Mathematics and Statistics Model set 1 by shaalaa.com 2024-2025 HSC Science (General) 12th Standard Board Exam Question Paper Solution | Shaalaa.com (2025)

Mathematics and Statistics [Model set 1 by shaalaa.com]

Marks: 80 Maharashtra State Board
HSC Science (General)
HSC Arts (English Medium)
HSC Science (Electronics)
HSC Science (Computer Science)
HSC Arts (Marathi Medium)

Academic Year: 2024-2025
Date: मार्च 2025

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General Instructions: The question paper is divided into four sections.

  1. Section A: Q.1 contains Eight multiple-choice types of questions, each carrying Two marks. Q.2 contains Four very short answer type questions, each carrying one mark.
  2. Section B: Q.3 to Q.14 contains Twelve short answer type questions, each carrying Two marks. (Attempt any Eight)
  3. Section C: Q.15 to Q.26 contain Twelve short answer type questions, each carrying Three marks. (Attempt any Eight)
  4. Section D: Q. 27 to Q.34 contain Eight long answer type questions, each carrying Four marks. (Attempt any Five)
  5. Use of Log table is allowed. Use of calculator is not allowed.
  6. Figures to the right indicate full marks.
  7. Use of graph paper is not necessary. Only rough sketch of graph is expected.
  8. For each multiple-choice type question, it is mandatory to write the correct answer along with its alphabet. e.g., (a) .............. /(b) ............... /(c) ............... /(d) ................ ,etc. No mark(s) shall be given if ONLY the correct answer or the alphabet of the correct answer is written. Only the first attempt will be considered for evaluation.
  9. Start answer to each section on a new page.

SECTION - A

[16]1|Select and write the correct answer for the following multiple-choice type of questions:

[2]1.i

Choose the correct option from the given alternatives :

Let f(x) and g(x) be differentiable for 0 ≤ x ≤ 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be ______.

1

3

2.5

–1

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.022000000000000002] Applications of Derivatives

[2]1.ii

The maximum value of z = 5x + 3y subject to the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0 is ______.

235

`235/9`

`235/19`

`235/3`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.017] Linear Programming

[2]1.iii

The area bounded by the curve y = x3, the X-axis and the Lines x = –2 and x = 1 is ______.

–9 sq.units

`- 15/4` sq.units

`15/4` sq.units

`17/4` sq.units

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.025] Application of Definite Integration

[2]1.iv

If `int_0^1 ("d"x)/(sqrt(1 + x) - sqrt(x)) = "k"/3`, then k is equal to ______.

`sqrt(2)(2sqrt(2) - 2)`

`sqrt(2)/3(2 - 2sqrt(2))`

`(2sqrt(2) - 2)/3`

`4sqrt(2)`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.024] Definite Integration

[2]1.v

Choose the correct option from the given alternatives :

`int_0^(pi/2) (sin^2x*dx)/(1 + cosx)^2` = ______.

`(4 - pi)/2`

`(pi - 4)/2`

`4 - pi/(2)`

`(4 + pi)/2`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.024] Definite Integration

[2]1.vi

Choose the correct option from the given alternatives :

If x = –1 and x = 2 are the extreme points of y = αlogx + βx2 + x`, then ______.

α = –6, β = `1/2`

α = –6, β = `-1/2`

α = 2, β = `-1/2`

α = 2, β = `1/2`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.022000000000000002] Applications of Derivatives

[2]1.vii

Select the correct option from the given alternatives:

The principal solutions of equation sin θ = `- 1/2` are ______.

`(5pi)/6, pi/6`

`(7pi)/6, (11pi)/6`

`pi/6, (7pi)/6`

`(7pi)/6, pi/3`

Concept: undefined - undefined
Chapter: [0.013000000000000001] Trigonometric Functions

[2]1.viii

Select the correct option from the given alternatives:

If cos pθ = cos qθ, p ≠ q, then ______.

θ = `(2npi)/(p +- q)`

θ = 2nπ

θ = 2nπ ± p

θ = 2nπ ± q

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.013000000000000001] Trigonometric Functions

[4]2|Answer the following questions:

[1]2.i

Find the general solution of the following equation:

sinθ = `1/2`.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.013000000000000001] Trigonometric Functions

[1]2.ii

Evaluate: `int_0^1 (x^2 - 2)/(x^2 + 1).dx`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.024] Definite Integration

[1]2.iii

Apply the given elementary transformation of the following matrix.

A = `[(1,0),(-1,3)]`, R1↔ R2

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.012] Matrics

[1]2.iv

State whether the following equation has a solution or not?

