Umashankar 1 694420
Umashankar 1 694420
Umashankar 1 694420
General Instructions:
1. This question paper contains five sections-A,B,C,D and E. Each part is
compulsory.
2. Section A has 5 MCQ’s questions of 1 mark each.
3. Section B has 2 Assertion and Reason type questions of 1 mark each.
4. Section C has 3 Short Answer type questions of 3 marks each.
5. Section D has 1 Long Answer type question of 5 marks.
6. Please check that this question paper contains 4 printed pages only.
7. There is no overall choice. However, internal choice has been provided. You
have to attempt only one of the alternatives in all such questions.
8. In Section E, Q12 is a case study-based problem having 2 subparts of 1 mark
each and third subpart of 2 marks.
9. Use of calculators is not permitted.
Section – A (5 X 1 = 5 Marks)
3𝑠𝑖𝑛2𝑥 2
, 𝑥<0
𝑥2
2
𝑥 + 2𝑥 + 𝑐 1
1. Let 𝑓(𝑥) = , 𝑥 ≥ 0, ≠ . If f is continuous at x = 0, then the
1 − 3𝑥 2 √3
1
{ 0, 𝑥=
√3
value of ‘c’ is
(A) 6 (B) 4
(C) – 4 (D) – 6
Page 1 of 4
𝑑𝑦
2. If sin y = x sin (a + y) then is equal to
𝑑𝑥
𝑠𝑖𝑛𝑎 𝑠𝑖𝑛2 (𝑎 + 𝑦)
(A) (B)
𝑠𝑖𝑛2 (𝑎 + 𝑦) 𝑠𝑖𝑛𝑎
𝑠𝑖𝑛2 (𝑎 + 𝑦)
(C) (D) none of these.
𝑐𝑜𝑠𝑎
(C) 2 (D) −1
2
(C) m/sec (D) None of these
3
Section – B (2 X 1 = 2 Marks)
Each of the following questions contains two statements. Assertion (A) and
Reason (R). Each of the questions has four alternative choices, only one of which is
the correct answer.
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true and R is not the correct explanation of A.
(C) A is true and R is false.
(D) A is false and R is true.
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𝑒𝑥 − 𝑒
6. Assertion (A) : 𝑓(𝑥) = , 𝑥 ≠ 1 is a continuous function at x = 1, if f(1) = e.
𝑥−1
7. Assertion (A) : A Jet aircraft of enemy is flying along the curve given by
y = x2 + 7. A soldier placed at (3, 7), wants to shoot down the
helicopter when it is near to him. The nearest distance is √5 .
Reason (R) : f’(x) = 0 and f’’(x) < 0 then ‘x’ is called the point of minima.
Section - C (3 X 3 = 9 Marks)
−1 𝑥 𝑑2𝑦 𝑑𝑦
8. If 𝑦 = 𝑒 𝑎𝑐𝑜𝑠 , −1 ≤ 𝑥 ≤ 1, show that (1 − 𝑥 2 ) ( ) − 𝑥 (𝑑𝑥 ) − 𝑎2 𝑦 = 0.
𝑑𝑥 2
9. The length ‘x’ of a rectangle is decreasing at the rate of 5 cm/sec and the
width ‘y’ is increasing at the rate of 4 cm/sec. When x = 8 cm and y = 6 cm
then find the rate of change of perimeter and area of the rectangle.
OR
Find the intervals in which the function 𝑓(𝑥) = 3𝑥 4 − 4𝑥 3 − 12𝑥 2 + 5 is
(a) strictly increasing (b) strictly decreasing.
Section – D (1 X 5 = 5 Marks)
11. A tank with a rectangular base and rectangular sides, open at the top is to be
constructed so that its depth is 2 m and volume is 8 m3. If building of tank
costs Rs.70 per sq. metre for the base and Rs.45 per sq. metre for sides, what is
the cost of least expensive tank?
Page 3 of 4
Section – E (1 X 4 = 4 Marks)
Case Study Question
Section E consists of 1 case study question of 4 mark with sub-parts of the
values 1,1 and 2 marks each respectively.
12. Rajesh wants to prepare a sweet box for Diwali at home. For making the lower
part of an open box, he takes a piece of a square cardboard of sides 18 cm by
cutting off equal squares from the corners and turning up the sides as shown
in the figure. Let ‘x’ metre be the length of a side of the removed squares.
(iii) Rajesh is interested in maximizing the volume of the box. So, what should
be the side of the square to be cut off to maximize the volume of the box?
OR [2M]
Find the maximum volume of the open box (in cm3).
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