Multiple Choice Question Bank (MCQ) Term - I: Class - XII
Multiple Choice Question Bank (MCQ) Term - I: Class - XII
Multiple Choice Question Bank (MCQ) Term - I: Class - XII
Class - XII
Multiple Choice Question Bank
[MCQ ] Term – I
MATHEMATICS [041]
Based on Latest CBSE Exam Pattern
for the Session 2021-22
1
कें द्रीय विद्यालय सगं ठन क्षेत्रीय कायाालय रायपरु
Kendriya Vidyalaya Sangathan Regional Office Raipur
I would like to extend my sincere gratitude to all the principals and the
teachers who have relentlessly striven for completion of the project of preparing
study materials for all the subjects. Their enormous contribution in making this
project successful is praiseworthy.
Happy learning and best of luck!
Vinod Kumar
(Deputy Commissioner)
2
कें द्रीय विद्यालय सगं ठन क्षेत्रीय कायाालय रायपरु
Kendriya Vidyalaya Sangathan Regional Office Raipur
Our Patorn
Vinod Kumar
Deputy Commissioner
KVS RO Raipur
3
CONTENT TEAM
COMPILED BY
4
INDEX
5
RELATIONS AND FUNCTIONS
1. If the relation R = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} defined on the set A = {1, 2,
3}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence
2. If the relation R = {(1, 2), (2, 1), (1, 3), (3, 1)} defined on the set A = {1, 2, 3}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence
3. If the relation R on the set N of all natural numbers defined as R = {(x, y) : y = x + 5 and (x < 4),
then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence
CASE STUDY-2
ONE – NATION
ONE – ELECTION
FESTIVAL
OF DEMOCRACY
GENERAL ELECTION – 2019
8
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and
voter turnout was about 67%, the highest ever.
Let I be the set of all citizens of India who were eligible to exercise their voting right in general election
held in 2019. A relation ‘R’ is defined on I as follows:
R = {(𝑉1,𝑉2)∶ 𝑉1,𝑉2 ∈𝐼 and both use their voting right in general election – 2019}
1. Two neighbors X and Y∈ I. X exercised his voting right while Y did not cast her vote in general election
– 2019. Which of the following is true?
a. (X,Y) ∈R
b. (Y,X) ∈R
c. (X,X) ∉R
d. (X,Y) ∉R
2. Mr.’𝑋’ and his wife ‘𝑊’both exercised their voting right in general election -2019, Which of the following
is true?
a. both (X,W) and (W,X) ∈ R
b. (X,W) ∈ R but (W,X) ∉ R
c. both (X,W) and (W,X) ∉ R
d. (W,X) ∈ R but (X,W) ∉ R
3. Three friends F1, F2 and F3 exercised their voting right in general election-2019, then which of the
following is true?
a. (F1,F2 ) ∈R, (F2,F3) ∈ R and (F1,F3) ∈ R
b. (F1,F2 ) ∈ R, (F2,F3) ∈ R and (F1,F3) ∉ R
c. (F1,F2 ) ∈ R, (F2,F2) ∈R but (F3,F3) ∉ R
d. (F1,F2 ) ∉ R, (F2,F3) ∉ R and (F1,F3) ∉ R
4. The above defined relation R is
a. Symmetric and transitive but not reflexive
b. Universal relation
c. Equivalence relation
d. Reflexive but not symmetric and transitive
5. Mr. Shyam exercised his voting right in General Election – 2019, then Mr. Shyam is related to which of
the following?
a. All those eligible voters who cast their votes
b. Family members of Mr.Shyam
c. All citizens of India
d. Eligible voters of India
CASE STUDY- 3
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji
observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be
the set of players while B be the set of all possible outcomes.A = {S, D}, B = {1,2,3,4,5,6}
9
1. Let 𝑅∶ 𝐵→𝐵 be defined by R = {(𝑥,): 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏 } is
a. Reflexive and transitive but not symmetric
b. Reflexive and symmetric and not transitive
c. Not reflexive but symmetric and transitive
d. Equivalence
2. Raji wants to know the number of functions from A to B. How many number of functions are possible?
a. 62
b. 26
c. 6!
d. 212
3. Let R be a relation on B defined by R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}. Then R is
a. Symmetric
b. Reflexive
c. Transitive
d. None of these three
4. Raji wants to know the number of relations possible from A to B. How many numbers of relations are
possible?
a. 62
b. 26
c. 6!
d. 212
5. Let 𝑅:𝐵→𝐵 be defined by R={(1,1),(1,2), (2,2), (3,3), (4,4), (5,5),(6,6)}, then R is
a. Symmetric
b. Reflexive and Transitive
c. Transitive and symmetric
d. Equivalence
CASE STUDY 4
Consider the mapping f : A → B is defined by f(x) = 𝑥 − 1/𝑥 – 2such that f is a bijection. Based on
the above information, answer the following questions:
1. Domain off is
(a) R – {2}
(b) R
(c) R – {1, 2}
(d) R – {0}
2. Range of f is
(a) R
(b) R – {1}
(c) R – {0}
(d) R – {1, 2}
3. If g : R – {2} → R – {1} is defined by g(x) = 2f(x) – 1, then g(x) in terms of x is
(a) 𝑥 + 2/𝑥
(b) 𝑥 + 1/𝑥 – 2
(c) 𝑥 − 2/𝑥
(d) 𝑥/𝑥 – 2
4. The function g defined above, is
(a) One-one
(b) Many-one
(c) into
(d) None of these
5. A function f(x) is said to be one-one if
(a) f(x1) = f(x2) ⇒– x1 = x2
(b) f(–x1) = f(–x2) ⇒– x1= x2
(c) f(x1) = f(x2) ⇒ x1= x2
(d) None of these
10
ASSERTION AND REASON
Read Assertion and reason carefully and write correct option for each question
1 Assertion (A)Let L be the set of all lines in a plane and R be the relation in L defined as R =
{(L1, L2) : L1 is perpendicular to L2}. R is not equivalence realtion.
Reason (R)R is symmetric but neither reflexive nor transitive
2 Assertion= {(T1, T2) : T1 is congruent to T2}. Then R is an equivalence relation.
Reason(R)Any relation R is an equivalence relation, if it is reflexive, symmetric and transitive
3 Assertion (A)The relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),(3, 3), (1, 2), (2, 3)} is reflexive but neither
symmetric nor transitive.
Reason (R)R is not symmetric, as (1, 2) ∈R but (2, 1) ∉R. Similarly, R is not transitive, as (1, 2) ∈R and (2, 3) ∈R
but (1, 3) ∉R.
4 Assertion (A) Show that the relation R in the set A of all the books in a library of a college, given
by R = {(x, y) :x and y have same number of pages} is not equivalence relation.
Reason (R) Since R is reflexive, symmetric and transitive.
5. Assertion (A) The relation R in R defined as R = {(a, b) :a≤b} is not equivalence relation.
Reason (R) Since R is not reflexive but it is symmetric and transitive.
6. Assertion (A) The relation R in R defined as R = {(a, b) :a≤𝑏2} is not equivalence relation.
Reason (R) Since R is not reflexive but it is symmetric and transitive.
7 Assertion (A)The relation R in the set Z of integers given by R = {(a, b) : 2 divides a – b} is reflexive and symmetric
Reason (R)R is reflexive, as 2 divides (a – a) for all a ∈Z.
8. Assertion (A) 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}. R is an equivalence relation
Reason (R) Since R is reflexive, symmetric but R is not transitive.
9. Assertion (A) Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3,
3), (3, 2)}. R is not equivalence relation.
Reason (R) R is not Reflexive relation but it is symmetric and transitive
10. Assertion (A) if n (A) = p and n(B) = q The number of relation from set A to B is 𝑝𝑞
Reason (R) The number of subset of A X B is 2𝑝𝑞
11. Assertion (A)A function f : X →Y is said to be one-one and onto (or bijective)
Reason (R) if f is both one-one and onto.
12. Assertion (A) The function f :N→N, given by f (x) = 2x, is one-one
Reason (R) The function f is one-one, for f (x) = f (y) ⇒2x = 2y⇒x = y. 13
Assertion (A) The function f :N→N, given by f (x) = 2x, is not onto Reason
(R) The function f is onto, for f (x) = f (y) ⇒2x = 2y⇒x = y.
14 Assertion (A) the function f :N→N, given by f (1) = f (2) = 1 and f (x) = x – 1, for every x > 2, is onto but not one-
one.
Reason (R) fis not one-one, as f (1) = f (2) = 1. But f is onto, as given any y ∈N, y ≠1, we can choose x as y + 1 such that f (y
+ 1) = y + 1 – 1 = y. Also for 1 ∈N, we have f (1) = 1.
15 Assertion (A) A one-one function f : {1, 2, 3} →{1, 2, 3} must be onto.
Reason (R) Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different elements of the co- domain {1, 2,
3} under f.
