Plus Two Maths Model Question Papers Paper 1 with Answers

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates:

  • In addition to the writing time, there is also a 15 minute “Cool off” period.
  • Use the ‘Cool off time’ to familiarize yourself with questions and to plan your responses.
  • Before answering questions, read them carefully.
  • Make sure you read the instructions carefully.
  • Selecting a question means answering all the sub-questions from that question.
  • The answer sheet should contain calculations, figures, and graphs.
  • Malayalam version of the questions is also provided.
  • Provide equations whenever necessary.
  • Examination halls do not allow electronic devices, except for non-programmable calculators.

Questions 1 to 7 carry 3 Scores each. Answer any six questions only. (6 × 3 = 18)

Question 1.
i) Let R be a relation on the set {1, 2, 3} given by R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)}. Which among the following element to be included to R so that R becomes Symmetric?
a) (3, 3)
b) (3, 2)
c) (1, 3)
d) (3, 1)
ii) If * is defined by a*b = a – b2 and ⊕ is defined by a ⊕ b = a2 + b , where a and b are integers. Then find the value of (3 ⊕ 4) * 5
Answer:
i) b) (3, 2)
ii) (3 ⊕ 4) * 5 = (32 + 4) * 5
= 13 * 5
= 13 – 52 = -12

Question 2.
If X + Y = [5029] and X – Y = [306−1]find 2X – 3Y.
Answer:
Plus Two Maths Model Question Papers Paper 1, 1

Question 3.
Find rate of change of area of a circle.
a) with respect to the radius, when r = 10cm
b) with respect to the time when the radius is increasing at the rate of 0.7 cm/s. Given that r = 5cm
Answer:
i) Area of a circle A = πr2
Plus Two Maths Model Question Papers Paper 1, 2

Question 4.
i) Integrate ∫(1+logx)2xdx
Plus Two Maths Model Question Papers Paper 1, 3
Answer:
Plus Two Maths Model Question Papers Paper 1, 4

Question 5.
Find the area of a circle with centre (0, 0) and radius ‘a’ using integration.
Answer:
Equation of the circle is x2 + y2 = a2
y = a2−x2−−−−−−√
Area of the circle will 4 time the area under the curve from 0 to a.
Area = 4∫a0ydx=4∫a0a2−x2−−−−−−√dx
Plus Two Maths Model Question Papers Paper 1, 5

Question 6.
Consider the differential equation dydx=x+yx
i) Write the order of the differential equation.
ii) Solve the above given differential equation.
Answer:
i) Order = 1.
Plus Two Maths Model Question Papers Paper 1, 6

Question 7.
The following table shows a brief description of the manufacturing process of a company. The time required in hours per unit of the product and maximum availability of machine is also given in the table:

Product The time required in hours/Unit on machine Profit per Unit in Rupees
Machine G Machine H
A 3 5 20
B 4 6 30
  Maximum available time 10 hrs/day Maximum available time 15 hrs/day  

i) Write the objective function.
ii) Whether it is a maximization case or a minimization case. Justify.
iii) Write the constraints.
Answer:
Let x units = Machine G and y units = Machine H
i) Objective function: Z = 20x + 30y
ii) It is a maximization problem.
iii) Constraints are:
3x + 4y ≤ 10; 5x + 6y ≤ 15; x, y ≥ 0

Questions 8 to 17 carry 4 scores each. Answer any 8 questions. (8 × 4 = 32)

Question 8.
i) A function f : A → B, where A = {1, 2, 3} and B = {4, 5, 6} defined by f(1) = 5, f(2) = 6, f(3) = 4, Check whether f is a bijection. If it is a bijection, write f-1 as set of ordered pairs.
ii) The operation table for an operation * is given below. Given that 1 is the identity element. Then which among the following is true regarding the element in the first column?

* 1 2 3
1   2 3
2   2 3
3   3 3

a) 3, 2, 2
b) 1, 2, 3
c) 1, 1, 2
d) 2, 2, 2
Check whether * is commutative.
Answer:
i) A = {1, 2, 3}; B = {4, 5, 6}
f(1) = 5, f(2) = 6, f(3) = 4
Hence f= {(1, 5),(2, 6),(3, 4)}
f-1 = {(5, 1), (6, 2), (4, 3)}
ii) b) 1, 2, 3
Since; 1*2 = 2 and 2*1 =2
3 * 2 = 3 and 2*3 = 3
Therefore * is commutative

