R.S. Aggarwal solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० chapter 20 - Additional Questions [Latest edition]

Assertion (A): \(\sqrt{10}\) is an irrational number.

Reason (R): If m is a natural number which is not a perfect square then \(\sqrt{m}\) is irrational.

Assertion (A): \[\frac{19}{500}\] is a terminating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a terminating decimal, if \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

Assertion (A): \[\frac{31}{90}\] is a nonterminating, repeating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a repeating decimal, if q has at least one factor other than 2 and 5.

Assertion (A): \[\sqrt{7}\] is an irrational number.

Reason (R): If p is prime then \[\sqrt{p}\] is irrational.

Assertion (A): \[\frac{22}{7}\] is a nonterminating, repeating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a terminating decimal, if \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

Assertion (A): \[\frac{1}{\sqrt{3}}\] is a rational number.

Reason (R): If p is prime then \[\frac{1}{\sqrt{p}}\] is irrational.

Assertion (A): π is irrational and \[\frac{22}{7}\] is rational.

Reason (R): \[\pi \neq \frac{22}{7}\]

Assertion (A): \[\sqrt{5}\] is an irrational number.

Reason (R): Let p be a prime number and a be a positive integer. Then, p divides \[a^{2} \Rightarrow\] p divides a.

Assertion (A): \[0.\overline{6}\] is a terminating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a terminating decimal, if \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

Assertion (A): The HCF of two numbers is 25 and their LCM is 60.

Reason (R): The HCF of two given numbers always divides their LCM exactly.

Assertion (A): The HCF of two numbers is 7 and their LCM is 70. If one of the numbers is 14 then the other number is 30.

Reason (R): For any two given positive integers a and b, we have \[\text{HCF},(a, b) \times \text{LCM},(a, b) = (a \times b).\]

Assertion (A): The number \[5^{n}\] cannot end with digit 0, where n is a natural number.

Reason (R): Prime factorisation of 5 has only two prime factors 1 and 5.

Based on the above information, answer the following questions.

1. How many books are arranged in each stack? 
2. How many stacks are used to arrange all the mathematics books? 
3. Determine the total number of stacks that will be used for arranging all the books. 
4. If the thickness of each book of history, science and mathematics is 1.8 cm, 2.2 cm and 2.5 cm respectively then find the height of each stack of history, science and mathematics books

Based on the above information, answer the following questions.

1. At what time after 8.00 a.m. will all the signals change simultaneously? 
2. At what time after 8.00 a.m. will the signals A and B change simultaneously? 
3. At what time after 8.00 a.m. will the signals B and C change simultaneously?
4. At what time after 8.00 a.m. will the signals A and C change simultaneously?

Assertion (A): A monic quadratic polynomial having 4 and –2 as its zeros is $$\left(x^{2}-2x-8\right)$$.

Reason (R): The monic quadratic polynomial having $$\alpha$$ and $$\beta$$ as its zeros, is given by $$x^{2}-(\alpha+\beta)x+\alpha\beta$$.

Assertion (A): The quadratic polynomial whose zeros are $$(3+\sqrt{2})$$ and $$(3-\sqrt{2})$$ is given by $$p(x)=(x^{2}-6x+7)$$.

Reason (R): If $$\alpha$$ is a zero of the quadratic polynomial $$p(x)$$ then $$(x-\alpha)$$ is a factor of $$p(x)$$.

Assertion (A): The polynomial $$p(x)=(x^{3}+x)$$ has one real zero.

Reason (R): A polynomial of nth degree has at most n zeros.

Assertion (A): If $$\alpha$$ and $$\beta$$ be the zeros of polynomial $$(x^{2}-6x+k)$$ such that $$(\alpha-\beta)=2$$ then $$k=8$$.

Reason (R): If $$\alpha$$ is a zero of the polynomial $$p(x)$$ then $$p(-\alpha)=0$$.

Assertion (A): If the zeros of the quadratic polynomial $$ax^{2}+bx+c$$ are both negative then a, b, c will have the same sign.

Reason (R): The zeros of a quadratic polynomial are either both positive or both negative.

Assertion (A): If one zero of the polynomial $$p(x)=(k^{2}+4)x^{2}+9x+4k$$ is the reciprocal of the other zero then $$k=2$$.

