Plus Two

Kerala Plus Two Maths Model Question Paper 5

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

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.
a. 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 inclu¬ded to R so that R becomes Symmetric?
i. (3,3)
ii. (3,2)
iii. (1,3)
iv. (3,1)

b. If * is defined by a * b=a-b² and ⊕ is de-fined by a ⊕ is b=a²+b, where a and b are integers. Then find the value of (3⊕4)*5.

Question 2.

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 0.7cm/s. Given that r=5cm.

Question 4.

Question 5.
Find area of a circle with centre (0, 0) and radius “a” using integration.

Question 6.
Consider the differential equation \(\frac { dy }{ dx } \) = \(\frac { x+y }{ x } \)
a. Write the order of the differential equation.
b. Solve above given differential equation.

Question 7.
Following Table shows a brief description about manufacturing process of a company. Time required in hours per unit of the product and maximum availability of machines is also given in the table

a. Write the objective function.
b. Whether it is a maximisation case or a minimisation case Justify,
c. Write the contraints.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.
a. Afunctionf: A → B whereA= {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 bijection, write f¹ as set of ordered pairs.2
b. The operation table for an operation * is given below. Given that I is the identify element, then which among the following is true regarding the elements in first column?

Question 9.

Question 10.
a. Find th relation between ‘a’ and ‘b’ so that the function defined by

b. “All continuous function are not differ-entiable”. Justify this statement with an example.

Question 11.
a. Find the equation of the tangent to the curve y = x² – 2x + 7 at (2,7)
b. Find the maximum value of the function.

Question 12.

Question 13.
Consider the differential equation \(\frac { xdy }{ dx } \) + y = \(\frac { 1 }{ { x }^{ 2 } } \)
a. Find the integrating factor
b. Solve the above differential equation.

Question 14.

Question 15.

Question 16.
a. Find the cartesian equation of a line passing through the origin and (5,-2,3)
b. The point P(x,y,z) lies in the First Octant and its distance from origin is 12 units. If the position vector of P makes angles 45°, 60° with x and y axes respectively, find co-ordinates of P.

Question 17.
Solve graphically :

Question 18 to 24 carry 6 scores each. Answer any 5 questions only

Question 18.

Question 19.
a. Without expanding prove that

b. Consider the following system of equations 2x – 3y + 5z = 11, 3x + 2y – 4z = -5, x + y-2z = -3
i. Express the system in the form Ax = B.
ii. Solve the system by matrix method a.

Question 20.

Question 21.
Evaluate the following.

Question 22.
Consider the parabolas y² = 4x and x² = 4y
a. Draw rough figure for the above parabolas.
b. Find the point of intersection of the two parabolas.
c. Find the area bounded by these two pa-rabolas.

Question 23.
a. Find the shortest distance between the lines whos vector equations are

b. If a plane meets positive x axis at a distance of 2 units from origin, positive y axis at a distance of 3 units from orign and positive z axiz at a distance of 4 units from origin. Find the equation of the plane.
c. Find the prependicular distance of (0,0,0) from the plane obtained in part (b).

Question 24.
a. 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.
b. Coloured balls are distributed in three boxes as shown in the following table.

A box is selected at random and a ball is taken out. If the ball taken is of red colour, What is the probabability that the other ball in the box is also of red colour?

ANSWERS

Answer 1.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.
Lex x = Machine G and y = Machine H
a. Objective function: Z= 20x + 30y
b. It is a maximization problem.
c. Constraints are:
3x + 4y < 10 ; 5x + 6y < 1 5 ; x > 0 ; y > 0

Answer 8.
a. A = {1,2,3}
B = {4,5,6}
f(1) = 5 ; f (2) = 6 ; f (3) = 4
Since it is one-one as well as onto, f is bijective.

∴ f = {(1,5), (2,6), (3,4)}
∴ f^-1 ={(5,1), (6,2), (4,3)}

b. (ii) 1,2,3
1 * 1 = 2,
2 * 1=2
Further 3 * 2 = 3
2 * 3 = 3
∴ * is commutative.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

Plus Two Maths Previous Year Question Papers and Answers

Kerala Plus Two Maths Model Question Paper 4

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

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions careully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.
a. The slope of the tangent to the curve given by
x = 1- cos θ , y = θ – sin θ at θ = \(\frac { \pi }{ 2 }\) is
(i) 0
(ii) -1
(iii) 1
(iv) Not defined

b. Find the intervals in which the function f (x) = x² – 4x + 6 is strictly decreasing.

Question 2.

Question 3.

Question 4.
a. Write the Cartesian equation of the straight line through the point (1, 2, 3) and along the vector 3\(\widehat { i }\) + \(\widehat { j }\) + 2\(\widehat { k }\).
b. Write a general point on this straight line.
c. Find the distance from (1, 2, 3) to the plane 2x + 3y – z + 2 = 0.

