text{ }D\left( 4,4,7 \right)\] We know that all sides of a square are equal. By using the formula,...
Show that the points (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of an isosceles right-angled triangle.
Given: The points \[\left( 0,\text{ }7,\text{ }10 \right),\text{ }\left( -1,\text{ }6,\text{ }6 \right)\text{ }and\text{ }\left( -4,\text{ }9,\text{ }6 \right)\] Isosceles right-angled triangle is a...
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
Given: The points \[\left( 1,\text{ }2,\text{ }3 \right),\text{ }\left( 2,\text{ }3,\text{ }1 \right)\text{ }and\text{ }\left( 3,\text{ }1,\text{ }2 \right)\] An equilateral triangle is a triangle...
Find the points on z-axis which are at a distance√21 from the point (1, 2, 3).
Given: The point \[\left( 1,\text{ }2,\text{ }3 \right)\] Distance \[=\text{ }\surd 21\] We know \[x\text{ }=\text{ }0\text{ }and\text{ }y\text{ }=\text{ }0\] on z-axis Let \[R\left( 0,\text{...
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Given: The points \[\left( 3,\text{ }1,\text{ }2 \right)\text{ }and\text{ }\left( 5,\text{ }5,\text{ }2 \right)\] We know \[x\text{ }=\text{ }0\text{ }and\text{ }z\text{ }=\text{ }0\] on y-axis Let...
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, -4)
Given: The points \[\left( 1,\text{ }5,\text{ }7 \right)\text{ }and\text{ }\left( 5,\text{ }1,\text{ }-4 \right)\] We know \[x\text{ }=\text{ }0\text{ }and\text{ }y\text{ }=\text{ }0\text{ }on\text{...
Determine the points in zx-plane which are equidistant from the points A(1, -1, 0), B(2, 1, 2) and C(3, 2, -1).
\[zx-plane\] We know \[y\text{ }=\text{ }0\text{ }in\text{ }xz-plane\] Let \[R\left( x,\text{ }0,\text{ }z \right)\] any point in \[xz-plane\] According to the question: \[RA\text{ }=\text{...
Determine the points in (i) xy-plane (ii) yz-plane
Given: The points \[A\left( 1,\text{ }-1,\text{ }0 \right),\text{ }B\left( 2,\text{ }1,\text{ }2 \right)\text{ }and\text{ }C\left( 3,\text{ }2,\text{ }-1 \right)\] (i) \[xy-plane\] We know \[z\text{...
Using distance formula prove that the following points are collinear: A(3, -5, 1), B(-1, 0, 8) and C(7, -10, -6)
(i) \[A\left( 3,\text{ }-5,\text{ }1 \right),\text{ }B\left( -1,\text{ }0,\text{ }8 \right)\text{ }and\text{ }C\left( 7,\text{ }-10,\text{ }-6 \right)\] Given: The points \[A\left( 3,\text{...
Using distance formula prove that the following points are collinear: (i) A(4, -3, -1), B(5, -7, 6) and C(3, 1, -8) (ii) P(0, 7, -7), Q(1, 4, -5) and R(-1, 10, -9)
(i) \[A\left( 4,\text{ }-3,\text{ }-1 \right),\text{ }B\left( 5,\text{ }-7,\text{ }6 \right)\text{ }and\text{ }C\left( 3,\text{ }1,\text{ }-8 \right)\] Given: The points \[A\left( 4,\text{...
Find the distance between the points P and Q having coordinates (-2, 3, 1) and (2, 1, 2).
Given: The points \[\left( -2,\text{ }3,\text{ }1 \right)\text{ }and\text{ }\left( 2,\text{ }1,\text{ }2 \right)\] By using the formula, The distance between any two points \[\left( a,\text{...
The difference of the squares of two natural numbers is 45 . The square of the smaller number is four times the larger number. Find the numbers.
Let the greater number be $x$ and the smaller number be $y$. According to the question: $\begin{array}{l} x^{2}-y^{2}=45 \\ y^{2}=4 x \end{array}$ From (i) and (ii), we get: $x^{2}-4 x=45$...