16 Assertion (A) Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function From A to
B. Then f is one-one.
Reason (R) Since the function f : N→N, given by f (x) = 2x, is not onto
11
17. Assertion (A) Let A and B be sets. Show that f : A × B →B × A such that f (a, b) = (b, a) is
bijective function
Reason (R) f is said to equivalence relation if f is reflexive , symmetric and transitive
18. Assertion (A) The number of all one-one functions from set A = {1, 2, 3} to itself is 6
Reason (R) if n (A) = p and n(B) = q The number of function from set A to B is 𝑝𝑞
19 Assertion (A) The Modulus Function f :R→R, given by f (x) = | x | is not one one and onto function
Reason (R) The Modulus Function f :R→R, given by f (x) = | x | is bijective function
20. Assertion (A) Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to
B. Then f is one-one.
Reason (R) f is bijective function
12
Question 8. The value of cos-1(1/2) + 2sin-1(1/2) is equal to
(a) π/4
(b) π/6
(c) 2π/3
(d) 5π/6
Question 9. Principal value of tan-1 (-1) is
(a) π/4
(b) −π/2
(c) 5π/4
(d) −π/4
Question 10. Principal value of sin-1(1/√2)
(a) π/4
(b) 3π/4
(c) 5π/4
(d) None of these
Question 11. sin-1 x = y Then
(a) 0 ≤ y ≤ π
(b) –π/2 ≤ y ≤ π/2
(c) 0 < y < π
(d) –π/2 < y < –π/2
Question 12. cos-1(cos 7π/6) is equal to
(a) 7π/6
(b) 5π/6
(c) π/3
(d) π/6
Question 13. sin[π/3 – sin-1(-1/2)] is equal to
(a) 1//2
(b) 1/3
(c) 1/4
(d) 1
Question 14. The principal value of cosec-1 (-2) is
(a) –2π/3
(b) π/6
(c) 2π/3
(d) –π/6
Question 15. The domain of the following f(x) = √(sin−1x)is.
(a) [0, 1]
(b) [-1, 1]
(c) [-2, 0]
(d) [0, 1]
Question 16. Which of the following is the principal value branch of cos -1 x?
(a) [−π/2, π/2]
(b) (0, π)
(c) [0, π]
(d) (0, π) – {π/2}
Question 17. Which of the following is the principal value branch of cosec -1 x?
(a) (−π/2, π/2)
(b) (0, π) – {π/2}
(c) [−π/2, π/2]
(d) [−π/2, π/2] – [0]
Question 18. If 3 tan-1 x + cot-1 x = π, then x equals
(a) 0
(b) 1
(c) -1
(d) 12
13
Question 19. The value of cos-1[cos(33π/5)] is
(a) 3π/5
(b) −3π/5
(c) π/10
(d) –π/10
Question 20. The domain of the function cos-1 (2x – 1) is
(a) [0, 1]
(b) [-1, 1]
(c) [-1, -1]
(d) [0, π]
Question 21. The domain of the function defined by f (x) = sin-1 √(x−1) is
(a) [1, 2]
(b) [-1, 1]
(c) [0, 1]
(d) None of these
Question 22. If cos(sin-12/5 + cos-1 x) = 0 then x is equal to
(a) 1/5
(b) 2/5
(c) 0
(d) 1
Question 23. The value of sin (2 tan-1 (0.75)) is equal to
(a) 0.75
(b) 1.5
(c) 0.96
(d) sin (1.5)
Question 24. The value of cos-1 (cos3π/2) is equal to
(a) π/2
(b) 3π/2
(c) 5π/2
(d) –7π/2
Question 25. The value of expression 2 sec-1 (2) + sin-1 (1/2) is
(a) π/6
(b) 5π/6
(c) 7π/6
(d) 1
Question 26. If sin-1(2a/1+a2) + cos-1(1−a2/1+a2) = tan-1(2x/1−x2) where a, x ∈ |0, 1| then the value of x is
(a) 0
(b) a2
(c) a
(d) 2a/1−a2
Question 27. The value of sin [cos-1(7/25)] is
(a) 25/24
(b) 25/7
(c) 24/25
(d) 7/24
Question 28. sin-1(−1/2)
(a) π/3
(b) –π/3
(c) π/6
(d) –π/6
Question 29. sec-1(−2/√3)
(a) π/6
(b) π/3
(c) 5π/6
(d) –2π/3
14
Question 30. cos-1(1/2)
(a) –π/3
(b) π/3
(c) π/2
(d) 2π/3
Question 31. cosec-1(−2/√3)
(a) –π/3
(b) π/3
(c) π/2
(d) –π/2
Question 32. cot-1(1)
(a) π/3
(b) π/4
(c) π/2
(d) 0
Question 33. cos-1(√3/2)
(a) 5π/6
(b) π/6
(c) 4π/9
(d) 2π/3
Question 34. cosec-1(2)
(a) π/6
(b) 2π/3
(c) 5π/6
(d) 0
Question 35. sec-1(2)
(a) π/6
(b) π/3
(c) 2π/3
(d) 5π/6
Question 36. tan-1(√3)
(a) π/6
(b) π/3
(c) 2π/3
(d) 5π/6
Question 37. cot-1(-√3)
(a) 5π/6
(b) π/3
(c) π/2
(d) π/4
Question 38. tan-1 (√3) + sec-1 (-2) – cosec-1 (2/√3)
(a) 5π/6
(b) 2π/3
(c) π/3
(d) 0
Question 39. cos-1 (−1/2) + 2sin-1 (−1/2)
(a) π/3
(b) 2π/3
(c) 3π/4
(d) 5π/8
15
CASE STUDY QUESTIONS
Case Study 1
A group of students of classXIIvisited India Gate onaneducation trip. Theteacher and students hadinterest in history as
well. The teacher narrated that India Gate, official name Delhi Memorial, originally called All-India War Memorial,
monumental sandstone arch in New Delhi, dedicated tothe troops of British India who died in wars fought between
1914 and 1919. The teacher also said that India Gate, which is located at the eastern end of the Raj path (formerly called
the Kingsway), is about 138 feet (42 metrs) in height.
1. What is the angle of elevation if they are standing at a distance of 42m away from the monument?
a) tan−1 1
b) sin−1 1
c) cos−1 1
d) sec−1 1
2. They want to see the tower at an angle of sec−1 2. So, they want to know the distance where they should
stand and hence find the distance.
a) 42 m b) 20.12 m c) 25.24 m d) 24.64 m
1
3. If the altitude of the Sun is at cos , then the height of the vertical tower that will cast a shadow of
−1
2
length 20 m is
a) 20√3 m b) 20/ √3 m c) 15/ √3 m d) 15√3 m
4. The ratio of the length of a rod and its shadow is 1:2. The angle of elevation of the Sun is
1 1 1 1
a) sin−1 b) cos−1 c) tan−1 d) cot −1
2 2 2 2
Case Study 2
A Satellite flying at height h is watching the top of the two tallest mountains in Uttarakhand and Karnataka, them being Nanda
Devi (height 7,816m) and Mullayanagiri (height 1,930 m). The angles of depression from the satellite, to the top of Nanda Devi and
Mullayanagiri are cot−1 √3 andtan−1 √3 respectively. If the distance between the peaks of the two mountains is 1937 km, and the
satellite is vertically above the midpoint of the distance between the two mountains.
16
17
1. The distance of the satellite from the top of Nanda Devi is
a) 1139.4 kmb) 577.52 kmc) 1937 kmd) 1025.36 km
2. The distance of the satellite from the top of Mullayanagiri is
a) 1139.4 kmb) 577.52 kmc) 1937 kmd) 1025.36 km
3. The distance of the satellite from the ground is
a) 1139.4 kmb) 577.52 kmc) 1937 kmd) 1025.36 km
4. What is the angle of elevation if a man is standing at a distance of 7816m from Nanda Devi?
3
a) sec −1 2b) cot −1 1c) sin−1 √ d) cos−1 1
2
𝟏
5.If a mile stone very far away from, makes 𝐜𝐨𝐬−𝟏 to the top of Mullanyangiri mountain. So, find the distance of this
mile stone from the mountain.
a) 1118.327 kmb) 566.976 kmc) 1937 kmd) 1025.36 km
Case Study 3
The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are
1
tan−1 and sec−1 √2, respectively.
