Following are HSC Board Mathematics and Statistics Model Question Paper 1 for Arts and Science.
Model Question Paper 1
Time 3 hours Max Marks 80
Note: (1) All questions are compulsory.
(2) The question paper consists of 30 questions divided in to four
sections A, B, C, D.
(3) Section A content questions of 1 mark each
Section B content questions of 2 mark each
Section C content questions of 3 mark each
Section D content questions of 4 mark each
(4) Use of logarithmic table is allowed.
(5) Use of calculator is not allowed.
(6) In LPP only rough sketch of graph is expected. Graph paper is not necessary
SECTION A
Select and write the most appropriate
answer from alternatives in each of the following questions
1. If p→q is true and p⋀q is false , then the truth values of p and q are
a) T,F b) T,T c) F,T d ) F,F
2. If the vectors $2\hat i - q\hat j + 3\hat k$ and $4\hat i - 5\hat j + 6\hat k$ are collinear then the value of q is
a) 5 b) 10 c) 5/2 d) 5/4
3. The two value of k for which the lines with direction ratios k,-6,-2 and k-1 ,k,4 are perpendicular to each other are
1. If p→q is true and p⋀q is false , then the truth values of p and q are
a) T,F b) T,T c) F,T d ) F,F
2. If the vectors $2\hat i - q\hat j + 3\hat k$ and $4\hat i - 5\hat j + 6\hat k$ are collinear then the value of q is
a) 5 b) 10 c) 5/2 d) 5/4
3. The two value of k for which the lines with direction ratios k,-6,-2 and k-1 ,k,4 are perpendicular to each other are
a )8,-1 b)2,3 c) 8,1 d)-8,-1
4. If the function
f(x) = $ (cosx)^\frac{1}{x}$ x≠0
= k, x=0
is continuous at x = 0, then the value of k is
a) 1 b)
-2 c) 3 d) 4
5. $\int\frac{dx}{x+x^n}$ is
a) $\frac{1}{n}log|x^n| $ + c b)$\frac{1}{1-n}log|x^{1-n}| $ +c
c) $\log|x^n+1| $ + c d)$\frac{1}{n}log|x^n+1| $ + c
6. The differential equation $y\frac{dy}{dx}$ + x = 0 represent family of
a) circle b) parabola c) ellipse d) hyperbola
SECTION B
8. Find the general solution of sin(x +⫪/5 ) = 0
9. In ∆ABC, show that $tan\frac{A}{2}tan\frac{B}{2} = \frac{a + b - c}{a + b + c}$
10. Find the value of ⋋ for which the points (6,-1, 2) (8,-7, ⋋) and (5, 2, 4) are collinear
10. Differentiate $sin^{-1}(2x\sqrt{1-x^2})$ w.r.t x
11. The displacement S of a particle at time is given by S = $ t^3 - t^2 -5t$ find the velocity and acceleration at time t = 2
14. Solve the differential question y-x$\frac{dy}{dx}$ = 0
5. $\int\frac{dx}{x+x^n}$ is
a) $\frac{1}{n}log|x^n| $ + c b)$\frac{1}{1-n}log|x^{1-n}| $ +c
c) $\log|x^n+1| $ + c d)$\frac{1}{n}log|x^n+1| $ + c
6. The differential equation $y\frac{dy}{dx}$ + x = 0 represent family of
a) circle b) parabola c) ellipse d) hyperbola
SECTION B
8. Find the general solution of sin(x +⫪/5 ) = 0
9. In ∆ABC, show that $tan\frac{A}{2}tan\frac{B}{2} = \frac{a + b - c}{a + b + c}$
10. Find the value of ⋋ for which the points (6,-1, 2) (8,-7, ⋋) and (5, 2, 4) are collinear
10. Differentiate $sin^{-1}(2x\sqrt{1-x^2})$ w.r.t x
11. The displacement S of a particle at time is given by S = $ t^3 - t^2 -5t$ find the velocity and acceleration at time t = 2
14. Solve the differential question y-x$\frac{dy}{dx}$ = 0
14. Find the area of the region bounded by
the curve y = $x^2$ , the
x- axis and given lines x=1,
and x=5
SECTION C
15. Using
truth table, prove the equivalence ∼p∧ q = (p∨ q)∧∼q
16.. Find the vector equation of the line passing
through the point (2,3,-4) and perpendicular to the XZ-plane .Hence find its
equation in Cartesian plane
17. Find the vector equation of the plane $ \bar r = ( 2\hat i + \hat k) +\lambda\hat i + \mu(\hat i +2\hat j - 3\hat k)$ in scalar product form
OR
17. Find the
equation of the plane passing through the intersection of the planes 3x=2y- z+1 = 0 and x+y+z-2 = 0 and the
point (2, 2, 1).
18. Find the value of k, if the function
f(x) = log (1+2x), for x ≠ 0
=
k, for x = 0
Is continuous at x = 0
19. Obtain the
probability distribution of the number of sixes in two tosses of
fair die
OR
19.Let X have p.m.f.
P(x) = $kx^2$ x = 1,2,3.4
= 0, otherwise
.Find
mean and variance of X
20. The probability that a certain kind of
component will survive a check
21. If three numbers are added the sum is 15. If the second is subtracted from the sum of first and third number the we get '5' and if twice the first number is added to the second and the third number is subtracted from the sum we get '4'. Use matrices to find the number.
test is 0.6. Find the probability
that exactly 2 of the next 4 tested components survive.
SECTION D
.22.In △ABC prove that (b + c -a) tan A/2 = (c + a -b) tan B/2 = tan ( a +b - c) tan C/2
22.Find the general solution of $\sqrt {3}$cosx - sinx = 0
23.Find p and q, if the equation p$x^2$ -8xy = 3$y^2$ = 14x + 2y + q =0 represent a pair of perpendicular lines.
OR
23.Find p and q, if the equation p$x^2$ -8xy = 3$y^2$ = 14x + 2y + q =0 represent a pair of perpendicular lines.
24.Find the volume of tetrahedron whose vertices are
A (-1, 2, 3), B (3, -2, 1), C
(2, 1, 3), and D (-1, -2, 4)
25.Solve by graphically
Maximize z = 15x + 30y subject to 3x + y ≤ 12, x + 2y ≤ 10, x ≥ 0, y ≥ 0.
26.If $x^{m}y^{n} = ( x + y)^{m + n}$ then show that $\frac{d^2 y}{dx^2}$ = 0.
27. A manufacture can sell
x items at the rate of ₹(330 -x ) each. The cost of
producing x items is ₹($x^2$+ 10x + 12 ) How many items must be sold so that his profit is maximum?
28.Find the area of the
region lying between the parabolas $y^2$ = x and $x^2$ = y.
29.Prove that
∫$\sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{2}$ +c
30.Solve the equation $x^{-1}cos^2ydy + y^{-1}co^2xdx$ = 0.
∫$\sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{2}$ +c
30.Solve the equation $x^{-1}cos^2ydy + y^{-1}co^2xdx$ = 0.
OR
30. In a culture of yeast,the active ferment double itself in 3 hours. Assuming that the quantity increases at a rate proportional to itself, determine the number of times it multiplies itself in 15 hours.
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