Find the distance between the following pairs of points: (i) P(1, -1, 0) and Q (2, 1, 2) (ii) A(3, 2, -1) and B (-1, -1, -1)
(i) \[P\left( 1,\text{ }-1,\text{ }0 \right)\text{ }and\text{ }Q\text{ }\left( 2,\text{ }1,\text{ }2 \right)\] Given: The points \[P\left( 1,\text{ }-1,\text{ }0 \right)\text{ }and\text{ }Q\text{...
What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?
Solution: It is given that a cube with 2 units edge, one of whose vertices coincides with the origin and the 3 edges passing through the origin, coincides with the positive direction of the axes...
Choose the correct answer from the given four options indicated against each of the Exercises if the distance between the points 
and 
is 
, then the value of 
is
(A) 5
(B) 
(C) 5
(D) none of these
Solution: Option(B) $\pm 5$ Explanation: Suppose $P$ be the point whose coordinate is $(a, 0,1)$ and $Q$ represents the point $(0,$, $(1,2) .$ It is given that, $\mathrm{PQ}=\sqrt{27}$ From the...
Choose the correct answer from the given four options indicated against each of the Exercises distance of the point 
from the origin 
is
(A) 
(B) 3
(C) 4
(D) 5
Solution: Option (A) $\sqrt{50}$ Explanation: Suppose $\mathrm{P}$ be the point whose coordinate is $(3,4,5)$ and $\mathrm{Q}$ represents the origin. From the distance formula it can be written as...
Choose the correct answer from the given four options indicated against each of the Exercises what is the length of foot of perpendicular drawn from the point 
on 
-axis
(A) 
(B) 
(C) 5
(D) none of these
Solution: Option(B) $\sqrt{34}$ Explanation: As it is known that $y$-axis lies on $x$ y plane and $y z$. Therefore, its distance from $x y$ and $y z$ plane is 0 . $\therefore$ By the basic...
Choose the correct answer from the given four options indicated against each of the Exercises the distance of point P (3, 4, 5) from the y z-plane is
(A) 3 units
(B) 4 units
(C) 5 units
(D) 550
Solution: (A) 3 units Explanation: From basic ideas of three-dimensional geometry, it is known that $x$-coordinate of a point is its distance from $y z$ plane. $\therefore$ The distance of Point $P...
Prove that the points (0, – 1, – 7), (2, 1, – 9) and (6, 5, – 13) are collinear. Find the ratio in which the first point divides the join of the other two.
Solution: It is given that the three points $A(0,-1,-7), B(2,1,-9)$ and $C(6,5,-13)$ are collinear So it can be written as $\begin{array}{l} A...
The mid-point of the sides of a triangle are (1, 5, – 1), (0, 4, – 2) and (2, 3, 4). Find its vertices. Also find the centroid of the triangle.
Solution: It is given that the mid-point of the sides of a triangle are $(1,5,-1),(0,4,-2)$ and $(2,3,$, 4). Suppose the vertices be $A\left(x_{1}, y_{1}, z_{1}\right), B\left(x_{2}, y_{2},...
Show that the three points 
and 
are collinear and find the ratio in which 
divides 
.
Solution: It is given that the three points are A $(2,3,4), \mathrm{B}(-1,2,-3)$ and $\mathrm{C}(-4,1,-10)$ We need to find collinear points, $\begin{array}{l}...
Let A 
and C 
be the vertices of a triangle. The internal bisector of the angle 
meets 
at the point 
. Find the coordinates of .
Solution: It is given $A(2,2,-3), B(5,6,9)$ and $C(2,7,9)$ are the vertices of a triangle. And it is also given that the internal bisector of the angle A meets BC at the point D....
If the origin is the centroid of a triangle ABC having vertices A (a, 1, 3), B (– 2, b, – 5) and C (4, 7, c), find the values of a, b, c.
Solution: It is given that the triangle ABC having vertices $A(a, 1,3), B(-2, b,-5)$ and $C(4,7, c)$ and origin is the centroid. The coordinates of the centroid for a triangle is given by the...
Find the coordinate of the points which trisect the line segment joining the points A 
and 
Solution: It is given the line segment joining the points are A $(2,1,-3)$ and $B(5,-8,3)$ Now suppose $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ be the points which...
Three vertices of a Parallelogram ABCD are A (1, 2, 3), B (– 1, – 2, – 1) and C (2, 3, 2). Find the fourth vertex D.