√3
19
ASSERTION AND REASON
Read Assertion and reason carefully and write correct option for each question
𝟏
Reason (R) = (𝐬𝐢𝐧 𝒙)−𝟏
𝐬𝐢𝐧 𝒙
3 5𝜋
10. Assertion (A) The principal value of cos−1 − √ =
2 6
Reason(R)Range of principal value branch of cos −1 𝑥 is [−𝜋 , 𝜋]
MATRICES
MULTIPLE CHOICE QUESTIONS
1 0]
1.If [x 1] [ = O, then x equals
−2 0
(a) 0
(b) -2
(c) -1
(d) 2
3 2
2.If A = [2 −3 ]4 , B = [2] , X = [1 2 ]3 and Y = [3], then AB+XY equals
2 4
(a) [28]
(b) [24]
20
(c) 28
(d) 24
3. Which of the given value of x and y make the following matrices equal
3x + 7 5 ],[ 0 y − 2 ]
[ y+1
2 − 3x 8 4
21
−1 , y=7
(a) x=
2
4. The number of all possible matrices of order 3x3 with each entry 0 or 1 is:
(a) 27
(b) 18
(c) 81
(d) 512
cos α − sin α
5. If A = [ ], and A + A′ = I , then the value of α is
π
sin α cos α
(a)
6
π
(b)
3
(c)π
3π
(d)
2
6. Matrix A and B will be inverse of each other only if
(a) AB=BA
(b) AB=BA=0
(c) AB=0, BA=I
(d) AB=BA=I
0 0 4
7. The matrix P = [0 4 0] is a
4 0 0
(a) square matrix
(b) diagonal matrix
(c) unit matrix
(d) None of these
8. If A and B are symmetric matrices of same order, then AB-BA is a
(a) Skew-symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity
0 −5 8
9. The matrix [ 5 0 17] is a
−8 −17 0
(a) Diagonal matrix
(b) Skew-symmetric matrix
(c) Symmetric matrix
(d) Scalar matrix
10. If a matrix has 6 elements, then number of possible orders of the matrix can be
(a) 2
(b) 4
(c) 3
(d) 6
(−i+2j)2 5
11.If A = [aij] is a 2 × 3 matrix, such that aij = then a23 is
1
(a)
5
2
22
(b)
5
9
(c)
5
16
(d)
5
23
12. Total number of possible matrices of order 2 × 3 with each entry 1 or 0 is
(a) 6
(b) 36
(c) 32
(d) 64
13. If A is a square matrix such that A2 = A, then (I + A)2 – 3A is
(a) I
(b) 2A
(c) 3I
(d) A
0 2
14. If A=[ ], then A2 is
2 0
0 4
(a) [ ]
4 0
4 0
(b) [ ]
4 0
0 4
(c) [ ]
0 4
4 0
(d) [ ]
0 4
15. The diagonal elements of a skew symmetric matrix are
(a) all zeroes
(b) are all equal to some scalar k(≠ 0)
(c) can be any number
(d) none of these
5 x
16. If A =[ ] and A = A′ then
y 0
(a) x = 0, y = 5
(b) x = y
(c) x + y = 5
(d) x – y= 5
17. If a matrix A is both symmetric and skew symmetric then matrix A is
(a) a scalar matrix
(b) a diagonal matrix
(c) a zero matrix of order n × n
(d) a rectangular matrix.
cos x sin x
18. If F(x) =[ ] , then F(x) F(y) is equal to
−sin x cos x
(a) F(x)
(b) F(xy)
(c) F(x + y)
(d) F(x – y)
0 2 1 0
19. The matrix A satisfies the equation [ ]A = [ ] then
−1 1 0 1
2 0
(a) [ ]
1 −1
1 −2
(b) [ ]
1 0
1 −1
(c) [ 1
2 ]
0
2
1 2
(d) [ ]
−1 4
24
0 0 1
20. The matrix A=[0 1 0] , then A6is equal to
10 0
(a) zero matrix
(b) A
(c) I
(d) none of these
3 1
21. If A= [ ], then A2 − 5A − 7I is
−1 2
(a) a zero matrix
(b) an identity matrix
(c) diagonal matrix
(d) none of these
22. A matrixhas 18elements,thenpossiblenumberof orders ofa matrixare (a) 3
(b) 4 (c) 6 (d) 5
23. If matrix A is of order m × n, and for matrix B, AB and BA both are defined, then order of matrix B is
(a) m × n
(b) n × n
(c) m × m
(d) n × m
2 −1 4
24. The matrix [ 1 0 −5] is
−4 5 7
(a) a symmetric matrix
(b) a skew-symmetric matrix
(c) a diagonal matrix
(d) none of these
3 −2 2
25. If A =[ ] , then the value of k if, 𝐴 = 𝑘𝐴 − 2𝐼 is
4 −2
(a) 0
(b) 8
(c) – 7
(d) 1
CASE STUDY QUESTIONS
1.A manufacture produces three stationery products Pencil, Eraser and Sharpener which he sells in two markets.
27
−2 2
(ii)[ ]
−4 −6
−2 2
(iii)[ ]
−6 −4
−6 −2
(iv)[ ]
2 4
(d) AC − BC is equal to
−4 −6
(i)[ ]
−4 4
−4 −4
(ii)[ ]
4 −6
−4 −4
(iii)[ ]
−6 4
−6 4
(iv)[ ]
−4 −4
(e) (a + b)B is equal to
0 8
(i)[ ]
10 2
2 10
(ii) [ ]
88 00
(iii)[ ]
2 10
2 0
(iv)[ ]
8 10
3. Two farmers Ramakishan and Gurucharan Singh cultivate only three varieties of rice namely Basmati,
Permal and Naura. The sale (in rupees) of these varieties of rice byboththe farmers inthe month of
September and October are given by the following matrices A and B.
(a) The total sales in September and October for each farmer in each variety can be represented as
(i) A+B
(ii) A–B
(iii) A>B
(iv) A<B
(b) What isthe value of A23?
(i) 10000
(ii) 20000
(iii) 30000
(iv) 40000
(c) The decrease in sales from September to October is given by .
(i) A+B
28
(ii) A–B
29
(iii) A>B
(iv) A<B
(d) If Ramkishan receives 2% profit on gross sales, compute his profit for each variety sold in October. (i) ₹ 100,
₹ 200 and ₹ 120
(ii) ₹ 100, ₹ 200 and ₹ 130
(iii) ₹ 100, ₹ 220 and ₹ 120
(iv) ₹ 110, ₹ 200 and ₹ 120
(e) If Gurucharan receives 2% profit on gross sales, compute his profit for each variety sold in September. (i) ₹
100, ₹ 200, ₹ 120
(ii) ₹ 1000, ₹ 600, ₹ 200
(iii) ₹ 400, ₹ 200, ₹ 120
(iv) ₹ 1200, ₹ 200, ₹ 120
4. Assume the following data regarding the number of USB cables and their types manufactured in the
company I, II and III per day.
Type A Type B Type C
I 40 30 50
II 20 80 10
III 40 60 5
(b) What does the element of 3rd row and 3rd column represents?
(i) Number of USB type ‘C’ = 5 Produced by company = III
(ii) Number of USB type ‘C’ = 50 Produced by company = III
(iii) Number of USB type ‘C’ = 40 Produced by company = III
(iv) Number of USB type ‘C’ = 5 Produced by company = I
(c) How many USB cables are produced by company I in 3 days?
(i) 120
(ii) 360
30
(iii) 90
31
(iv) 150
(d) How many USB cables are produced by all the companies in 2 days?
(i) 670
(ii) 560
(iii) 870
(iv) 1050
(e) How many USB cables of C-type are produced by company II?
(i) 10
(ii) 5
(iii) 50
(iv) 60
1.In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R).
Mark the correct choice as.
(A) Both A and R are true and R is the correct explanation of A
(B) Both A and R are true but R is not the correct explanation of A
(C) A is true but R is false
(D) A is False and R is True.
i)Assertion (A) : If A is a square matrix such that A2 = A , then ( I + A )2 – 3A = I
Reason (R) : AI = IA = A
7 0 0
Ii)Assertion (A) :[0 7 0 ] is a scalar matrix .
0 0 7
Reason (R) : If all the elements of the principal diagonal are equal , it is called a scalar matrix.
iii) Assertion (A) : (A + B )2≠ A2 + 2AB + B2
Reason (R) : Generally AB ≠ BA
iv) A and B are two matrices such that AB and BA are defined
Assertion (A) : (A + B ) ( A – B ) = A2 – B2
Reason (R) : ( A + B ) ( A – B ) = A2 – AB + BA – B2
v) Let A and B be the two symmetric matrices of order 3
Assertion (A) : A(BA) and (AB)A are symmetric matrices
Reason (R) : AB is symmetric matrix if matrix multiplication of A with B is commutative .
0 2b −2
vi) Assertion (A) : If the matrix P = [ 3 1 3] is a symmetric matrix ,
3a 3 3
−2 3
then a = and b =
3 2
P’
Reason (R) : If P is a symmetric matrix , Then = -P
vii) Assertion (A) : If A is a symmetric matrix, then B’AB is also symmetric
Reason (R) : (ABC)’ = C’B’A’
viii) Assertion (A) : If A and B are symmetric matrices , then AB – BA is a skew-symmetric matrix.
