Subject: Mathematics
Class XI
Time Allowed: 3 hours
Max. Marks: 80
GENERAL INSTRUCTIONS
(i) This Question paper contains- five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions.
(ii) Section A has 18 MCQ’s and 2 Assertion Reasoning based Questions of 1 mark each.
(iii) Section B has 5 Very short Answer (VSA) – type questions of 2 mark each.
(iv) Section C has 6 Short Answer (SA) – type questions of 3 mark each.
(v) Section D has 4 Long Answer (LA) – type questions of 5 mark each.
(vi) Section E has 3 source based / case based /integrated units of assessment (4 mark each) with sub parts.
SECTION A
Multiple choice questions
Q1. The value of tan2θ sec2θ (cot2θ – cos2θ) is
a) 0 b) 1 c) -1 d) ½
Q2. Value of
a) 2 b) 3 c) 1 d) 0
Q3. The value of sin 7650 is 1/√n. Value of n is
a) 2 b) 3 c) 4 d) 1
Q5 The minimum value of f(x) = 3cos x + 4sin x + 8 is
a) 5 b) 9 c) 7 d) 3
Q6. The set A = {14, 21, 28, 35, 42, ….. 98} in set builder form is
a) A = {𝑥: 𝑥 = 7n, n ∈ N and 1 ≤ n ≤ 15}
b) A = {𝑥: 𝑥 = 7n, n ∈ N and 2 ≤ n ≤ 14}
c) A = {𝑥: 𝑥 = 7n, n ∈ N and 3 ≤ n ≤ 113}
d) A = {𝑥: 𝑥 = 7n, n ∈ N and 4 ≤ n ≤ 12}
Q7. For any two sets A and B, A Ç (A ∩ B)’ is equal to
(a) A (b) B (c) f (d) A Ç B
Q8. If A and B are two non-empty sets, then ( A – B) È (B – A) equals
a) (A ∪ B) – B b) A- (A ∩ B) (c) (A ∪ B) – (A ∩ B) d) (A ∩ B) ∪ (A ∪ B)
Q9. Let U be the universal set containing 700 elements. If A, B are subsets of U such that
N(A) = 200, n(B) = 300, n(A ∩ B) = 100, then n(A’ ∩ B’) is
a) 400 b) 600 c) 300 d) None of these
Q10. Domain of √(a2 – x2), a > 0 is given by
a) (-a, a) b) [-a, a] c) [0, a] d) (-a, 0]
Q11. Let A B = {1,2,3},B = {2,3,4}, then which of the following is a function from A to B?
a) {(1,2) , (1,3) , (2,3) , (3,3)}
b) {(1,3) , (2,4)}
c) {(1,3) , (2,2) , (3,3)}
d) {(1,2) , (2,3) , (3,2) , (3,4)}
Q12. The figure shows a relation R between the sets P and Q
Then the relation R in roster form is
a) {(9, 3), (4, 2), (25, 5)}
b) {(9, -3), (4, -2), (25, -5)}
c) {(9, -3), (9, 3), (4, 2), (25, 5), (25, -5)}
d) None of the above
Q13. If A = {1, 2, 6} and R be the relation defined on A by R = {(a, b):a ∈ A, b ∈ A and a divides b}, then range of R is equal to
a) {a, b} b) {2, 6} c) {1, 2, 6} d) None of these
Q14. The conjugate of a complex number is 1/(i – 1), then the complex number is
Q15. If 4x + i(3x – y) = 3 + i(-6), where x and y are real numbers, then the values of x and y are
a) x = 3, y = 4 b) x = ¾, y = 33/4 c) x = 33, y = 4 d) x = 4i, y = 3
Q16. The modulus of the complex number (I – i)-2 + (1 + i)-2 is equal to
a) 1 b) 2 c) 3 d) 0
Q17. Solution set of the inequations 2x – 1 ≤ 3 and 3x + 1 ³ ≥-5 is
a) (-2, 2) b) [-2, 2] c) (- ∞, -2) ∪ (2, ∞) d) (-∞, -2] ∪ [2, ∞}
Q18. If |x + 2| ≤ 9 , then
a) x ∈ (-7, 11) b) x ∈ [-11, 7] c) x ∈ (-∞, -7) ∪ (11, ∞) d) x ∈ (-∞, -7) ∪ [11, ∞)
Assertion Reasoning Based Questions
Q19. Given below are two statements: one is labeled as Assertion A and the other is labeled as Reason R Assertion (A): Roots of quadratic equation x2 + 3x + 5 = 0 is:
Reason (R): If x2 – x + 2 = 0 is a quadratic equation, then its roots are:
In the light of the above statements, choose the most appropriate answer from the options given below.