Solution: It is given that the three consecutive vertices of a parallelogram ABCD are A $(1,2,3), B(-1,$, $-2,-1)$ and $C(2,3,2)$ Suppose the fourth vertex be $D(x, y, z)$. By using midpoint...
The mid-points of the sides of a triangle are (5, 7, 11), (0, 8, 5) and (2, 3, – 1). Find its vertices.
Solution: It is given that the mid-points of the sides of a triangle are $(5,7,11),(0,8,5)$ and $(2,3,-$ 1). Suppose the vertices be $A\left(x_{1}, y_{1}, z_{1}\right), B\left(x_{2}, y_{2},...
Find the centroid of a triangle, the mid-point of whose sides are 
and 
4).
Solution: It is given that: Mid-points of sides of triangle $\mathrm{DEF}$ are: $\mathrm{D}(1,2,-3), \mathrm{E}(3,0,1)$ and $\mathrm{F}(-1,1,-4)$ Using the geometry of centroid, It is known that the...
Find the third vertex of triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, –2, –5).
Solution: It is given the centroid is origin and two vertices are $(2,4,6)$ and $(0,-2,-5)$ Suppose the third vertex be $(x, y, z)$ The coordinates of the centroid for a triangle is given by the...
Show that the triangle 
with vertices 
and 
is right angled.
Solution: The given vertices are $A(0,4,1), B(2,3,-1)$ and $C(4,5,0)$ We need to prove right angled triangle, consider $\begin{array}{l}...
Three consecutive vertices of a parallelogram ABCD are A (6, – 2, 4), B (2, 4, – 8), C (–2, 2, 4). Find the coordinates of the fourth vertex.
Solution: The three consecutive vertices of a parallelogram $A B C D$ are as given $A(6,-2,4), B$ (2, $4,-8), C(-2,2,4)$ Suppose the forth vertex be $D(x, y, z)$ Midpoint of diagonal $A...
Show that the point 
and 
are collinear.
Solution: The given points are $A(1,-1,3), B(2,-4,5)$ and $(5,-13,11)$. We need to prove collinear, $\begin{array}{l} \mathrm{AB}=\sqrt{(1-2)^{2}+(-1+4)^{2}+(3-5)^{2}}=\sqrt{1+9+4}=\sqrt{14} \\...
Show that if 
, then the point 
is at a distance 1 unit from the origin.
Solution: It is given that $x^{2}+y^{2}=1 \Rightarrow 1-x^{2}-y^{2}=0$ Distance of the point $\left(x, y \sqrt{1-x^{2}-y^{2}}\right)$ from origin is $=$ $\begin{array}{l}...
Find the distance from the origin to (6, 6, 7).
Solution: The distance from the origin to (6, 6, 7) $=\sqrt{{6^2}+{6^2}+{7^2}}$ $=\sqrt{{36}+{36}+{49}}$ $=\sqrt{121}$ $=11$ units
How far apart are the points (2, 0, 0) and (–3, 0, 0)?
Solution: The points $(2, 0, 0)$ and $(–3, 0, 0)$ are at a distance of $=$ $|2 − (−3)| = 5$ units.
Let A, B, C be the feet of perpendiculars from a point P on the xy, yz and zx planes respectively. Find the coordinates of A, B, C in each of the following where the point P is
(i) (4, – 3, – 5).
Solution: (i) $(4, – 3, – 5):- A (4, −3, 0), B (0, −3, −5), C (4, 0, −5)$
Let A, B, C be the feet of perpendiculars from a point P on the xy, yz and zx planes respectively. Find the coordinates of A, B, C in each of the following where the point P is
(i) (3, 4, 5)
(ii) (–5, 3, 7)
Solution: (i) $(3, 4, 5):- A (3, 4, 0), B (0, 4, 5), C (3, 0, 5)$ (ii) $(–5, 3, 7):- A (−5, 3, 0), B (0, 3, 7), C (−5, 0, 7)$
Let A, B, C be the feet of perpendiculars from a point P on the x, y, z-axis respectively. Find the coordinates of A, B and C in each of the following where the point P is :
(i) (4, – 3, – 5)
Solution: (i) $(4, – 3, – 5):- A (4, 0, 0), B (0, −3, 0), C (0, 0, −5)$
Let A, B, C be the feet of perpendiculars from a point P on the x, y, z-axis respectively. Find the coordinates of A, B and C in each of the following where the point P is :
(i) A = (3, 4, 2)
(ii) (–5, 3, 7)
Solution: (i) $(3, 4, 2):- A (3, 0, 0), B (0, 4, 0), C (0, 0, 2)$ (ii) $(–5, 3, 7):- A (−5, 0, 0), B (0, 3, 0), C (0, 0, 7)$
Name the octant in which each of the following points lies.