Reason (R): If $$(x-\alpha)$$ is a factor of the polynomial $$p(x)$$ then $$\alpha$$ is a zero of $$p(x)$$.

Assertion (A): If the sum of the zeros of the quadratic polynomial $$p(x)=kx^{2}+2x+3k$$, is equal to the product of its zeros then $$k=\frac{-2}{3}$$.

Reason (R): If $$\alpha$$ and $$\beta$$ be the zeros of a quadratic polynomial $$p(x)=ax^{2}+bx+c$$ then $$(\alpha+\beta)=\frac{-b}{a}$$ and $$\alpha\beta=\frac{c}{a}$$.

Assertion (A): If on dividing the polynomial $$p(x)=x^{2}-3ax+3a-7$$ by $$(x+1)$$, we get 6 as remainder then $$a=4$$.

Reason (R): When a polynomial $$p(x)$$ is divided by $$(x-\alpha)$$ then the remainder is $$p(\alpha)$$.

Assertion (A): A monic cubic polynomial having 3, 2, –1 as its zeros is $$p(x)=x^{3}+4x^{2}-x+6$$.

Reason (R): The monic cubic polynomial having $$\alpha,\beta,\gamma$$ as its zeros is given by $$x^{3}-(\alpha+\beta+\gamma)x^{2}+(\alpha\beta+\beta\gamma+\gamma\alpha)x-(\alpha\beta\gamma)$$.

Based on the above information, answer the following questions.

1. The graph of a rainbow y = f(x) is shown in the figure given below. Write the number of zeros of the curve.
   
2. If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeros will it have? 
3. The polynomial x² – 2x – (7a + 3) represents a rainbow. If –4 is a zero of it, find the value of a. 
4. If α and β are the zeros of the quadratic polynomialp (x) = x² – rx – s then find the value of α² + β².

Based on the above information, answer the following questions.

1. How many zeros are there for the polynomial represented by the given graph?
2. Write the zeros of the polynomial. 
3. If the zeros of a polynomial x² + (p + 1)x + q are 2 and –3 then determine the values of p and g. 
4. If the square of the difference between the zeros of the polynomial x² + ax + 45 is 144 then find the value of a.

Based on the given information, answer the following questions.

1. Find the zeros of the polynomial p(t) = 20t – 16t².
2. Which of the following types of graphs represents p(t)?
   
3. What would be the value of h at `t = 3/2`? Interpret the result.
4. How much distance has the dolphin covered before hitting the water level again?

Assertion (A): The system of equations \[2x+3y-6=0\] and \[4x+6y-12=0\] has infinitely many solutions.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[2x-3y-5=0\] and \[4x-6y+3=0\] has no solutions.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has no solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[x+2y+2=0\] and \[3x+2y-2=0\] has a unique solution.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[3x-6y=0\] and \[x-2y-6=0\] has infinitely many solutions.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of equations \[2x-5y+4=0\], \[2x+y-8=0\] has a unique solution.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] is inconsistent, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The homogeneous system of equations \[2x+3y=0\], \[4x+6y=0\] has an infinite number of solution.

Reason (R): The homogeneous system of equations \[a_{1}x+b_{1}y=0,\ a_{2}x+b_{2}y=0\] has an infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}.\]

Assertion (A): The homogeneous system of equations \[3x-4y=0\], \[2x-3y=0\] has a unique solution x = 0 and y = 0.

Reason (R): The homogeneous system of equations \[a_{1}x+b_{1}y=0,\ a_{2}x+b_{2}y=0\] has a unique solution x = 0 and y = 0 when \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

Assertion (A): The system of linear equations \[3x+5y-4=0\], \[15x+25y-25=0\] is inconsistent.

Reason (R): The pair of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] is inconsistent, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of linear equations \[5x-9y+6=0\], \[10x-18y+12=0\] is inconsistent.

Reason (R): The pair of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] is inconsistent, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

Assertion (A): The system of linear equations \[2x-3y+1=0\], \[4x-6y+3=0\] is inconsistent.

Reason (R): The system of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

Assertion (A): The system of linear equations \[x-2y+6=0\], \[2x-4y+9=0\] has a unique solution.

Reason (R): The system of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

Assertion (A): The system of linear equations \[x+y-9=0\], \[x-y+3=0\] has a unique solution.

Reason (R): The system of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}.\]

Based on the above information, answer the following questions.

