Mathematics (041) · Class XII · Chapter-wise Board Questions

A relation R on set A = {1, 2, 3} defined as R = {(1, 1), (2, 2), (1, 2)} is

1. Reflexive only
2. Reflexive and Transitive
3. Symmetric and Transitive
4. Transitive only

Assertion (A) : A function f : \mathbb{N} \to \mathbb{N} given by f(x) = x^{3} + 2, \forall\, x \in \mathbb{N} is one-one but not onto.

Reason (R) : Since \forall\, y \in \mathbb{N} (Codomain), there does not exist x = (y - 2)^{1/3} in \mathbb{N} (Domain) such that f(x) = x^{3} + 2 = y.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A) : A relation R on the set {1, 2, 3} defined as R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} is an equivalence relation.

Reason (R) : A relation that is reflexive, symmetric and transitive is an equivalence relation.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A) : Let R be a relation on set A of all books in a school library given by R = \{(x, y) : x, y \in A, x and y have the same number of pages\} and S is a relation on set A from the same library such that S = \{(p, q) : p and q are Mathematics books\}. Also, R \cap S \ne \phi. Then, R \cap S is an equivalence relation.

Reason (R) : The intersection of two equivalence relations on a set is an equivalence relation on the set.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A) : Let Z be the set of integers. A function f:Z\to Z defined as f(x)=3x-5,\ \forall x\in Z is a bijective.

Reason (R) : A function is a bijective if it is both surjective and injective.

1. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

For real x, let f(x)=x^3+5x+1. Then :

1. f is one-one but not onto on R
2. f is onto on R but not one-one
3. f is one-one and onto on R
4. f is neither one-one nor onto on R

If f:N\to W is defined as

then f is :

1. injective only
2. surjective only
3. a bijection
4. neither surjective nor injective

Assertion (A) : Let A=\{x\in R:-1\le x\le1\}. If f:A\to A be defined as f(x)=x^2, then f is not an onto function.

Reason (R) : If y=-1\in A, then x=\pm\sqrt{-1}\notin A.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A) : Let f(x) = e^x and g(x) = \log x. Then (f + g)\, x = e^x + \log x where domain of (f + g) is R.

Reason (R) : \text{Dom}(f + g) = \text{Dom}(f) \cap \text{Dom}(g).

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Consider the function f : R \longrightarrow R, defined as f(x) = x^3.

Assertion (A) : f(x) is a one-one function.

Reason (R) : f(x) is a one-one function, if co-domain = range.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

A function f : R_+ \to R (where R_+ is the set of all non-negative real numbers) defined by f(x) = 4x + 3 is :

1. one-one but not onto
2. onto but not one-one
3. both one-one and onto
4. neither one-one nor onto

Let f : R_+ \to [-5, \infty) be defined as f(x) = 9x^2 + 6x - 5, where R_+ is the set of all non-negative real numbers. Then, f is :

1. one-one
2. onto
3. bijective
4. neither one-one nor onto

Let R_+ denote the set of all non-negative real numbers. Then the function f : R_+ \to R_+ defined as f(x) = x^2 + 1 is :

1. one-one but not onto
2. onto but not one-one
3. both one-one and onto
4. neither one-one nor onto

Assertion (A) : The relation R = \{(x, y) : (x + y) \text{ is a prime number and } x, y \in N\} is not a reflexive relation.

Reason (R) : The number ‘2n’ is composite for all natural numbers n.

1. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

A function f:\mathbb{R}\to\mathbb{R} defined as f(x)=x^2-4x+5 is :

1. injective but not surjective.
2. surjective but not injective.
3. both injective and surjective.
4. neither injective nor surjective.

Let A = {3, 5}. Then number of reflexive relations on A is

1. 2
2. 4
3. 0
4. 8

Check whether f : R - \{3\} \to R defined as f(x) = \frac{x-2}{x-3} is onto or not.

Check whether f : Z \times Z \to Z \times Z (where Z is the set of integers) defined as f(x, y) = (2y, 3x) is injective or not.

A relation R on A = \{1, 2, 3\} is defined as R = \{(1, 1)\ (3, 3), (1, 2)\}. Is R a symmetric relation ? Justify. Write the smallest relation set R_1 such that R \cup R_1 becomes an equivalence relation on the set \{1, 2, 3\}.

Students are trying to fix thumb pins on a graph sheet on the class notice board. A relation R is defined on the set A of thumb pins as

R = {(P, Q) : P, Q \in A and distance between P and Q is less than 3 cm}

Determine if R is an equivalence relation.

Let f:A\to B be defined by f(x)=\frac{x-2}{x-3}, where A=\mathbb{R}-\{3\} and B=\mathbb{R}-\{1\}.

Discuss the bijectivity of the function.

A function f:A\to B defined as f(x)=2x is both one-one and onto. If A=\{1,2,3,4\}, then find the set B.

Prove that the greatest integer function f : \mathbb{R} \to \mathbb{R}, given by f(x) = [x], is neither one-one nor onto.

Let A = \mathbb{R} - \{3\} and B = \mathbb{R} - \{1\}. A function f : A \to B is defined by f(x) = \left(\frac{x - 2}{x - 3}\right). Find whether f is one-one and onto.

Let n be a fixed positive integer. A relation R is defined in set Z such that R = \{(x, y) : (x - y) \text{ is divisible by n},\ x, y \in Z\}. Determine if R is an equivalence relation.

If R is a relation on set of natural numbers N such that

R = \{(x, y) : x, y \in N, x + 2y = 25\},

then find the Domain and Range of the relation R. Also, determine if it is an equivalence relation.

Determine whether the function f : \mathbb{R} \to \mathbb{R} defined by f(x) = 2x^3 - 7, \forall\, x \in \mathbb{R} (Real numbers) is bijective or not.

Let R be a relation defined over N, where N is set of natural numbers, defined as “mRn if and only if m is a multiple of n, m, n \in N.” Find whether R is reflexive, symmetric and transitive or not.

If f:R^+\to R is defined as f(x)=\log_a x (a > 0 and a\ne 1), prove that f is a bijection.

(R^+ is a set of all positive real numbers.)

Let A = {1, 2, 3} and B = {4, 5, 6}. A relation R from A to B is defined as R = {(x, y) : x + y = 6, x\in A,\ y\in B}.

(i) Write all elements of R.

(ii) Is R a function ? Justify.

(iii) Determine domain and range of R.

A student wants to pair up natural numbers in such a way that they satisfy the equation 2x + y = 41,\ x, y \in N. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.

Show that the function f : N \to N, where N is a set of natural numbers, given by f(n) = \begin{cases} n - 1, & \text{if } n \text{ is even} \\ n + 1, & \text{if } n \text{ is odd} \end{cases} is a bijection.

Show that the function f:\mathbb{R}\to\mathbb{R} defined by f(x)=4x^3-5,\ \forall\ x\in\mathbb{R} is one-one and onto.

Let R be a relation defined on a set N of natural numbers such that R=\{(x,y):xy\text{ is a square of a natural number, }x,y\in\mathbb{N}\}. Determine if the relation R is an equivalence relation.

A relation R on set A = {1, 2, 3, 4, 5} is defined as R = \{(x, y) : |x^2 - y^2| \lt 8\}. Check whether the relation R is reflexive, symmetric and transitive.

A function f is defined from R \to R as f(x) = ax + b, such that f(1) = 1 and f(2) = 3. Find function f(x). Hence, check whether function f(x) is one-one and onto or not.

Case Study -1

A city’s traffic management department is planning to optimize traffic flow by analyzing the connectivity between various traffic signals. The city has five major spots labelled A,B,C,D, and E.

The department has collected the following data regarding one-way traffic flow between spots:

1. Traffic flows from A to B, A to C, and A to D.
2. Traffic flows from B to C and B to E.
3. Traffic flows from C to E.
4. Traffic flows from D to E and D to C.

The department wants to represent and analyze this data using relations and functions. Use the given data to answer the following questions:

I. Is the traffic flow reflexive? Justify. [1]

II. Is the traffic flow transitive? Justify. [1]

III A. Represent the relation describing the traffic flow as a set of ordered pairs. Also state the domain and range of the relation.