Question 5.

Question 6.
Solve the system of linear equations :
x + 2y + z = 8
2x + y – z = 1
x – y + z = 2
x + 2y + z = 8
2x + y – z = 1

Question 7.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.

Question 9.

Question 10.
i. Find the equation of the plane through the points (3,-1,2), (5,2,4) and (-1,-1,6).
ii. Find the perpendicular distance from the point (6,5,9) to this plane.

Question 11.

a. Express the euqations of the lines into vector form.
b. Find the shortest distance between the lines.

Question 12.

Question 13.
a. Find the equation of a plane with intercepts 2, 3 and 4 on X, Y and Z axes respectively.
b. Find the distance of the point (-1,-2, 3) from the plane r.(2\(\widehat { i }\) – \(\widehat { j }\)) + 4 \(\widehat { k }\)  = 4.

Question 14.

Question 15.

Question 16.

Question 17.
a. For two independent events A and B, which of the following pair of events need not be independent?
i. A’, B’
ii. A, B’
iii. A’, B
iv. A-B, B-A

Question 18 to 24 carry 6 scores each. Answer any 5 questions only

Question 18.
Consider the following L.P.P.
Maximize Z = 3x+2y
Subject to the constraints
x+2y < 10
3x+y < 15
x, y > 0
a. Draw its feasible region.
b. Find the comer points of the feasible region.
c. Find the maximum value of Z.

Question 19.
a. y=a cos x+b sin x is the solution of the differential equation

Question 20.

Question 21.

Question 22.

Question 23.

Question 24.
a. If \(\overline { a } \),\(\overline { b } \),\(\overline { c } \),\(\overline { d } \) respectively are the position vectors representing the vertices A, B, C, D of a parallelogram, then write \(\overline { d } \) in terms of \(\overline { a } \), \(\overline { b } \) and \(\overline { c } \) .

b. Find the projection vector of \(\overline { b } \) = \(\widehat { i }\) + 2 \(\widehat { j }\) + \(\widehat { k }\) along the vector \(\overline { a } \) = 2i +j + 2k. Also write \(\overline { b } \) as the sum of a vector along \(\overline { a } \) and a vector perpendicular to \(\overline { a } \) .

c. Find the area of a parallelogram for which the vectors 2 \(\widehat { i }\) + \(\widehat { j }\) and 3 \(\widehat { i }\) + \(\widehat { j }\) + 4 \(\widehat { k }\) are adjacent sides.

ANSWERS

Answer 1.

The point x = 2 divides the real line into two disjoint intervals namly (-∞, 2) and (2,∞). In the interval(-∞,2), f'(x)=2x-4 < 0
∴ f is strictly decreasing in this interval.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

Plus Two Maths Previous Year Question Papers and Answers

Kerala Plus Two Maths Model Question Paper 3

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

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions careully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.
a. Let f : R → R be a function defined by f (x) = x³ + 5. Then f¹ (x) is
i. (x+5)^1/3
ii. (x-5)^1/3
iii. (5-x)^1/3
iv. 5-x

b. Let * be a binary operation defined on Q
a*b = a-b + ab. Check whether
i. It is commutative?
ii. Is * associative ?

Question 2.

Question 3.

Question 4.

Question 5.
Prove that the function f given by f (x)= log sin x is strictly increasing on ( 0,\(\frac { \pi }{ 2 } \))

Question 6.

Question 7.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.
a. Show that the relation R in set of real numbers defined as R = {(a,b): a < b²} is neither reflexive nor symmetric not transitive.
b. Show that the operation * on Q, defined by a*b = a+b-ab is commutative, and ex-its and identity elements find it.

Question 9.
a. The principal value of the expression cos-1 cos (680) is …………..

Question 10.

Question 11.

Question 12.

Question 13.

Question 14.

Question 15.
a. Find the distance between the planes x-y + z-5 = 0 and 2x-2y + 2z = 0.
b. Write the vector equation corresponding to Cartesian equation of a line

Question 16.
Find the shortest distance between the lines

Question 17.

Question 18 to 25 carry 6 scores each. Answer any 5 questions only

Question 18.

Question 19.
a. Use differential to approximate (0.999)^1/10
b. A window is in the form of rectangle sur-mounted by a semicircular opening. The total perimeter of the windows is 1 Om. Find the dimensions of the window to admit maximum light through the whole opening.

Question 20.
Find the area lying above x axis and included between the circle x² + y² = 8x and inside of the parabola y² = 4x. Also draw a neat diagram.

Question 21.
Evaluate:

Question 22.
Minimize and maximize Z = 5x + 10 y subject to x + 2y < 120, x + y > 60, x – 2y > 0, x, y > 0
a. Draw the feasible region
b. Find the comer points
c. Find the maximum and minimum profit.

