Dikshit tution
CLASS XII PRACTICE PAPER -1 Max Marks : 80
SUB: MATHEMATICS (CODE-041) DURATION :3HOURS
General Instructions: Read the following instructions very carefully and strictly follow them:
1. This Question paper contains 38 questions. All questions are compulsory.
2. This Question paper is divided into five Sections - A, B, C, D and E.
3. In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19
and 20 are Assertion-Reason based questions of 1 mark each.
4. In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2
marks each.
5. In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks
each.
6. In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks
each.
7. In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
8. There is no overall choice. However, an internal choice has been provided in 2 questions in
Section B, 3 questions in Section C, 2 questions in Section D and one subpart each in 2
questions of Section E.
9. Use of calculators is not allowed.
SECTION-A [1 20 =20]
(This section comprises of multiple-choice questions (MCQs) of 1 mark each)
Select the correct option (Question 1 - Question 18):
Q1. For which of the given values of 𝑥 and 𝑦, the following pair of matrices are equal?
3𝑥 + 7 5 0 𝑦−2
[ ],[ ]
𝑦+1 2 − 3𝑥 8 4
−1 −2 −1 −2
(a) 𝑥 = ,𝑦 = 7 (b) no such x and y possible (c) 𝑦 = 7, 𝑥 = (d) 𝑥 = ,𝑦 =
3 3 3 3
Q2. Assume 𝑿, 𝒀, 𝒁, 𝑾 and 𝑃 are matrices of order 2 × 𝑛, 3 × 𝑘, 2 × 𝑝, 𝑛 × 3 and 𝒑 × 𝒌, respectively.
Then the restriction on 𝒏, 𝒌 and 𝒑 so that 𝑷𝒀 + 𝑾𝒀 will be defined are:
(a) 𝑘 = 3, 𝑝 = 𝑛(b) 𝒌 is arbitrary, 𝒑 = 2 (c) 𝑝 is arbitrary, 𝑘 = 3 (d) 𝑘 = 2, 𝑝 = 3
Q3. The interval in which the function y=𝑥 2 𝑒 −𝑥 is increasing is:
(a) (-∞, ∞) (b )(-2, 0) (c) (2, ∞) (d) (0, 2)
Q4. A and B are two matrices such that 𝐴𝐵 = 𝐴 and 𝐵𝐴 = 𝐵 then B2 is
(a) A (b) B (c) 0 (d) I
𝑑𝑦
Q5. The general solution of the differential equation log(
𝑑𝑥
) + 𝑥 = 0 is
x x
(a) y e c (b) y e c (c) y ex c (d) y ex c
Q6. If A is a square matrix of order 3 such that |A| = - 5 then value of |−AA′ | is
(a) 125 (b) - 125 (c) 25 (d) – 25
Q7. If A, B are symmetric matrices of same order, then AB – BA is a
(a) Skew symmetric matrix (b) Symmetric matrix (c) Zero matrix (d) Identity matrix
Q8. Two events A and B will be independent, if
(a) A and B are mutually exclusive
(b) P(A′B′) = [1 – P(A)] [1 – P(B)]
(c) P(A) = P(B)
(d) P(A) + P(B) = 1
Q9. A vector in the direction of vector 𝚤ˆ − 2𝚥ˆ + 2𝑘ˆ that has magnitude 15 is
(a) −5𝚤ˆ − 10𝚥ˆ − 10𝑘ˆ (b) 5𝚤ˆ + 10𝚥ˆ + 10𝑘ˆ (c) −5𝚤ˆ + 10𝚥ˆ + 10𝑘ˆ (d) 5𝚤ˆ − 10𝚥ˆ + 10𝑘ˆ
Q10. If |𝑎⃗| = 3, |𝑏⃗⃗| = 4 and |𝑎⃗ + 𝑏⃗⃗| = 5, then |𝑎⃗ − 𝑏⃗⃗| =
(A) 3 (B) 4 (C) 5 (D) 8
Q11. The region represented by graph of the inequality 2x + 3y > 6 is
(a) half plane that contains the origin
(b) half plane that neither contains the origin nor the points on the line 2x + 3y=6
(c) whole XOY-plane excluding the points on the line 2x + 3y = 6
