HSC Science (General)
HSC Arts (English Medium)
HSC Science (Electronics)
HSC Science (Computer Science)
HSC Arts (Marathi Medium)
Academic Year: 2022-2023
Date & Time: 26th July 2023, 11:00 am
Duration: 3h
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General Instructions:
The question paper is divided into FOUR sections:
- Section A: Q. No. 1 contains Eight multiple choice type of questions, each carrying Two marks. Q. No. 2 contains Four very short answer type-questions, each carrying One mark.
- Section B: Q. No. 3 to Q. No. 14 contain Twelve short answer type questions, each carrying Two marks. (Attempt any Eight)
- Section C: Q. No. 15 to Q. No. 26 contain Twelve short answer type questions, each carrying Three marks. (Attempt any Eight)
- Section D: Q. No. 27 to Q. No. 34 contain Eight long qnswer type questions, each carrying Four marks. (Attempt any Five)
- Use of the 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 of question, it is mandatory to write the correct answer along with its alphabet. e.g. (a) ....../ (b) ....../ (c) ....../ (d) ...... 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.
The dual statement of p ∧ ~q is equivalent to ______.
~p ∧ q
p ↔ q
~p ∨ q
~p → ~q
Chapter:
If `|bara|` = 3 and `|barb|` = 4, then the value of λ for which `bara + lambda barb` is perpendicular to `bara - lambda barb` is ______.
`9/16`
`3/4`
`3/2`
`4/3`
Chapter:
The acute angle between the lines `(x - 1)/1 = (y - 2)/-1 = (z - 3)/2 and (x - 1)/2 = (y - 2)/1 = (z - 3)/2` is ______.
60°
30°
45°
90°
Chapter:
The vector equation of the plane passing through point `A(bara)` and parallel to `barb` and `barc` is ______.
`(barr - bara) xx (barb xx barc) = bar0`
`(barr - bara) · (barb + barc) = 0`
`(barr - bara) · (barb xx barc) = 0`
`(barr - bara) xx (barb - barc) = bar0`
Chapter:
If x = at4, y = 2at2, then `(dy)/(dx)` = ______.
`1/t^2`
t2
2t2
`-1/t^2`
Chapter:
The area bounded by the curve y = 2x the Y-axis, the X-axis and x = 3 is ______.
3 sq. units
6 sq. units
9 sq. units
12 sq. units
Chapter:
The differential equation `y dy/dx + x = 0` represents family of ______.
circles
parabolas
ellipses
hyperbolas
Chapter: [13] Differential Equations
If `f(x){:(= kx^2(1 - x)",", "for" 0 < x < 1),(= 0",", "otherwise"","):}`
is the probability distribution function of a random variable X, then the value of k is ______.
12
10
−9
−12
Chapter:
Write the domain of the inverse secant function.
Chapter:
Find the value of k if the lines represented by kx2 + 4xy – 4y2 = 0 are perpendicular to each other.
Chapter: [4] Pair of Straight Lines
Obtain the differential equation by eliminating the arbitrary constant from the equation y2 = 4ax.
Chapter:
Write the following compound statement symbolically.
Nagpur is in Maharashtra and Chennai is in Tamil Nadu.
Chapter: [1] Mathematical Logic
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Write the following compound statement symbolically.
If ΔABC is right-angled at B, then m∠A + m∠C = 90°.
Chapter: [1] Mathematical Logic
Find the co-factors of the elements a11 and a21 of matrix A = `[(1, 2),(3, 4)]`.
Chapter:
Find the solution of cos θ = `1/2`, where 0 ≤ θ < 2π.
Chapter:
Find the joint equation of the line passing through the origin having slopes 2 and 3.
Chapter: [4] Pair of Straight Lines
If the vectors `- 3hati + 4hatj - 2hatk , hati + 2hatk` and `hati - phatj` are coplanar, then find the value of p.
Chapter:
Find the direction cosines of the following vector:
`2hati + 2hatj - hatk`
Chapter:
Find the equation of the tangent to the curve at the point on it.
y = x2 + 2ex + 2 at (0, 4)
Chapter: [9] Applications of Derivatives
Find the area of the region bounded by the curve y = x2, the X-axis and the lines x = 1, x = 3.
Chapter:
Write the integrating factor (I.F.) of the differential equation `(dy)/(dx) + y = e^-x`.
Chapter:
Given X ~ B(n, p) if p = 0.6 and E(X) = 6, find n and Var(X).
Chapter: [15] Binomial Distribution
If one of the lines given by ax2 + 2hxy + by2 = 0 bisects an angle between the coordinate axes, then show that (a + b)2 = 4h2.
Chapter: [4] Pair of Straight Lines
Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be the third point on the line AB dividing the segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.
Chapter:
Find the position vector of point P such that OP is inclined to the X-axis at 45° and to the Y-axis at 60° and OP = 12 units.
Chapter:
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Find the vector equation of the line passing through the point having position vector `2hati + hatj - 3hatk` and perpendicular to vectors `hati + hatj + hatk and hati + 2hatj - hatk`.
Chapter:
The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.
Chapter: [6] Line and Plane
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
Chapter: [8] Differentiation
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Chapter: [9] Applications of Derivatives
Solve the differential equation `(dy)/(dx) = (4x + y + 1)^2`.
Chapter:
The probability distribution of X is as follows:
| x | 0 | 1 | 2 | 3 | 4 |
| P[X = x] | 0.1 | k | 2k | 2k | k |
Find:
- k
- P[X < 2]
- P[X ≥ 3]
- P[1 ≤ X < 4]
- P(2)
Chapter: [14] Probability Distributions
Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.
Chapter: [15] Binomial Distribution
Construct the truth table for the statement pattern (p → q) ∧ [(q → r) → (p → r)] and interpret your result.
Chapter:
Solve the following equations by using the method of inversion:
x − y + z = 4, 2x + y − 3z = 2, x + y + z = 2
Chapter:
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`.
Chapter: [3] Trigonometric Functions
Solve the following LPP by graphical method:
Maximize: z = 3x + 5y
Subject to: x + 4y ≤ 24
3x + y ≤ 21
x + y ≤ 9
x ≥ 0, y ≥ 0
Also find the maximum value of z.
Chapter: [7] Linear Programming
If y = f(x) is a differentiable function of x on an interval I and y is one-one onto and `(dy)/(dx) ≠ 0` on I, then prove that `(dx)/(dy) = 1/(((dy)/(dx)))`, where `(dy)/(dx) ≠ 0`. Hence, prove that `d/dx (cot^-1 x) = -1/(1 + x^2)`.
Chapter:
The volume of the spherical ball is increasing at the rate of 4π cc/sec. Find the rate at which the radius and the surface area are changing when the volume is 288 π cc.
Chapter: [9] Applications of Derivatives
Prove that `int 1/(x^2 - a^2) dx = 1/(2a) log |(x - a)/(x + a)| + c`.
Hence evaluate `int 1/(x^2 - 3) dx`.
Chapter:
Evaluate:
`int_0^(pi/2) cos^3x dx`
Chapter: [11] Definite Integration
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