OR

III B. Does the traffic flow represent a function? Justify your answer. [2]

Case Study – 1

A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition.

7 students out of 80 are selected for the project and are categorized into two sets such that :

Girl students belong to Set A = \{G_1, G_2, G_3, G_4\},

Boy students belong to Set B = \{B_1, B_2, B_3\}.

Based on the above information, answer the following questions :

(i) How many relations are possible from Set A \to Set B ? [1]

(ii) Let R be a relation from A \to B such that R = \{(G_1, B_1), (G_2, B_2), (G_3, B_2), (G_4, B_3), (G_1, B_2)\}. Is R an injective function ? Justify your answer. [1]

(iii) (a) Let the relation R from A \to A be such that R = {(x, y), x, y \in A, x and y are students from the same colony in the city}. Verify if R is an equivalence relation. [2]

OR

(iii) (b) Verify if any function f : B \to A is bijective. Give reason to support your answer. [2]

A class-room teacher is keen to assess the learning of her students the concept of “relations” taught to them. She writes the following five relations each defined on the set A = {1, 2, 3} :

R_1=\{(2,3),(3,2)\}
R_2=\{(1,2),(1,3),(3,2)\}
R_3=\{(1,2),(2,1),(1,1)\}
R_4=\{(1,1),(1,2),(3,3),(2,2)\}
R_5=\{(1,1),(1,2),(3,3),(2,2),(2,1),(2,3),(3,2)\}

The students are asked to answer the following questions about the above relations :

(i) Identify the relation which is reflexive, transitive but not symmetric.

(ii) Identify the relation which is reflexive and symmetric but not transitive.

(iii) (a) Identify the relations which are symmetric but neither reflexive nor transitive.

OR

(iii) (b) What pairs should be added to the relation R_2 to make it an equivalence relation ?

A school is organizing a debate competition with participants as speakers S = \{S_1, S_2, S_3, S_4\} and these are judged by judges J = \{J_1, J_2, J_3\}. Each speaker can be assigned one judge. Let R be a relation from set S to J defined as R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y,\ x \in S,\ y \in J\}.

Based on the above, answer the following :

(i) How many relations can be there from S to J ? [1]

(ii) A student identifies a function from S to J as f = \{(S_1, J_1), (S_2, J_2), (S_3, J_2), (S_4, J_3)\} Check if it is bijective. [1]

(iii) (a) How many one-one functions can be there from set S to set J ? [2]

OR

(iii) (b) Another student considers a relation R_1 = \{(S_1, S_2), (S_2, S_4)\} in set S. Write minimum ordered pairs to be included in R_1 so that R_1 is reflexive but not symmetric. [2]

Case Study – 2

Let A be the set of 30 students of class XII in a school. Let f:A\to N, N is a set of natural numbers such that function f(x) = Roll Number of student x.

On the basis of the given information, answer the following :

(i) Is f a bijective function ? [1]

(ii) Give reasons to support your answer to (i). [1]

(iii) (a) Let R be a relation defined by the teacher to plan the seating arrangement of students in pairs, where
R=\{(x,y):x,y are Roll Numbers of students such that y=3x\}.
List the elements of R. Is the relation R reflexive, symmetric and transitive ? Justify your answer. [2]

OR

(iii) (b) Let R be a relation defined by
R=\{(x,y):x,y are Roll Numbers of students such that y=x^3\}.
List the elements of R. Is R a function ? Justify your answer. [2]

Case Study – 2

During the festival season, there was a mela organized by the Resident Welfare Association at a park, near the society. The main attraction of the mela was a huge swing installed at one corner of the park. The swing is traced to follow the path of a parabola given by x^2 = y.

Based on the above information, answer the following questions :

(i) Let f : N \longrightarrow R is defined by f(x) = x^2. What will be the range ? [1]

(ii) Let f : N \longrightarrow N is defined by f(x) = x^2. Check if the function is injective or not. [1]

(iii) (a) Let f : \{1, 2, 3, 4 \ldots\} \longrightarrow \{1, 4, 9, 16 \ldots\} be defined by f(x) = x^2. Prove that the function is bijective. [2]

OR

(iii) (b) Let f : \mathbb{R} \longrightarrow \mathbb{R} is defined by f(x) = x^2. Show that f is neither injective nor surjective. [2]

Students of a school are taken to a railway museum to learn about railways heritage and its history.

An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by

R=\{(l_1,l_2):l_1\text{ is parallel to }l_2\}

On the basis of the above information, answer the following questions :

(i) Find whether the relation R is symmetric or not.

(ii) Find whether the relation R is transitive or not.

(iii) If one of the rail lines on the railway track is represented by the equation y=3x+2, then find the set of rail lines in R related to it.

Students of a school are taken to a railway museum to learn about railways heritage and its history.

An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by

R=\{(l_1,l_2):l_1\text{ is parallel to }l_2\}

Let S be the relation defined by S=\{(l_1,l_2):l_1\text{ is perpendicular to }l_2\} check whether the relation S is symmetric and transitive.

Case Study-I

An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let B=\{b_1,b_2,b_3\} and G=\{g_1,g_2\}, where B represents the set of Boys selected and G the set of Girls selected for the final race.

Based on the above information, answer the following questions :

(I) How many relations are possible from B to G ? [1]

(II) Among all the possible relations from B to G, how many functions can be formed from B to G ? [1]

(III) Let R:B\to B be defined by R=\{(x,y):x\text{ and }y\text{ are students of the same sex}\}. Check if R is an equivalence relation. [2]

OR

(III) A function f:B\to G be defined by f=\{(b_1,g_1),(b_2,g_2),(b_3,g_1)\}. Check if f is bijective. Justify your answer. [2]

Let A=N\times N, where N is the set of natural numbers and a relation R is defined on A such that (a,b)R(c,d)\ if\ and\ only\ if\ a^2+d^2=b^2+c^2.
Check whether R is an equivalence relation or not?

A relation R:(-\infty,\infty)\to[0,\infty) is defined as R=\{(x,y):y=x^2+5\}
Draw its graph and check whether it is a function or not.
If the given relation is a function, then examine whether the function is injective, and whether it is surjective or not.

For Visually Impaired: A relation R:(-\infty,\infty)\to[0,\infty) is defined as R=\{(x,y):y=x^2+7\}.
Check whether it is a function or not. If the given relation is a function, then examine whether the function is injective, surjective, or both, and conclude whether it is bijective or not.

A relation R is defined on Z, the set of integers, as

R = \{(x, y) : |x - y| \text{ is divisible by a prime number 'p'},\ x, y \in Z\}

check whether R is an equivalence relation or not.

A function f : R - \left\{\frac{3}{5}\right\} \longrightarrow R - \left\{\frac{3}{5}\right\} is defined as f(x) = \frac{3x + 2}{5x - 3}. Show that f is one-one and onto.

Show that f : \mathbb{R} \to \mathbb{R} defined as f(x) = \frac{x}{\sqrt{1 + x^2}} is one-one but not onto.

Show that a function f : R \to R defined by f(x) = \frac{2x}{1+x^2} is neither one-one nor onto. Further, find set A so that the given function f : R \to A becomes an onto function.

A relation R is defined on N \times N (where N is the set of natural numbers) as :

(a, b)\ R\ (c, d) \Leftrightarrow a - c = b - d

Show that R is an equivalence relation.

A relation R on set A = {– 4, – 3, – 2, – 1, 0, 1, 2, 3, 4} be defined as R = {(x, y) : x + y is an integer divisible by 2}. Show that R is an equivalence relation. Also, write the equivalence class [2].

Let A = R – {5} and B = R – {1}. Consider the function f : A \to B, defined by f(x) = \frac{x-3}{x-5}. Show that f is one-one and onto.

Check whether the relation S in the set of real numbers R defined by S = {(a, b) : where a - b + \sqrt{2} is an irrational number} is reflexive, symmetric or transitive.

