HSC Science (General)

HSC Arts (English Medium)

HSC Science (Electronics)

HSC Science (Computer Science)

Academic Year: 2016-2017

Date: July 2017

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The inverse of the matrix `[[1,-1],[2,3]]` is ...............

(A) `1/5[[3,-1],[-2,1]]`

(B) `1/5[[3,1],[-2,1]]`

(C) `1/5[[-3,1],[-2,1]]`

(D) `1/5[[3,-1],[2,-1]]`

Chapter: [0.02] Matrices

If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`

(A) 100

(B) 101

(C) 110

(D) 109

Chapter: [0.015] Vectors [0.07] Vectors

If a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.

(A) `0,1/sqrt2,-1/sqrt2`

(B) `0,-1/sqrt2,-1/sqrt2`

(C) `1,1/sqrt2,1/sqrt2`

(D) `0,-1/sqrt2,1/sqrt2`

Chapter: [0.08] Three Dimensional Geometry

`barr=(hati-2hatj+3hatk)+lambda(2hati+hatj+2hatk)` is parallel to the plane `barr.(3hati-2hatj+phatk)=10`, find the value of p.

Chapter: [0.1] Plane

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

Chapter: [0.03] Trigonometric Functions

Write the negations of the following statements:

a.`forall n in N, n+7>6`

b. The kitchen is neat and tidy.

Chapter: [0.01] Mathematical Logic

Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.

Chapter: [0.08] Three Dimensional Geometry

If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.

Chapter: [0.07] Vectors

If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.

Chapter: [0.03] Trigonometric Functions

Write the converse, inverse and contrapositive of the following statement.

“If it rains then the match will be cancelled.”

Chapter: [0.01] Mathematical Logic

Find p and q if the equation px^{2} – 8xy + 3y^{2} + 14x + 2y + q = 0 represents a pair of prependicular lines.

Chapter: [0.013999999999999999] Pair of Straight Lines

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).

Chapter: [0.1] Plane

Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r=(mbarb+nbara)/(m+n)` . Hence find the position vector of R which divides the line segment joining the points A(1, –2, 1) and B(1, 4, –2) internally in the ratio 2 : 1.

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

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The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`

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

Find the cartesian equation of the line passing throught the points A(3, 4, -7) and B(6,-1, 1).

Chapter: [0.04] Pair of Straight Lines

Find the vector equation of a line passing through the points A(3, 4, –7) and B(6, –1, 1).

Chapter: [0.1] Plane

Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0

Chapter: [0.03] Trigonometric Functions

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=[[1,-1,2],[3,0,-2],[1,0,3]]` verify that A (adj A) = |A| I.

Chapter: [0.02] Matrices

A company manufactures bicycles and tricycles each of which must be processed through machines A and B. Machine A has maximum of 120 hours available and machine B has maximum of 180 hours available. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B.

If profits are Rs. 180 for a bicycle and Rs. 220 for a tricycle, formulate and solve the L.P.P. to determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.

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

If θ is the measure of acute angle between the pair of lines given by `ax^2+2hxy+by^2=0,` then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|,a+bne0`

Chapter: [0.04] Pair of Straight Lines

find the acute angle between the lines

x^{2} – 4xy + y^{2} = 0.

Chapter: [0.04] Pair of Straight Lines

Given f (x) = 2x, x < 0

= 0, x ≥ 0

then f (x) is _______ .

discontinuous and not differentiable at x = 0

continuous and differentiable at x = 0

discontinuous and differentiable at x = 0

continuous and not differentiable at x = 0

Chapter: [0.12] Continuity

If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`

(A) 1

(B) 2

(C) –1

(D) –2

Chapter: [0.15] Integration

The function f (x) = x^{3} – 3x^{2} + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing

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

Differentiate 3^{x} w.r.t. log_{3}x

Chapter: [0.12] Continuity [0.13] Differentiation

Check whether the conditions of Rolle’s theorem are satisfied by the function

f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3]

Chapter: [0.14] Applications of Derivative

Evaluate: `int sqrt(tanx)/(sinxcosx) dx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

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Find the area of the region bounded by the curve x^{2} = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.

Chapter: [0.16] Applications of Definite Integral

Given X ~ B (n, p)

If n = 10 and p = 0.4, find E(X) and var (X).

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

If the function `f(x)=(5^sinx-1)^2/(xlog(1+2x))` for x ≠ 0 is continuous at x = 0, find f (0).

Chapter: [0.12] Continuity

The probability mass function for X = number of major defects in a randomly selected

appliance of a certain type is

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

P(X = x) | 0.08 | 0.15 | 0.45 | 0.27 | 0.05 |

Find the expected value and variance of X.

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

Suppose that 80% of all families own a television set. If 5 families are interviewed at random, find the probability that

a. three families own a television set.

b. at least two families own a television set.

Chapter: [0.19] Probability Distribution

Find the approximate value of cos (60° 30').

(Given: 1° = 0.0175c, sin 60° = 0.8660)

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

The rate of growth of bacteria is proportional to the number present. If, initially, there were

1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½

hours.

[Take `sqrt2` = 1.414]

Chapter: [0.14] Applications of Derivative

**Prove that: **

`{:(int_(-a)^a f(x) dx = 2 int_0^a f(x) dx",", "If" f(x) "is an even function"),( = 0",", "if" f(x) "is an odd function"):}`

Chapter: [0.024] Definite Integration

If f (x) is continuous on [–4, 2] defined as

f (x) = 6b – 3ax, for -4 ≤ x < –2

= 4x + 1, for –2 ≤ x ≤ 2

Show that a + b =`-7/6`

Chapter: [0.12] Continuity

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Probability distribution of X is given by

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

P(X = x) | 0.1 | 0.3 | 0.4 | 0.2 |

Find P(X ≥ 2) and obtain cumulative distribution function of X

Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution

Solve the differential equation `dy/dx -y =e^x`

Chapter: [0.17] Differential Equation

If y = f (x) is a differentiable function of x such that inverse function x = f ^{–1}(y) exists, then

prove that x is a differentiable function of y and

`dx/dy=1/(dy/dx)`, Where `dy/dxne0`

Hence if `y=sin^-1x, -1<=x<=1 , -pi/2<=y<=pi/2`

then show that `dy/dx=1/sqrt(1-x^2)`, where `|x|<1`

Chapter: [0.13] Differentiation

Evaluate: `∫8/((x+2)(x^2+4))dx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

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