(i) What are the fixed charges?

(ii) What are the charges per kilometre?

(iii) If fixed charge are ₹20 and charges per kilometre are ₹10 then how much would Arun have to pay for travelling a distance of 10 km?

(iv) Find the total amount paid by Arun for travelling 10 km from home to office and 25 km from office to home. [Fixed charges and charges per kilometre are as in (i) and (ii).]

Based on the above information, answer the following questions.

1. Represent the above information algebraically (in terms of p and q). 
2. What is the prize amount for hockey? 
3. Prize amount for which game is higher and by how much? 
4. What will be the total prize amount if there are 2 students each from the two games?

Read the following passage:

Based on the above information, answer the following questions:

1. Represent the information given above in terms of x and y.
2. Find the monthly fee paid by a poor child.
   OR
   Find the difference in the monthly fee paid by a poor child and a rich child.
3. If there are 10 poor and 20 rich children in batch II, what is the total monthly collection of fees from batch II?

Assertion (A): The quadratic equation $$x^{2}+3x+4=0$$ has both real roots.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has both real roots, if $$(b^{2}-4ac)\geq 0$$.

Assertion (A): The polynomial $$p(x)=x^{2}+3x+3$$ has two real zeros.

Reason (R): A quadratic polynomial can have at most two real zeros.

Assertion (A): The polynomial $$x^{2}+4x$$ has two real zeros.

Reason (R): The zeros of the polynomial $$x^{2}+ax\ (a\neq 0)$$ are 0 and $$a$$. 

Assertion (A): If $$(5 + \sqrt{7})$$ is a root of a quadratic equation with rational coefficients then the other root is $$(5 - \sqrt{7})$$.

Reason (R): The surd roots of a quadratic equation with rational coefficients occur in conjugate pairs.

Assertion (A): If one root of the quadratic equation $$4x^{2}-10x+(k-4)=0$$ is the reciprocal of the other then the value of $$k$$ is 8.

Reason (R): The roots of the quadratic equation $$x^{2}-x+1=0$$ are real.

Assertion (A): If the equation $$x^{2}-kx+1=0$$ has no real roots then $$-2<k<2$$.

Reason (R): The equation $$ax^{2}+bx+c=0$$ has real roots, if $$(b^{2}-4ac)\geq 0$$.

Assertion (A): The quadratic equation $$\sqrt{3}x^{2}+11x+6\sqrt{3}=0$$ has both real roots.

Reason (R): The quadratic equation $$ax^{2}+bx+c=0$$ has both imaginary roots, if $$(b^{2}-4ac)<0$$.

Assertion (A): If –5 is a root of $$2x^{2}+2px-15=0$$ and the equation $$p(x^{2}+x)+k=0$$ has equal roots then $$k=\frac{7}{8}$$.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has equal roots, if $$(b^{2}-4ac)=0$$.

Assertion (A): If the quadratic equation $$px^{2}-4x+4=0$$ has real roots them $$p\leq 1$$.

Reason (R): The quadratic equation $$ax^{2}+bx+c=0$$ has both real roots, if $$(b^{2}-4ac)<0$$.

Assertion (A): The quadratic equation $$x^{2}+5kx+16=0$$ has no real roots, if $$\frac{-8}{5}<k<\frac{8}{5}$$.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has no real root, if $$(b^{2}-4ac)<0$$.

Assertion (A): The quadratic equation $$x^{2}-x-6=0$$ has –2 and 3 as its roots.

Reason (R): If $$ax^{2}+bx+c=0$$ then $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$$

Assertion (A): The quadratic equation $$x^{2}-4x+3=0$$ has both real roots.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has both real roots, if $$(b^{2}-4ac)\geq 0$$.

Assertion (A): The quadratic equation $$3x^{2}-5x+4=0$$ has no real root.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has no real root, if $$(b^{2}-4ac)<0$$.

Based on the given information, answer the following questions.

1. Taking x as the number of rows in the beginning, represent the above situation by a quadratic equation. 
2. How many rows are there in the original arrangement? 
3. How many seats are there in the auditorium in the beginning? 
4. How many seats are there in the auditorium after rearrangement?

 Based on the above information, answer the following questions.

1. Write an algebraic equation depicting the above information. 
2. Write the corresponding quadratic equation in standard form. 
3. What should be the new dimensions of the enlarged photo? 
4. Can any rational value of x make the new area equal to 220 cm²?