(i) (2, – 4, – 7)
(ii) (– 4, 2, – 5).
Solution: (i) $(2, – 4, – 7)$:- 8th Octant, (ii) $(– 4, 2, – 5)$:- 6th Octant.
Name the octant in which each of the following points lies.
(i) (– 4, 2, 5)
(ii) (–3, –1, 6)
Solution: (i) $(– 4, 2, 5)$:- 2nd Octant, (ii) $(–3, –1, 6)$:- 3rd Octant,
Name the octant in which each of the following points lies.
(i) (4, –2, –5)
(ii) (4, 2, –5)
Solution: (i) $(4, –2, –5)$:- 8th Octant, (ii) $(4, 2, –5)$:- 5th Octant,
Name the octant in which each of the following points lies.
(i) (1, 2, 3),
(ii) (4, – 2, 3),
Solution: (i) $(1, 2, 3)$:- 1st Octant, (ii) $(4, – 2, 3)$:- 4th Octant,
Locate the following points:
(i) (– 2, – 4, –7)
(ii) (– 4, 2, – 5).
Solution: (i) $(– 2, – 4, –7)$:- 7th octant, (ii) $(– 4, 2, – 5)$:- 6th octant.
Locate the following points:
(i) (1, – 1, 3),
(ii) (– 1, 2, 4)
Solution: (i) $(1, – 1, 3)$:- 4th octant, (ii) $(– 1, 2, 4)$:- 2nd octant,
Construct a 2 × 2 matrix A = [ai j] whose elements ai j are given by: ai j = e2ix sin x j
Given \[{{a}_{i\text{ }j}}~=\text{ }{{e}^{2ix}}~sin\text{ }x\text{ }j\] Let \[A\text{ }=\text{ }{{[{{a}_{i\text{ }j}}]}_{2\times 2}}\] So, the elements in a \[2\text{ }\times \text{ }2\] matrix are...
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Let the limit given below
Solution: As per the given question,
Let f(x) be a function defined by given below limit . Then show that lim x-> 0 f(x) doest not exist.
Solution: As per the given question,
Show that the given limit does not exist
Solution: As per the given question,
Solve:
Solution: As per the given question, So, let \[x\text{ }=\text{ }2\text{ }-\text{ }h,\text{ }where\text{ }h\text{ }=\text{ }0\] Substituting the value of \[x,\]we get
Show that the given limit does not exist
Solution: As per the given question,
The coordinates of a point are (3, -2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.
Given: Point \[\left( 3,\text{ }-2,\text{ }5 \right)\] The Absolute value of any point \[\left( x,\text{ }y,\text{ }z \right)\] is given by, \[\surd ({{x}^{2}}~+\text{ }{{y}^{2}}~+\text{...
Find the distances of the point P (-4, 3, 5) from the coordinate axes.
Given: The point \[P\text{ }\left( -4,\text{ }3,\text{ }5 \right)\] The distance of the point from x-axis is given as: The distance of the point from y-axis is given as: The distance of the point...
Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.
Given: Points are \[\left( 5,\text{ }0,\text{ }2 \right)\text{ }and\text{ }\left( 3,\text{ }-2,\text{ }5 \right)\] We need to find the lengths of the edges of the parallelepiped formed For point...
Planes are drawn parallel to the coordinates planes through the points (3, 0, -1) and (-2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.
Given: Points are \[\left( 3,\text{ }0,\text{ }-1 \right)\text{ }and\text{ }\left( -2,\text{ }5,\text{ }4 \right)\] We need to find the lengths of the edges of the parallelepiped formed. For point...
A cube of side 5 has one vertex at the point (1, 0, 1), and the three edges from this vertex are, respectively, parallel to the negative x and y-axes and positive z-axis. Find the coordinates of the other vertices of the cube.