Reason (R) : (AB)’ = B’ A’
32
DETERMINANTS
MULTIPLE CHOICE QUESTIONS
1. If A is a square matrix of order 3, such that A(adjA) = 10I, then |adj A| is equal to
(a) 1
(b) 10
(c) 100 (d)
1000
Ax2 x3 1 Ax By Cz
(a) ∆ + ∆1 = 0
(b) ∆ ≠ ∆1
(c) ∆ = x∆1
(d) ∆ – ∆1 = 0
3. Let A be a square matrix of order 2 × 2, then |KA| is equal to (a)
K|A| (b) K2|A| (c) K3|A| (d) 2K|A|
a11 a12 a13
4. If ∆ =|a21 a22 a23 | and Aij is cofactor of aij, then the value of ∆ is given by (a) a11A31 + a12A32 + a13A33
a31 a32 a33
(b) a11A11 + a12A21 + a13A31 (c)
a21A11 + a22A12 + a23A13
(d) a11A11 + a21A21 + a31A31
5. If A and B are invertible matrices then which of the following is not correct
(a) AdjA = |A|.A–1
(b) det(A–1) = (det A)–1
(c) (AB)–1 = B–1A–1
(d) (A + B)–1 = A–1 + B–1
6. Let A be a non-angular square matrix of order 3 × 3, then |A . adj A| is equal to
(a) |A|3
(b) |A|2
(c) |A|
(d) 3|A|
7. Let A be a square matrix of order 3 × 3 and k a scalar, then |kA| is equal to
(a) k |A|
(b) |k| |A|
(c) k3 |A|
(d) none of these
a a2 1 + a3
8. If a, b, c are all distinct, and|b b2 1 + b3 | = 0 = 0, then the value of abc is
c c2 1 + c3
(a) 0
(b) –1
(c) 3
(d) –3
x+1 x+2 x+a
9. If a, b, c are in AP, then the value of |x + 2 x+3 x + b| is
33
x+3 x+4 x+c
(a) 4
(b) –3
34
(c) 0
(d) abc
10. If A is a skew-symmetric matrix of order 3, then the value of |A| is
(a) 3
(b) 0
(c) 9
(d) 27
2 3 2
11. If |x x x| + 3 = 0, then the value of x is
4 9 1
(a) 3
(b) 0
(c) -1
(d) 1
12. Let A = [200 50] and B = [50 40], then |AB| is equal to
10 2 2 3
(a) 460
(b) 200
(c) 3000
(d) -7000
a 0 0
13. If A = [0 a 0], then det(adj A) equals
0 0 a
(a) a27
(b) a9
(c) a6
(d) a2
14. If A is any square matrix of order 3x3 such that |A| = 3, then the value of |adj A| is (a)3
1
(b)
3
(c) 9
(d) 27
5| = |6 −2
15. If |2x |, then the value of x is
8 x 7 3
(a) 3
(b) ±3
(c) ±6
(d) 6
16. The area of a triangle with vertices (−3,0), (3,0)and (0, k) is 9 sq. units. Then, the value of k will be
(a) 9
(b) 3
(c) -9
(d) 6
17. If A and B are invertible matrices, then which of the following is not correct?
(a) adj A = |A|. A−1
(b) det(A−1) = (det(A))−1
(c) (AB)−1 = B−1A−1
(d) (A + B)−1 = B−1 + A−1
1 2
18. Adjoint of matrix [ ] is
3 4
4 2
(a) [ ]
3 1
4 −2
(b) [ ]
35
−3 1
36
1 2
(c) [ ]
3 4
1 −2
(d) [ ]
−3 4
2 −3
19. If A = [ ], then A−1 will be
1 2
3 4
3
(a) [ ]
−3
17 4
14 3
(b) [ ]
17 −3 2
−1 4 3
(c) [ ]
17 −3 2
1 4 3
(d) [ ]
17 −3 −2
20.For any square matrix A, AAT is a
(a) Unit matrix
(b) Symmetric matrix
(c) Skew symmetric matrix
(d) Diagonal matrix
21. Which of the following is not true?
(a) Every skew-symmetric matrix of odd order is non-singular
(b) If determinant of a square matrix is non-zero, then it is non singular
(c) Adjoint of symmetric matrix is symmetric
(d) Adjoint of a diagonal matrix is diagonal
22. If a matrix A is such that 3A3 + 2A2 + 5A + I = O then its inverse is (a) −(3A2 + 2A
+ 5I)
(b) (3A2 + 2A + 5I)
(c) (3A2 − 2A + 5I)
(d) None of these
23.If the order of matrix A is m x p and the order of B is p x n. Then the order of matrix AB is?
(a) m × n
(b) n × m
(c) n × p
(d) m × p
1 4
24. What is x if [ ] is a singular matrix?
2 x
(a) 5
(b) 6
(c) 7
(d) 8
a b g h i
38
information given above, answer the following questions:
(a) The equations in terms of X and Y are (i)
x – y = 50, 2x – y = 550
(ii) x – y = 50, 2x + y = 550
(iii) x + y = 50, 2x + y = 550
(iv) x + y = 50, 2x + y = 550
(b) Which of the following matrix equation is represented by the given information:
1 −1 x 50
(i)[ ] [y] = [ ]
21 11 550
(ii)[ ] [y]x= [ 50 ]
2 1 550
1 1 x 50
(iii)[ ] [y] = [ ]
2 −1 550
x
1 1
(iv)[ ] [y] = [ −50 ]
2 1 −550
(c) The value of x (length of rectangular field) is
(i) 150 m
(ii) 400 m
(iii) 200 m
(iv) 320 m
(d) The value of y (breadth of rectangular field) is
(i) 150 m
(ii) 200 m
(iii) 430 m
(iv) 350 m
(e) How much is the area of rectangular field?
(i) 60000 sq m.
(ii) 30000 sq m.
(iii) 30000 m
(iv) 3000 m
2. Raja purchases 3 pens, 2 pencils and 1 mathematics instrument box and pays ₹ 41 to the shopkeeper. His
friends, Daya and Anil purchases 2 pens, 1 pencil, 2 instrument boxes and 2 pens, 2 pencils and 2 mathematical
instrument boxes respectively. Daya and Anil pays ₹ 29 and ₹ 44 respectively. Based on the above information
answer the following:
(a) The cost of one pen is
(i) ₹2
(ii) ₹5
(iii) ₹ 10
(iv) ₹ 15
(b) The cost of one pen and one pencil is
(i) ₹5
(ii) ₹ 10
39
(iii) ₹ 15
(iv) ₹ 17
40
(c) The cost of one pen and one mathematical instrument box is
(i) ₹7
(ii) ₹ 10
(iii) ₹ 15
(iv) ₹ 18
(d) The cost of one pencil and one mathematical instrumental box is
(i) ₹5
(ii) ₹ 10
(iii) ₹ 15
(iv) ₹ 20
(e) The cost of one pen, one pencil and one mathematical instrumental box is
(i) ₹ 10
(ii) ₹ 15
(iii) ₹ 22
(iv) ₹ 25
3. The management committee of a residential colony decided to award some of its members (say x) for honesty,
some (say y) for helping others and some others (say z) for supervising the workers to kept the colony neat and
clean. The sum of all theawardees is 12. Three times the sum of awardees for cooperation and supervision
added to two times the number of awardees for honesty is 33. The sum of the number of awardees for honesty
and supervision is twice the number of awardeesfor helping.
(i) Value of x + y + z is
(a) 3
(b) 5
(c) 7
(d) 12
(i) Value of x − 2 y is
(a) z
(b) -z
(c) 2z
(d) -2z
(i) The value of z is
(a) 3
(b) 4
(c) 5
(d) 6
(iv) The value of x + 2 y is
(a) 9
(b) 10
41
(c) 11
(d) 12
42
(v) The value of 2x + 3y + 5z is
(a) 40
(b) 43
(c) 50
(d) 53
4. Read the following text and answer the following questions on the basis of the same:
Two schools Oxford and Navdeep want to award their selected students on the values of sincerity, truthfulness
and helpfulness. Oxford wants to award Ex each, y each and z each for the three respective values to 3,2 and 1 students
respectively with a total award money of 1600. Navdeep wants to spend 2300 to award its4, 1and3 studentsonthe
respectivevalues(bygivingthe sameamount tothethree valuesasbefore).The total amount of the award for one
prize on each is ₹900.
(i) Value of x + y + z is
(a) 800
(b) 900 (c)
1000 (d)
1200
(ii) Value of 4x + y + 3z is (a)
1600
(b)2300
(c) 900
(d) 1200
(iii) The value ofy is
(a) 200
(b) 250
(c) 300
(d) 350
(iv) The value of 2x + 3 y = ⋯ … … … ..
(a) 1000
(b)1100
(c) 1200
(d)1300
(v) The value of y − x = ⋯ … … … …
(a) 100
(b) 200
(c) 300
(d) 400
43
ASSERTION AND REASON
1.In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R).
Mark the correct choice as.
(A) Both A and R are true and R is the correct explanation of A
(B) Both A and R are true but R is not the correct explanation of A
(C) A is true but R is false
(D) A is False and R is True.
i)Let A be a 2 × 2 matrix
Assertion (A) : adj (adj A) = A
Reason (R) :|adj A| = |A|
1 0 0
2 0 0 2
1
, then A = 0 0
-1
ii)Assertion (A) : if A =[0 3 0] 3
0 0 4 1
[0 0 4 ]
Reason (R): The inverse of an invertible diagonal matrix is a diagonal matrix.
iii) Assertion (A) : if every element of a third order determinant of value ∆ is multiplied by 5 , then the value of
new determinant is 125∆
Reason (R) : If k is a scalar and A is an n × n matrix , then |kA| = kn|A|
1 3 γ+2
iv) Assertion (A) : If the matrix A = [2 4 8 ] is singular , then γ = 4
3 5 10
Reason (R) : If A is a singular matrix , then |A| = 0
2 −3
v) Given A= [ ]
−4 7
Assertion (A) : 2A-1 = 9I – A
1
Reason (R) : A-1 = (adjA)
|A|
2 3 −1 1
vi) Assertion (A) : If A = [
−2] A
and = kA , then k =
5 9
1
Reason(R): |A−1 |=
|A|
vii) Consider the system of equations: x + y + z = 2 , 2x + y - z = 3
and 3x + 2y + kz = 4
Assertion (A) : The system of equations has unique solution if k≠ 0
Reason(R): The system of equations has unique solution if |A| = 1
viii) Consider the system of equations: x + 2y + 5z = 10, x - y - z = - 2 and 2x + 3y - z = -11 Assertion(A)
: The system of equations has unique solution if x = -1 , y = -2 and z = 3 Reason(R): If |A| = 0
then the system of linear equations has no solutions.
