Answer: (a) 96 Solution: Suppose the tens and the units digits of the required no. be $\mathrm{x}$ and $\mathrm{y}$, respectively. The required number $=(10 x+y)$ As per the question, we have:...
The sum of the digits of a two digit number is $15 .$ The number obtained by interchanging the digits exceeds the given number by $9 .$ The number is
The graphs of the equations $5 x-15 y=8$ and $3 x-9 y=\frac{24}{5}$ are two lines which are
(a) coincident
(b) parallel
(c) intersecting exactly at one point
(d) perpendicular to each other
Answer: (a) coincident Solution: The correct option is (a). The given system of equations can be written as follows: $5 x-15 y-8=0$ and $3 x-9 y-\frac{24}{5}=0$ Given equations are of the following...
The graphs of the equations $2 \mathrm{x}+3 \mathrm{y}-2=0$ and $\mathrm{x}-2 \mathrm{y}-8=0$ are two lines which are
(a) coincident
(b) parallel
(c) intersecting exactly at one point
(d) perpendicular to each other
Answer: Solution: The given system of equations are as follows: $2 x+3 y-2=0$ and $x-2 y-8=0$ They are of the following form: $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$ Here, $a_{1}=2,...
The graphs of the equations 6x – 2y + 9 = 0 and 3x – y + 12 = 0 are two lines which are
(a) coincident
(b) parallel
(c) intersecting exactly at one point
(d) perpendicular to each other
Answer: (b) parallel Solution: The given system of equations are as follows: $6 x-2 y+9=0$ and $3 x-y+12=0$ They are of the following form: $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$...
$$\begin{tabular}{|l|l|} \hline Assertion (A) & Reason $(\mathrm{R})$ \\ \hline The system of equations $\mathrm{x}+\mathrm{y}-8=0$ and $\mathrm{x}-\mathrm{y}-2=0$ has a unique solutions. & $\begin{array}{l}\text { The system of equations } \\ \mathrm{a}_{1} \mathrm{x}+\mathrm{b}_{1} \mathrm{y}+\mathrm{c}_{1}=0 \\ \text { and } \mathrm{a}_{2} \mathrm{x}+\mathrm{b}_{2} \mathrm{y}+\mathrm{c}_{2}-0 \\ \text { has a unique solution when } \\ & \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\end{array}$ \\ \hline \end{tabular}$$
The correct answer is: (a) / (b)/ (c)/ (d).
Answer: (c) Solution: The correct answer is option (C). It is clear that, Reason (R) is false. Upon solving $x+y=8$ and $x-y=2$, we obtain: $x=5$ and $y=3$ Therefore, the given system has a unique...
5 years hence, the age of a man shall be 3 times the age of his son while 5 years earlier the age of the man was 7 times the age of his son. The present age of the man is
(a) 45 years
(b) 50 years
(c) 47 years
(d) 40 years
Answer: (d) 40 years Solution: Suppose the present age of the man be $\mathrm{x}$ years. And his son's present age be $y$ years. 5 years later: $\begin{array}{l} (x+5)=3(y+5) \\ \Rightarrow x+5=3...
In a cyclic quadrilateral $\mathrm{ABCD}$, it is being given that $\angle \mathrm{A}=(\mathrm{x}+\mathrm{y}+10)^{0}, \angle \mathrm{B}=(\mathrm{y}+20)^{0}$, $\angle \mathrm{C}=(\mathrm{x}+\mathrm{y}-30)^{0}$ and $\angle \mathrm{D}=(\mathrm{x}+\mathrm{y})^{0} .$ Then, $\angle \mathrm{B}=?$
(a) $70^{\circ}$
(b) $80^{\circ}$
(c) $100^{0}$
(d) $110^{\circ}$
Answer: (b) $80^{\circ}$ Solution: Correct option is (b). In a cyclic quadrilateral $\mathrm{ABCD}$ : $\begin{array}{l} \angle A=(x+y+10)^{0} \\ \angle B=(y+20)^{0} \\ \angle C=(x+y-30)^{0} \\...
In a $\triangle \mathrm{ABC}, \angle \mathrm{C}=3 \angle \mathrm{B}=2(\angle \mathrm{A}+\angle \mathrm{B})$, then $\angle \mathrm{B}=?$
(a) $20^{0}$
(b) $40^{0}$
(c) $60^{0}$
(d) $80^{0}$
Answer: (b) $40^{0}$ Solution: Suppose $\angle \mathrm{A}=\mathrm{x}^{0}$ and $\angle \mathrm{B}=\mathrm{y}^{0}$ $\therefore \angle \mathrm{A}=3 \angle \mathrm{B}=(3 \mathrm{y})^{0}$ Now, $\angle...
If a pair of linear equations is inconsistent, then their graph lines will be
(a) parallel
(b) always coincident
(c) always intersecting
(d) intersecting or coincident
Answer: (a) parallel Solution: If a pair of linear equations in two variables is inconsistent, then no solution exists as they have no common point. And, since there is no common solution, their...
If a pair of linear equations is consistent, then their graph lines will be
(a) parallel
(b) always coincident
(c) always intersecting
(d) intersecting or coincident
Answer: (d) intersecting or coincident Solution: If a pair of linear equations is consistent, then the two graph lines either intersect at a point or coincidence.
