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Mathematics and Statistics Shaalaa.com Model Set 1 2020-2021 HSC Science (General) 12th Board Exam Question Paper Solution

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Mathematics and Statistics [Shaalaa.com Model Set 1]
Marks: 80 Academic Year: 2020-2021
Date: April 2021
Duration: 3h
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  1. The question paper is divided into four sections.
  2. Section A: Q. No. 1 contains 8 multiple-choice type of questions carrying two marks each.
  3. Section A: Q. No. 2 contains 4 very short answer type of questions carrying One mark each.
  4. Section B: Q. No. 3 to Q. No. 14 contains Twelve short answer type of questions carrying Two marks each. (Attempt any Eight).
  5. Section C: Q. No.15 to Q. No. 26 contains Twelve short answer type of questions carrying Three marks each. (Attempt any Eight).
  6. Section D: Q.No. 27 to Q. No. 34 contains Five long answer type of questions carrying Four marks each. (Attempt any Five).
  7. Use of log table is allowed. Use of calculator is not allowed.
  8. Figures to the right indicate full marks.
  9. For each MCQ, correct answer must be written along with its alphabet.
    e.g., (a) ..... / (b ) .... / ( c ) .... / ( d) ..... Only first attempt will be considered for evaluation.
  10. Use of graph paper is not necessary. Only rough sketch of graph is expected:
  11. Start answers to each section on a new page. 

Section A
[16] 1 | Select and write the most appropriate answer from the given alternatives for each sub-question:
[2] 1.i

Which of the following statement is true

3 + 7 = 4 or 3 – 7 = 4

If Pune is in Maharashtra, then Hyderabad is in Kerala

It is false that 12 is not divisible by 3

The square of any odd integer is even

Concept: Truth Value of Statement
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2] 1.ii

If A = `[(cos alpha, sin alpha),(-sin alpha, cos 10 alpha)]`, then A10 = ______

`[(cos10  alpha, -sin10  alpha),(sin10  alpha, cos10  alpha)]`

`[(cos10  alpha, sin10  alpha),(-sin10  alpha, cos10  alpha)]`

`[(cos10  alpha, sin10  alpha),(-sin10  alpha, -cos10  alpha)]`

`[(cos10  alpha, -sin10  alpha),(-sin10  alpha, -cos10  alpha)]`

Concept: Inverse of Matrix
Chapter: [0.012] Matrics
[2] 1.iii

Bernoulli distribution is a particular case of binomial distribution if n = ______

4

10

2

1

Concept: Bernoulli Trials and Binomial Distribution
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
[2] 1.iv

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(("c")/x^3",", "for"  x = 1","  2","  3","),(0",", "otherwise"):}` then E(X) = ______

`343/297`

`294/251`

`297/294`

`294/297`

Concept: Probability Distribution of Discrete Random Variables
Chapter: [0.027000000000000003] Probability Distributions
[2] 1.v

The separate equations of the lines represented by `3x^2 - 2sqrt(3)xy - 3y^2` = 0 are ______ 

`x + sqrt(3)y` = 0 and `sqrt(3)x + y` = 0

`x - sqrt(3)y` = 0 and `sqrt(3)x - y` = 0

`x - sqrt(3)y` = 0 and `sqrt(3)x + y` = 0

`x + sqrt(3)y` = 0 and `sqrt(3)x - y` = 0

Concept: Equation of a Line in Space
Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line
[2] 1.vi

Let I1 = `int_"e"^("e"^2)  1/logx  "d"x` and I2 = `int_1^2 ("e"^x)/x  "d"x` then 

I1 = `1/3 "I"_2`

I1 + I2 = 0 

I1 = 2I2 

I1 = I2 

Concept: Methods of Evaluation and Properties of Definite Integral
Chapter: [0.024] Definite Integration
[2] 1.vii

If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______

f(x) − log x + c

f(x) + log x + c

log x − f(x) + c

`1/5x^5` f(x) + c

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2] 1.viii

If the foot of the perpendicular drawn from the origin to the plane is (4, −2, 5), then the equation of the plane is ______ 

4x + y + 5z = 14

4x − 2y − 5z = 45

x − 2y − 5z =10

4x + y + 6z = 11

Concept: Equation of a Plane
Chapter: [0.016] Line and Plane
[1] 2 | Answer the following questions:
[1] 2.i

State the truth value of `sqrt(3)` is not an irrational number

Concept: Truth Value of Statement
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[1] 2.ii

Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1 sqrt(3))`

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
[1] 2.iii

Solve each of the following inequations graphically using XY-plane:

4x - 18 ≥ 0

Concept: Linear Programming Problem (L.P.P.)
Chapter: [0.017] Linear Programming
[1] 2.iv

The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. Find the velocity when 𝑡 = 2 sec

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
Section B
[2] 3 | Attempt any Eight:

Write the converse and contrapositive of the following statements.

