Let's learn Mathematics

Q. A function is defined in (0,3) in the following way- Is f(x) continuous at x=2? Solution:

Choose the correct alternative: Q. If two rows or columns of a determinant are identical then value of the determinant becomes: 0 1 2 Can't say Solution: If two rows or columns of a determinant are identical then value of the determinant becomes 0.

Choose the correct alternative: Q. In an LPP Only the objective function is linear. Only the constraints are linear. The objective function as well as the constraints are linear. None of the above. Solution: In an LPP, the objective function as well as constraints are linear.

Choose the correct alternative: Q. The characteristics of logarithm of the number 375.67 is: 1 2 0 3 Solution: When there are 'n' digits before the decimal point of any number then the value of the characteristics is (n-1). Here, there are 3 digits before the decimal point, so the characteristics value is (3-1)=2. Therefore, the characteristics of logarithm of the number 375.67 is 2.

Q. Find the inverse of the matrix: Solution: First of all, we will find out the determinant of the matrix A. Therefore, the inverse of A, A⁻¹ exists. Now we will find the  cofactors of the elements and will form the cofactor matrix B.

Q. Find the derivative of y=5ˣx⁵ Solution:

Q. Find the derivative of y=(3x+8)⁶ Solution:

Question. Solution:

Q. Solve by Cramer's rule: 2x+y+z=1 x-y+2z=-1 3x+2y-z=4 Solution:     Since the delta value is non zero, so the solution exists.

Q. A, B and C are any three sets, prove that  A∪(B∩C)=(A∪B)∩(A∪C) Solution: Let us assume that A={1,2,3,4,5} B={3,5,7,9,10} C={2,4,6,8,10} Now, B∩C={10} A∪(B∩C)={1,2,3,4,5,10}=LHS And, A∪B={1,2,3,4,5,7,9,10} A∪C={1,2,3,4,5,6,8,10} (A∪B)∩(A∪C)={1,2,3,4,5,10}=RHS Therefore LHS=RHS( hence proved)

Q. In a classroom of 120 students, 80 read Assam Tribune, 55 read the Hindu, 25 read the both. How many students read neither Assam Tribune nor the Hindu? Solution: Let us assume, n(A)= The set of students who read Assam Tribune. n(H)= The set of students who read the Hindu. Given, n(A)=80, n(H)=55, n(A∩H)=25, to find: n(A∪H)=? we know that, n(A∪H)=n(A)+n(H)-n(A∩H)              =80+55-25              =135-25              =110 so, the no. of students who read neither Assam Tribune nor the Hindu=120-110=10.

Q. A man borrows Rs. 12,000 and promises to pay back in 20 instalments, each of value Rs 30 more than the last. Find the first instalment. Solution: Let us assume that a=first installment, d=common difference, n= number of installments. Here, Sₙ=12,000, n=20, d=30. To find a=? Therefore the first installment is Rs. 315

Q. The sum of three consecutive terms in A.P. is 24 product is 440, find the terms. Solution: Let us assume that the numbers are a-d, a, a+d Given, (a-d)+a+(a+d)=24 ⇒3a=24 ⇒a=24/3 ⇒a=8 and (a-d)*a*(a+d)=440 ⇒(8-d)*8*(8+d)=440 ⇒(8-d)(8+d)=440/8 ⇒8²-d²=55   (since (a+b)(a-b)=a²-b²) ⇒64-d²=55 ⇒-d²=55-64 ⇒-d²=-9 ⇒d²=9 ⇒d²=3² ⇒d=±3 If d=3, then the numbers are 8-3,8,8+3 => 5,8,11 If d=-3, then the numbers are 8-(-3), 8, 8+(-3)=>8+3, 8, 8-3 =>11, 8, 5

Q. Find the sum of n terms of the following series: 9+99+999+...... Solution:  

http://easymaths.science.blog/2020/01/10/arithmetic-progression/

http://easymaths.science.blog/2020/01/10/geometric-progression/

Question. The total cost of output x is given by  . Find: Cost when output is 4 units. Average cost of output of 10 units. Marginal cost when output is 3 units. Solution: 1. When the output is 4 units then the value of cost is 2. Average cost=AC When output is 10 units then average cost is 3. Marginal cost= MC Marginal cost when output is 3 units