Check whether the relation S in the set of all real numbers (\mathbb{R}) defined by S=\{(a,b):a\le b^{3}\} is reflexive, symmetric or transitive.

A relation R is defined on a set of real numbers \mathbb{R} as

R=\{(x,y):x\cdot y\text{ is an irrational number}\}.

Check whether R is reflexive, symmetric and transitive or not.

A function f: [-4, 4] \to [0, 4] is given by f(x) = \sqrt{16 - x^2}. Show that f is an onto function but not a one-one function. Further, find all possible values of ‘a’ for which f(a) = \sqrt{7}.

Show that a function f : \mathbb{R} \to \mathbb{R} defined as f(x) = \frac{5x - 3}{4} is both one-one and onto.

If N denotes the set of all natural numbers and R is the relation on N\times N defined by (a,b)\,R\,(c,d), if ad(b+c)=bc(a+d). Show that R is an equivalence relation.

Let f:\mathbb{R}-\left\{-\frac{4}{3}\right\}\to\mathbb{R} be a function defined as f(x)=\frac{4x}{3x+4}. Show that f is a one-one function. Also, check whether f is an onto function or not.

Show that the relation S in set \mathbb{R} of real numbers defined by

is neither reflexive, nor symmetric, nor transitive.

Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by

R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Hence, find the elements of equivalence class [1].

Identify the function whose graph is given below:

For Visually Impaired: Find the range of \frac{\pi}{2}+\sec^{-1}\left(\frac{x}{2}\right)
(A) [0,\pi]   (B) [0,\pi]-\left\{\frac{\pi}{2}\right\}   (C) \left[\frac{\pi}{2},\frac{3\pi}{2}\right]   (D) \left[\frac{\pi}{2},\frac{3\pi}{2}\right]-\{\pi\}

1. \cos^{-1}\left(\frac{x}{2}\right)
2. \cos^{-1}(2x)
3. \pi-\cos^{-1}x
4. \pi+\cos^{-1}x

Domain of the function: \sec^{-1}(2x) is

1. (-\infty,\infty)
2. \left(-\frac12,\frac12\right)
3. \left(-\infty,-\frac12\right)\cup\left(\frac12,\infty\right)
4. \left(-\infty,-\frac12\right]\cup\left[\frac12,\infty\right)

Value of \cot^{-1}\left(2\cos\left(\frac12\sin^{-1}\left(\frac{\sqrt3}{2}\right)\right)\right)

1. \frac{\pi}{6}
2. \frac{\pi}{4}
3. \frac{\pi}{3}
4. \frac{\pi}{2}

Identify the function shown in the graph

For Visually Impaired: Inverse Trigonometric Function, whose domain is \left[-\frac{1}{3},\frac{1}{3}\right], is ...
(A) \cos^{-1}x   (B) \cos^{-1}\left(\frac{x}{3}\right)   (C) \cos^{-1}(3x)   (D) 3\cos^{-1}x

1. \sin^{-1}x
2. \sin^{-1}(2x)
3. \sin^{-1}\left(\frac{x}{2}\right)
4. 2\sin^{-1}x

Assertion (A): Value of the expression \sin^{-1}\left(\frac{\sqrt3}{2}\right)+\tan^{-1}1-\sec^{-1}(\sqrt2) is \frac{\pi}{4}.

Reason (R): Principal value branch of \sin^{-1}x is \left[-\frac{\pi}{2},\frac{\pi}{2}\right] and that of \sec^{-1}x is [0,\pi]-\left\{\frac{\pi}{2}\right\}.

1. Both (A) and (R) are true and (R) is the correct explanation of (A).
2. Both (A) and (R) are true but (R) is not the correct explanation of (A).
3. (A) is true but (R) is false.
4. (A) is false but (R) is true.

If 2\cos^{-1}x = y, then

1. 0 \le y \le \pi
2. -\pi \le y \le \pi
3. 0 \le y \le 2\pi
4. -\pi \le y \le 0

The domain of f(x) = \cos^{-1}(2x - 5) is :

1. [-1, 1]
2. [4, 6]
3. [-7, -3]
4. [2, 3]

The following graph represents :

1. y = \cos^{-1} x
2. y = \sec^{-1} x
3. y = \tan^{-1} x
4. y = \text{cosec}^{-1} x

The principal value of \sec^{-1}\left(\sqrt{2}\right) + 2\,\text{cosec}^{-1}\left(-\sqrt{2}\right) is :

1. -\frac{\pi}{2}
2. -\frac{\pi}{4}
3. \frac{\pi}{4}
4. \frac{\pi}{2}

For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?

1. \tan^{-1} : R \to \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)
2. \sec^{-1} : R - (-1, 1) \to [0, \pi] - \left\{\frac{\pi}{2}\right\}
3. \cot^{-1} : R \to (0, \pi)
4. \text{cosec}^{-1} : R - (-1, 1) \to \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

The value of \sin^{-1}\left(\sin\frac{3\pi}{5}\right) is :

1. \frac{3\pi}{5}
2. \frac{\pi}{5}
3. \frac{7\pi}{5}
4. \frac{2\pi}{5}

[\tan^{-1}\sqrt{3} - \sec^{-1}(-2)] is equal to :

1. \pi
2. -\frac{\pi}{3}
3. \frac{\pi}{3}
4. \frac{2\pi}{3}

The graph of a trigonometric function is as shown. Which of the following will represent graph of its inverse ?

1. Graph (A)
2. Graph (B)
3. Graph (C)
4. Graph (D)

If y = \sin^{-1}x,\ -1 \le x \le 0, then the range of y is

1. \left(\frac{-\pi}{2}, 0\right)
2. \left[\frac{-\pi}{2}, 0\right]
3. \left[\frac{-\pi}{2}, 0\right)
4. \left(\frac{-\pi}{2}, 0\right]

The principal value of \sin^{-1}\left(\sin\left(-\frac{10\pi}{3}\right)\right) is :

1. -\frac{2\pi}{3}
2. -\frac{\pi}{3}
3. \frac{\pi}{3}
4. \frac{2\pi}{3}

The principal value of \cot^{-1}\left(-\frac{1}{\sqrt3}\right) is :

1. -\frac{\pi}{3}
2. -\frac{2\pi}{3}
3. \frac{\pi}{3}
4. \frac{2\pi}{3}

The following graph is a combination of :

1. y = \sin^{-1} x and y = \cos^{-1} x
2. y = \cos^{-1} x and y = \cos x
3. y = \sin^{-1} x and y = \sin x
4. y = \cos^{-1} x and y = \sin x

\left[\sec^{-1}(-\sqrt{2}) - \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\right] is equal to :

1. \frac{11\pi}{12}
2. \frac{5\pi}{12}
3. -\frac{5\pi}{12}
4. \frac{7\pi}{12}

The given graph illustrates :

1. y=\tan^{-1}x
2. y=\text{cosec}^{-1}x
3. y=\cot^{-1}x
4. y=\sec^{-1}x

Domain of f(x)=\cos^{-1}x+\sin x is :

1. \mathbb{R}
2. (-1,1)
3. [-1,1]
4. \phi

Assertion (A) : Set of values of \sec^{-1}\left(\frac{\sqrt3}{2}\right) is a null set.

Reason (R) : \sec^{-1}x is defined for x\in\mathbb{R}-(-1,1).

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The domain of f(x) = \cos^{-1}(2x) is :

1. [-1, 1]
2. \left[0, \frac{1}{2}\right]
3. [-2, 2]
4. \left[-\frac{1}{2}, \frac{1}{2}\right]

Assertion (A) : Domain of y=\cos^{-1}(x) is [-1,1].

Reason (R) : The range of the principal value branch of y=\cos^{-1}(x) is [0,\pi]-\left\{\frac{\pi}{2}\right\}.

1. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A) : \cos^{-1}\left(\cos\frac{13\pi}{6}\right) is equal to \frac{\pi}{6}.

Reason (R) : The range of the principal value branch of the function y=\cos^{-1}x is [0,\pi].