(d) entire XOY plane
Q12. ∫𝑒 𝑥 𝑠𝑒𝑐 𝑥 (1 + 𝑡𝑎𝑛 𝑥) 𝑑𝑥=⋯
(a) 𝑒 𝑥 𝑐𝑜𝑠 𝑥 + 𝑐 (b) 𝑒 𝑥 𝑠𝑒𝑐 𝑥 + 𝑐 (c) 𝑒 𝑥 𝑠𝑖𝑛 𝑥 + 𝑐 (d) 𝑒 𝑥 𝑡𝑎𝑛 𝑥 + 𝑐
2𝜋
Q13. ∫0 cosec 7 𝑥𝑑𝑥 =
(a) 0 (b) 1 (c) 4 (d) 2𝜋
Q14. The number of arbitrary constants in the particular solution of a differential equation of third order
is /are
(a) 3 (b) 2 (c) 1 (d) 0
1
Q15. If cos [tan−1 {cot (sin−1 2)}] = ⋯
1 1
(a) 1 (b) 0 (c) (d) 2
√2
Q16. The corner points of the feasible region in the graphical representation of a linear programming
problem are (2,72), (15,20) and (40,15). If z = 18x+9y be the objective function, then:
(a) z is maximum at (2,72), minimum at (15,20)
(b) z is maximum at (15,20), minimum at (40,15)
(c) z is maximum at (40,15), minimum at (15,20)
(d) z is maximum at (40,15), minimum at (2,72)
𝑑2 𝑦
Q17. If x = t2, y= t3, then =
𝑑𝑥 2
3 3 3 3𝑡
(a) (b) (c) (d)
2 4𝑡 2𝑡 2
Q18. The area bounded by the line y = x, x-axis and lines x = – 1 to x = 2, is
1 3 5
a) 0 sq. units b) 2 sq units c) 2 sq units d) 2 sq units
ASSERTION – REASON BASED QUESTIONS
Directions: Each of these questions contains two statements, Assertion and Reason. Each of these
questions also has four alternative choices, only one of which is the correct answer. You have to select
one of the codes (a), (b), (c) and (d) given below.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.
x 2 − 5x + 6 x − 3 , for x ≠ 3
Q19. Consider the function 𝑓(𝑥)={ is continuous at x = 3
k, for x = 3
Assertion (A): The value of k is 4
Reason (R): If f(x) is continuous at a point a then lim 𝑓(𝑥) =𝑓(𝑎)
𝑥→𝑎
cos 𝑥+ 𝑠𝑖𝑛𝑥 𝜋 𝜋 𝑑𝑦
Q20. Assertion (A) : If 𝑦 = 𝑡𝑎𝑛−1 ( ) ,− <𝑥< then =1
cos 𝑥−𝑠𝑖𝑛𝑥 4 4 𝑑𝑥
cos 𝑥+ 𝑠𝑖𝑛𝑥 𝜋
Reason(R) : = tan(𝑥 + )
cos 𝑥−𝑠𝑖𝑛𝑥 4
SECTION B
VERY SHORT ANSWER TYPE QUESTIONS(VSA)
(Each question carries 2 marks)
3𝜋 3𝜋
Q21. Evaluate : sin−1 (sin ) + cos −1 (cos ) + tan−1(1)
4 4
2𝑥 + 2, if 𝑥<2
Q22. Function f is defined as 𝑓(𝑥) = { 𝑘, if 𝑥=2 Find the value of k for which the
3𝑥, if 𝑥>2
function f is continuous at x = 2.
OR
𝑑𝑦 𝜋
If 𝑥 = 𝑎(𝜃 − sin 𝜃) and 𝑦 = 𝑎(1 + cos 𝜃), find at 𝜃 =
𝑑𝑥 3
Q23. Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate
Q24. Find the projection of the vector 𝑎⃗ = 2𝚤ˆ + 3𝚥ˆ + 2𝑘ˆ on the vector 𝑏⃗⃗ = 𝚤ˆ + 2𝚥ˆ + 𝑘ˆ.
OR
If 𝑎⃗ = 4𝚤ˆ − 𝚥ˆ + 𝑘ˆ and 𝑏
⃗⃗ = 2𝚤ˆ − 2𝚥ˆ + 𝑘ˆ , then find a unit vector along the vector 𝑎⃗ × 𝑏⃗⃗.
Q25. If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are unit vectors such that 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = ⃗0⃗, then write the value of 𝑎⃗ ⋅ 𝑏⃗⃗ + 𝑏⃗⃗ ⋅ 𝑐⃗ + 𝑐⃗ ⋅ 𝑎⃗.
SECTION C
SHORT ANSWER TYPE QUESTIONS(SA)
(Each question carries 3 marks)
3
Q26. Find the intervals in which the function 𝑓(𝑥) = 2 𝑥 4 − 4𝑥 3 − 45𝑥 2 + 51 is
strictly increasing and strictly decreasing
Q27. The volume of a cube is increasing at the rate of 9 cm3/s. How fast is its surface area increasing
when the length of an edge is 10 cm?