Question. The total cost c(x) of a firm is C(x)=0.005x³-0.7x²-30x+3000, where x is the output. Determine average cost (AC)and marginal cost (MC). Solution: Here, C(x)=total cost=TC Average cost=  Marginal cost=

Q. Solve the quadratic equation:. 2x²-13x+15=0 Solution: 2x²-13x+15=0 ⇒2x²-(10+3)x+15=0 ⇒2x²-10x-3x+15=0 ⇒2x(x-5)-3(x-5)=0 ⇒(x-5)(2x-3)=0 either, x-5=0⇒x=5 or, 2x-3=0⇒2x=3⇒x=3/2 so, the solutions are 5 and 3/2

Question: Solution: ( We can end our answer here, or can go for further simplification.)

Question. solution,

Solution: Let us assume, Now we will substitute this value to the given question,

Q. The third and sixth terms of a series in A.P  are 13 and 31 respectively. Find 20th term. Solution: Let us assume that first term=a and common difference=d. Given, third term=t₃=13 ⇒a+(3-1)*d=13 ⇒a+2d=13...................(i) and sixth term=t₆=31 ⇒a+(6-1)*d=31 ⇒a+5d=31...............(ii) (ii)-(i)⇒(a+5d)-(a+2d)=31-13 ⇒a+5d-a-2d=18 ⇒a-a+5d-2d=18 ⇒0+3d=18 ⇒3d=18 ⇒d=18÷3 ⇒d=6 Substituting the value of d=6 to the equation (i),we get (i)⇒a+2*6=13 ⇒a+12=13 ⇒a=13-12 ⇒a=1 20th term=t₂₀=a+(20-1)*d =1+19*6 =1+114 =115.

Q.  The 4th and 7th terms of a series in G.P are 48 and 384 respectfully. Find the first term and the common ratio. Solution: Let us assume that first term=a and common ratio=r. Given,4th term=48. ⇒ar⁽⁴⁻¹⁾⁼48 ⇒ar³=48...........................................(i) and, 7th term=384 ⇒ar⁽⁷⁻¹⁾=384 ⇒ar⁶=384.......................................(ii) (now we will divide equation (ii)by (i) ) (ii)÷(i)⇒(ar⁶)÷(ar³)=384÷48 ⇒r³=8 (by the formula of indices we get it. here first of all, a is cancelled out as it is present in both numerator and denominator.then the remaining value of power of r is following by this formula. That means 6-3=3) (Now in the LHS the value is in the power of r and the power is 3, so we will try to express the RHS value in the power of 3. We will concentrate to express the value in the power of 3. We know that 2³=8. So we will write the RHS value in the form 2³=8) ⇒r³=2³ ⇒r=2 (because again from the indices formula,  , If you compare the v...

 If all the elements of a row or column is zero then the value of the determinant is zero. If we exchange all the row and column of the determinant then value of the determinant will not change, it will remain same. If any two row or column of a determinant are identical then value of the determinant is zero. If we exchange only any two row or column of the determinant then the value of the new determinant is (-1) times of the actual value of the determinant.

In this problem we directly can't substitute the value x=2 to the problem because if we do so then then value of the function become 0÷0 form which is undefined. Therefore we remove the factor that is (x-2) which is causing this problem. To remove (x-2) we represent the quadratic equation in factorized form. When we factorized the quadratic equation then in the numerator and denominator we get the value (x-2) present in both. We can cancel this value. After cancellation, there remain the value. Now if we substitute x=2 to this remaining function then it will not give 0÷0 form. So it is safe to substitute the value x=2 to the remaining function. After substituting the value we get the value of the limit .