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A) : The range of the function f(x)=2\sin^{-1}x+\frac{3\pi}{2}, where x\in[-1,1], is \left[\frac{\pi}{2},\frac{5\pi}{2}\right].

Reason (R) : The range of the principal value branch of \sin^{-1}(x) is [0,\pi].

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Assertion (A) : Maximum value of (\cos^{-1} x)^2 is \pi^2.

Reason (R) : Range of the principal value branch of \cos^{-1}x is \left[\frac{-\pi}{2}, \frac{\pi}{2}\right].

1. Both (A) and (R) are true and (R) is the correct explanation of (A).
2. Both (A) and (R) are true, but (R) is not the correct explanation of (A).
3. (A) is true, but (R) is false.
4. (A) is false, but (R) is true.

Assertion (A) : Range of [\sin^{-1} x + 2\cos^{-1} x] is [0, \pi].

Reason (R) : Principal value branch of \sin^{-1} x has range \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Assertion (A) : All trigonometric functions have their inverses over their respective domains.

Reason (R) : The inverse of \tan^{-1}x exists for some x\in\mathbb{R}.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

\sin\left[\frac{\pi}{3}+\sin^{-1}\left(\frac{1}{2}\right)\right] is equal to

1. 1
2. \frac{1}{2}
3. \frac{1}{3}
4. \frac{1}{4}

Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d) as given below.

Assertion (A) : The principal value of \cot^{-1}(\sqrt{3}) is \frac{\pi}{6}.

Reason (R) : Domain of \cot^{-1} x is \mathbb{R} - \{-1, 1\}.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Evaluate \tan\left(\tan^{-1}(-1)+\frac{\pi}{3}\right)

Find the domain of \cos^{-1}(3x-2)

Simplify : \tan^{-1}\left(\frac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x}\right),\ 0 \lt x \lt \frac{\pi}{4}.

Evaluate : \tan\left(\sin^{-1}1 - \cos^{-1}\left(-\frac{1}{2}\right)\right)

Find the value of \sin[\cot^{-1}\sqrt{2}\,(\cos(\tan^{-1} 1))].

Evaluate :

Evaluate \sin\left[\tan^{-1}\tan\left(\frac{3\pi}{4}\right)\right].

Evaluate : \tan^{-1}\left[2\sin\left(2\cos^{-1}\frac{\sqrt{3}}{2}\right)\right]

Simplify \sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right).

Find domain of \sin^{-1}\sqrt{x-1}.

Find the domain of the function f(x)=\cos^{-1}(x^2-4).

Find the domain of f(x) = \sin^{-1}(-x^2).

Find the principal value of \cos^{-1}\left(-\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right).

Prove that :

Find the value of \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\frac{\pi}{2}\right)\right].

Find the domain of the function f(x) = \sin^{-1}(x^2 - 4). Also, find its range.

Evaluate :

Express \tan^{-1}\left(\frac{\cos x}{1-\sin x}\right), where \frac{-\pi}{2} \lt x \lt \frac{\pi}{2} in the simplest form.

Find the principal value of \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right).

Find value of k if

Find the value of \cos^{-1}\left(\frac{1}{2}\right)-\tan^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\text{cosec}^{-1}(-2).

Evaluate :

Evaluate \sin^{-1}\left(\sin\frac{3\pi}{4}\right) + \cos^{-1}(\cos\pi) + \tan^{-1}(1).

Draw the graph of \cos^{-1} x, where x \in [-1, 0]. Also, write its range.

Evaluate : 3\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) + 2\cos^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cos^{-1}(0)

Draw the graph of f(x) = \sin^{-1} x, x \in \left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]. Also, write range of f(x).

Find the domain of y=\sin^{-1}(x^2-4).

Evaluate :

Write the domain and range (principle value branch) of the following functions :

f(x)=\tan^{-1}x

Simplify :

Case Study – 3

If a function f : X \to Y defined as f(x) = y is one-one and onto, then we can define a unique function g : Y \to X such that g(y) = x, where x \in X and y = f(x),\ y \in Y. Function g is called the inverse of function f.

The domain of sine function is R and function sine : R \to R is neither one-one nor onto. The following graph shows the sine function.

Let sine function be defined from set A to [-1, 1] such that inverse of sine function exists, i.e., \sin^{-1}x is defined from [-1, 1] to A.

On the basis of the above information, answer the following questions :

(i) If A is the interval other than principal value branch, give an example of one such interval. [1]

(ii) If \sin^{-1}(x) is defined from [-1, 1] to its principal value branch, find the value of \sin^{-1}\left(-\frac{1}{2}\right) - \sin^{-1}(1). [1]

(iii) (a) Draw the graph of \sin^{-1}x from [-1, 1] to its principal value branch. [2]

OR

(iii) (b) Find the domain and range of f(x) = 2\sin^{-1}(1 - x). [2]

Find the domain of g(x) = \cos^{-1}(x^2 - 1). Hence, find the value of x for which g(x) = \frac{\pi}{3}.

Also, write the range of \cos^{-1}x other than its principal branch.

If A=\begin{bmatrix}2&1&-2\\3a&0&4\\b&2c&1\end{bmatrix} is a symmetric matrix, then value of \sqrt{6a+2b+3c} is:

1. -2
2. 2
3. \pm2
4. 4

If \begin{bmatrix}1&x\end{bmatrix}\begin{bmatrix}1&0\\0&2\end{bmatrix}\begin{bmatrix}4\\-1\end{bmatrix}=O, then value of x is

1. -2
2. -1
3. 1
4. 2

If for three matrices A=[a_{ij}]_{m\times 4}, B=[b_{ij}]_{n\times 3} and C=[c_{ij}]_{p\times q} products AB and AC both are defined and are square matrices of same order, then value of m,n,p and q are:

1. m=q=3\ \text{and}\ n=p=4
2. m=2,q=3\ \text{and}\ n=p=4
3. m=q=4\ \text{and}\ n=p=3
4. m=4,p=2\ \text{and}\ n=q=3

If the matrix A=\begin{bmatrix}0&r&-2\\3&p&t\\q&-4&0\end{bmatrix} is skew-symmetric, then value of \frac{q+t}{p+r} is....

1. -2
2. 0
3. 1
4. 2

Which of the following cannot be the order of a row-matrix ?

1. 2 \times 1
2. 1 \times 2
3. 1 \times 1
4. 1 \times n

Which of the following properties is/are true for two matrices of suitable orders ?

(i) (A + B)' = A' + B'    (ii) (A - B)' = B' - A'

(iii) (AB)' = A'B'    (iv) (kAB)' = kB'A' (k is a scalar)

1. (i) only
2. (i), (ii) and (iii)
3. (i) and (ii)
4. (i) and (iv)

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?

1. AB
2. AB + BA
3. (A + B)^2
4. A – B

If A and B are square matrices of same order, then which of the following statements is/are always true ?

(i) (A + B)(A - B) = A^2 - B^2
(ii) AB = BA
(iii) (A + B)^2 = A^2 + AB + BA + B^2
(iv) AB = 0 \Rightarrow A = 0 \text{ or } B = 0

1. Only (i) and (iii)
2. Only (ii) and (iii)
3. Only (iii)
4. Only (iii) and (iv)

If A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix} is a symmetric matrix, then the value of 3a + b + c is

1. 2
2. 6
3. 4
4. 0

If A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix} and A + A' = I, then the value of x \in \left[0, \frac{\pi}{2}\right] is

1. 0
2. \frac{\pi}{4}
3. \frac{\pi}{3}
4. \frac{\pi}{2}

If A^2 = 4A + 3I and A^{-1} = xA + yI, then the value of (x + y) is :

1. -1
2. 1
3. \frac{5}{3}
4. 7

If A and B are skew-symmetric matrices of same order, then AB' + BA' is a/an :

1. symmetric matrix
2. skew-symmetric matrix
3. null matrix
4. identity matrix

If \begin{bmatrix}4\\1\\3\end{bmatrix}A = \begin{bmatrix}-4 & 8 & 4\\-1 & 2 & 1\\-3 & 6 & 3\end{bmatrix}, then order of A must be :

1. 3 \times 1
2. 1 \times 3
3. 1 \times 1
4. 3 \times 3

If a square matrix A is such that A^2 = A and (I - A)^3 = xA + I, then value of x must be :

1. 7
2. 5
3. -7
4. -1

If A = [a_{ij}] is a 2 \times 2 matrix whose elements are given by a_{ij} = \frac{|i - 3j|}{2}, then A' is :

1. \begin{bmatrix} 1 & \frac{5}{2} \\ \frac{1}{2} & 2 \end{bmatrix}
2. \begin{bmatrix} 1 & \frac{1}{2} \\ \frac{5}{2} & 2 \end{bmatrix}
3. \begin{bmatrix} 2 & \frac{5}{2} \\ \frac{1}{2} & 1 \end{bmatrix}
4. \begin{bmatrix} 2 & \frac{1}{2} \\ \frac{5}{2} & 1 \end{bmatrix}

Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.

