Mathematics and Statistics Set 1 2018-2019 HSC Science (General) 12th Standard Board Exam Question Paper Solution

The principal solutions of cot x = -`sqrt3`  are .................

`pi/6 ,(5pi)/6`

`(5pi)/6 , (7pi)/6`

`(5pi)/6,(11pi)/6`

`pi/6,(11pi)/6`

The acute angle between the two planes x+y+2z = 3 and 3x -2y +2z = 7 ________.

`"sin"^-1( 5/sqrt102)`

`"cos"^-1(5/sqrt102)`

`"sin"^-1(15/sqrt102)`

`"cos"^-1(15/sqrt102)`

The direction ratios of the line which is perpendicular to the lines with direction ratios –1, 2, 2 and 0, 2, 1 are _______.

–2, –1, –2

2, 1, 2

2, –1, –2

–2, 1, –2

If f(x) = `(1+2"x")^(1/"x)"` for x≠0 is continuous at x = 0 then f(0) =  _______.

e

e²

0

2

`int "dx"/(9"x"^2 + 1)= ______. `

`1/3 "tan"^-1(2"x") +"c"`

`1/3 "tan"^-1"x" +"c"`

`1/3 "tan"^-1(3"x") +"c"`

`1/3 "tan"^-1(6"x") +"c"`

If y = ae^5x + be^-5x then the differential equation is _________.

`("d"^2"y")/"dx"^2 = 25"y"`

`("d"^2"y")/"dx"^2 = -25"y"`

`("d"^2"y")/"dx"^2 = -5"y"`

`("d"^2"y")/"dx"^2 = 5"y"`

Write the truth values of the following statements:
(1) 2 is a rational number and`sqrt2` is an irrational number. .
(2) 2 + 3 = 5 or `sqrt 2 +sqrt3 = sqrt5`

Find the volume of the parallelopiped, if the coterminus edges are given by the vectors `2hat"i" + 5hat"j" -4 hat"k", 5hat"i" +7hat"j"+5 hat "k" , 4hat"i" +5hat"j" - 2 hat"k"`.                               

Find the value of p, if the vectors `hat"i" - 2hat"j" + hat"k", 2hat"i" -5hat"j"+"p" hat "k" , 5hat"i" -9hat"j" + 4 hat"k"` are coplanar.

 Show that the points A (-7 , 4 , -2),B (-2 , 1 , 0)and C (3 ,-2 ,2) are collinear.

 Write the equation of the plane 3x + 4y -2z = 5 in the vector form.

 If `"y" ="x"^"x" , "find"  "dy"/"dx"`.

 Find the equation of tangent to the curve y = x² +4x + 1 at (-1 , -2).

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

Evaluate : `int _0^(pi/2) "sin"^ 2  "x"  "dx"`

 In , ΔABC prove that 

`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/2)`                               

Show that `"sin"^-1(5/13) + "cos"^-1(3/5) = "tan"^-1(63/16)`

Let `A (bara)` and `B (barb)` are any two points in the space and `"R"(bar"r")` 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) `

 The equation of a line is 2x -2 = 3y +1 = 6z -2 find the direction ratios and also find the vector equation of the line. 

Discuss the continuity of the function 

f(x) = `("log"(2+"x") - "log"(2-"x"))/("tan""x")` , for x ≠ 0

 = 1  for x = 0 at the point x =0

Find the variance of X.                                  

For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1) 

`f(x) = x^2/18, -3 < x < 3`

            = 0,             otherwise

Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result.

If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of the first and third numbers, we get 8, and if three times the first number is added to the sum of the second and third numbers, we get 4. Find the numbers using matrices. 

 In ,Δ ABC with usual notations prove that 
b² = c² +a² - 2 ca cos B

 In , ΔABC with usual notations prove that

(a-b)² cos² `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`

 Find 'p' and 'q' if the equation px² - 8xy +3y² +14 x +2y +q = 0 represents a pair of perpendicular lines.

 Maximize: z = 3x + 5y  Subject to

x +4y ≤ 24                3x + y  ≤ 21 

x + y ≤ 9                     x ≥ 0 , y ≥0

If X = f(t) and Y = g(t) Are Differentiable Functions of t ,  then prove that y is a differentiable function of x and

`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`

Hence find `"dy"/"dx"` if x = a cos² t and y = a sin² t.

f(x) = (x-1)(x-2)(x-3) , x ε[0,4], find if 'c' LMVT can be applied

 A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle. 

 Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log  |"x" +sqrt("x"^2 +"a"^2) | + "c"`

 Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`

 Solve the differential equation: 

`"dy"/"dx" + "y"  "sec"  "x" = "tan""x"`

 Solve the differential equation:

`("x" + "y") "dy"/"dx" = 1`