Question 9.
i) If Sin-1 x = y, then
(a) 0 ≤ y ≤ π
b) −π2 ≤ y ≤ π2
c) 0 < y < π
d) −π2 ≤ y ≤ π2
ii) Find the principal value of sin-1 (12)
iii) sin-1x = 34 find the value of cos-1 x
Answer:
i) −π2 ≤ y ≤ π2
ii) sin-1 12 = π6
iii) sin-1x + cos-1x = π2
34 + cos-1x = π2 ⇒ cos-1x = π2 – 34

Question 10.
i) Find the relation between ‘a’ and ‘b’ so that the function defined by
f(x) = {ax+1,bx+3,x≤3x>3 is continuous at x = 3
ii) “All continuous function are not differentiable.” Justify your answer with an example.
Answer:
i) Since f(x) is continuous at x = 3
limx→3− f(x) = limx→3+ f(x) = f(3)
limx→3 (ax + 1) = limx→3 (bx + 3) = 3a +1
⇒ 3a + 1 = 3b + 3 = 3a +1
⇒ 3a + l = 3b + 3 ⇒ 3a – 3b – 2 = 0

ii) Consider the function f(x) = |x|.
Let us check the continuity and differentiability at x = 0.
Plus Two Maths Model Question Papers Paper 1, 7
Left derivative * Right derivative
Therefore not differentiable at x = 0.

Question 11.
i) Find the equation to the tangent to the curve y = x2 – 2x + 7 at (2, 7)
ii) Find the maximum value of the function?
Answer:
f(x) = sin x + cos x, 0 < x < π2
i) y = x2 – 2x + 7 ⇒ dydx = 2x – 2
Slope at x = 2 is 2 × 2 – 2 – 2
Equation of the tangent is
(y – y1) = m(x – x1)
⇒ (y – 7) = 2(x- 2)
⇒ y – 7 = 2x – 4
⇒ 2x – y + 3 = 0

ii) c) f(x) = sin x + cos x
f'(x) = cos x – sin x
f”(x) = -sinx-cosx
For turning points;
f'(x) = cos x – sin x = 0 ⇒ tan x = 1
Plus Two Maths Model Question Papers Paper 1, 8

Question 12.
Integrate ∫x+22×2+6x+5dx
Answer:
Put x + 2 = A(4x + 6) + B
4A = 1 ⇒ A = 14
Plus Two Maths Model Question Papers Paper 1, 9
Plus Two Maths Model Question Papers Paper 1, 10

Question 13.
Consider the differential equation
x dydx + y = 1×2
i) Find the integrating factor.
ii) Solve the above differential equation.
Answer:
Plus Two Maths Model Question Papers Paper 1, 11

Question 14.
lf the vectors PQ¯¯¯¯¯¯¯¯ = -3i + 4j + 4k and PR¯¯¯¯¯¯¯¯ = -5i + 2j + 4k are the sides of a ΔPQR
i) Find the angle between PQ¯¯¯¯¯¯¯¯ and PR¯¯¯¯¯¯¯¯
ii) Find the length of the median through the vertex P.
Answer:
Plus Two Maths Model Question Papers Paper 1, 12
ii)
Plus Two Maths Model Question Papers Paper 1, 13
With respect to the initial P the position vector of Q and R will be PQ¯¯¯¯¯¯¯¯ = -3i + 4j + 4k
and PR¯¯¯¯¯¯¯¯ = -5i + 2j + 4k respectively.
Since M is the midpoint of QR. The position vector of M will be
Plus Two Maths Model Question Papers Paper 1, 14

Question 15.
i) If a¯ = 5i – j – 3k and b¯ = i + 3j + 5k, then show that the vectors a¯+b¯,a¯−b¯ are perpendicular.
ii) If a¯ = i – 2j + 3k, b¯ = 2i + 3j – 4k and
c¯ = i – 3j + 5k, then check whether a¯, b¯, c¯ are coplanar.
Answer:
Plus Two Maths Model Question Papers Paper 1, 15

Question 16.
i) Find the Cartesian equation of the line passing through origin and (5, -2, 3) ii) The point P(x, y, z) lies in the first octant and its distance from the origin is 12 units. If the position vector of P makes angles 45°, 60°with x and y axes respectively, find coordinates of P.
Answer:
Plus Two Maths Model Question Papers Paper 1, 16
Since P lies in the first octant, we take n = 12
Therefore the coordinate of P is
Plus Two Maths Model Question Papers Paper 1, 17

