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
Advertisements
General Instructions: The question paper is divided into four sections.
- 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.
- Section B: Q.3 to Q.14 contains Twelve short answer type questions, each carrying Two marks. (Attempt any Eight)
- Section C: Q.15 to Q.26 contain Twelve short answer type questions, each carrying Three marks. (Attempt any Eight)
- Section D: Q. 27 to Q.34 contain Eight long answer type questions, each carrying Four marks. (Attempt any Five)
- Use of Log table is allowed. Use of calculator is not allowed.
- Figures to the right indicate full marks.
- Use of graph paper is not necessary. Only rough sketch of graph is expected.
- 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.
- 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`
VIEW SOLUTION
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
Advertisements
[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:
- when the number of pupils questioned remains at 4.
- 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
Advertisements
[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