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]]`
Concept: Matrices - Inverse of a Matrix Existance
If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`
(A) 100
(B) 101
(C) 110
(D) 109
Concept: Scalar Triple Product of 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`
Concept: Direction Cosines and Direction Ratios of a Line
`barr=(hati-2hatj+3hatk)+lambda(2hati+hatj+2hatk)` is parallel to the plane `barr.(3hati-2hatj+phatk)=10`, find the value of p.
Concept: Plane - Equation of Plane Passing Through the Given Point and Parallel to Two Given Vectors
If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.
Concept: Angle Between Line and a Plane
Write the negations of the following statements:
a.`forall n in N, n+7>6`
b. The kitchen is neat and tidy.
Concept: Mathematical Logic - Sentences and Statement in Logic
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Concept: Direction Cosines and Direction Ratios of a Line
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.
Concept: Basic Concepts of Vector Algebra
If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.
Concept: Properties of Inverse Trigonometric Functions
Write the converse, inverse and contrapositive of the following statement.
“If it rains then the match will be cancelled.”
Concept: Mathematical Logic - Sentences and Statement in Logic
Find p and q, if the equation `px^2-8xy+3y^2+14x+2y+q=0` represents a pair of prependicular lines.
Concept: Pair of Straight Lines - Condition for Perpendicular 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).
Concept: Plane - Equation of Plane Passing Through the Intersection of Two Given Planes
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.
Concept: Equation of a Line in Space
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
Concept: Trigonometric Functions - Solution of a Triangle
Find the cartesian equation of the line passing throught the points A(3, 4, -7) and B(6,-1, 1).
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
Find the vector equation of a line passing through the points A(3, 4, –7) and B(6, –1, 1).
Concept: Vector and Cartesian Equation of a Plane
Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0
Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
find the symbolic fom of the following switching circuit, construct its switching table and interpret it.
Concept: Mathematical Logic - Application - Introduction to Switching Circuits
If `A=[[1,-1,2],[3,0,-2],[1,0,3]]` verify that A (adj A) = |A| I.
Concept: Determinants - Adjoint Method
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.
Concept: Graphical Method of Solving 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`
Concept: Acute Angle Between the Lines
find the acute angle between the lines
x2 – 4xy + y2 = 0.
Concept: Acute Angle Between the Lines
Given f (x) = 2x, x < 0
= 0, x ≥ 0
then f (x) is _______.
(A) discontinuous and not differentiable at x = 0
(B) continuous and differentiable at x = 0
(C) discontinuous and differentiable at x = 0
(D) continuous and not differentiable at x = 0
Concept: Continuity - Discontinuity of a Function
If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`
(A) 1
(B) 2
(C) –1
(D) –2
Concept: Properties of Definite Integrals
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Concept: Increasing and Decreasing Functions
Differentiate 3x w.r.t. log3x
Concept: Exponential and Logarithmic Functions
Differentiate 3x w.r.t. log3x
Concept: Exponential and Logarithmic Functions
Check whether the conditions of Rolle’s theorem are satisfied by the function
f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3]
Concept: Mean Value Theorem
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Concept: Methods of Integration - Integration by Substitution
Find the area of the region bounded by the curve x2 = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.
Concept: Area of the Region Bounded by a Curve and a Line
Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).
Concept: 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).
Concept: Continuity - Continuity of a Function at a Point
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.
Concept: Variance of Binomial Distribution (P.M.F.)
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.
Concept: Conditional Probability
Find the approximate value of cos (60° 30').
(Given: 1° = 0.0175c, sin 60° = 0.8660)
Concept: Approximations
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]
Concept: Rate of Change of Bodies Or Quantities
Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.
= 0, if f (x) is an odd function.
Concept: Methods of Integration - Integration by Parts
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`
Concept: Algebra of Continuous Functions
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Concept: Methods of Integration - Integration by Parts
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
Concept: Random Variables and Its Probability Distributions
Solve the differential equation `dy/dx -y =e^x`
Concept: General and Particular Solutions of a 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`
Concept: Derivative - Derivative of Inverse Function
Evaluate: `∫8/((x+2)(x^2+4))dx`
Concept: Methods of Integration - Integration Using Partial Fractions
Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2016 - 2017
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