Question 23.
a. Find the distance of the point (-1, -5, -10) from the point of intersection of the line

Question 24.
a. Two numbers are selected at random (with-out replacement) from the Pt six positive integers.

Let X denote the larger of the two numbers obtained. Find E(X) and Var(X)

b. A card from a pack of 52 cards is lost from the remaining X cards of the pack, two cards are drawn and are found to be both spades. Find the probability of the lost card being a spade.

ANSWERS

Answer 1.
a. ii
b. a * b = a – b + ab
b*a = b- a + ba = b – a + ab
∴ * is not commutative.
(a,b) * c = d * c = d- c + dc
= a- b + ab – c + ac – bc + abc
= a- b + ab – c + ac – bc + abc
= a – b – c + ab – bc + ca + abc
a * (b * c) = a * d = a- d + ad
= a – (b – c + be) + a (b – c + be)
= a- b + c – bc + ab – ac + abc
= a – b + c ab – be + ca + abc
∴ * is not associative.

Answer 2.

Answer 3.
We are giving that

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.
a. R = {(a,b): a < b²}
Relation R is defined in the set of real numbers.
i. Reflexive
Consider a ∈ R
If a ∈ R ⇒ a < a² which is false (a, a) ∈ R
R is not reflexive.

ii. Symmetric
Let a,b ∈ R and
(a,b) ∈ R ⇒ a < b² and b < a²,
which is false ⇒ (a,b) ∈ R, but (b,a) ∈ R
∴ R is not symmetric.

iii. Transitive
Let a, b, c ∈ R

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.
The given equation of the circle x² + y² = 8x can be expressed as (x – 4)² + y² = 16. Thus, the centre of the circle is (4,0) and radius is 4. Its intersection with the parabola y² = 4x gives
x² + 4x = 8x
or
x² – 4x = 0
or
x(x-4) = 0
or
x = 0, x = 4
Thus, the point of intersection of these two curves are 0 (0,0) and P (4,4) above the x-axis.

From the above the required area of the region OPQCO included between these two curves above x-axis is
= (area of the region OCPO)
+ (area of the region PCQP)

Answer 21.

Answer 22.
a. The feasible region determine! by the constraints,
x + 2y < 120, x + y > 60, x – 2y > 0,
x > 0 and y > 0
is as follows.

b. The comer points of the feasible are region are A(60,0), C(60,30) and D (40,20).
The values of Z at these comer points are as follows.

a. The minimum value of Z is 300 at (60,0) and the maximum value of Z is 600 at all the points on the line segment joining (120,0) and (60,30)

Answer 23.

Answer 24.
a. The two positive integers can be sele-cted from the fist six positive integers without replacement in 6 x 5 = 30 ways.

X represents the larger of the two numbers obtained. Therefore, X can take the value of 2,3,4,5 or 6.

For X=2, the possible observations are (1,1)and(2,1)
∴ P (x = 2) = \(\frac { 2 }{ 30 } \) = \(\frac { 1 }{ 15 } \)

For X = 3 the possible observations are (1,3), (2,3), (3,1) and (3,2).
∴ p (x = 3) = \(\frac { 4 }{ 30 } \) = \(\frac { 2 }{ 15 } \)

For x = 4 the possible observations are

(1,4), (2,4), (3,4), (4,3), (4,2) and (4,1).
∴ p (x = 4) = \(\frac { 6 }{ 30 } \) = \(\frac { 1 }{ 5 } \)

For X = 5, the possible observations are (1.5) , (2,5), (3,5), (4,5), (5,4), (5,3), (5,2) and (5,1).
∴ p (x = 5) = \(\frac { 8 }{ 30 } \) = \(\frac { 4 }{ 15 } \)

For X = 6, the possible observations are (1.6), (2,6), (3,6), (4,6), (5,6), (6,4), (6,3), (6,2) and (6,1)
∴ p (x = 6) = \(\frac { 10 }{ 30 } \) = \(\frac { 1 }{ 3 } \)

Therefore, the required probability distribution is as follows.

b. Let E and E, be the respective events of choosing a spade card and a card which is not spade. Out of 52 cards, 13 cards are spade and 39 cards are not spade.

