HSC Science (General)
HSC Arts (English Medium)
HSC Science (Electronics)
HSC Science (Computer Science)
HSC Arts (Marathi Medium)
Academic Year: 2024-2025
Date & Time: 3rd July 2025, 11:00 am
Duration: 3h
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General instructions:
The question paper is divided into FOUR sections.
- Section A: Q.1 contains Eight multiple choice type questions carrying Two marks each.
Q.2 contains Four very short answer type questions carrying One mark each. - Section B: Q.3 to Q.14 contain 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 of questions, only the first attempt will be considered for evaluation.
- Start answer to each section on a new page.
Inverse of statement pattern (p ∨ q) → (p ∧ q) is ________ .
(p ∧ q) → (p ∨ q)
∼ (p ∨ q) → (p ∧ q)
(∼p ∧ ∼q) → (∼p ∨ ∼q)
(∼p ∨ ∼q) → (∼p ∧ ∼q)
Chapter: [1] Mathematical Logic
In ΔABC, if a = 2, b = 3 and sin A = `2/3`, then ∠B = ______.
`π/2`
`π/3`
`π/4`
`π/6`
Chapter:
If `vec(AB) = 2hati - 4hatj + 7hatk` and initial point A ≡ (1, 5, 0) then terminal point B is ______.
(1, 3, 7)
(7, 3, 1)
(1, 7, 3)
(3, 1, 7)
Chapter:
The angle between the lines `vecr = (hati + 2hatj + 3hatk) + λ(2hati - 2hatj + hatk)` and `vecr = (hati + 2hatj + 3hatk) + µ(hati + 2hatj + 2hatk)` is ______.
`π/4`
`π/2`
`π/3`
0
Chapter:
If y is a function of x and log (x + y) = xy then the value of `(dy/dx)` at x = 0 is ______.
1
–1
2
0
Chapter:
If the displacement of a particle at time t is given by S = 2t3 – 5t2 + 4t – 3, then its acceleration at time t = 1 is ______.
2
8
10
14
Chapter:
The solution of the D.E. sec2x. tan ydx + sec2y tan xdy = 0 is ______.
tan x. cot y = c
cot x – cot y = c
tan x. tan y = c
cot x – tan y = c
Chapter:
If X is waiting time in minutes for a bus and its p.d.f. is given by
f(x) `{:(= 1/5",", "for" 0 ≤ x ≤5","),(= 0",", "otherwise"):}`
then the probability that waiting time is between 1 and 3 is ______.
`1/5`
`2/5`
`3/5`
`4/5`
Chapter:
Find the magnitude of the vector `veca = 3hati + hatj + 7hatk`.
Chapter:
Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p
Chapter: [1] Mathematical Logic
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Find the cofactors of the elements of the matrix `[(2, -3),(3, 5)]`.
Chapter:
Find the polar coordinates of the point whose cartesian coordinates are `(-sqrt(2), -sqrt(2))`.
Chapter:
In ΔABC, if a = 2, b = 3, c = 4, then prove that the triangle is obtuse angled.
Chapter:
If `veca = 3hati - hatj + 2hatk, hatb = 2hati + hatj - hatk`, then find `|veca xx vecb|`.
Chapter:
Find the cartesian equation of the plane passing through the point A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
Chapter: [6] Line and Plane
Find `dy/dx`, if `xsqrt(x) + ysqrt(y) = asqrt(a)`.
Chapter: [8] Differentiation
Show that the tangent to the curve y = x3 – 6x2 + x + 3 at the point (0, 3) is parallel to the line y = x + 5.
Chapter:
Check whether the conditions of Rolle’s theorem are satisfied by the function f(x) = x2 – 4x + 3, x ∈ [1, 3].
Chapter:
If X ~ B (n, p) and E(X) = 6 and Var (X) = 4.2, then find n and p.
Chapter: [15] Binomial Distribution
Find the combined equation of pair of lines passing through (2, 3) and perpendicular to the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0.
Chapter:
Show that the acute angle θ between the lines represented by ax2 + 2hxy + by2 = 0 is given by, tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`.
Chapter:
By vector method prove that the medians of a triangle are concurrent.
Chapter:
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Find the vector equation of the plane passing through the point A (–1, 2, –5) and parallel to the vectors `4hati - hatj + 3hatk` and `hati + hatj - hatk`.
Chapter:
Find the shortest distance between the lines, `(x - 1)/2 = (y - 2)/3 = (z - 3)/4` and `(x - 2)/3 = (y - 4)/4 = (z - 5)/5`.
Chapter:
Water is being poured at the rate of 27 m3/sec into a cylindrical vessel of base radius 3 m. Find the rate at which the water level is rising.
Chapter:
Solve the following differential equation:
`dy/dx + y/x = x^3 - 3`
Chapter: [13] Differential Equations
Obtain the differential equation by eliminating the arbitrary constants from y = c1 cos (log x) + c2 sin (log x).
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[1 ≤ X < 4]
Chapter:
In a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four correct answers just by guessing?
Chapter:
Give an alternative equivalent simple circuit for the following circuit:
Chapter: [1] Mathematical Logic
The sum of three numbers is 2. If twice of the second number is added to the sum of first and third number we get 0, Adding five times the first number to twice the sum of second and third number we get 7. Find the numbers using matrix method.
Chapter:
Using properties of scalar triple product, prove that `[(bar"a" + bar"b", bar"b" + bar"c", bar"c" + bar"a")] = 2[(bar"a", bar"b", bar"c")]`.
Chapter:
Solve the following L.P.P. using graphical method:
Maximize, z = 9x + 13y
Subject to, 2x + 3y ≤ 18,
2x + y ≤ 10
x ≥ 0, y ≥ 0
Chapter:
If x = f(t) and y = g(t) are differentiable functions of t, so that y is a function of x and `dx/dt ≠ 0`, then prove that `dy/dx = ((dy/dt))/((dx/dt))`. Hence find `dy/dx`, if y = at2 and x = 2at.
Chapter:
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Chapter: [10] Indefinite Integration
Evaluate: `int_0^(1/2) dx/((1 - 2x^2) * sqrt(1 - x^2))`
Chapter:
Solve the following :
Find the area of the region lying between the parabolas y2 = 4x and x2 = 4y.
Chapter: [12] Application of Definite Integration
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