44
CONTINUITY AND DIFFERENTIABILITY
𝑥2
Q1. If f(x) =2x and 𝑔(𝑥) = + 1 then which of the following can be a discontinuous function?
2
𝑓(𝑥)
(A) F(x)+g(x) (B)f(x)-g(x) (C)f(x).g(x) (D)
𝑔(𝑥)
4−𝑥2
Q2. The function 𝑓(𝑥) = is
4𝑥−𝑥 3
(A) Discontinuous at only one point (B)Discontinuous at only two point (C)Discontinuous at only three point
(D)none of the above
Q3. The function 𝑓(𝑥) = 𝑒|𝑥| is
(A)Continuous everywhere but not differentiable at x=0
(B)Continuous and differentiable everywhere
(C) Not continuous at x=0
(D)None of the above
1
Q4. If 𝑓(𝑥) = 𝑥2 sin ( ) where 𝑥 ≠ 0 then the value of the function f at x = 0,so that the function is
𝑥
continuous at x = 0 is
(A) 0 (B) -1 (C) 1 (D) none of these
Q5. The derivative of 𝑐𝑜𝑠 (2𝑥 − 1) with respect to 𝑐𝑜𝑠−1𝑥
−1 2
−1 2
(A) 2 (B) 2
(C) (D)1 − 𝑥2
2√1−𝑥 𝑥
Q6. If 𝑦 = √𝑠𝑖𝑛𝑥 + 𝑦,then 𝑑𝑦
is equal to
𝑑𝑥
𝑑𝑥 3
√3 √3 1
(A) (B)− (C) (D) 1
2 2
Q10. If 𝑥𝑦2 = 𝑒𝑥−𝑦 dy
,then d𝑥 is
1+𝑥 1−𝑙𝑜𝑔𝑥 𝑥 𝑥
(A) (B) (C) (D)
1+𝑙𝑜𝑔𝑥 1+𝑙𝑜𝑔𝑥 1+𝑙𝑜𝑔𝑥 (1+𝑙𝑜𝑔𝑥)2
𝑑𝑦
Q12. Ifsin(𝑥 + 𝑦) = log(𝑥 + 𝑦) then is
𝑑𝑥
(A)2 (B) -2 (C)1 (D) -1
45
𝑑𝑦
Q13. If t= ex and y= t2-1 then at t = 1 is
𝑑𝑥
1 1
(A) (B) (C) 2 (D) 2𝑒2
2𝑒2 2
1 𝑑𝑦
Q14. If 8𝑓(𝑥) + 6𝑓 ( ) = 𝑥 + 5 and 𝑦 = 𝑥 2𝑓(𝑥) then the value of at x= -1 is
𝑥 𝑑𝑥
1 −1
(A)0 (B) (C) (D) 1
14 14
46
𝑑𝑦 𝜋
Q15. If = 𝑙𝑜𝑔√𝑡𝑎𝑛𝑥 , then the value of at 𝑥 = is
𝑑𝑥 4
1
(A) ∞ (B) 1 (C) 0 (D)
2
𝑑𝑦
Q16. If 𝑠𝑖𝑛𝑦 = acos(𝑎 + 𝑦) then is equal to
𝑑𝑥
𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥 𝑑𝑦
Q17. If 𝑦 = 𝑡𝑎𝑛 −1 [ ],then is equal to
𝑐𝑜𝑠𝑥−𝑠𝑖𝑛𝑥 𝑑𝑥
1
(A) (B)0 (C) 1 (D)-1
2
1−𝑥 2 𝑑𝑦
Q18. If 𝑦 = 𝑠𝑖𝑛 −1 [ ] is equal to
1+𝑥2 𝑑𝑥
−2 2 1 2
(A) (B) (C) (D)
1+𝑥2 1+𝑥2 2−𝑥2 2−𝑥2
(A)-2x+9 for all x∈R (B)2x-9 if 4<x<5 (C)-2x+9 if 4<x<5 (D)none of these Q21. If 𝑓 (𝑥 ) = √𝑥 2 − 10𝑥 + 25, then the derivative of
f(x) in the interval [0,7] is (A)1 (B)-1 (C)0 (D)none of these Q22.
Derivative of sin 𝑥 w.r.t cos 𝑥 is
(A)− cot 𝑥 (B)cot x (C) tan 𝑥 (D)none of these
𝑑𝑦
Q23. If 𝑦 = log|3𝑥|, 𝑥 ≠ 0, then is
𝑑𝑥
3 1 1
(A) (B) (C) (D) none of these
𝑥 𝑥 3𝑥
𝑑𝑦
Q24. If |𝑥|< 1 and 𝑦 = 1 + 𝑥 + 𝑥 2 + ⋯ 𝑡𝑜∞, then is
𝑑𝑥
1 1
(A) (1−𝑥2)2 (B) (1+𝑥2)2 (C)(1 − 𝑥 2) 2 (D)none of these
𝑥+1 𝑥−1 𝑑𝑦
Q25. If 𝑦 = 𝑠𝑒𝑐 −1 ( ) + 𝑠𝑖𝑛 −1 ( ), then is
𝑥−1 𝑥+1 𝑑𝑥
𝑘 , 𝑥 =2
(A)5 (B) 1 (C) 7 (D) 10
𝑥2−16
Q30. If 𝑓(𝑥) = { , 𝑥≠ 4
𝑥−4 is continuous at x = 4, find k
𝑘 , 𝑥 =4
47
(A)3 (B) 5 (C)10 (D)8
48
1. What will be the height of the plant after 2 days ?
a. 4cm
b. 6cm
c. 8cm
d. 10cm
𝑑𝑦
2. For what value of x , =0
𝑑𝑥
a. 3
b. 4
c. 5
d. 2
𝑑𝑦
3. For the value of x where = 0 the height of the plant is maximum. What is the maximum height of the plant ?
𝑑𝑥
a. 4cm
b. 6cm
c. 8cm
d. 10cm
𝑑2𝑦
4. What is the value of 𝑑𝑥2 at x=2 ?
a. -2
b. -4
c. -5
d. -1
𝑑2𝑦
5. If y = 𝑒𝑥𝑠𝑖𝑛𝑥 what is .
𝑑𝑥 2
a. 𝑒𝑥(𝑠𝑖𝑛𝑥 +cosx)
b. 2𝑒𝑥𝑐𝑜𝑠𝑥
c. 2 𝑒𝑥𝑠𝑖𝑛𝑥
d. none of these
CASE STUDY-2
A potter made a mud vessel , where the the shape of the pot is based on f(x) = |x-3| + |x-2|, where f(x) represents height of
the pot.
49
1. When x>4 what will be the height in terms of x ?
a. x-2
50
b. x-3
c. 2x-5
d. 5-2x
𝑑𝑦
2. Whatis at x=3 ?
𝑑𝑥
a. 2
b. -2
c. Function is not differentiable
d. 1
3. When the value of x lies between (2,3) then the function is
a. 2x-5
b.5-2x
c. 1
d. 5
4. If the potter is trying to make a pot using the function f(x)= [x] will he get a pot or not ?
a. Yes, because it is a continuous function
b. Yes, because it is not a continuous function
c. No, because it is a continuous function
d. No, because it is not a continuous function
5. What is the value of derivative of f(x) =[x] at the point x=8 ?
a. 1
b. f(x) is not differentiable at x=8
c. 0
d. none of these
CASE STUDY- 3
𝑥 𝑥
Amanufacturer cansell x itemsata priceof rupees (5- ) each.The cost priceofxitemsis rupees( + 500) .
100 5
a. 5x + ( )
100
𝑥2
b. 10x + ( )
100
𝑥2
c. 5x - ( 100
)
𝑥2
d. 10x - ( 100
)
2. The value of profit function P(x) willbe:
𝑥2
a. (24)x +( 100
) +500
5
24 𝑥2
b. ( )x +( ) -500
5 100
5 𝑥2
c. (24)x -(
51
100 )+
500
24 𝑥2
d. ( )x -( ) -500
5 100
52
24 𝑥
b. +
5 50
24 𝑥
c. − −
5 50
24 𝑥
d. − +
5 50
4. The second derivative of profit function P(x)
1
a.
50
−1
b.
50
c. 1
d. 0
5. For what value of P(x), P’(x) =0 a.
120
b. 60
c. 240
d. 24
CASE STUDY- 4
A gardener wants to construct a rectangular bed of garden in a circular patch of land. He takes the
maximum perimeter of the rectangular region as possible. Refer the image. Radius of the circular patch of land is a.
In the rectangular region he wants to plant flowers.
53
5. The value of x at which P’(x) =0 is
𝑎
a.
2
𝑎
b.