Reason (R) : Product of two diagonal matrices of same order is commutative.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If matrix A = \begin{bmatrix}-p & q\\ r & p\end{bmatrix} is such that A^2 = I, then :

1. 1 + p^2 + qr = 0
2. 1 - p^2 - qr = 0
3. 1 - p^2 + qr = 0
4. 1 + p^2 - qr = 0

If A is a square matrix such that A^2 = A, then (A - I)^3 - A is equal to :

1. I
2. -I
3. A
4. A^2

Let A = \begin{bmatrix}0 & -3 & 4\\ 1 & 0 & 2\end{bmatrix} and B = \begin{bmatrix}-3 & 0 & 1\\ 2 & 4 & 0\end{bmatrix}. If A + B + C = O, then matrix C is :

1. \begin{bmatrix}-3 & -3 & 5\\ 3 & 4 & 2\end{bmatrix}
2. \begin{bmatrix}3 & 3 & 5\\ -3 & -4 & -2\end{bmatrix}
3. \begin{bmatrix}3 & 3 & -5\\ -3 & -4 & -2\end{bmatrix}
4. \begin{bmatrix}-3 & -3 & -5\\ 3 & 4 & 2\end{bmatrix}

If A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} satisfies A' + A = I, then the value of \theta is :

1. \frac{\pi}{6}
2. \frac{\pi}{3}
3. \frac{2\pi}{3}
4. \frac{5\pi}{6}

If A is a square matrix, then which of the following is not true ?

1. A + A' is a symmetric matrix
2. AA' and A'A are symmetric matrices
3. A + A' is a skew symmetric matrix
4. A - A' is a skew symmetric matrix

If A = \begin{bmatrix} 1 & -2 & 3 \\ 3 & 2 & -1 \end{bmatrix}, B = \begin{bmatrix} 2 & 3 \\ -1 & 2 \\ 4 & -5 \end{bmatrix} and C = [C_{ij}] = AB, then C_{12} is :

1. 22
2. -16
3. 4
4. 0

If A=\begin{bmatrix}-1&0&0\\0&1&0\\0&0&1\end{bmatrix}, then A^{-1} is

1. \begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}
2. \begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}
3. \begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}
4. \begin{bmatrix}-1&0&0\\0&1&0\\0&0&1\end{bmatrix}

Let A=\begin{bmatrix}1&-2&-1\\0&4&-1\\-3&2&1\end{bmatrix}, B=\begin{bmatrix}-2\\-5\\-7\end{bmatrix}, C=\begin{bmatrix}9&8&7\end{bmatrix}, which of the following is defined ?

1. Only AB
2. Only AC
3. Only BA
4. All AB, AC and BA

If A=\begin{bmatrix}7&0&x\\0&7&0\\0&0&y\end{bmatrix} is a scalar matrix, then y^x is equal to

1. 0
2. 1
3. 7
4. \pm 7

If A = \begin{bmatrix}1 & 12 & 4y\\ 6x & 5 & 2x\\ 8x & 4 & 6\end{bmatrix} is a symmetric matrix, then (2x + y) is

1. -8
2. 0
3. 6
4. 8

Which of the following can be both a symmetric and skew-symmetric matrix ?

1. Unit Matrix
2. Diagonal Matrix
3. Null Matrix
4. Row Matrix

Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify 4AB + 3(AB + BA) - 4BA, where A and B are both matrices of order 2 \times 2. It is known that A \ne B \ne I and A^{-1} \ne B.

Their answers are given as :

Abhay : 6 AB
Bina : 7 AB – BA
Chhaya : 8 AB
Devesh : 7 BA – AB

Who answered it correctly ?

1. Abhay
2. Bina
3. Chhaya
4. Devesh

If A and B are square matrices of order m such that A^2 - B^2 = (A - B)(A + B), then which of the following is always correct ?

1. A = B
2. AB = BA
3. A = 0 or B = 0
4. A = I or B = I

Assertion (A) : A = diag [3 5 2] is a scalar matrix of order 3 \times 3.

Reason (R) : If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.

1. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

If A and B are square matrices of same order such that AB = A and BA = B, then A^2+B^2 is equal to :

1. A + B
2. BA
3. 2 (A + B)
4. 2BA

The matrix \begin{bmatrix}0&1&-2\\-1&0&-7\\2&7&0\end{bmatrix} is a :

1. diagonal matrix
2. symmetric matrix
3. skew symmetric matrix
4. scalar matrix

If A=\begin{bmatrix}5&0&0\\0&5&0\\0&0&5\end{bmatrix}, then A^3 is :

1. 3\begin{bmatrix}5&0&0\\0&5&0\\0&0&5\end{bmatrix}
2. \begin{bmatrix}125&0&0\\0&125&0\\0&0&125\end{bmatrix}
3. \begin{bmatrix}15&0&0\\0&15&0\\0&0&15\end{bmatrix}
4. \begin{bmatrix}5^3&0&0\\0&5&0\\0&0&5\end{bmatrix}

If A=\begin{bmatrix}1&2&3\\-4&3&7\end{bmatrix} and B=\begin{bmatrix}4&3\\-1&2\\0&5\end{bmatrix}, then the correct statement is :

1. Only AB is defined.
2. Only BA is defined.
3. AB and BA, both are defined.
4. AB and BA, both are not defined.

Let both AB' and B'A be defined for matrices A and B. If order of A is n \times m, then the order of B is :

1. n \times n
2. n \times m
3. m \times m
4. m \times n

If A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}, then A is a/an :

1. scalar matrix
2. identity matrix
3. symmetric matrix
4. skew-symmetric matrix

Sum of two skew-symmetric matrices of same order is always a/an :

1. skew-symmetric matrix
2. symmetric matrix
3. null matrix
4. identity matrix

What is the total number of possible matrices of order 3\times3 with each entry as \sqrt2 or \sqrt3 ?

1. 9
2. 512
3. 615
4. 64

The matrix A=\begin{bmatrix}\sqrt3&0&0\\0&\sqrt2&0\\0&0&\sqrt5\end{bmatrix} is a/an :

1. scalar matrix
2. identity matrix
3. null matrix
4. symmetric matrix

If \begin{bmatrix}2x-1&3x\\0&y^2-1\end{bmatrix}=\begin{bmatrix}x+3&12\\0&35\end{bmatrix}, then the value of (x-y) is :

1. 2 or 10
2. -2 or 10
3. 2 or -10
4. -2 or -10

If the matrix A = [a_{ij}]_{2\times 2} is such that a_{ij} = \begin{cases}1, & \text{if } i \ne j\\ 0, & \text{if } i = j\end{cases}, then A + A^2 is equal to :

1. \begin{bmatrix}1&0\\0&1\end{bmatrix}
2. \begin{bmatrix}0&1\\1&0\end{bmatrix}
3. \begin{bmatrix}1&1\\1&1\end{bmatrix}
4. \begin{bmatrix}0&0\\0&0\end{bmatrix}

For a non-singular matrix X, if X^2 = I, then X^{-1} is equal to :

1. X
2. X^2
3. I
4. O

If a matrix has 36 elements, the number of possible orders it can have, is :

1. 13
2. 3
3. 5
4. 9

If \begin{bmatrix} x+y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, then the value of \left(\frac{24}{x} + \frac{24}{y}\right) is :

1. 7
2. 6
3. 8
4. 18

If A and B are two non-zero square matrices of same order such that (A + B)^2 = A^2 + B^2, then :

1. AB = O
2. AB = -BA
3. BA = O
4. AB = BA

If the sum of all the elements of a 3 \times 3 scalar matrix is 9, then the product of all its elements is :

1. 0
2. 9
3. 27
4. 729

If A = [a_{ij}] be a 3 \times 3 matrix, where a_{ij} = i - 3j, then which of the following is false ?

