HSC Science (General)

HSC Arts (English Medium)

HSC Science (Electronics)

HSC Science (Computer Science)

Academic Year: 2015-2016

Date & Time: 26th February 2016, 11:00 am

Duration: 3h

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The negation of p ∧ (q → r) is ______________.

p ∨ ( ~q ∨ r )

~p ∧ ( q → r )

~p ∧ ( ~q → ~r )

~p ∨ ( q ∧ ~r )

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

If `sin^-1(1-x) -2sin^-1x = pi/2` then x is

- -1/2
- 1
- 0
- 1/2

Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions

The joint equation of the pair of lines passing through (2,3) and parallel to the coordinate axes is

- xy -3x - 2y + 6 = 0
- xy +3x + 2y + 6 = 0
- xy = 0
- xy - 3x - 2y - 6 = 0

Chapter: [0.04] Pair of Straight Lines

Find (AB)^{-1} if

`A=[(1,2,3),(1,-2,-3)], B=[(1,-1),(1,2),(1,-2)]`

Chapter: [0.02] Matrices

Find the vector equation of the plane passing through a point having position vector `3 hat i- 2 hat j + hat k` and perpendicular to the vector `4 hat i + 3 hat j + 2 hat k`

Chapter: [0.1] Plane

If `bar p = hat i - 2 hat j + hat k and bar q = hat i + 4 hat j - 2 hat k` are position vector (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2:1

Chapter: [0.015] Vectors [0.07] Vectors

Find k, if one of the lines given by 6x^{2} + kxy + y^{2} = 0 is 2x + y = 0

Chapter: [0.04] Pair of Straight Lines

If the lines

`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`

are at right angle then find the value of k

Chapter: [0.09] Line

Examine whether the following logical statement pattern is a tautology, contradiction, or contingency.

[(p→q) ∧ q]→p

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

By vector method prove that the medians of a triangle are concurrent.

Chapter: [0.015] Vectors [0.07] Vectors

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

Chapter: [0.09] Line

In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`

Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions

Minimize `z=4x+5y ` subject to `2x+y>=7, 2x+3y<=15, x<=3,x>=0, y>=0` solve using graphical method.

Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems

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The cost of 4 pencils, 3 pens and 2 erasers is Rs. 60. The cost of 2 pencils, 4 pens and 6 erasers is Rs. 90 whereas the cost of 6 pencils, 2 pens and 3 erasers is Rs. 70. Find the cost of each item by using matrices.

Chapter: [0.02] Matrices

Find the volume of tetrahedron whose coterminus edges are `7hat i+hatk; 2hati+5hatj-3hatk and 4 hat i+3hatj+hat k`

Chapter: [0.08] Three Dimensional Geometry

Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Show that every homogeneous equation of degree two in x and y, i.e., ax^{2} + 2hxy + by^{2} = 0 represents a pair of lines passing through origin if h^{2}−ab≥0.

Chapter: [0.04] Pair of Straight Lines

If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line

Find the vector equation of the plane passing through the points `hati +hatj-2hatk, hati+2hatj+hatk,2hati-hatj+hatk`. Hence find the cartesian equation of the plane.

Chapter: [0.1] Plane

Find the general solution of `sin x+sin3x+sin5x=0`

Chapter: [0.03] Trigonometric Functions

if the function

`f(x)=k+x, for x<1`

`=4x+3, for x>=1`

id continuous at x=1 then k=

(a) 7

(b) 8

(c) 6

(d) -6

Chapter: [0.12] Continuity

The equation of tangent to the curve y=`y=x^2+4x+1` at

(-1,-2) is...............

(a) 2x -y = 0 (b) 2x+y-5 = 0

(c) 2x-y-1=0 (d) x+y-1=0

Chapter: [0.06] Conics

Given that X ~ B(n= 10, p). If E(X) = 8 then the value of

p is ...........

(a) 0.6

(b) 0.7

(c) 0.8

(d) 0.4

Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution

The displacement 's' of a moving particle at time 't' is given by s = 5 + 20t — 2t^{2}. Find its acceleration when the velocity is zero.

Chapter: [0.14] Applications of Derivative

Find the area bounded by the curve y^{2} = 4ax, x-axis and the lines x = 0 and x = a.

Chapter: [0.16] Applications of Definite Integral

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The probability distribution of a discrete random variable X is:

X=x | 1 | 2 | 3 | 4 | 5 |

P(X=x) | k | 2k | 3k | 4k | 5k |

find P(X≤4)

Chapter: [0.19] Probability Distribution

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Ify 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: [0.13] Differentiation

The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo similar operations,

(a) None will recover

(b) Half of them will recover.

Chapter: [0.19] Probability Distribution

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Discuss the continuity of the following functions. If the function have a removable discontinuity, redefine the function so as to remove the discontinuity

`f(x)=(4^x-e^x)/(6^x-1)` for x ≠ 0

`=log(2/3) ` for x=0

Chapter: [0.12] Continuity

**Prove that: **

`int sqrt(a^2 - x^2) dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a)+c`

Chapter: [0.023] Indefinite Integration [0.15] Integration

A body is heated at 110°C and placed in air at 10°C. After 1 hour its temperature is 60°C. How much additional time is required for it to cool to 35°C?

Chapter: [0.17] Differential Equation

Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`

Chapter: [0.15] Integration

Evaluate: `int (1+logx)/(x(2+logx)(3+logx))dx`

Chapter: [0.15] Integration

If `y=cos^-1(2xsqrt(1-x^2))`, find dy/dx

Chapter: [0.13] Differentiation

Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.

Chapter: [0.17] Differential Equation

A wire of length l is cut into two parts. One part is bent into a circle and other into a square. Show that the sum of areas of the circle and square is the least, if the radius of circle is half the side of the square.

Chapter: [0.14] Applications of Derivative

The following is the p.d.f. (ProbabiIity Density Function) of a continuous random variable X :

`f(x)=x/32,0<x<8`

= 0 otherwise

(a) Find the expression for c.d.f. (Cumulative Distribution Function) of X.

(b) Also find its value at x = 0.5 and 9.

Chapter: [0.19] Probability Distribution

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