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

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

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`

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

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)`

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`

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`

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`

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

Find the general solution of the following equation:

sinθ = `1/2`.

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

Apply the given elementary transformation of the following matrix.

A = `[(1,0),(-1,3)]`, R₁↔ R₂

State whether the following equation has a solution or not?

2sinθ = 3

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. 

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

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

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)`.

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)`

Solve the following :

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

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

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

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

 f(x) = sin x, for 0 ≤ x ≤ `π/2`

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

p → (p ∨ ∼ q)

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

Solve graphically: 2x – 3 ≥ 0

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

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.

Differentiate the following w.r.t. x: x^e + x^x + e^x + e^e 

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

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.

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

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.

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

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

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

cos⁸xcotx

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

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

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

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.

Evaluate the following integrals as limit of a sum:

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

Evaluate the following integrals:

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

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.

Using the truth table prove the following logical equivalence.

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

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)`.

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.

Solve the following differential equation:

(x² + y²)dx - 2xy dy = 0

Find the second order derivatives of the following : e^4x. cos 5x

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))`