“If a function is differentiable then it is continuous”

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
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[2] 4

Find the principal solutions of tan x = `-sqrt(3)`

Concept: Trigonometric Equations and Their Solutions
Chapter: [0.013000000000000001] Trigonometric Functions
[2] 5

Find the combined equation of the following pair of lines passing through (2, 3) and parallel to the coordinate axes.

Concept: Combined Equation of a Pair Lines
Chapter: [0.013999999999999999] Pair of Straight Lines
[2] 6

Find k, if the sum of the slopes of the lines represented by x2 + kxy − 3y2 = 0 is twice their product. 

Concept: Homogeneous Equation of Degree Two
Chapter: [0.013999999999999999] Pair of Straight Lines
[2] 7

Find the differential equation of family of all ellipse whose major axis is twice the minor axis

Concept: Formation of Differential Equations
Chapter: [0.026000000000000002] Differential Equations
[2] 8

Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0

Concept: Differential Equations
Chapter: [0.026000000000000002] Differential Equations
[2] 9

Find the derivative of the inverse of function y = 2x3 – 6x and calculate its value at x = −2

Concept: Derivatives of Inverse Functions
Chapter: [0.021] Differentiation
[2] 10

A car is moving in such a way that the distance it covers, is given by the equation s = 4t2 + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
[2] 11

`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2] 12

Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
[2] 13

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
[2] 14

If `bar("a"), bar("b")` and `bar("c")` are position vectors of the points A, B, C respectively and `5bar("a") - 3bar("b") - 2bar("c") = bar(0)`, then find the ratio in which the point C divides the line segement BA

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
Section C
[3] 15 | Attempt any Eight:

Write the converse, inverse, and contrapositive of the following statement.

"If it snows, then they do not drive the car"

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3] 16

In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
[3] 17

Maximize z = 10x + 25y subject to x + y ≤ 5, 0 ≤ x ≤ 3, 0 ≤ y ≤ 3

Concept: Linear Programming Problem (L.P.P.)
Chapter: [0.017] Linear Programming
[3] 18

The probability that certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

Concept: Mean of Binomial Distribution (P.M.F.)
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
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[3] 19

A random variable X has the following probability distribution :

X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine :
(i) k
(ii) P(X < 3)
(iii) P( X > 4)

Concept: Probability Distribution of Discrete Random Variables
Chapter: [0.027000000000000003] Probability Distributions
[3] 20

If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`

Concept: Logarithmic Differentiation
Chapter: [0.021] Differentiation
[3] 21

Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing

Concept: Increasing and Decreasing Functions
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
[3] 22

A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
[3] 23

`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[3] 24

If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[3] 25

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
[3] 26

Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar

Concept: Vector Triple Product
Chapter: [0.015] Vectors
Section D
[4] 27 | Attempt any Five:

If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third numbers, we get 8. If three times the first number is added to the sum of second and third numbers, we get 4. Find the numbers using matrices.

Concept: Elementary Transformations
Chapter: [0.02] Matrices
[4] 28

Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`

Concept: Inverse Trigonometric Functions
Chapter: [0.013000000000000001] Trigonometric Functions
[4] 29

Find the area of the region lying between the parabolas 4y2 = 9x and 3x2 = 16y

Concept: Area Between Two Curves
Chapter: [0.025] Application of Definite Integration [0.16] Applications of Definite Integral
[4] 30

Evaluate: `int_0^1 (1/(1 + x^2)) sin^-1 ((2x)/(1 + x^2))  "d"x`

Concept: Methods of Evaluation and Properties of Definite Integral
Chapter: [0.024] Definite Integration
[4] 31

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours  `("Given"  sqrt(2) = 1.414)`

Concept: Application of Differential Equations
Chapter: [0.026000000000000002] Differential Equations
[4] 32

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin2x

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [0.021] Differentiation [0.13] Differentiation
[4] 33

Show that the lines `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1)` and `(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/4` intersect each other.also find the coordinates of the point of intersection

Concept: Equation of a Plane
Chapter: [0.016] Line and Plane
[4] 34

Let `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
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Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2020 - 2021

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