Which term of the series 1,3,9,27,... is 6561? 7th 8th 9th 10th Solution: 3.9th term This is a geometric progression.  Let us assume first term=a, common ratio=r, nth term=tₙ Given, a=1, r=3÷1=3, tₙ=6561 For a geometric progression, tₙ=ar⁽ⁿ⁻¹⁾ (We are to use this formula to find the answer, we know the value of first term(a), common ratio(r) and the nth term. The unknown value is n.  We substitute all the values to the formula) tₙ=ar⁽ⁿ⁻¹⁾ ⇒6561=1*3⁽ⁿ⁻¹⁾ ⇒6561=3⁽ⁿ⁻¹⁾   (now we will express 6561 in the power of 3, because in the RHS the value is in power of 3.so by expressing the LHS value in power of 3,we can equate both sides and by the concept of indices we can find the value of n) ⇒3⁸=3⁽ⁿ⁻¹⁾ ⇒8=n-1 (if aˣ=aⁿ then x=n) ⇒8+1=n ⇒9=n ⇒n=9.

Which of the following can't be defined as a set? The set of Indian Nobel laureates. The set of meritorious students of Assam. The set of even numbers less than 500. None of the above. Solution:2 Explanation: The set of meritorious students of Assam is not a set because the concept of merit differ person to person. I may find X as a meritorious student but you may not find him/her meritorious.

Choose the correct alternatives. The characteristics of logarithm of the number 375.67 is i) 1    ii) 2   iii) 0   iv)3      Ans:  b)2 In an LPP: i) only the objective function is linear. ii) only the constraints are linear. iii) the objective function as well as the constraints are linear. iv) none of the above. Ans: iii) the objective function as well as the constraints are linear. If two rows or columns of a determinant are identical then the value of the determinant becomes    i)  0   ii) 1  iii) 2   iv) can't say.    Ans:  i) 0 log(a/b )+log(b/c)+log(c/a )=?     i) 1  ii) 0  iii) -1  iv)  2    Ans:ii) 0 If TC, MC,AC and x represent the total cost, marginal cost, average cost and output respectively, then which of the following are true?               i) d/dx(TC)=MC  ii) TC/x=AC...

A man borrows Rs. 4,500 and promises to pay back in 30 installments, each of value Rs. 10 more than the last. Find the 1st and last installments.      Solution: Here the summation amount is 4500. S=4500. d=10 n=30. We assume that first installment=a. We will use the summation formula.     S=(n÷2)×{2a+(n-1)×d} =>4500=(30÷2)×{2a+(30-1)×10} =>4500=15×{2a+29×10} =>4500=15×{2a+290} =>4500÷15=2a+290 =>300=2a+290 =>300-290=2a =>10=2a =>10÷2=a =>5=a So the first installment=Rs. 5 The last installment is  t₃₀=a+(30-1)×d         =5+29×10         =5+290         =295.

Sometimes we get these types of number series viz 1,3,5,7,9,........ Or 2,4,6,8,10.....or 3,6,9,12,15,....or 10,20,30,40,50,......and so on. Here, if we closely observe the number series then we find that the differences between the first term and second term is same with the one of the second term and third term. The difference is same with all the pair of the numbers. If you notice the first example then the difference between 1 and 3 is 2, 3 and 5 is also 2, 7  and 9 is also 2. In the second example the difference is also 2 between the pairs of numbers. In the third example the difference is 3 between the pairs of numbers. In the fourth example the difference is 10. These type of arrangement of numbers is called arithmetic progression. So when the differences between the number of pairs is same then it is called arithmetic progression or A.P. The  difference is called common difference and it is denoted by d. The first term of the series is denoted by a. S...

Set theory is defined as a collection of well defined objects. These objects are called elements. Here the main emphasis lies on the word 'well defined'. What is the meaning of well defined? The meaning of well defined is that everyone should agree with the concept that the particular object or element is included in the collection. If the opinion differ from person to person about the inclusion of that particular object or element then that collection can't be defined as a set. Let us take some examples so that you can clearly understand the definition. The set of all the students of JNU. The set of the rivers of India. The set of oceans of the world. The set of IIT's of India. The set of national parks of India. All these are called set because here everyone agree about the particular object. In the first example, you can easily say whether the students X,Y,Z are from JNU or not. Either he or she is a student there or not. There is no biasness about that. ...