a. Both A and R are correct and R is the correct explanation of A
b. Both A and R are correct but R is NOT the correct explanation of A
c. A is correct but R is not correct
d. A is not correct but R is correct
Q20. Given below are two statements: one is labeled as Assertion A and the other is labeled as Reason R
Assertion A: The range of the function f(x) = 2 – 3x, x ∈ R, x > 0 is R
Reason R: The range of the function f(x) = x2 + 2, is [2, ∞)
In the light of the above statements, choose the most appropriate answer from the options given below.
a. Both A and R are correct and R is the correct explanation of A
b. Both A and R are correct but R is NOT the correct explanation of A
c. A is correct but R is not correct
d. A is not correct but R is correct
SECTION B
Q21. Describe in roster form:
a) A = {x:x2 + 7x – 8 = 0, x ∈ R}
b) B = {x: x is a prime number and a divisor of 60}
Q22. If = f(x) = x2 – 3x + 4, then find the values of x satisfying the equation f(x) = f(2x + 1)
OR
Let A = {9, 10, 11, 12, 13} and let f: A -> N be defined by f(n) = the highest prime factor of n. Find the range of f.
Q23. Prove that: cos4x = 1 – 8sin2x cos2x
OR
Prove that:
Q24. Express the following in the form of x + iy :
Q25. Find the value of: (a + b)4 – (a – b)4
SECTION C
Q26. Let A and B be sets. If A ∩ X = Ø = B ∩ X and A ∪ X = B ∪ X for some set X, show that A = B.
Q27. Find the domain and range of following:
OR
Find the domain and range of following:
Q28. Prove the following:
OR
If cos (α + β) = 4/5 and sin (α – β) = 5/13, where α lies between 0 and π/4, find the value of tan 2α.
Q29. If tan x = ¾, π < x < 3 π/2. Find the value of sin x/2 and tan x/2
Q30. Express the following numbers in the polar form:
Q31. Solve the following in equations:
SECTION D
Q32. (i) Given A = {-1, 0, 2, 5, 6, 11} and B = {-2, -1, 0, 18, 28, 108} and f(x) = x2 – x – 2,
Is f(A) = B, Find f(A)
(ii) If f(x) = (x – 1)/(x + 1), x ≠ – 1. Find the value of f(f(x))
Q33. Prove that:
OR
Prove that:
Q34. (i) If x = 3 + 2i, find the value of x4 – 4x3 + 4x2 + 8x + 44.
(ii) Find the values of x and y for which the complex numbers -3 + ix2y and x2 + y + 4i are conjugate of each other.
Q35. Solve graphically:
SECTION E
CASE STUDY BASED QUESTIONS
Q36. The school organized a farewell party for 100 students and school management decided three types of drinks will be distributed in farewell party i.e. Milk (M), Coffee (C) and Tea (T). Organizer reported that 10 students had all the three drinks M, C, T. 20 students had M and C; 30 students had C and T; 25 students had M and T. 12 students had M only; 5 students had C only; 8 Students had T only. Based on the above information, answer the following questions.
(i) Find the number of students who did not take any drink.
(ii) Find the number of students who prefer Milk and Coffee but not tea.
Q37. Function as a Relation A relation f from a non-empty set A to a non-empty set B is said to be a function, if every element of set A has one and only one image in set B. In other words, we can say that a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element or component.
If f is a function from a set A to a set B, then we write
F: A – > B
and it is read as f is a function from A to B or f maps A to B
Based on the above information, answer the following questions.
(i) If f(x) = (x2 – 1/x2), find the value of f(x) + f(1/x)
(ii) Find the range of: f(x) = 11 – 7 sin x
Q38. Trigonometry (from Greek trigon, “triangle” and metron, “measure”) is a branch of mathematics that studies relationships between side lengths and angles of triangles. Measurement of angles can be done in DEGREES (English System), GRADES (French System) AND RADIANS (Circular System). Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, navigation and many other fields.
Based on the concept of trigonometry answer the below given questions-:
i) The minute hand of a clock is 2 cm long. How far does its tip move in 20 minutes (Take π= 22/7)?
(ii) The angles subtended at the centre by the equal arcs of two wipers of the car as shown above are 650 and 1100, the ratio of their radii will be?