1. a_{11} \lt 0
2. a_{12} + a_{21} = -6
3. a_{13} \gt a_{31}
4. a_{31} = 0

If F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} and [F(x)]^2 = F(kx), then the value of k is :

1. 1
2. 2
3. 0
4. – 2

Assertion (A) : For any symmetric matrix A, B′AB is a skew-symmetric matrix.

Reason (R) : A square matrix P is skew-symmetric if P′ = – P.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If A = [a_{ij}] is an identity matrix, then which of the following is true ?

1. a_{ij} = \begin{cases}0, & \text{if } i = j\\ 1, & \text{if } i \ne j\end{cases}
2. a_{ij} = 1, \forall\, i, j
3. a_{ij} = 0, \forall\, i, j
4. a_{ij} = \begin{cases}0, & \text{if } i \ne j\\ 1, & \text{if } i = j\end{cases}

If A and B are two skew symmetric matrices, then (AB + BA) is :

1. a skew symmetric matrix
2. a symmetric matrix
3. a null matrix
4. an identity matrix

Assertion (A) : Every scalar matrix is a diagonal matrix.

Reason (R) : In a diagonal matrix, all the diagonal elements are 0.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If \begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix} is a scalar matrix, then the value of a + 2b + 3c + 4d is :

1. 0
2. 5
3. 10
4. 25

Given that A^{-1} = \frac{1}{7}\begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}, matrix A is :

1. 7\begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}
2. \begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}
3. \frac{1}{7}\begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}
4. \frac{1}{49}\begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}

If A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}, then the value of I - A + A^2 - A^3 + \ldots is :

1. \begin{bmatrix} -1 & -1 \\ 4 & 3 \end{bmatrix}
2. \begin{bmatrix} 3 & 1 \\ -4 & -1 \end{bmatrix}
3. \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
4. \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Given that [1 \;\; x]\begin{bmatrix} 4 & 0 \\ -2 & 0 \end{bmatrix} = 0, the value of x is :

1. -4
2. -2
3. 2
4. 4

If A=\begin{bmatrix}a & c & -1\\ b & 0 & 5\\ 1 & -5 & 0\end{bmatrix} is a skew-symmetric matrix, then the value of 2a-(b+c) is :

1. 0
2. 1
3. -10
4. 10

If \begin{bmatrix}x & 2 & 0\end{bmatrix}\begin{bmatrix}5\\ -1\\ x\end{bmatrix}=\begin{bmatrix}3 & 1\end{bmatrix}\begin{bmatrix}-2\\ x\end{bmatrix}, then value of x is :

1. -1
2. 0
3. 1
4. 2

Find the matrix A^2, where A=[a_{ij}] is a 2\times2 matrix whose elements are given by a_{ij}=\text{maximum}(i,j)-\text{minimum}(i,j) :

1. \begin{bmatrix}0 & 0\\ 0 & 0\end{bmatrix}
2. \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}
3. \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}
4. \begin{bmatrix}1 & 1\\ 1 & 1\end{bmatrix}

For two matrices A and B, given that A^{-1}=\frac{1}{4}B, then inverse of (4A) is :

1. 4B
2. B
3. \frac{1}{4}B
4. \frac{1}{16}B

If X, Y and XY are matrices of order 2\times3, m\times n and 2\times5 respectively, then number of elements in matrix Y is :

1. 6
2. 10
3. 15
4. 35

If for a square matrix A, A^2-3A+I=O and A^{-1}=xA+yI, then the value of x+y is :

1. -2
2. 2
3. 3
4. -3

If A=\begin{bmatrix}3&4\\5&2\end{bmatrix} and 2A+B is a null matrix, then B is equal to :

1. \begin{bmatrix}6&8\\10&4\end{bmatrix}
2. \begin{bmatrix}-6&-8\\-10&-4\end{bmatrix}
3. \begin{bmatrix}5&8\\10&3\end{bmatrix}
4. \begin{bmatrix}-5&-8\\-10&-3\end{bmatrix}

If A=\begin{bmatrix}0&1\\-1&0\end{bmatrix} and (3I+4A)(3I-4A)=x^2I, then the value(s) x is/are :

1. \pm\sqrt7
2. 0
3. \pm5
4. 25

If A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, then A^{2023} is equal to

1. \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
2. \begin{bmatrix} 0 & 2023 \\ 0 & 0 \end{bmatrix}
3. \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
4. \begin{bmatrix} 2023 & 0 \\ 0 & 2023 \end{bmatrix}

If \begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix} = P + Q, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to

1. \begin{bmatrix} 2 & 5/2 \\ 5/2 & 4 \end{bmatrix}
2. \begin{bmatrix} 0 & -5/2 \\ 5/2 & 0 \end{bmatrix}
3. \begin{bmatrix} 0 & 5/2 \\ -5/2 & 0 \end{bmatrix}
4. \begin{bmatrix} 2 & -5/2 \\ 5/2 & 4 \end{bmatrix}

If A = \begin{bmatrix} 1 & 4 & x\\ z & 2 & y\\ -3 & -1 & 3\end{bmatrix} is a symmetric matrix, then the value of x + y + z is :

1. 10
2. 6
3. 8
4. 0

A and B are skew-symmetric matrices of same order. AB is symmetric, if :

1. AB = O
2. AB = -BA
3. AB = BA
4. BA = O

For what value of x \in \left[0, \frac{\pi}{2}\right], is A + A' = \sqrt{3}\, I, where A = \begin{bmatrix} \cos x & \sin x\\ -\sin x & \cos x\end{bmatrix} ?

1. \frac{\pi}{3}
2. \frac{\pi}{6}
3. 0
4. \frac{\pi}{2}

If x\begin{bmatrix}1\\2\end{bmatrix}+y\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}4\\9\end{bmatrix}, then :

1. x=1,\ y=2
2. x=2,\ y=1
3. x=1,\ y=-1
4. x=3,\ y=2

The product \begin{bmatrix}a & b\\-b & a\end{bmatrix}\begin{bmatrix}a & -b\\b & a\end{bmatrix} is equal to :

1. \begin{bmatrix}a^2+b^2 & 0\\0 & a^2+b^2\end{bmatrix}
2. \begin{bmatrix}(a+b)^2 & 0\\(a+b)^2 & 0\end{bmatrix}
3. \begin{bmatrix}a^2+b^2 & 0\\a^2+b^2 & 0\end{bmatrix}
4. \begin{bmatrix}a & 0\\0 & b\end{bmatrix}

If A is a square matrix and A^2=A, then (I+A)^2-3A is equal to :

1. I
2. A
3. 2A
4. 3 I

If a matrix A=[1\ \ 2\ \ 3], then the matrix AA' (where A' is the transpose of A) is :

1. 14
2. \begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}
3. \begin{bmatrix}1&2&3\\2&3&1\\3&1&2\end{bmatrix}
4. [14]

If for a square matrix A, A^2-A+I=O, then A^{-1} equals

1. A
2. A + I
3. I – A
4. A – I

If A=\begin{bmatrix}1&0\\2&1\end{bmatrix}, B=\begin{bmatrix}x&0\\1&1\end{bmatrix} and A=B^2, then x equals

1. \pm1
2. -1
3. 1
4. 2

If matrix A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} and A^2 = kA, then the value of k is :

1. 1
2. -2
3. 2
4. -1

It is given that X\begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}. Then matrix X is :

1. \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
2. \begin{bmatrix} 0 & -1 \\ 1 & 1 \end{bmatrix}
3. \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}
4. \begin{bmatrix} 1 & -1 \\ 1 & -1 \end{bmatrix}

Solve the matrix equation for x and y.