√2
c. 2a
𝑑. √2 a
DIRECTION: In the following questions, a statement of assertion (A) is followed by a statement of reason (R) . Mark the correct choice
as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
(b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
(e) Both Assertion (A) and reason (R) are false.
𝜋
1. Assertion(A): f(x) = 𝑡𝑎𝑛2𝑥 is continuous at x= .
2
Reason(R):𝑥2 𝜋
is continuous at x= .
2
2. Assertion(A): f(x) = |sin x|is continuous for all x ∈ R
Reason(R): sin x and |x| are continuous at on R.
3. Assertion(A): f(x) = |sin x|is continuous x=0.
Reason(R): |sin x| is differentiable at x=0.
𝑘𝑥 , 𝑥 < 0
4. Consider the function f(x) = 𝑓(𝑥) = {|𝑥| Which is continuous at x=0.
3𝑥 ≥ 0
𝑑𝑦 𝑑
9. Assertion(A): If y = 𝑥𝑥 then can be found by applying the formula of (𝑥 𝑛) = n 𝑥𝑛−1
𝑑𝑥 𝑑𝑥
Reason (R): Using logarithm the derivative of y= 𝑥𝑥 can be found.
𝑑𝑦
10.Assertion(A): If y = 𝑡𝑎𝑛5𝑥 then = 5 𝑡𝑎𝑛4𝑥
𝑑𝑥
𝑑 𝑑𝑦 𝑦(1−𝑥)
Reason (R): (𝑥𝑛) = n 𝑥𝑛−1. 1 s on(A): If 𝑒𝑥+𝑦 = 𝑥𝑦 then is .
𝑑𝑥 2 e
11. Assertion(A): If y= log √𝑡𝑎𝑛𝑥, then the value of . r
A t
Reason (R): The value of log 1 is not defined. s i
54
𝑑𝑦
at 𝜋 is ∞.
𝑑𝑥 4
x
=
𝑑𝑥 𝑥(𝑦−1)
Reason (R): The value of log e = 1
55
APPLICATION OF DERIVATIVES
Q.14. The point on the curve y2 = x, where the tangent makes an angle of π/4 with x-axis is (a)(½, ¼)
(b) ( ¼ , ½ ) (c) (4, 2) (d) (1, 1)
Q.15. The slope of normal to the curve y = 2x2 + 3 sin x at x = 0 is (a) -1/3
56
Q. 16. The line y = x + 1 is a tangent to the curve y2 = 4x at the point (a) (1, 2)
(b) ( 2 , 1) ( -1, 2 ) (d) ( -1 , -2 )
Q. 17. The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to the line x + 3y = 8 is
57
(a) x + 3y = 8 (b) x + 3y + 8 = 0 (c) x + 3y ± 8 = 0 (d) None of These
Q. 18. The tangent to the curve y = e2x at the point (0, 1) meets x-axis at
−1 1 2
(a) ( , 0) (b) ( , 0) (c) ( , 0) (d) None these
2 2 3
Q..19. The slope of tangent to the curve x = t2 + 3t − 8 and y = 2t2 − 2t − 5 at t = 2 is (a) 7/6 (b)
6/7 (c) -7/6 (d) -6/7
Q.20. The abscissaof the point on the curve 3y = 6x − 5x3, the normal at which passes through the origin is (a) 1 (b) 2
(,c) -1 (d) -2
𝜋
Q. 21 The Equation normal to the curve y=x + Sinx + Cosx at x = is
2
a) x = 2 b) x = 𝜋 c) x+𝜋=0 d) 2x =
Q.22 2
The Point on the curve y = x - 3x + 2 where tangent is perpendicular to y=x is
a) ( ½, ¼) b) (¼ , ½) c) ( 4 ,2 ) d) ( 1 ,1 )
Q.23 The point on the curve y2 = x where tangent makes 450 angle with x-axis is
(a) (0,0) (b) (2,16) (c) (3, 9) (d) none of these
Q.24 The angle between the curves y2 = x and x2 = y at (1,1)is:
4 3
(a) tan-1 (b) tan-1 (c ) 900 (d) 450
3 4
Q.25 At what point the slope of the tangent to the curve x2+y2− 2x −3 is zero?
a) (3, 0), (−1, 0) (b)(3,0),(1,2) (c) (−1, 0),(1,2) (d) (1, 2),(1, − 2)
2 3 0
Q.26 If the curve ay + x = 7 and x = y cut each other at 90 at ( 1 , 1) , then value of a is :
a) 1 b) -6 c) 6 d) 0
Q.27 The equation of normal x = acos θ , y=a sin θ at the point θ= 𝜋/4 is
3 3
a) -2 b) 0 c) 2 d) can’t be determine
Q.37 The maximum value of y = sinx. cosx is
58
1 1
(a) (b) (c) √2 (d) 2√2
4 2
Q.38. If the function f(x) = x3+ax2+bx+1 is maximum at x=0 and x=1 then :
59
2 3 3
(a) a = , b = 0 (b) a=- , b=0 (c) a = 0 , b = (d) None of These
3 2 2
1. Yash wants to prepare a handmade gift box for his friend’s birthday at his home. For making
lower part of the box, he took a square piece of paper of each side equal to 10 cm.
61
2. A tank with rectangular base of length x metre, breath y metre and rectangular side, open
at the top is to be constructed so that the depth is 1 m and volume is 9𝑚3.If buildingof
tankisRs 70persquaremetrefor the baseand Rs45persquare metrefor the sides?
62
(iv) What is the value of A”(2.5)
15 30 15 30
(a) − (b) − (c) (d) −
√18.75 √18.75 √18.75 √80
(a)75√18.75𝑐𝑚2 (b)10√18.75𝑐𝑚2
(c)75√5cm2 (d)75√7𝑐𝑚2
4. A piece of wire of length 25cm is to be cut into pieces one of which is to bent into the form
of a square and other into the form of a circle.
(iii) If we talk about total length of wires then what is the relation between x and y?
(a)X+y=25 (b) x+y=28 (c) x+y =26 (d) x+y =27
𝑑𝐴
(iv) When = 0, then find the value of y
𝑑𝑦
50 100 25 50𝜋
(a) (b) (c) (d)
𝜋+4 𝜋+4 𝜋+4 𝜋+4
63
5. Thesumofthelength hypotenuseand asideofaright-angled triangleisgivenby AC+BC = 10
64
Based on the above information answer the following questions:
(i) Base BC =?
100−𝑐2 100+𝑐2 𝑐2−100 10−𝑐2
(a) (b) (c) (d)
20 20 20 20
𝑑𝑆
(ii) If ‘S’ be the area of the triangle, then find the value of ?
𝑑𝑐
100−3𝑐2 100−3𝑐2 3𝑐2−100 100+3𝑐2
(a) (b) (c) ) (d)
20 40 20 40
𝑑𝑆
20√3 15√3
(𝑎) 10√3 (b) (c)
5√3
(d)
3 3 2 3 3
𝑑 𝑆 10√3
(iv) Find the values of at c =
𝑑𝑐2 3
3 √3 −√3 1
(a)- (𝑏) (c) (c)
2 2 2 2
6. The front gate of a building is in the shape of a trapezium as shown below. Its three sides other than base are
10m each.The height of the gate is hmeter. Onthebasis ofthis information and figure given below answer
the following questions:
b.(10-x)√100 + 𝑥 2
c.(10+x)√100 − 𝑥 2
66
d. (10-x)√100 − 𝑥2
𝑑𝐴
(ii) .The value of is
𝑑𝑥
2𝑥2+10𝑥−100
a. √100−𝑥2
2𝑥2−10𝑥−100
b. √100−𝑥2
2𝑥2+10𝑥+100
c. √100−𝑥2
−2𝑥2−10𝑥+100
d.
√100−𝑥2
b. 5
c. 20
d. 15
𝑑𝐴
(iv). If at the value of x where =0 area of trapezium is maximum then what is maximum area of trapezium ?
𝑑𝑥
a. 25√3 sqm
b. 100√3 sqm
c. 75√3 sqm
d. 50√3 sqm
2𝐴
(v). If the area of trapezium is maximum then the value of 𝑑 is2
𝑑𝑥
a. positive
b. negative
c. 0
d. none of these
67
C. A is true but R is false
D. A is false but R is true
68
2. . Assertion (A) Tangent to the curve 𝒚 = 𝟐𝒙𝟑 + 𝒙𝟐 + 𝟐 at the point (-1,0) is parallel to the line y = 4x+3
Reason (R): Slope of the tangent at (-1,0) is 4 equal to the slope of the given line .
A. A is false but R is true
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. Both A and R are true and R is the correct explanation of A
3. Assertion(A): Function f(x)== 𝐱 𝟑 − 𝟑𝐱𝟐 + 𝟑𝐱 + 𝟐 isalwaysincreasing.
Reason(R):Derivative f’(x) is always negative.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
𝝅
4. Assertion(A): Y = sinx is increasing in the interval ( , 𝝅)
𝟐
Reason(R): 𝒅𝒚 is negative in the given interval.
𝒅𝒙
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
5. Assertion(A): 𝒚 = 𝒆𝒙 is always strictly increasing function.
Reason (R): 𝒅𝒚 = 𝒆𝒙 > 0 for all real values of x.