2sinθ = 3

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.013000000000000001] Trigonometric Functions

SECTION - B

[2]3|Attempt any EIGHT of the following questions:

A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is `1/100`. What is the probability that he will win a prize at least once.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution

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[2]4

Find k, the slope of one of the lines given by kx2 + 4xy – y2 = 0 exceeds the slope of the other by 8.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.013999999999999999] Pair of Straight Lines

[2]5

Find the position vector of midpoint M joining the points L(7, –6, 12) and N(5, 4, –2).

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.015] Vectors [0.07] Vectors

[2]6

Find the vector equation of the lines passing through the point having position vector `(-hati - hatj + 2hatk)` and parallel to the line `vecr = (hati + 2hatj + 3hatk) + λ(3hati + 2hatj + hatk)`.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line

[2]7

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following:

y = `sqrt(x)`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.021] Differentiation

[2]8

Solve the following :

Find the area enclosed between the circle x2 + y2 = 1 and the line x + y = 1, lying in the first quadrant.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.025] Application of Definite Integration

[2]9

Find the co-factor of the element of the following matrix:

`[(-1, 2),(-3, 4)]`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.012] Matrics

[2]10

Verify which of the following is p.d.f. of r.v. X:

f(x) = sin x, for 0 ≤ x ≤ `π/2`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.027000000000000003] Probability Distributions

[2]11

Using the rule of negation write the negation of the following with justification.

p → (p ∨ ∼ q)

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.011000000000000001] Mathematical Logic

[2]12

Find the Cartesian equation of the plane passing through A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.016] Line and Plane

[2]13

Solve graphically: 2x – 3 ≥ 0

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.017] Linear Programming

[2]14

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.013999999999999999] Pair of Straight Lines

SECTION - C

[3]15|Attempt any EIGHT of the following questions:

In a large school, 80% of the pupil like Mathematics. A visitor to the school asks each of 4 pupils, chosen at random, whether they like Mathematics.

Find the probability that the visitor obtains answer yes from at least 2 pupils:

  1. when the number of pupils questioned remains at 4.
  2. when the number of pupils questioned is increased to 8.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.027999999999999997] Binomial Distribution

[3]16

Differentiate the following w.r.t. x: xe + xx + ex + ee

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.021] Differentiation

[3]17

Find `(dy)/(dx)`, if x3 + x3y + xy2 + y3 = 81

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.021] Differentiation

[3]18

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution

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[3]19

Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.023] Indefinite Integration [0.15] Integration

[3]20

It is observed that it rains on 12 days out of 30 days. Find the probability that it it will rain at least 2 days of given week.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.027999999999999997] Binomial Distribution

[3]21

Find the inverse of the following matrix by the adjoint method.

`[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.012] Matrics

[3]22

Integrate the following functions w.r.t.x:

cos8xcotx

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.023] Indefinite Integration [0.15] Integration

[3]23

Integrate the following functions w.r.t.x:

`(2sinx cosx)/(3cos^2x + 4sin^2 x)`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.023] Indefinite Integration [0.15] Integration

[3]24

Find two unit vectors each of which is perpendicular to both `baru` and `barv` where `baru = 2hati + hatj - 2hatk`, `barv = hati + 2hatj - 2hatk`.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.015] Vectors

[3]25

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hati - hatj + 3hatk` and `- 5hati + 2hatj - 5hatk` in the ratio 3:2 is internally.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.015] Vectors [0.07] Vectors

[3]26

Evaluate the following integrals as limit of a sum:

\[\int\limits_0^2 (3x^2 - 1)\cdot dx\]

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.024] Definite Integration

SECTION - D

[4]27|Attempt any FIVE of the following questions:

Evaluate the following integrals:

`int (7x + 3)/sqrt(3 + 2x - x^2).dx`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.023] Indefinite Integration [0.15] Integration

[4]28

Find the vector and Cartesian equations of the line passing through the point (–1, –1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z − 2.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.016] Line and Plane

[4]29

Using the truth table prove the following logical equivalence.

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

[4]30

Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative

[4]31

If `bara = hati - 2hatj`, `barb = hati + 2hatj, barc = 2hati + hatj - 2hatk`, then find (i) `bara xx (barb xx barc)` (ii) `(bara xx barb) xx barc`. Are the results same? Justify.

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.015] Vectors

[4]32

Solve the following differential equation:

(x2 + y2)dx - 2xy dy = 0

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.026000000000000002] Differential Equations

[4]33

Find the second order derivatives of the following : e4x. cos 5x

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.021] Differentiation [0.13] Differentiation

[4]34

If | x | < 1, then prove that

`2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))`

VIEW SOLUTION

Concept: undefined - undefined
Chapter: [0.013000000000000001] Trigonometric Functions

Mathematics and Statistics Model set 1 by shaalaa.com 2024-2025 HSC Science (General) 12th Standard Board Exam Question Paper Solution | Shaalaa.com (2025)
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