CBSE Class 12 Mathematics Mid-Term Exam Paper PDF

Roll Number SET A INDIAN SCHOOL MUSCAT HALF YEARLY EXAMINATION SUBJECT : MATHEMATICS CLASS: XII...

Roll Number SET A INDIAN SCHOOL MUSCAT HALF YEARLY EXAMINATION SUBJECT : MATHEMATICS CLASS: XII Sub.Code:041 Time Allotted: 3 Hrs. 22.09.2019 Max.Marks: 80 General Instructions: (i) All questions are compulsory. (ii) This question paper contains 36 questions. (iii) Question 1- 20 in Section A are MCQ/Very short-answer type questions carrying 1 mark each. (iv) Question21-26 in Section B are short-answer type questions carrying 2 marks each. (v) Question 27-32 in Section C are long-answer-I type questions carrying 4 marks each. (vi) Question 33-36 in Section D are long-answer-II type questions carrying 6 marks each. SECTION A 1. If f,g : R→ 𝑅 be two functions defined as f(x) = |𝑥| + 𝑥 and g(x) = |𝑥| − 𝑥,for all x in R,find 1 fog(-5). 2. Find the value of cos −1 cos ( ). 7𝜋 1 6 3. Find the value of 1 1 1 tan−1 (1) + cos −1 (− ) + sin−1 ( ) 2 2 4. Find the area bounded by the curve y = cosx, between x = 0 and x = 2𝜋. 1 5. Evaluate: ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥 1 1 1 6 Evaluate: ∫−1 [𝑥]𝑑𝑥 1 2𝜋 7. Evaluate : ∫0 𝑠𝑖𝑛𝑥 𝑑𝑥 1 1−𝑐𝑜𝑠2𝑥 8. Evaluate: ∫ 1+𝑐𝑜𝑠2𝑥 𝑑𝑥 9. 1 Find the area bounded by the lines y = x and x = 1 in the first quadrant. 10. A point C in the domain of a function f at which either 𝑓 ′ (𝑐) = 0 or f is not differentiable is 1 called ---------------. Page 1of 4  𝑎𝑥 2 + 1 , 𝑥 > 1 1 f(x) = { is differentiable at x = 1, then find the value of a. 11. 𝑥 + 𝑎 ,𝑥 ≤ 1 1 a) 2 b) 1 c) 0 d) 2 1 12. 𝑥𝑆𝑖𝑛 ,𝑥 ≠ 0 1 f(x) ={ 𝑥 is continuous at x = 0. Find k. 𝑘, 𝑥=0 a) 8 b) 1 c) -1 d) 0 13. 𝑑2 𝑥 If 𝑦 = 𝑥 + 𝑒 𝑥 , then 𝑑𝑦 2 = ------------- 1 1 −𝑒 𝑥 −𝑒 𝑥 𝑎) b) c) d) 𝑒 𝑥 (1+𝑒 𝑥 )2 (1+𝑒 𝑥 )2 (1+𝑒 𝑥 )3 14. Let R be the relation in the set N given by R = {(𝑎, 𝑏): 𝑎 = 𝑏 − 2, 𝑏 > 6}.Choose the correct 1 answer. A) (2,4) ∈ R B) (3,8) ∈ R C) (6,8) ∈ R D) (8,7)∈ R 1 15. Let f : R→ R be defined as f(x) = 𝑥 4.Choose the correct answer. a)F is one- one onto b) f is many-one onto c) f is one-one but not onto d) f is neither one-one nor onto. 16. The interval in which 𝑦 = 𝑥 2 𝑒 −𝑥 is increasing is 1 𝑎) (−∞, ∞) b) (−2, 0) c) (2, ∞) d) (0, 2) 1 17. The line y = x + 1 is a tangent to the curve y2 = 4x at the point a) (1, 2) b) ( 2 , 1) c) ( 1, -2) d) (-1, 2) 1 18. Choose the correct principal value branch of the range of 𝑦 = tan−1 𝑥. 𝜋 𝜋 𝜋 𝜋 𝑎) [− 2 , 2 ] b) (− 2 , 2 ) c) [0, 𝜋] d) (0, 𝜋) 1 19. Find the area bounded by f(x) = |𝑥| , between x = -3 and x = 3. a) 0 b) 18 sq.units c) 9 sq.units d) 6 sq.units 1 20. Find the derivative of Sin(𝑥)3with respect to Cos(𝑥)3. 𝑎) − 𝑡𝑎𝑛(𝑥 3 ) b) -𝑐𝑜𝑡(𝑥 3 ) c) 𝑐𝑜𝑡(𝑥 3 ) d) 𝑡𝑎𝑛(𝑥 3 ) SECTION B 21. 1 Prove that tan−1 (2) + tan−1 (11) = tan−1 (4) 2 3 2 OR 1 4 Evaluate: 𝑠𝑖𝑛 ( cos −1 ) 2 5 22. Find the value of k ,if the following function is continuous at 1 2 𝑘(𝑥 2 − 2) , 𝑥 ≤ 1 f(x) = { 4𝑥 + 1 , 𝑥 > 1 23. 𝑑𝑦 Find 𝑑𝑥 𝑖𝑓, y = sin−1 ( 1+𝑥 2 ) 1−𝑥 2 0 { (𝜋−2𝑥)2 2 𝑥+2 Evaluate: ∫ 2𝑥 2 + 6 𝑥+5 dx 32. 4 Page 3of 4  SECTION D 33. Let f: N→ 𝑹 be a function defined as f(x) = 4𝑥 2 +12x +15. show that f:N→ 𝑺, where S is the 6 range of f is invertible.Find the inverse of f. OR Show that the relation R in the set N of Natural numbers given by R = {(𝑎, 𝑏): |𝑎 − 𝑏| 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3} is an equivalence relation. Find the area of the region enclosed between the two circles 𝑥 2 + 𝑦 2 = 4 𝑎𝑛𝑑 34. 6 (𝑥 − 2)2 + 𝑦 2 = 4 OR Using integration find the area of region bounded by the triangle whose vertices are (1,0),(2,2) and (3,1). 35. Evaluate: ∫ √tan 𝑥 + √cot 𝑥 𝑑𝑥 6 36. Show that the right circular cone of least curved surface and given volume has an altitude equal to 6 √2 times the radius of the base. Page 4of 4