𝒅𝒙
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
𝟐𝒙
6. Assertion(A): 𝒚 = 𝐥𝐨𝐠(𝟏 + 𝒙) − ,x>-1 is a decreasing function of x throughout its
𝟐+𝒙
domain
Reason (R ): 𝒅𝒚 > 0 for all x>-1
𝒅𝒙
69
C. A is true but R is false
D. A is false but R is true
𝝅
8. Function f(x) = logcosx is strictly increasing on (𝟎, )
𝟐
Reason( R): Slope of tangent on the above curve is negative in the given interval.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
9. Assertion (A): Slope of the tangent to the curve y = 𝟑𝒙𝟒 − 𝟒𝒙 at x=4 is 764
𝒅𝒚
Reason (R): The value of = 𝟏𝟐𝒙𝟑 − 𝟒 is 764 at x= 4
𝒅𝒙
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
10. Assertion(A): Tangents to the curve y = 𝟕𝒙𝟑 + 𝟏𝟏 at the points where x = 2 and x
= – 2 are parallel.
Reason(R): Slope of tangents at both the points are equal.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
11. Assertion(A): At the (3,27) on the curve 𝒚 = 𝒙𝟑, slope of the tangent is equal to y
coordinate of the point.
𝒅𝒚
Reason (R): = 𝟑𝒙𝟐 = 𝟐𝟕 𝒂𝒕 𝒙 = 𝟑
𝒅𝒙
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
12. Assertion(A): The line y = x + 1 is a tangent to the curve 𝒚𝟐= 4x at the point (1,2).
Reason (R) : Slope of tangent to the given curve at the given point is 1 and the point
also satisfies equation of the line.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
70
13. Assertion (A): x = 0 is the point of local maxima of the function f given by 𝒇 =
𝟑𝒙𝟒 + 𝟒𝒙𝟑 − 𝟏𝟐𝒙𝟐 +12
Reason(R): 𝒇′(𝒙) = 𝟎 𝒂𝒕 𝒙 = 𝟎 𝒂𝒏𝒅 𝒂𝒍𝒔𝒐 𝒇′′(𝒙) < 0 𝑎𝑡 𝑥 = 0
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
14. Assertion (A): Maximum value of the function f(x) =(𝟐𝒙 − 𝟏)𝟐 + 𝟑 is 3.
Reason(R): f(x)≥ 𝟑 for all real values of x.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
15. Assertion f(x) = 𝒆𝒙 do not have maxima and minima
Reason ( R) : f ’(x) =𝒆𝒙 ≠ 𝟎 for all real values of x.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
LINEAR PROGRAMMING
72
Constraints 𝑥 + 2𝑦 ≤ 70 , 2𝑥 + 𝑦 ≤ 95 , 𝑥, 𝑦 ≥ 0is obtained, is
(a) (30, 25) (b) (20, 35 ) (c ) (35 ,20 ) (d) (40 , 15)
73
19. The corner points of the feasible region determined by the following
System Of linear inequalities: 2𝑥 + 𝑦 ≤ 10 , 𝑥 + 3𝑦 ≤ 15 ,
𝑥, 𝑦 ≥ 0 are (0,0),(5,0), (3,4) and (0, 5 ) .
Let 𝑍 = 𝑝𝑥 + 𝑞𝑦, where 𝑝 , 𝑞 > 0.Condition on 𝑝 and 𝑞so that the maximum of 𝑍
occurs at both ( 3, 4 ) and ( 0, 5) is
(a) 𝑝 = 𝑞 (b) 𝑝 = 2𝑞 (c) 𝑝 = 3𝑞 (d) 𝑞 = 3𝑝
20. Solution set of inequations 𝑥 − 2𝑦 ≥ 0, 2𝑥 − 𝑦 ≤ −2 , 𝑥 ≥ 0, 𝑦 ≥ 0 is
(a) First quadrant (b) infinite
(c ) Empty (d) closed half plane
CASE STUDY QUESTIONS
I. A small firm manufacturers gold rings and chains. The total number of rings and chains manufactured
per day is atmost 24 . it takes 1 hour to make ring and 30 minutes to make a chain . The maximum number of hours
available per day is 16 . If the profit on a ring is Rs.300 and that on a chain is Rs.190 . Firm is concerned about earning
maximum profit on the number of rings(𝑥) and chains(𝑦) that have to be manufactured per day .
Using the above information give the answer of the following questions.
(i) To get maximum profit how many first class tickets should be sold –
(a) 20 (b) 180 (c) 160 (d) 40
(ii) Difference between the maximum profit and minimum profit is equal to
(a) 8000 (b) 56000 (c) 64000 (d) none of the above
(iii) Corner points of feasible region are
(a) (20,180) (b) (20,0) (c) (40,0) (d) all theabove
(iv) Minimum profit is equal to
(a) 8000 (b) 6000 (c) 64000 (d) none of the above
(v) The objective function is
(a) 400𝑥 + 300𝑦 (b) 300𝑥 + 400𝑦 (c) 𝑥 + 𝑦 (d) none of the above
74
(b) A is true, R is true; R is not a correct explanation for A.
(c) A is true: R is false.
(d) A is false: R is true.
1. Assertion (A) Maximum value of 𝑍 = 11𝑥 + 7𝑦 , subject to constraints 2𝑥 + 𝑦 ≤ 6, 𝑥 ≤ 2 , 𝑥 ≥ 0 , 𝑦 ≥ 0 will be obtained at (0,6).
Reason (R)In a bounded feasible region, it always exist a maximum and minimum value.
2. Assertion (A)The linear programming problem, maximize 𝑍 = 2𝑥 + 3𝑦
subject to constraints 𝑥 + 𝑦 ≤ 4 , 𝑥 ≥ 0 , 𝑦 ≥ 0
It gives the maximum value of Z as 8 .
Reason (R)To obtain maximum value of Z, we need to compare value of Z at all the corner points of the feasible region .
3. Assertion (A) For an objective function 𝑍 = 4𝑥 + 3𝑦 , corner points are (0,0), (25,0) , (16,16) and (0,24) . Then optimal
values are 112 and 0 respectively .
Reason (R) Themaximum or minimum values of an objective function is known as optimal value of LPP . These values are
obtained at corner points .
4. Assertion (A) Objective function 𝑍 = 13𝑥 − 15𝑦 , is minimized subject to constraints 𝑥 + 𝑦 ≤ 7 , 2𝑥 − 3𝑦 + 6 ≥ 0 , 𝑥 ≥ 0 , 𝑦 ≥ 0 occur
at corner point (0,2) .
Reason (R) If the feasible region of the given LPP is bounded , then the maximum or minimum values of an objective function
occur at corner points .
5. Assertion (A) Maximise𝑍 = 3𝑥 + 4𝑦, subject to constraints : 𝑥 + 𝑦 ≤ 1 , 𝑥 ≥ 0 , 𝑦 ≥ 0 . Then
maximum value of Z is 4 .
Reason (R) If the shaded region is not bounded then maximum value cannot be determined.
ANSWERS
RELATIONS AND FUNCTIONS
ANSWERS OF MCQ
5- Answer: (c) 24
75
23- Answer: (c) bijective
24- Answer/Explanation
Answer: (d) Explanation: (d), not reflexive, as l1 R l2
⇒ l1 ⊥ l1 Not true
Symmetric, true as l1 R l2 ⇒ l2R h
Transitive, false as l1 R l2, l2 R l3
⇒ l1 || l3 . l1 R l2.
25- Answer/Explanation
Answer: c
Explanation: (c), here (1,2) e R, (2,1) € R, if transitive (1,1) should belong to R.
26- Answer/Explanation
Answer: b
Explanation: (b), A relation R is an identity relation in set A if for all a ∈ A, (a, a) ∈ R.
27- Answer/Explanation
Answer: c
Explanation: (c), total injective mappings/functions= 4 P3 = 4! = 24.
CASE STUDY- 1
1- Sol. (a) reflexive
Explanation. Clearly, (1, 1), (2, 2), (3, 3), ∈ R. So, R is reflexive on A. Since, (1, 2) ∈ R but (2, 1)
∉ R. So, R is not symmetric on A. Since, (2, 3), ∈ R and (3, 1) ∈ R but (2, 1) ∉ R. So, R is not
transitive on A.
2- Sol. (b) Symmetric
Explanation. Since, (1, 1), (2, 2) and (3, 3) are not in R. So, R is not reflexive on A. Now, (1, 2) ∈
R ⇒ (2, 1) ∈ R and (1, 3) ∈ R ⇒ (3, 1) ∈ R. So, R is symmetric Clearly, (1, 2) ∈ R and (2, 1) ∈ R
but (1, 1) ∉ R. So, R is not transitive.
3- Sol. (c) transitive
Explanation. We have, R = {(x, y) : y = x + 5 and x < 4}, where x, y ∈ N.
∴ R = {(1, 6), (2, 7), (3, 8)} Clearly, (1, 1), (2, 2) etc. are not in R.
So, R is not reflexive. Since, (1, 6) ∈ R but (6, 1) ∉ R.
So, R is not symmetric. Since, (1, 6) ∈ R and there is not order pair in R which has 6 as the first
element.
