HSC Science (General)
HSC Arts (English Medium)
HSC Science (Electronics)
HSC Science (Computer Science)
HSC Arts (Marathi Medium)
Academic Year: 2025-2026
Date & Time: 21st February 2026, 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 tyре 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 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 question, only the first attempt will be considered for evaluation.
- Start answer to each section on a new page.
The converse of contrapositive of ∼p → q is ______.
q → p
∼q → p
p → ∼q
∼q → ∼p
Chapter: [1] Mathematical Logic
If A = `[(2, -4),(3, 1)]`, then the adjoint of matrix A is ______.
`[(-1, 3),(-4, 1)]`
`[(1, 4),(-3, 2)]`
`[(1, 3),(4, -2)]`
`[(-1, -3),(-4, 2)]`
Chapter:
If tan–1 (2x) + tan–1 (3x) = `π/4`, then x = ______.
–1
`1/3`
`1/6`
`3/2`
Chapter: [3] Trigonometric Functions
The angle between the line `vecr = (hati + 2hatj + hatk) + lambda(hati + hatj + hatk)` and the plane `vecr * (2hati - hatj + hatk) = 8` is ______.
`sin^-1 (sqrt(2)/3)`
`sin^-1 (sqrt(3)/2)`
`sin^-1 (1/2)`
`sin^-1 (1/sqrt(2))`
Chapter:
If y = sec (tan−1x), then `dy/dx` at x = 1 is ______.
`1/2`
1
`1/sqrt(2)`
`sqrt(2)`
Chapter: [8] Differentiation
The approximate value of the function f(x) = x3 – 3x + 5 at x = 1.99 is ______.
6.09
6.91
7.09
7.91
Chapter:
`int_1^2 (1)/(x^2) e^(1/x) * dx` = ______.
`sqrt(e) + 1`
`sqrt(e) - 1`
`sqrt(e)(sqrt(e) - 1)`
`(sqrt(e) - 1)/e`
Chapter:
If the p.d.f. of a continuous r.v. X is
`f(x) = {{:((x + 2)/18",", "for" -2 < x < 4),(= 0",", "otherwise"):}`
then P(|X| < 1) = ______.
`1/9`
`2/9`
`1/27`
`2/27`
Chapter:
For the principal value, evaluate of the following:
`cos^-1 1/2 + 2 sin^-1 (1/2)`
Chapter:
Write the degree of the differential equation
(y")2 + 3(y")3 + 3xy' + 5y = 0
Chapter:
Construct the switching circuit of the following:
(∼ p ∧ q) ∨ (p ∧ ∼ r)
Chapter: [1] Mathematical Logic
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In ΔABC, prove that a(b cos C – c cos B) = b2 – c2.
Chapter:
Find the general solution of the following equation:
4 cos2 θ = 3
Chapter: [3] Trigonometric Functions
Find k, if the sum of the slopes of the lines represented by x2 + kxy – 3y2 = 0 is twice their product.
Chapter: [4] Pair of Straight Lines
Find the value of p, for which the vectors `veca = 3hati + 2hatj + 9hatk` and `vecb = hati + phatj + 3hatk` are perpendicular to each other.
Chapter:
Find the vector equation of the line passing through the points A(1, 2, 3) and B(2, 3, 4).
Chapter:
Find `dy/dx`, if `sqrt(x) + sqrt(y) = sqrt(a)`.
Chapter: [8] Differentiation
Find `(d^2y)/(dx^2)`, if y = x3 + 7x2 – 2x – 9.
Chapter:
Test whether the function f(x) = x3 + 6x2 + 12x – 7 is increasing or decreasing for all x ∈ R.
Chapter:
A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of 3 cm/sec. How fast the area enclosed is increasing when the radius is 8 cm?
Chapter:
`int sqrt(1 + sin2x) dx`
Chapter: [10] Indefinite Integration
Given that X ~ B(n, p), if n = 10, E(X) = 8, find Var(X).
Chapter: [14] Probability Distributions
Examine whether the statement pattern (p ∧ q) and (∼ p ∨ ∼ q) is a tautology or contradiction or contingency.
Chapter:
In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.
Chapter: [3] Trigonometric Functions
If two of the vertices of a triangle are A (3, 1, 4) and B(– 4, 5, –3) and the centroid of the triangle is at G (–1, 2, 1), then find the coordinates of the third vertex C of the triangle.
Chapter:
If D, E, F are the midpoints of the sides BC, CA, AB of a triangle ABC, prove that `bar"AD" + bar"BE" + bar"CF" = bar0`.
Chapter:
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Show that the lines `vecr = (hati + hatj - hatk) + lambda(2hati - 2hatj + hatk)` and `vecr = (4hati - 3hatj + 2hatk) + µ(hati - 2hatj + 2hatk)` intersect each other.
Chapter:
Find the cartesian equation of the plane `vecr = (hati - hatj) + λ(hati + hatj + hatk) + µ(hati - 2hatj + 3hatk)`.
Chapter:
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x then prove that y = f(g(x)) is a differentiable function of x and `(dy)/(dx) = (dy)/(du) * (du)/(dx)`.
Chapter:
Verify Lagrange’s mean value theorem for the following function:
f(x) = log x, on [1, e]
Chapter: [9] Applications of Derivatives
Evaluate the following : `int sinx/(sin 3x).dx`
Chapter: [10] Indefinite Integration
Solve the D.E. 3ex tan y dx + (1 + ex) sec2y dy = 0.
Chapter:
Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.
Chapter: [14] Probability Distributions
A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.
Chapter: [15] Binomial Distribution
If A = `[(1, 2),(3, 4)]`, prove that A · (adj A) = (adj A) · A = |A| · I
Chapter:
Show that every homogeneous equation of degree two in x and y, i.e., ax2 + 2hxy + by2 = 0, represents a pair of lines passing through the origin, if h2 – ab ≥ 0.
Chapter:
Let `A(bara)` and `B(barb)` 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 `barr = (mbarb + nbara)/(m + n)`.
Chapter:
Solve the L.P.P. graphically:
Minimize: z = 5x + 2у,
Subject to: 5x + y ≥ 10, x + y ≥ 6,
x ≥ 0, y ≥ 0
Chapter:
Evaluate: `int (3x^2 + 4x - 5)/((x^2 - 1)(x + 2)) dx`
Chapter:
Prove that: `int_a^b f(x)dx = int_a^b f(a + b - x) * dx`. Hence, find `int_(π/6)^(π/3) sin^2x * dx`
Chapter:
Solve the following:
Find the area enclosed between the circle x2 + y2 = 1 and the line x + y = 1, lying in the first quadrant.
Chapter: [12] Application of Definite Integration
Solve the following differential equation:
`x^2 dy/dx = x^2 + xy + y^2`
Chapter: [13] Differential Equations
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