Given: A cube has side \[4\]having one vertex at \[\left( 1,\text{ }0,\text{ }1 \right)\] Side of cube \[=\text{ }5\] We need to find the coordinates of the other vertices of the cube. So let the...
Find the image of: (-4, 0, 0) in the xy-plane
\[\left( -4,\text{ }0,\text{ }0 \right)\] Since we need to find its image in \[xy-plane,\] sign of its \[z-coordinate\]will change So, Image of point \[\left( -4,\text{ }0,\text{ }0 \right)\text{...
Find the image of: (i) (5, 2, -7) in the xy-plane (ii) (-5, 0, 3) in the xz-plane
(i) \[\left( 5,\text{ }2,\text{ }-7 \right)\] Since we need to find its image in \[xy-plane,\] a sign of its \[z-coordinate\] will change So, Image of point \[\left( 5,\text{ }2,\text{ }-7...
Find the image of: (i) (-2, 3, 4) in the yz-plane (ii) (-5, 4, -3) in the xz-plane
(i) \[\left( -2,\text{ }3,\text{ }4 \right)\] Since we need to find its image in \[yz-plane,\] a sign of its \[x-coordinate\]will change So, Image of point \[\left( -2,\text{ }3,\text{ }4...
Name the octants in which the following points lie: (i) (2, -5, -7) (ii) (-7, 2, -5)
(i) \[\left( 2,\text{ }-5,\text{ }-7 \right)\] In this case, since \[z\text{ }and\text{ }y\] are negative and \[x\] is positive then the octant will be \[XOY\prime Z\prime \] (ii) \[\left( -7,\text{...
Name the octants in which the following points lie: (i) (-5, -4, 7) (ii) (-5, -3, -2)
(i) \[\left( -5,\text{ }-4,\text{ }7 \right)\] In this case, since \[x\text{ }and\text{ }y\]are negative and \[z\]is positive then the octant will be \[X\prime OY\prime Z\] (ii) \[\left( -5,\text{...
Name the octants in which the following points lie: (i) (4, -3, 5) (ii) (7, 4, -3)
(i) \[\left( 4,\text{ }-3,\text{ }5 \right)\] In this case, since \[y\]is negative and \[x\text{ }and\text{ }z\] are positive then the octant will be \[XOY\prime Z\] (ii) \[\left( 7,\text{ }4,\text{...
Name the octants in which the following points lie: (i) (5, 2, 3) (ii) (-5, 4, 3)
(i) \[\left( 5,\text{ }2,\text{ }3 \right)\] In this case, since \[x,\text{ }y\text{ }and\text{ }z\] all three are positive then octant will be \[XOYZ\] (ii) \[\left( -5,\text{ }4,\text{ }3...
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases: conjugate axis is 7 and passes through the point (3, -2)
conjugate axis is \[7\]and passes through the point \[\left( 3,\text{ }-2 \right)\] Given: Conjugate axis \[=\text{ }7\] Passes through the point \[\left( 3,\text{ }-2 \right)\] Conjugate axis is...
Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases: (i) the distance between the foci = 16 and eccentricity = √2 (ii) conjugate axis is 5 and the distance between foci = 13
(i) the distance between the \[foci\text{ }=\text{ }16\text{ }and\text{ }eccentricity\text{ }=\text{ }\surd 2\] Given: Distance between the foci \[=\text{ }16\] Eccentricity \[=\text{ }\surd 2\] Let...
Find the centre, eccentricity, foci and directions of the hyperbola x^2 – 3y^2 – 2x = 8
\[{{x}^{2}}-\text{ }3{{y}^{2}}-\text{ }2x\text{ }=\text{ }8\] Given: The equation \[=>\text{ }{{x}^{2}}-\text{ }3{{y}^{2}}-\text{ }2x\text{ }=\text{ }8\] Let us find the centre, eccentricity,...
Find the centre, eccentricity, foci and directions of the hyperbola (i) 16x^2 – 9y^2 + 32x + 36y – 164 = 0 (ii) x^2 – y^2 + 4x = 0
(i) \[16{{x}^{2}}-\text{ }9{{y}^{2}}~+\text{ }32x\text{ }+\text{ }36y\text{ }-\text{ }164\text{ }=\text{ }0\] Given: The equation \[=>\text{ }16{{x}^{2}}-\text{ }9{{y}^{2}}~+\text{ }32x\text{...