Question 17.
Solve graphically Maximise Z = 5x + 3y Subject to the constraints 3x + 5y ≤ 15; 5x + 2y ≤ 10; x ≥ 0, y ≥ 0.
Answer:
In the figure the shaded region OABC is the fesible region. Here the region is bounded.
The corner points are
O(0, 0), A(2, 0), B(2019,4519), C(0, 3)
Plus Two Maths Model Question Papers Paper 1, 18
Given Z = 5x + 3y

Corner points Value of Z
O Z = 0
A Z = 5(2) + 3(0) = 10
B Z = 5(2019)+3(4519)=23519
C Z = 5(0) + 3(3) = 9

Since maximum value of Z occurs at B, the soluion is z = 23519, (2019, 4519)

Questions 18 to 24 carry 6 scores each. Answer any 5 questions. (5 × 6 = 30)

Question 18.
A = ⎡⎣⎢3−2−43−2−5−112⎤⎦⎥
a) Find AT
b) Express A as the sum of a symmetric and skew symmetric matrix.
ii) If AT = [cosx−sinxsinxcosx], verift that AT A = I
Answer:
Plus Two Maths Model Question Papers Paper 1, 19
Plus Two Maths Model Question Papers Paper 1, 20
Plus Two Maths Model Question Papers Paper 1, 21

Question 19.
i) Without expanding prove that
∣∣∣∣x+yz1y+zx1z+xy1∣∣∣∣ = 0
ii) Consider the following system of equations
2x – 3y + 5z = 11; 3x + 2y – 4z = -5; x + y – 2z = -3
a) Express the system in Ax = B form.
b) Solve the system by matrix method.
Answer:
Plus Two Maths Model Question Papers Paper 1, 22
Plus Two Maths Model Question Papers Paper 1, 23

Question 20.
Find dydx of the following
i) x2 + 2xy + 2y2 = 1
ii) yx = 2x
iii) x = cos θ; y = sin θ at θ = π4
Answer:
i) x2 + 2xy + 2y2 = 1
Differentiating w.r.to x;
Plus Two Maths Model Question Papers Paper 1, 24

ii) yx = 2x
Take log on both sides;
x log y = x log 2
Differentiating w.r.to x;
Plus Two Maths Model Question Papers Paper 1, 25

Question 21.
Evaluate the following
Plus Two Maths Model Question Papers Paper 1, 26
Answer:
Plus Two Maths Model Question Papers Paper 1, 27
Plus Two Maths Model Question Papers Paper 1, 28
Plus Two Maths Model Question Papers Paper 1, 29

Question 22.
Consider the parabolas y2 = 4x, x2 = 4y
i) Draw a rough figure for the above parabolas.
ii) Find the point of intersection of the two parabolas.
iii) Find the area bounded by these two parabolas.
Answer:
Plus Two Maths Model Question Papers Paper 1, 30
ii) solving the two conics we get the point of intersection as (0, 0) and (4, 4).
iii) Area of the enclosed region
Plus Two Maths Model Question Papers Paper 1, 31

Question 23.
i) Find the shortest distance between the lines whose vector equations are
r¯ = (i + 2j + 3k) + λ(i – 3j + 2k) and r¯ =(4i + 5j + 6k) + μ(i – 3j + 2k)
ii) If a plane meets positive x axis at a distance of 2 units from the origin, positive y axis at a distance of 3 units from the origin and positive z axis at a distance of 4 units from the origin. Find the equation of the plane.
iii) Find the perpendicular distance of (0, 0, 0) from the plane obtained in part (ii)
Answer:
i) a1¯¯¯¯¯ = i + 2j + 3k; b¯ = i – 3j + 2k
a2¯¯¯¯¯ = 4i + 5j + 6k Both lines are parallel.
Plus Two Maths Model Question Papers Paper 1, 32
Plus Two Maths Model Question Papers Paper 1, 33

Question 24.
i) A die is thrown twice let the event A be ‘odd number on first throw’ and B be ‘odd number on the second throw’ check whether A and B are independent.
ii) Coloured balls are distributed in three boxes as shown in the following table:

Box Red Black
1 2 0
II 0 2
III 1 1

A box is selected at random and a ball is taken out. If the ball taken is of red colour, What is the probability that the other ball in the box is also of red colour?
Answer:
i) n(S) = 36
A = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
B = {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (1, 5), (2, 5), (3,5), (4, 5), (5, 5), (6, 5)}
(A ∩ B) = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}
Plus Two Maths Model Question Papers Paper 1, 34
Therefore A and B are independent.

ii) Let B1, B2, B3 are the event of getting the boxes.
P(B1) = P(B2) = P(B3) = 13
Let E be the event of getting a red ball.
Plus Two Maths Model Question Papers Paper 1, 35

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