Plus Two Maths Previous Year Question Papers and Answers

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

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:

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

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 \(\oplus\) is defined by a \(\oplus\) b = a² + b , where a and b are integers. Then find the value of (3 \(\oplus\) 4) * 5
Answer:
i) b) (3, 2)
ii) (3 \(\oplus\) 4) * 5 = (32 + 4) * 5
= 13 * 5
= 13 – 5² = -12

Question 2.
If X + Y = \(\left[\begin{array}{ll}
5 & 2 \\
0 & 9
\end{array}\right]\) and X – Y = \(\left[\begin{array}{cc}
3 & 6 \\
0 & -1
\end{array}\right]\)find 2X – 3Y.
Answer:

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

Question 4.
i) Integrate \(\int \frac{(1+\log x)^{2}}{x} d x\)

Answer:

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 = \(\sqrt{a^{2}-x^{2}}\)
Area of the circle will 4 time the area under the curve from 0 to a.
Area = \(4 \int_{0}^{a} y d x=4 \int_{0}^{a} \sqrt{a^{2}-x^{2}} d x\)

Question 6.
Consider the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\)
i) Write the order of the differential equation.
ii) Solve the above given differential equation.
Answer:
i) Order = 1.

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:

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?

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) \(-\frac{\pi}{2}\) ≤ y ≤ \(\frac{\pi}{2}\)
c) 0 < y < π
d) \(-\frac{\pi}{2}\) ≤ y ≤ \(\frac{\pi}{2}\)
ii) Find the principal value of sin^-1 (\(\frac{1}{2}\))
iii) sin^-1x = \(\frac{3}{4}\) find the value of cos^-1 x
Answer:
i) \(-\frac{\pi}{2}\) ≤ y ≤ \(\frac{\pi}{2}\)
ii) sin^-1 \(\frac{1}{2}\) = \(\frac{\pi}{6}\)
iii) sin^-1x + cos^-1x = \(\frac{\pi}{2}\)
\(\frac{3}{4}\) + cos^-1x = \(\frac{\pi}{2}\) ⇒ cos^-1x = \(\frac{\pi}{2}\) – \(\frac{3}{4}\)

Question 10.
i) Find the relation between ‘a’ and ‘b’ so that the function defined by
f(x) = \(\left\{\begin{array}{ll}
a x+1, & x \leq 3 \\
b x+3, & x>3
\end{array}\right.\) 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
\(\lim _{x \rightarrow 3^{-}}\) f(x) = \(\lim _{x \rightarrow 3^{+}}\) f(x) = f(3)
\(\lim _{x \rightarrow 3}\) (ax + 1) = \(\lim _{x \rightarrow 3}\) (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.

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 < \(\frac{\pi}{2}\)
i) y = x² – 2x + 7 ⇒ \(\frac{d y}{d x}\) = 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

Question 12.
Integrate \(\int \frac{x+2}{2 x^{2}+6 x+5} d x\)
Answer:
Put x + 2 = A(4x + 6) + B
4A = 1 ⇒ A = \(\frac{1}{4}\)

Question 13.
Consider the differential equation
x \(\frac{d y}{d x}\) + y = \(\frac{1}{x^{2}}\)
i) Find the integrating factor.
ii) Solve the above differential equation.
Answer:

Question 14.
lf the vectors \(\overline{P Q}\) = -3i + 4j + 4k and \(\overline{P R}\) = -5i + 2j + 4k are the sides of a ΔPQR
i) Find the angle between \(\overline{P Q}\) and \(\overline{P R}\)
ii) Find the length of the median through the vertex P.
Answer:

ii)

With respect to the initial P the position vector of Q and R will be \(\overline{P Q}\) = -3i + 4j + 4k
and \(\overline{P R}\) = -5i + 2j + 4k respectively.
Since M is the midpoint of QR. The position vector of M will be

Question 15.
i) If \(\bar{a}\) = 5i – j – 3k and \(\bar{b}\) = i + 3j + 5k, then show that the vectors \(\bar{a}+\bar{b}, \bar{a}-\bar{b}\) are perpendicular.
ii) If \(\bar{a}\) = i – 2j + 3k, \(\bar{b}\) = 2i + 3j – 4k and
\(\bar{c}\) = i – 3j + 5k, then check whether \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are coplanar.
Answer:

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:

Since P lies in the first octant, we take n = \(\frac{1}{2}\)
Therefore the coordinate of P is

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(\(\frac{20}{19}, \frac{45}{19})\), C(0, 3)

Given Z = 5x + 3y

Since maximum value of Z occurs at B, the soluion is z = \(\frac{235}{19}\), (\(\frac{20}{19}\), \(\frac{45}{19}\))

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

Question 18.
A = \(\left[\begin{array}{ccc}
3 & 3 & -1 \\
-2 & -2 & 1 \\
-4 & -5 & 2
\end{array}\right]\)
a) Find A^T
b) Express A as the sum of a symmetric and skew symmetric matrix.
ii) If A^T = \(\left[\begin{array}{cc}
\cos x & \sin x \\
-\sin x & \cos x
\end{array}\right]\), verift that A^T A = I
Answer:

Question 19.
i) Without expanding prove that
\(\left|\begin{array}{ccc}
x+y & y+z & z+x \\
z & x & y \\
1 & 1 & 1
\end{array}\right|\) = 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:

Question 20.
Find \(\frac{d y}{d x}\) of the following
i) x² + 2xy + 2y² = 1
ii) y^x = 2^x
iii) x = cos θ; y = sin θ at θ = \(\frac{\pi}{4}\)
Answer:
i) x² + 2xy + 2y² = 1
Differentiating w.r.to x;

ii) y^x = 2^x
Take log on both sides;
x log y = x log 2
Differentiating w.r.to x;

Question 21.
Evaluate the following

Answer:

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:

ii) solving the two conics we get the point of intersection as (0, 0) and (4, 4).
iii) Area of the enclosed region

Question 23.
i) Find the shortest distance between the lines whose vector equations are
\(\bar{r}\) = (i + 2j + 3k) + λ(i – 3j + 2k) and \(\bar{r}\) =(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) \(\overline{a_{1}}\) = i + 2j + 3k; \(\bar{b}\) = i – 3j + 2k
\(\overline{a_{2}}\) = 4i + 5j + 6k Both lines are parallel.

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:

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)}

Therefore A and B are independent.

ii) Let B₁, B₂, B₃ are the event of getting the boxes.
P(B₁) = P(B₂) = P(B₃) = \(\frac{1}{3}\)
Let E be the event of getting a red ball.

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

Kerala Plus Two Maths Previous Year Question Paper March 2017 with Answers

Time: 2½ Hours
Cool off time : 15 Minutes

General Instructions to Candidates

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2½ hrs.
• You are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and the only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures, and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Question 1.
a. Let R be a relation defined on A = {1,2, 3} by R = {(1,3), (3, 1), (2, 2)}.R is
a. Reflexive
b. Symmetric
c. Transitive
d. Reflexive but not transitive

b. Find fog and gof if f(x) = |x + 1| and g(x) = 2x – 1.

c. Let * be a binary operation defined on N x N by (a, b) * (c, d) = (a + c, b + d). Find the identity element for * if it exists.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.
Slope of the normal to the curve y² = 4x at (1, 2) is
a. 1
b. \(\frac { 1 }{ 2 } \)
c. 2
d.-1

b. Find the interval in which 2x³ + 9x² + 12x – 1 is strictly increasing.
OR
a. The rate of change of volume of a sphere with respect to its radius when the radius is 1 unit
a. 4π
b. 2π
c. π
d. \(\frac { \pi }{ 2 } \)

b. Find two positive numbers whose sum is 16 and the sum of whose cubes are minimum.

Question 7.

Question 8.

Question 9.

Question 10.

Question 11.

Question 12.
a. The angle between the vectors \(\widehat { i } \) + \(\widehat { j } \) and \(\widehat { j } \) + \(\widehat { k } \) is
(i) 60°
(ii) 30°
(iii) 45°
(iv) 90°

Question 13.
a. The line x – 1 = y = z is perpendicular to the line

b. Find the shortest distance between the lines

Question 14.

Question 15.
Consider the linear programming problem :
Maximize Z = 50x + 40y
Subject to the constraints
x + 2y > 10
3x + 4y < 24
x > 0, y > 0
a. Find the feasible region.
b. Find the comer points of the feasible region.
c. Find the maximum value of Z.
Maximize Z = 50x + 40y
Subject to the constraints
x + 2y > 10
3x + 4y < 24
x > 0, y > 0

Question 16.
If A and B are two events such that A ⊂ B and P(A) ≠ then P (A/B) is
a. \(\frac { P(A) }{ P(B) } \)
b. \(\frac { P(B) }{ P(A) } \)
c. \(\frac { 1 }{ P(A) } \)
d. \(\frac { 1 }{ P(B) } \)

b. There are two identical bags. Bag I contains 3 red and 4 black balls while Bag II contains 5 red and 4 black balls. One ball is drawn at random from one of the bags.
i. Find the probability that the ball is drawn is red.
ii. If the ball drawn is red what is the probability that it was drawn from the bag I?
OR
Consider the following probability distribution of a random variable X.