Based on the given information, answer the following questions.

1. Write a quadratic equation in x to represent the above information. 
2. Find the value of x.
3. Find the perimeter of rectangle PQRS. 
4. For what value of x will the area of the path of uniform width be 93600 m²?

Assertion (A): The numbers $$a, b, c$$ are in AP if and only if $$2b = (a + c).$$

Reason (R): The sum of the first $$n$$ odd natural numbers is $$n^{2}.$$

Assertion (A): The sum of first n natural numbers is given by $$S_{n} = \frac{1}{2}n(n + 1).$$

Reason (R): In an AP with first term a and common difference d, the sum of n terms is given by $$S_{n} = \frac{n}{2} \times [2a + (n - 1)d].$$

Assertion (A): The sum of first n even natural numbers is n(n + 1).

Reason (R): The sum of n terms of an AP with first term a and last term $$l$$ is $$S_{n} = \frac{n}{2},(a + l).$$

Assertion (A): The 9th term of the AP 37, 33, 29, 25, ... is –5.

Reason (R): The nth term of the AP with first term a and common difference d is given by $$T_{n} = a + (n - 1)d.$$

Assertion (A): The 10th term from the end of the AP 16, 11, 6, 1, ..., –79 is –34.

Reason (R): The nth term from the end of an AP with last term $$l$$ and common difference d is $${\{l - (n - 1)d}\}.$$

Assertion (A): 47 + 45 + 43 + ... + 11 is an AP having 19 terms.

Reason (R): In an AP with first term a and common difference d, the nth term is given by $$T_{n} = a + (n - 1)d.$$

Assertion (A): In an AP, the first term is 6, the last term is 78 and the sum of the given terms is 630. Then, there are 16 terms in the given AP.

Reason (R): If a is the first term, $$l$$ is the last term and n is the number of terms in an AP then $$S_{n} = \frac{n}{2} (a + l).$$

Assertion (A): The next term of the AP $$\sqrt{3},\ \sqrt{12},\ \sqrt{27},\ \ldots$$ is $$\sqrt{48}.$$

Reason (R): In an AP, we have $$T_{n + 1} = (T_{n} + d).$$

Assertion (A): In an AP, it is given that $$T_{9} = 39$$ and $$T_{15} = 63.$$ Then, its first term is 7 and the common difference is 4.

Reason (R): In an AP with first term a and common difference d, the nth term is given by $$T_{n} = a + (n - 1)d.$$

Assertion (A): The sum of 14 terms of the AP 7 + 10 + 13 + ... is 375.

Reason (R): The sum of n terms of an AP with first term a and common difference d is given by $$S_{n} = \frac{n}{2} \cdot {\{2a + (n - 1)d}\}.$$

Based on the given information, answer the following questions.

1. Find the number of pots placed in the 10th row.
2. Find the difference in the number of pots placed in the 5th row and the 2nd row.
3. If Sanjana wants to place 100 pots in total then find the total number of rows formed in the arrangement.
4. If Sanjana has sufficient space for 12 rows then how many total number of pots are placed by her with the same arrangement?

Based on the above information, answer the following questions.

1. In which year will the factory produce 29200 sets?
2. Find the production during the 8th year.
3. Find the production during the first 3 years.
4. Find the difference in production during the 7th year and the 4th year.

Based on the above information, answer the following questions.

1. If the first circular row has 30 seats then how many seats will there be in the 10th row?
2. For 1500 seats, how many rows must the auditorium have?
3. If 1500 seats are to be arranged in the auditorium, how many seats are still left to be put after the 10th row?
4. If there were 17 rows in the auditorium, how many seats would be in the middle row?

Based on the given information, answer the following questions.

1. Find the amount paid by him in the 10th instalment.
2. Find the total amount paid by him in the first 10 instalments.
3. In how many instalments would he clear his total loan?
4. What amount will he be able to clear in his first 45 instalments?

Based on the given figure, answer the following questions.

1. Identify the AP for the number of squares in each row.
2. Identify the AP for the number of triangles in each row.
3. If each shaded square is of side 2 cm then find the shaded area when 15 rows have been designed.
4. Write a formula for finding the total number of triangles in n number of rows. Hence, find \(S_{10}\).