2
Q28. Evaluate: ∫−1 |𝑥 3 − 𝑥|𝑑𝑥
OR
𝑥+3
Find ∫ √5−4𝑥−2𝑥 2 𝑑𝑥
Q29. Let 𝑎⃗ = 4𝚤ˆ + 5𝚥ˆ − 𝑘ˆ, 𝑏⃗⃗ = 𝚤ˆ − 4𝚥ˆ + 5𝑘ˆ and 𝑐⃗ = 3𝚤ˆ + 𝚥ˆ − 𝑘ˆ. Find a vector 𝑑⃗ which is perpendicular
to both 𝑐⃗ and 𝑏⃗⃗ and 𝑑⃗ ⋅ 𝑎⃗ = 21.
OR
𝑥+1 𝑦+3 𝑧+5 𝑥−2 𝑦−4 𝑧−6
Show that the lines = = and = = intersect. Also find their point of
3 5 7 1 3 5
intersection.
Q30. Consider the following Linear Programming Problem: Minimize 𝑍 = 𝑥 + 2𝑦
Subject to 2𝑥 + 𝑦 ≥ 3, 𝑥 + 2𝑦 ≥ 6, 𝑥, 𝑦 ≥ 0. Show graphically that the minimum of 𝑍 occurs at
more than two points.
1 1 1
Q31. A problem is given to three students whose probabilities of solving it are 3 , 4 and 6 respectively. If
the events of solving the problem are independent, find probabilities that at least one of them
solves it.
OR
A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find
the probability that it is actually a six.
SECTION D
LONG ANSWER TYPE QUESTIONS(LA)
(Each question carries 5 marks)
1 −1 2 −2 0 1
Q32. Use product [0 2 −3 9 2 −3] to solve the system of equations:
] [
3 −2 4 6 1 −2
x – y + 2 z = 1, 2 y- 3 z = 1, 3 x – 2 y +4 z = 2
Q33. Draw the graph of y=|x+1| and find the area bounded by it with x- axis, x = −4 and x=2.
sin3 𝑡 cos3 𝑡 𝑑𝑦
Q34. If x and y are connected parametrically by the equations and 𝑥 = ,𝑦 = find
√cos 2𝑡 √cos 2𝑡 𝑑𝑥
OR
sin( a 1) x sin x
, x0
x
Determine a, b, c so that f(x) = is continuous at x = 0.
c , x0
x bx 2 x
, x0
bx 3/2
Q35. Find the shortest distance between the lines.
𝑥−1 𝑦−2 𝑧−3 𝑥−2 𝑦−4 𝑧−5
= = 𝑎𝑛𝑑 = =
2 3 4 3 4 5
OR
Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:
𝑥 − 8 𝑦 + 19 𝑧 − 10 𝑥 − 15 𝑦 − 29 𝑧 − 5
= = 𝑎𝑛𝑑 = =
3 −16 7 3 8 −5
SECTION E
(3 case study questions carry 4 marks each)
Q36. Vani and Mani are playing Ludo at home while it was raining outside. While rolling the dice
Vani's brother Varun observed and noted the possible outcomes of the throw every time belongs to
the set {1,2,3,4,5,6}. Let 𝐴 be the set of players while 𝐵 be the set of all possible outcomes.A =
{ Vani, Mani }, 𝐵 = {1,2,3,4,5,6}.
Answer the following questions:
a. Let 𝑅: 𝐵 → 𝐵, be defined by 𝑅 = {(𝑥, 𝑦) : 𝑦 is divisible by 𝑥}.Verify that whether R is reflexive,
symmetric and transitive. (2marks)
b. Is it possible to define an onto function from A to B? Justify. (1mark)
c. Which kind of relation is R defined on B given by
R= {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}? (1mark)
OR
Find the number of possible relations from A to B .
Q37. Read the following passage and answer the questions given below: 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
h meter. On the basis of this information and figure given below answer the following questions:
I. Find area 𝐀 of the gate expressed as a function of 𝐱.
𝑑𝐴
II. Find value of 𝑑𝑥 .
III. Find x and show that area is maximum
OR
Find maximum area of trapezium.
Q38. Case-Study 3:
A biased die is tossed and respective probabilities for various faces to turn up are the following :
Face 1 2 3 4 5 6
Probability 0⋅1 0 ⋅ 24 0 ⋅ 19 0 ⋅ 18 0 ⋅ 15 K
Based on the above information, answer the following questions:
(a) What is the value of K ? 2 marks
(b) If a face showing an even number has turned up, then what is the probability that it is the face with
2 or 4 ?
2marks