Same is the case for (2, 7) and (3, 8). So, R is transitive.
CASE STUDY- 2
1. (d) (X,Y) ∉R
2. (a) both (X,W) and (W,X) ∈ R
3. (a) (F1,F2 ) ∈R, (F2,F3) ∈ R and (F1,F3) ∈ R
4. (c) Equivalence relation
5. (a) All those eligible voters who cast their votes
CASE STUDY- 3
1. (a) Reflexive and transitive but not symmetric
2. (a) 62
3. (d) None of these three
4. (d)212
5. (b) Reflexive and Transitive
76
CASE STUDY- 4
1- Sol. (a) R – {2}
Explanation. For f(x) to be defined x – 2 ≠ 0 i.e. x ≠ 2 ∴ Domain of f = R – {2}
2- Sol. (b) R – {1}
Explanation. Let y = f(x), then y = 𝒙 − 𝟏/𝒙 – 𝟐
xy – 2y = x – 1 ⇒xy– x = 2y – 1 ⇒ x = 𝟐𝒚 − 1/𝑦 – 1
Since, x ∈ R – {2}, therefore y ≠ 1 Hence, range of f = R – {1}
3- Sol. (d) 𝑥/𝑥 – 2
4- Sol. (a) One-one
Explanation. We have, g(x) = 𝑥 /𝑥 – 2
Let g(x1) = g(x2) ⇒x1 /𝑥1 – 2 = 𝑥2 /𝑥2 – 2⇒ x1x2– 2 x1 = x1 x2– 2 x2⇒ 2 x1 = 2 x2⇒ x1 = x2 Thus,
g(x1) = g(x2) ⇒ x1= x2 Hence, g(x) is one-one.
5- Sol. (c) f(x1) = f(x2) ⇒ x1 = x2
1- Answer- (a) 1
2- Answer- (d) π/3
3- Answer- (b) 3 cos x -1
Case Study 3
1. d 2. a 3. d 4. c 5. a 6. c 7. a 8. b 9. a 10. C
78
MATRICES
ANSWERS OF MCQ
1. (d) 2,
1 0] [
Explanation: [x 1] [ =0 0]
−2 0
[x − 2 0] = [0 0]
x-2=0
x=2
2. (a) [28]
3. (b) Not possible to find
4. (d) 512,
Explanation: Total elements are 6 and each entry can be done in 2 ways.
Hence, total possibilities = 29 = 512
π
5. (b)
3
Explanation: A + A′ = I
cos α − sin α cos α sin α 1 0
[ ]+[ ]=[ ]
sin α cos α −sin α cos α 0 1
2cos α 0 1 0
[ ]=[ ]
0 2 cos α 0 1
2cos α = 1
1
cos α =
2
π
α=
3
6. (d) AB=BA=I
7. (a) square matrix
8. (a) Skew-symmetric matrix,
Explanation: (AB − BA)′ = (AB)′ − (BA)′
= B′A′ − A′B′
= BA − AB
= −(AB − BA)
9. (b) Skew-symmetric matrix
10. (b) 4,
Explanation: 6 → 1 × 6, 2 × 3, 3 × 2, 6 × 1.
16
11.(d)
5
12. (d) 64,
Explanation: Total elements are 6 and each entry can be done in 2 ways. Hence,
total possibilities = 26 = 64.
13. (a) I
Explanation: (I + A)2 – 3A = I2 + IA + AI + A2 – 3A = I + A + A + A – 3A = I
4 0
14. (d)[ ]
0 4
15. (a) all zeroes
Explanation: Iin skew symmetric matrix, aij = –aji
⇒ aii = – aii ⇒ 2aii = 0
⇒ aii = 0, i.e. diagonal elements are zeroes.
16. (b) x = y
5 x 5 y
Explanation: [y ] = [ ] ⇒x = y
0 x 0
17. (c) a zero matrix of order n × n
79
Explanation: aij = aji, aij = –aji and aii = 0
80
18. (c) F(x + y)
cos x sin x cos x sin x cos(x + y) sin(x + y)
Explanation: [ ][ ] =[ ]
−sin x cos x −sin x cos x − sin(x + y) cos(x + y)
1
−1
19. (c) [2 1 ]
0
2
20. (c) I
0 0 1 0 0 1 1 0 0
Explanation: A2 = [0 1 0] [0 1 0] = [0 1 0] = I
1 0 0 1 0 0 0 0 1
A6 = (a2)3=I
=[ ]
0 −14
22. (c) 6
Explanation: 18 → 1 × 18, 2 × 9, 3 × 6, 6 × 3, 9 x 2, 18 x 1.
23. (d) n × m
24. (d) none of these
25. (d) 1
3 −2 3 −2 3 −2 1 0
Explanation: [ ][ ] = 𝑘 [ 4 −2 ] − 2 [ 0 1]
4 −2 4 −2
1 −2 [ ]=[3k − 2 −2
4 −4 ]
4 −2k −2
3𝑘 − 2 = 1
Hence, 𝑘 = 1
40 30 50
4. (a) (ii)[20 80 10]
40 60 5
82
(b) (iv) Number of USB type ‘C’ = 5 Produced by company = I
(c) (ii) 360
(d) (i) 670
(e) (i) 10
DETERMINANTS
ANSWERS OF MCQ
1. (c) 100
△1= | x3 y3 z3 | = | x3 y3 z3 |
xyz xyz
xyz zxy xyz 1 1 1
Ax2 x3 1
△1= |By2 y3 1|
Cz2 z3 1
△1=△
3. (b) K2|A|
4. (d) a11A11 + a21A21 + a31A31
Explanation: as value of determinant is sum of the product of elements of any row and column and their
respective cofactor
5. (d) (A + B)–1 = A–1 + B–1
6. (a) |A|3
Explanation: as |A . adj A| = |A|n, where A is matrix of order n × n.
7. (c) k3 |A|
8. (b) –1
a a2 1 + a3
Explanation: we have |b b2 1 + b3| = (1 + abc)(a – b)(b – c)(c – a) = 0.
c c2 1 + c3
Also a ≠ b ≠ c ⇒ 1 + abc = 0
⇒ abc = –1.
83
9. (c) 0
10. (b) 0
11. (c) -1
84
2 3 2
Explanation: As, |x x x| + 3 = 0
4 9 1
On expanding along first row,
2(x − 9x) − 3(x − 4x) + 2(9x − 4x) + 3 = 0 x = −1
12. (d) -7000
Explanation: As,AB = [200 50] [50 40]
10 2 2 3
10000 + 100 8000 + 150
AB = [ ]
500 + 4 400 + 6
13. (c) a6
Explanation: as,det(A) = a3 det(adj A)
= (a3 )3−1 = a6 14. (c) 9
Explanation: as, |adj A| = |A|3−1
15. (c) ±6
5| = |6 −2
Explanation: as |2x |
8 x 7 3
2x2 − 40 = 18 + 14
x = ±6
16. (b) 3
x1 y1 1
1
Explanation: as, ∆= |x 2 2 y2 1|
x3 y3 1
1 −3 0 1
9= | 3 0 1|
2
0 k 1
k=3
17. (d) (A + B)−1 = B−1 + A−1
1 −2
18. (d) [ ]
−3
1 4
4
3
19. (b) [ ]
17 −3 2
20.(b) Symmetric matrix
21. (a) Every skew-symmetric matrix of odd order is non-singular
22. (a) −(3A2 + 2A + 5I)
23.(a) m × n
24.(d) 8
25. (d) 3 x 3
86
(d) (iv) ₹ 20
(e) (iii) ₹ 22
3. (i) (d) 12
Explanation: as, x + y + z = 12 2x + 3y +
3z = 33
x − 2y + z = 0
1 1 1 x 12
A = [2 3 3] , X = [ y ] , B = X = [33]
1 −2 1 z 0
|A| = 3
1 1 9 −3 0
A−1 = adj A = 3 [ 1 0 −1]
|A| −7 3 1
X = A−1 B = 1 [ 19 −3
0
0 12
−1] [33] = [4]
3
3
−7 3 1 0
5
x = 3, y = 4, z = 5
x + y + z = 12
(ii) (b) -z
(iii) (c) 5
(iv) (c) 11
(v) (b)43
4. (i) (b) 900
Explanation: as, 3x + 2y + z = 1600 4x + y + 3z =
2300
x + y + z = 900
3 2 1 x 1600
A = [4 1 3] , X = [y] , B = X = [2300]
1 1 1 z 900
|A| = −5
1 1 −2 −1 5
A−1 = adj A = [−1 2 −5]
|A| −5
3 −1 −5
APPLICATION OF DERIVATIVES
ANSWERS OF MCQ
1. a 2. b 3. a 4. a 5. a 6. b 7. d 8. a 9. a 10. a 11. b 12. d 13. a 14. a 15. a 16. a
17. c 18. a 19. b 20. A 21. d 22. b 23. b 24. b 25. d 26. c 27. c 28. c 29. a 30. b 31. d
32. a 33. d 34. b 35. c 36. c 37. b 38. b 39. c 40. c 41. b 42. a 43. c
88
1. b 2. d 3. a 4. a 5. c
89
KENDRIYA VIDYALAYA SANGATHAN W
RAIPUR REGION