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

Kerala Plus Two Maths Previous Year Question Paper March 2018 with Answers

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

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Question 1 to 7 carry 3 scores each. Answer any six questions. (6 × 3 = 18)

Question 1.
If f(x) = \(\frac{x}{x-1}\), x ≠ 1
i) Find fof (x)
ii) Find the inverse of f.
Answer:

Question 2.
Using elementary row operations, find the inverse of the matrix \(\left[\begin{array}{cc}
1 & 2 \\
2 & -1
\end{array}\right]\)
Answer:

Question 3.
i) f(x) is a strictly increasing function, if f'(x) is ………..
a) positive
b) negative
c) 0
d) None of these,
ii) Show that the function f given by f(x) = x³ – 3x² + 4x, x ∈ R is strictly increasing.
Answer:
i) a) Positive.
ii) f(x) = x³ – 3x² + 4x
= 3 (x² – 2x + 1 – 1 + \(\frac{4}{3}\)) = 3((x + 1)2 + \(\frac{1}{3}\)) > 0

Question 4.
i) \(\int_{0}^{a}\)f(a – x)dx = …………

Answer:

Question 5.
Find the area of the region bounded by the curve y² = x’, x-axis and the lines x = 1 and x = 4.
Answer:

Question 6.
Find the general solution of the differential equation x \(\frac{d y}{d x}\) + 2y = x² log x

Solution is

Question 7.
A manufacturer produces nuts and bolts. It takes 1 hour of work on Machine A and 3 hours on Machine B to produce a package of nuts. It takes 3 hours on Machine A and 1 hour on Machine B to produce a package of bolts. He earns profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. Formulate the above LPp if the Machine operates for at most 12 hours a day.
Answer:
Let x packet of nuts and y packets of bolts.
Maximise: Z=17. 5x + 7y
Subject to
x + 3y ≤ 12; 3x + y ≤ 12; x, y ≥ O

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

Question 8.
Let A = N × N and ‘*’ be a binary operation on A defined by
(a, b)*(c, d)=(a + c, b + d)
i) Find (1, 2) * (2, 3)
ii) Prove that * is commutative.
iii) Prove that * is associative.
Answer:
i) (1, 2)*(2, 3)=(1 + 2, 2 + 3) = (3, 5)
ii) (c, d)*(a, b) = (c + a, d + b)
= (a + c, b + d) = (a, b)* (c, d)
iii) (a, b)*[(c, d)*(e, f)] = (a, b)*(c + e, d + f)
= (a + c + e, b + d + f)
[(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f)
= (a + c + e, b + d + f)

Question 9.
i) Identify the function from the above graph

a) tan^-1x
b) sin^-1x
c) cos^-1x
d) cos ec^-1x
ii) Find the domain and range of the function represented in above graph.
iii) Prove that \(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{2}{11}=\tan ^{-1} \frac{3}{4}\)
Answer:
i) sin^-1x

Question 10.
i) \(\frac{d\left(a^{x}\right)}{d x}\) = ………..
a) a^x
b) log(a^x)
c) a^x log a
d) xa^x-1
ii) Find \(\frac{d y}{d x}\) if x^y = y^x
Answer:
i) a^x loga
ii) Given; y^x = x^y, taking log on both sides;
x log y = y log x,
Differentiating with respect to x;

Question 11.
i) Find the slope of the tangent to the curve y = (x – 2)² at x = 1.
ii) Find a point at which the tangent to the curve y = (x – 2)² is parallel to the chord joining the point A (2, 0) and B (4, 4)
iii) Find the equation of the tangent to the above curve and parallel to the line AB.
y² = 4ax, a > 0 and x² = 4ay, a > 0
Answer:
i) \(\frac{d y}{d x}\) = 2(x – 2) ⇒ Slope = 2(1 – 2) = -2
ii) Slope of AB = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{4-0}{4-2}=2\)
Here 2(x – 2) = 2 ⇒ x = 3, y = 1
iii) Equation of the tangent line is
y – 1 = 2 (x – 3) ⇒ 2x – y = 5

Question 12.
\(\int_{0}^{2}\)(x² +1)dx as the limit of a sum.
Answer:
Here; f(x) = x² + 1; a = 0, b = 2 ⇒ h = \(\frac{2}{n}\)
\(\int_{0}^{2}\)(x² +1)dx
\(\lim _{h \rightarrow 0}\) h[f(0) + f(0 + h) + f(0 + 2h)+…+ f(0 + (n – 1)h)]
\(\lim _{h \rightarrow 0}\) h (1 + (h² + 1)+((2h) ² + 1)+…+[(n – 1)h²] + 1]

Question 13.
Consider the following figure:

i) Find the point of intersection ‘P’ of the circle x² + y² = 50 and the line y = x
ii) Find the area of the shaded region.
Answer:
i) point of intersection is x² + x² = 50 ⇒ x² = 25 ⇒ ±5; y = ±5
point P is (5, 5)

Question 14.
i) The degree of the differential equation \(x y\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+x^{4}\left(\frac{d y}{d x}\right)^{3}-y \frac{d y}{d x}=0\) is ………
a) 4
b)3
c)2
d)1
ii) Find the general solution of the differential equation
sec² x tan ydx + sec² y tan xdy = 0
Answer:
i) c
ii) Given; sec²x tan ydx + sec²y tan xdy = 0

log tanx = -log tany + c
log tan x + log tan y = c

Question 15.
i) Prove that for any vector \(\bar{a}, \bar{b}, \bar{c}\)
\([\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}]=2\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]\)
ii) Show that if \(\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}\) are coplanar then \(\bar{a}, \bar{b}, \bar{c}\) are also coplanar.
Answer:

Question 16.
i) Find the equation of a plane which makes x,y,z intercepts respectively as 1, 2, 3.
ii) Find the equation of a plane passing through the point (1, 2, 3) which is parallel to above plane.
Answer:
\(\frac{x}{1}+\frac{y}{2}+\frac{z}{3}\) ⇒ 6x + 3y + 2z = 6
ii) Equation of the parallel plane passing
6(x – 1) + 3( y – 2) + 2(z – 3) = 0
⇒ 6x + 3y + 2z – 18 = 0

Question 17.
Solve the LPP graphically:
Minimise z = -3x + 4y
Subject to constraints:
x + 2y ≤ 8; 3x + 2y ≥ l2; x, y ≥ 0
Answer:
x + 2y = 8

3x + 12y = 12

The corner points are O(0, 0), A(4, 0), B(2, 3), C(0, 4)

Z attains minimum at (4,0).

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

Question 18.
i) Find x and y if \(x\left[\begin{array}{l}
2 \\
3
\end{array}\right]+y\left[\begin{array}{c}
-1 \\
1
\end{array}\right]=\left[\begin{array}{c}
10 \\
5
\end{array}\right]\)
ii) Express the matrix \(\left[\begin{array}{ccc}
2 & -2 & -4 \\
-1 & 3 & 4 \\
1 & -2 & -3
\end{array}\right]\) as the sum of a symmetric and a skew symmetric matrices.
Answer:
Given; 2x – y = 10; 3x + y = 5
5x = 15 ⇒ x = 3; y = -4

Question 19.
i) Prove that \(\left|\begin{array}{ccc}
a & b & c \\
a+2 x & b+2 y & c+2 z \\
x & y & z
\end{array}\right|=0\)
ii) If A = \(\left[\begin{array}{ccc}
1 & -1 & 2 \\
0 & 2 & -3 \\
3 & -2 & 4
\end{array}\right]\), B = \(\left[\begin{array}{ccc}
-2 & 0 & 1 \\
9 & 2 & -3 \\
6 & 1 & -2
\end{array}\right]\)
a) Prove that B = A^-1
b) Using Asolve the system of linear equations given below:
x – y + 2z = 1; 2y – 3z = 1; 3x – 2y + 4z = 2
Answer:

iii) Matrix form of the system of linear equation is

Question 20.
i) Prove that the function defined by f(x) = cos x² is a continuous function.
ii) a) If y = \(e^{a \cos ^{-1} x}\), -1 ≤ x ≤ 1, show that
\(\frac{d y}{d x}=\frac{-a e^{a \cos ^{-1} x}}{\sqrt{1-x^{2}}}\)
b) Hence prove that
\(\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0\)
Answer:
i) f(x) = cosx; g(x) = x² both are continuous.
Composition two continuous function is again continuous.
fog(x) = f(g(x)) = cos(x²)

Question 21.
Evaluate the following:

Answer:

Question 22.
i) lf \(\bar{a}\) = 3i + 2j + 2k, \(\bar{b}\) = i + 2j – 2k
a) Find \(\bar{a}\) + \(\bar{b}\), \(\bar{a}\) – \(\bar{b}\)
b) Find a unit vector perpendicular to both \(\bar{a}\) + \(\bar{b}\), \(\bar{a}\) – \(\bar{b}\)
ii) Consider the points A(1, 2, 7); B(2, 6, 3); C (3, 10, -1)
a) Find \(\overline{A B}, \overline{B C}\)
b) Prove that A, B, C are collinear points.
Answer:

Question 23.
i) Find the angle between the lines \(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\)
ii) Find the shortest distance between the pair of lines
\(\bar{r}\) = (i + 2j + 3k) + λ(i – 3j + 2k)
\(\bar{r}\) = (4i + 5j +6k) + μ(2i + 3j + k)
Answer:

Question 24.
i) The probability distribution of a random variable is given by P(x). What is Σp(x)?
ii) The following is a probability distribution function of a random variable.

a) Find k
b) P(x > 3)
c) P(-3 < x < 4)
d) P(x < -3)
Answer:
Σp(x) = 1

ii) a) k + 2k + 3k + 4k + 5k + 7k
+ 8k + 9k + 10k + 11k + 12k = 1
72k = 1 ⇒ k = \(\frac{1}{72}\)
b) P(x > 3) = P(4) + P(5) = 23k = \(\frac{23}{72}\)
c) P(-3 < x < 4) = P(-2) + P(-1) + P(0) + P(1) + P > (2)+ P > (3) = 43k = \(\frac{43}{72}\)
d) P(x < -3) = P(-5) + P(-4) = 3k = \(\frac{3}{72}\)

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Kerala Plus Two Maths Model Question Paper 2

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

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions careully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.

