HSC Science (General)

HSC Arts (English Medium)

HSC Science (Electronics)

HSC Science (Computer Science)

Academic Year: 2013-2014

Date: March 2014

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Which of the following represents direction cosines of the line :

(a)`0,1/sqrt2,1/2`

(b)`0,-sqrt3/2,1/sqrt2`

(c)`0,sqrt3/2,1/2`

(d)`1/2,1/2,1/2`

Chapter: [0.08] Three Dimensional Geometry

`A=[[1,2],[3,4]]` ans A(Adj A)=KI, then the value of 'K' is

2

- 2

10

-10

Chapter: [0.02] Matrices

The general solution of the trigonometric equation tan^{2} θ = 1 is ____________

`theta =npi+-(pi/3),n in z`

`theta =npi+-pi/6, n in z`

`theta=npi+-pi/4, n in z`

`0=npi, n in z`

Chapter: [0.03] Trigonometric Functions

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment AB.

Chapter: [0.07] Vectors

The Cartestation equation of line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

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

Equation of a plane is `vecr (3hati-4hatj+12hatk)=8`. Find the length of the perpendicular from the origin to the plane.

Chapter: [0.1] Plane

Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, - 4, 5.

Chapter: [0.08] Three Dimensional Geometry

Write the dual of the following statements: (p ∨ q) ∧ T

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

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

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

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Chapter: [0.016] Line and Plane [0.09] Line

Prove that three vectors `bara, barb and barc ` are coplanar, if and only if, there exists a non-zero linear combination `xbara + ybarb + z barc = bar0`.

Chapter: [0.07] Vectors

Using truth table prove that ∼p ˄ q ≡ (p ˅ q) ˄ ∼p

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

In any ΔABC, with usual notations, prove that b^{2} = c^{2} + a^{2} – 2ca cos B.

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

Show that the equation `x^2-6xy+5y^2+10x-14y+9=0 ` represents a pair of lines. Find the acute angle between them. Also find the point of intersection of the lines.

Chapter: [0.04] Pair of Straight Lines

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Express the following equations in the matrix form and solve them by method of reduction :

2x- y + z = 1, x + 2y + 3z = 8, 3x + y - 4z =1

Chapter: [0.02] Matrices

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

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

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

If A, B, C, D are (1, 1, 1), (2, 1, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Chapter: [0.015] Vectors [0.07] Vectors

Find the equation of the plane passing through the line of intersection of planes 2x – y + z = 3 and 4x – 3y + 5z + 9 = 0 and parallel to the line `(x + 1)/2 = (y + 3)/4 = (z - 3)/5`

Chapter: [0.1] Plane

Minimize :Z=6x+4y

Subject to : 3x+2y ≥12

x+y ≥5

0 ≤x ≤4

0 ≤ y ≤ 4

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

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

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

If y =1 - cos θ , x = 1 - sin θ , then ` dy/dx at " "θ =pi/4` is ________

Chapter: [0.13] Differentiation

The integrating factor of linear differential equation `dy/dx+ysecx=tanx` is

(a)secx- tan x

(b) sec x · tan x

(c)sex+tanx

(d) secx.cotx

Chapter: [0.17] Differential Equation

The equation of tangent to the curve y = 3x^{2} - x + 1 at the point (1, 3) is

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

Chapter: [0.06] Conics

Examine the continuity of the function

f(x) =sin x- cos x, for x ≠ 0

=- 1 ,forx=0

at the poinl x = 0

Chapter: [0.12] Continuity

Verify Rolle's theorem for the function

f(x)=x^{2}-5x+9 on [1,4]

Chapter: [0.14] Applications of Derivative

Evaluate : `intsec^nxtanxdx`

Chapter: [0.15] Integration

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The probability mass function (p.m.f.) of X is given below:

X=x | 1 | 2 | 3 |

P (X= x) | 1/5 | 2/5 | 2/5 |

find E(X^{2})

Chapter: [0.19] Probability Distribution

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

Chapter: [0.18] Statistics

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

**Obtain the differential equation by eliminating the arbitrary constants from the following equation:**

y = A cos (log x) + B sin (log x)

Chapter: [0.026000000000000002] Differential Equations

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

Chapter: [0.023] Indefinite Integration [0.15] Integration

An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.

Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative

Prove that :

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

Chapter: [0.026000000000000002] Differential Equations [0.17] Differential Equation

If the function f (x) is continuous in the interval [-2, 2],find the values of a and b where

`f(x)=(sinax)/x-2, for-2<=x<=0`

`=2x+1, for 0<=x<=1`

`=2bsqrt(x^2+3)-1, for 1<x<=2`

Chapter: [0.12] Continuity

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

Chapter: [0.17] Differential Equation

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

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

If x^{p}y^{q} = (x + y)^{p+q} then Prove that `dy/dx = y/x`

Chapter: [0.021] Differentiation [0.13] Differentiation

Find the area of the sector of a circle bounded by the circle x^{2} + y^{2} = 16 and the line y = x in the ftrst quadrant.

Chapter: [0.16] Applications of Definite Integral

**Prove that:**

`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`

Chapter: [0.023] Indefinite Integration [0.15] Integration

A random variable X has the following probability distribution :

X=x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

P[X=x] | k | 3k | 5k | 7k | 9k | 11k | 13k |

(a) Find k

(b) find P(O <X< 4)

(c) Obtain cumulative distribution function (c. d. f.) of X.

Chapter: [0.19] Probability Distribution

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