If A=\begin{bmatrix}2&3\\-1&2\end{bmatrix}, then show that A^2-4A+7I=0.

For two matrices, P=\begin{bmatrix}2&0&0\\0&1&2\end{bmatrix} and Q=\begin{bmatrix}1&0\\0&2\\1&0\end{bmatrix}, find PQ and QP, whichever is defined. Also, find the inverse of the product matrix, if it exists.

If A = \begin{bmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix}, then compute A^{2} - 7A + 10I.

Express the matrix A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix} as the sum of a symmetric matrix and a skew symmetric matrix.

Let A = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix} and C = \begin{bmatrix} 3 & 4 & 2 \\ 12 & 16 & 8 \\ -6 & -8 & -4 \end{bmatrix} be two matrices. Then, find the matrix B if AB = C.

A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹ 150 (Chemistry), ₹ 175 (Physics) and ₹ 180 (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is ₹ 35,000, what profit did he earn after the sale of two days ?

Find a matrix A such that

Also, find A^{-1}.

Given a square matrix A of order 3 such that A^{2}=\begin{bmatrix}3&-4&4\\0&-1&0\\-2&2&-3\end{bmatrix}, show that A^{3}=A^{-1}.

If A=\begin{bmatrix}1&2&3\\3&-2&1\\4&2&1\end{bmatrix}, then show that A^3-23A-40I=O.

Given that P = \begin{bmatrix}2 & -1\\3 & 4\end{bmatrix}, Q = \begin{bmatrix}5 & 2\\7 & 4\end{bmatrix} and R = \begin{bmatrix}2 & 5\\3 & 8\end{bmatrix}, find a matrix S such that PQ - RS is a null matrix.

If f(\alpha) = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}, prove that f(\alpha)\cdot f(-\beta) = f(\alpha - \beta)

If A = \begin{bmatrix} 1 & 0 & 2\\ 0 & 2 & 1\\ 2 & 0 & 3\end{bmatrix}, then show that A^3 - 6A^2 + 7A + 2I = O.

If P and Q are matrices of same order such that P=\begin{bmatrix}2&1\\0&1\end{bmatrix}, |Q|=3, then find |adj(PQ)|

1. -3
2. 1
3. 6
4. 36

If A is a square matrix of order 4 and |adj\,A|=27, then A\,(adj\,A) is equal to

1. 3
2. 9
3. 3\,I
4. 9\,I

The inverse of the matrix \begin{bmatrix}3&0&0\\0&2&0\\0&0&5\end{bmatrix} is...

1. \begin{bmatrix}0&0&3\\0&2&0\\5&0&0\end{bmatrix}
2. \begin{bmatrix}\frac13&0&0\\0&\frac12&0\\0&0&\frac15\end{bmatrix}
3. \begin{bmatrix}-\frac13&0&0\\0&-\frac12&0\\0&0&-\frac15\end{bmatrix}
4. \begin{bmatrix}-3&0&0\\0&-2&0\\0&0&-5\end{bmatrix}

Value of the determinant \begin{vmatrix}\cos 67^o&\sin 67^o\\\sin 23^o&\cos 23^o\end{vmatrix} is

1. 0
2. \frac12
3. \frac{\sqrt3}{2}
4. 1

If \Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix} and \Delta_2 = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix}, then

1. \Delta_1 = 2\Delta_2
2. \Delta_2 = -2\Delta_1
3. \Delta_1 = \Delta_2
4. \Delta_2 = -\Delta_1

One of the values of x for which \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1 is

1. 0
2. \frac{\pi}{4}
3. \frac{\pi}{3}
4. \frac{\pi}{2}

For a square matrix A, (3A)^{-1} =

1. 3A^{-1}
2. 9A^{-1}
3. \frac{1}{3}A^{-1}
4. \frac{1}{9}A^{-1}

If \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86, then the sum of all possible values of a is

1. 4
2. 5
3. –4
4. 9

If B(\text{adj } B) = \begin{bmatrix}\frac{1}{3} & 0 & 0\\0 & \frac{1}{3} & 0\\0 & 0 & \frac{1}{3}\end{bmatrix}, then the value of \det(B^{-1}) =

1. \frac{1}{3}
2. \frac{1}{9}
3. 3
4. 9

If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :

1. \frac{4}{7}
2. -\frac{3}{7}
3. 4
4. -4

If A is a non-singular matrix, then which of the following is not true ?

1. adj A is singular
2. (\text{adj } A)^{-1} = (\text{adj } A^{-1})
3. |A| \ne 0
4. A^{-1} exists

If the area of \Delta ABC with vertices A(3, 1), B(-2, 1) and C(0, k) is 5 sq. units, then values of k are :

1. 3, 1
2. -1, 3
3. -1, 2
4. 0, 2

If \begin{vmatrix} 3x & x+2 \\ 3(x-1) & x-2 \end{vmatrix} = \begin{vmatrix} 1 & 5 \\ 3 & 3 \end{vmatrix}, then the value of x is :

1. 2
2. \frac{1}{3}
3. 5
4. 3

If A and B are square matrices of order 3 such that |A| = 4 and |B| = 7, then |2AB| is :

1. 112
2. 56
3. 28
4. 224

If A is a square matrix of order 2 such that \det(A)=4, then \det(4\,\text{adj}\,A) is equal to :

1. 16
2. 64
3. 256
4. 512

If A and B are invertible matrices, then which of the following is not correct ?

1. (A+B)^{-1}=B^{-1}+A^{-1}
2. (AB)^{-1}=B^{-1}A^{-1}
3. \text{adj}(A)=|A|\,A^{-1}
4. |A|^{-1}=|A^{-1}|

If M and N are square matrices of order 3 such that det (M) = m and MN = mI, then det (N) is equal to :

1. -1
2. 1
3. -m^2
4. m^2

If \begin{vmatrix}2x&5\\12&x\end{vmatrix}=\begin{vmatrix}6&-5\\4&3\end{vmatrix}, then the value of x is :

1. 3
2. 7
3. \pm7
4. \pm3

If A=[a_{ij}] is a 3\times3 diagonal matrix such that a_{11}=1, a_{22}=5 and a_{33}=-2, then |A| is :

1. 0
2. -10
3. 10
4. 1

If \begin{bmatrix}4+x&x-1\\-2&3\end{bmatrix} is a singular matrix, then the value of x is :

1. 0
2. 1
3. -2
4. -4

If A and B are two square matrices each of order 3 with |A|=3 and |B|=5, then |2AB| is :

1. 30
2. 120
3. 15
4. 225

Let A be a square matrix of order 3. If |A|=5, then |\text{adj}\,A| is :

1. 5
2. 125
3. 25
4. -5

The value of the determinant \begin{vmatrix}\cos 75^\circ & \sin 75^\circ\\ \sin 15^\circ & \cos 15^\circ\end{vmatrix} is :

1. 1
2. zero
3. \frac{1}{2}
4. \frac{\sqrt{3}}{2}

The cofactor of the element a_{32} in the determinant \Delta = \begin{vmatrix}1&2&3\\2&3&1\\3&1&2\end{vmatrix} is :

1. \pm 5
2. -5
3. 5
4. 0

If A is an identity matrix of order n, then A (Adj A) is a/an :

1. identity matrix
2. row matrix
3. zero matrix
4. skew symmetric matrix

\begin{vmatrix} x+1 & x-1 \\ x^2+x+1 & x^2-x+1 \end{vmatrix} is equal to :

1. 2x^3
2. 2
3. 0
4. 2x^3 - 2

Assertion (A) : For matrix A = \begin{bmatrix} 1 & \cos\theta & 1 \\ -\cos\theta & 1 & \cos\theta \\ -1 & -\cos\theta & 1 \end{bmatrix}, where \theta \in [0, 2\pi], |A| \in [2, 4].