Question 2.

Question 3.
Consider the differential equation \(\frac { { d }^{ 2 }{ y } }{ { dx }^{ 2 } } +y=0\)
a. write its order and degree
b. Verify that y = a cos x + b sin x, where a, b ∈ R is a solution of the given differential equation

Question 4.
Consider the differential equation x \(\frac { dy }{ dx } \) + 2y = x2 ; x ≠ 0
a. What is its integrating factor?
b. Obtain its general solution.

Question 5.
The foot of the perpendicular drawn from origin to a plane is (4,-2, 5).
a. How far is the plane from the origin?
b. Find a unit vector perpendicular to that plane.
c. Obtain the equation of the plane in general form.

Question 6.

Question 7.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.

Question 9.

Question 10.

Question 11.
Consider the points A (2, 2, -1), B (3, 4, 2) and C (7, 0, 6). Find the vector and Cartesian equation of the plane passing through these points.

Question 12.
a. Find the cartesian equation of the plane through the point (1,2,-3) and perpen-dicular to the vector 2\(\widehat { i } \) – \(\widehat { j } \) + 2\(\widehat { k } \).
b. Find the angle between the above plane and the line \(\frac { x-1 }{ 2 } \) = \(\frac { y-3 }{ 3 } \) = \(\frac { z }{ 6 } \)

Question 13.
a. Let R be the relation on the set N of . natural numbers given by R = {(a,b): a – b > 2, b > 3}.
Choose the correct answer.
A. (4,1) ∈ R
B. (5,8) ∈ R
C. (8,7) ∈ R
D.(10,6) ∈ R

b. If f (x) = 8x³ and g (x) = x^1/3, find g (f(x)) and f (g (x)).

c. Let * be a binary operation on the set Q of rational numbers defined by a*b= \(\frac { ab }{ 3 } \).

Check whether * is commutative and associative?

Question 14.

The probability distribution of a random variable X taking values 1, 2, 3, 4, 5 is given.
a. Find the value of P.
b. Find the mean of X.
c. Find the variance of X.

Question 15.
a. Consider the family of all circles having their centre at the point (1,2). Write the equation of the family.
Write the corresponding differential equation.
b. Write the integrating factor of the differential equation,

Question 16.
Consider the functions: f (x) = |x|-1 and g(x) = 1- | x|
a. Sketch their graphs and shade the closed region between them.
b. Find the area of their shaded region.

Question 17.
a. What is the value of sin^-1 (sin 160°)?

Question 18 to 24 carry 6 scores each. Answer any 5 questions only

Question 18.

Question 19.
integrate the following :

Question 20.
a. Choose the correct statement related to the matrices A = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)

Question 21.
a. Which of the following functions is always increasing?
i. x + sin 2x
ii. x – sin 2x
iii. 2x + sin 3x
iv. 2x – sin x
b. The radius of a cylinder increases at a rate of 1 cm/s and its height decreases at a rate of 1 cm/s. Find the rate of change of its volume when the radius is 5 cm and the height is 15 cm.
If the volume should not change even when the radius and height are changed, what is the relation between the radius and height?
c. Write the equation of tangent at (1,1) on the curve 2x² + 3y² = 5.

Question 22.
Consider the linear programming problem :
Minimize Z = 3x + 9y
Subject to the constraints :
x+3y < 60 ; x + y > 10 ; x < y ; x > 0 , y > 0
a. Draw its feasible region.
b. Find the vertices of the feasible region.
c. Find the minimum value of Z subject to the given constraints.
Minimize Z = 3x + 9y
Subject to the constraints :
x + 3y < 60 ; x + y > 10 ; x < y ; x > 0, y > 0

Question 23.
Consider the following figure :

a. Find the point of intersection P, of the circle, x²+ y² = 32 and the line y = x.
b. Express the area of the shaded portion as a Sum of two definite integrals.
c. Find the area of the shaded portion.

Question 24.

ANSWERS

Answer 1.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

c. 1. f (x) is continuous at (-2, 2)
2. f (x) is differentiable at (-2, 2)
3. If f (b) = f (a) then there exist a point at (-2, 2) such that f¹ (c) = 0
f (b) = f (-2) = (-2)² + 2 = 6
f (a) = f (2) = 2² + 2 = 6
∴ Then there exist a point C such that
f¹ (c) = 0
f¹ (x) = 2x
f¹ (x) = 2c = 0
∴ c = 0 at (-2,2)
∴ Rolle’s theorem verified.

Plus Two Maths Previous Year Question Papers and Answers

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