Reason (R) : \cos\theta \in [-1, 1],\ \forall\, \theta \in [0, 2\pi].

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc, then the value of k is :

1. 0
2. 1
3. 2
4. 4

Let A = \begin{bmatrix} a & b\\ c & d\end{bmatrix} be a square matrix such that adj A = A. Then, (a + b + c + d) is equal to :

1. 2a
2. 2b
3. 2c
4. 0

If \begin{vmatrix} 1 & 3 & 1\\ k & 0 & 1\\ 0 & 0 & 1\end{vmatrix} = \pm 6, then the value of k is :

1. 2
2. -2
3. \pm 2
4. \mp 2

If A = \begin{bmatrix} 2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 5\end{bmatrix}, then A^{-1} is :

1. \begin{bmatrix} \frac12 & 0 & 0\\ 0 & \frac13 & 0\\ 0 & 0 & \frac15\end{bmatrix}
2. 30\begin{bmatrix} \frac12 & 0 & 0\\ 0 & \frac13 & 0\\ 0 & 0 & \frac15\end{bmatrix}
3. \frac{1}{30}\begin{bmatrix} 2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 5\end{bmatrix}
4. \frac{1}{30}\begin{bmatrix} \frac12 & 0 & 0\\ 0 & \frac13 & 0\\ 0 & 0 & \frac15\end{bmatrix}

If A = \begin{bmatrix} -2 & 0 & 0 \\ 1 & 2 & 3 \\ 5 & 1 & -1 \end{bmatrix}, then the value of |A(\text{adj. } A)| is :

1. 100 I
2. 10 I
3. 10
4. 1000

If A is a square matrix of order 3 such that the value of |\text{adj}\cdot A|=8, then the value of |A^T| is :

1. \sqrt{2}
2. -\sqrt{2}
3. 8
4. 2\sqrt{2}

If inverse of matrix \begin{bmatrix}7 & -3 & -3\\ -1 & 1 & 0\\ -1 & 0 & 1\end{bmatrix} is the matrix \begin{bmatrix}1 & 3 & 3\\ 1 & \lambda & 3\\ 1 & 3 & 4\end{bmatrix}, then value of \lambda is :

1. -4
2. 1
3. 3
4. 4

If A=\begin{bmatrix}0&1&-1\\1&2&1\\0&3&-2\end{bmatrix}, then the value of |A\,\text{adj}(A)| is :

1. -1
2. 1
3. 2
4. 3

If |A|=2, where A is a 2\times2 matrix, then |4A^{-1}| equals :

1. 4
2. 2
3. 8
4. \frac{1}{32}

Let A be a 3\times3 matrix such that |\text{adj }A|=64. Then |A| is equal to :

1. 8 only
2. -8 only
3. 64
4. 8 or -8

If \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{bmatrix} is non-singular matrix and a \in A, then the set A is

1. \mathbb{R}
2. \{0\}
3. \{4\}
4. \mathbb{R} - \{4\}

If |A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is

1. 1
2. -1
3. 2
4. 0

If (a, b), (c, d) and (e, f) are the vertices of \Delta ABC and \Delta denotes the area of \Delta ABC, then \begin{vmatrix} a & c & e \\ b & d & f \\ 1 & 1 & 1 \end{vmatrix}^2 is equal to

1. 2\Delta^2
2. 4\Delta^2
3. 2\Delta
4. 4\Delta

If A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 3\end{bmatrix}, then the value of |A| + |\text{adj } A| is equal to :

1. 12
2. 9
3. 3
4. 27

Let A be the area of a triangle having vertices (x_1, y_1), (x_2, y_2) and (x_3, y_3). Which of the following is correct ?

1. \begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{vmatrix} = \pm A
2. \begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{vmatrix} = \pm 2A
3. \begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{vmatrix} = \pm \frac{A}{2}
4. \begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{vmatrix}^2 = A^2

The value of \begin{vmatrix}x+y & y+z & z+x\\z & x & y\\1 & 1 & 1\end{vmatrix} is

1. 0
2. 1
3. x+y+z
4. 2(x+y+z)

If \begin{vmatrix}\alpha&3&4\\1&2&1\\1&4&1\end{vmatrix}=0, then the value of \alpha is

1. 1
2. 2
3. 3
4. 4

If A is a square matrix of order 3 and |A| = 6, then the value of |\text{adj } A| is :

1. 6
2. 36
3. 27
4. 216

For which value of x, are the determinants \begin{vmatrix} 2x & -3 \\ 5 & x \end{vmatrix} and \begin{vmatrix} 10 & 1 \\ -3 & 2 \end{vmatrix} equal ?

1. \pm 3
2. -3
3. \pm 2
4. 2

The value of the cofactor of the element of second row and third column in the matrix \begin{bmatrix} 4 & 3 & 2 \\ 2 & -1 & 0 \\ 1 & 2 & 3 \end{bmatrix} is :

1. 5
2. -5
3. -11
4. 11

Let A and B be two square matrices of order 3 such that det (A) = 3 and det (B) = − 4. Find the value of det (− 6AB).

Two lines are represented by equations 5x - 7y - 2 = 0 and 7x - 5y - 3 = 0 in the xy-plane. Using matrix method, find their point of intersection.

Let 2x+5y-1=0 and 3x+2y-7=0 represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.

To celebrate their child's birthday, a family decided to distribute fruits among children in three orphanages. The fruit vendor packed three different types of baskets, according to the quantity required at each orphanage, details are given as follows:
Basket A contains 5 kg Apples, 10 kg Oranges and 5 kg Pomegranate
Basket B contains 10 kg Apples, 10 kg Oranges and 5kg Pomegranate
Basket C contains 8 kg Apples, 15 kg Oranges and 7kg Pomegranate
Cost of basket A, B and C are respectively ₹2750, ₹3500, and ₹4100.

Based on the given information, answer the following questions:

(i) Write the matrix form of the equations involved in this situation and find the adjoint of the coefficient matrix involved. [2]

(ii) Using inverse of the matrix, calculate the price of each fruit per kg. [2]

Case Study – 2

A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

On the basis of the above information, answer the following questions :

(i) Write the equations representing the various dimensions and express them as the matrix equation AX = B. [1]

(ii) Find if A^{-1} exists. Justify your answer. [1]

(iii) (a) Find A^{-1}. [2]

OR

(iii) (b) Find A^{2} + 7I. [2]

Case Study – 1

Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers.

Based upon the above information, answer the following questions :

(i) Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form AX = B. [1]

(ii) Find |A| and confirm if it is possible to find A^{-1}. [1]

(iii) (a) Find A^{-1}, if possible, and write the formula to find X. [2]

OR

(iii) (b) Find A^2 - 8I, where I is an identity matrix. [2]

Case Study – 3

A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs.

Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022 – 23, the school offered monthly scholarship of ₹ 3,000 each to some girl students and ₹ 4,000 each to meritorious achievers in academics as well as sports.

In all, 50 students were given the scholarships and monthly expenditure incurred by the school on scholarships was ₹ 1,80,000.

Based on the above information, answer the following questions :

(i) Express the given information algebraically using matrices. [1]

(ii) Check whether the system of matrix equations so obtained is consistent or not. [1]

(iii) (a) Find the number of scholarships of each kind given by the school, using matrices. [2]

OR

(iii) (b) Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school ? [2]

Case Study – 3

An architect is developing a plot of land for a commercial complex. When asked about the dimensions of the plot, he said that if the length is decreased by 25 m and the breadth is increased by 25 m, then its area increases by 625 m². If the length is decreased by 20 m and the breadth is increased by 10 m, then its area decreases by 200 m².

On the basis of the above information, answer the following questions :

(i) Formulate the linear equations in x and y to represent the given information.

(ii) Find the dimensions of the plot of land by matrix method.

Case Study-II

Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250.

Based on the above information, answer the following questions :

(I) Convert the given above situation into a matrix equation of the form AX = B. [1]

(II) Find |A|. [1]

(III) Find A^{-1}. [2]

OR

(III) Determine P=A^2-5A. [2]