The pair of equations $2 x+3 y=5$ and $4 x+6 y=15$ has
(a) a unique solution
(b) exactly two solutions
(c) infinitely many solutions
(d) no solution
Answer: (d) no solution Here, $a_{1}=3, b_{1}=2 k, c_{1}=-2, a_{2}=2, b_{2}=5$ and $c_{2}=1$ $\therefore \frac{a_{1}}{a_{4}}=\frac{3}{2}, \frac{b_{1}}{b_{L}}=\frac{2 k}{5}$ and...
The pair of equations $x+2 y+5=0$ and $-3 x-6 y+1=0$ has
(a) a unique solution
(b) exactly two solutions
(c) infinitely many solutions
(d) no solution
Answer: (d) no solution Solution: We can write the given system of equations as: $x+2 y+5=0$ and $-3 x-6 y+1=0$ Given equations are of the following form: $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2}...
For what value of $k$ do the equations $k x-2 y=3$ and $3 x+y=5$ represent two lines intersecting at a unique point?
(a) $k=3$
(b) $k=-3$
(c) $k=6$
(d) all real values except -6
Answer: (d) all real values except -6 Solution: We can write the given system of equations as follows: $\mathrm{kx}-2 y-3=0$ and $3 \mathrm{x}+\mathrm{y}-5=0$ Given equations are of the following...
For the system of equations to have no solution, we must have:
(a) $\frac{-5}{4}$
(b) $\frac{2}{5}$
(c) $\frac{3}{2}$
(d) $\frac{15}{4}$
Answer: (d) $\frac{15^{2}}{4}$ Solution: We can write the given system of equations as follows: $3 x+2 k y-2=0$ and $2 x+5 y+1=0$ Given equations are of the following form: $a_{1} x+b_{1} y+c_{1}=0$...
The system $x+2 y=3$ and $5 x+k y+7=0$ have no solution when?
(a) $k=10$
(b) $\mathrm{k} \neq 10$
(c) $k=\frac{-7}{3}$
(d) $k=-21$
Answer: (a) $\mathrm{k}=10$ We can write the given system of equations as follows: $x+2 y-3=0$ and $5 x+k y+7=0$ The given equations are of the following form: $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2}...
The system $x-2 y=3$ and $3 x+k y=1$ have a unique solution only when ?
(a) $k=-6$
(b) $k \neq-6$
(c) $k=0$
(d) $k \neq 0$
Answer: (b) $\mathrm{k} \neq-6$ Solution: The correct option is (b). We can write the given system of equations as follows: $x-2 y-3=0$ and $3 x+k y-1=0$ given equations are of the following form:...
The system of $\mathrm{kx}-\mathrm{y}=2$ and $6 \mathrm{x}-2 \mathrm{y}=3$ has a unique solution only when
(a) $\mathrm{k}=0$
(b) $k \neq 0$
(c) $\mathrm{k}=3$
(d) $k \neq 3$
Answer: (d) $k \neq 3$. Solution: The given system of equations are $\begin{array}{l} \mathrm{kx}-\mathrm{y}-2=0\dots \dots(i) \\ 6 \mathrm{x}-2 \mathrm{y}-3=0\dots \dots(ii) \end{array}$ Here,...
If $\frac{2}{x}+\frac{3}{y}=6$ and $\frac{1}{x}+\frac{1}{2 y}=2$ then
(a) $x=1, y=\frac{2}{3}$
(b) $x=\frac{2}{3}, y=1$
(c) $x=1, y=\frac{3}{2}$
(d) $x=\frac{3}{2}, y=1$
Answer: (b) $x=\frac{2}{3}, y=1$ Solution: The given system of equations are $\begin{array}{l} \frac{2}{x}+\frac{3}{y}=6\dots (i) \\ \frac{1}{x}+\frac{1}{2 y}=2\dots (ii) \end{array}$ Multiplying...
If $2^{x+y}=2^{x-y}=\sqrt{8}$ then the value of $\mathrm{y}$ is
(a) $\frac{1}{2}$
(b) $\frac{3}{2}$
(c) 0
(d) none of these
Answer: (e) 0 Solution: $\begin{array}{l} \because 2^{x+y}=2^{x-y}=\sqrt{8} \\ \therefore \mathrm{x}+\mathrm{y}=\mathrm{x}-\mathrm{y} \\ \Rightarrow \mathrm{y}=0 \end{array}$
If $29 x+37 y=103$ and $37 x+29 y=95$ then
(a) x=1, y=2
(b) x=2, y=1
(c) x=3, y=2
(d) x=2, y=3
Answer: $(a) x=1, y=2$ Solution: The given system is $29 x+37 y=103\dots \dots(i)$ $37 x+29 y=95\dots \dots(ii)$ Adding equation(i) and equation(ii), we get $66 x+66 y=198$ $\Rightarrow x+y=3\dots...
If $4 x+6 y=3 x y$ and $8 x+9 y=5 x y$ then
(a) x=2, y=3
(b) x=1, y=2
(c) x=3, y=4
(d) x=1, y=-1
Answer: (c) $x=3, y=4$ Solution: The given system of equations are $\begin{array}{l} 4 x+6 y=3 x y\dots (i) \\ 8 x+9 y=5 x y\dots (ii) \end{array}$ Dividing equation(i) and equation(ii) by $x y$, we...
If $\frac{3}{x+y}+\frac{2}{x-y}=2$ and $\frac{9}{x+y}-\frac{4}{x-y}=1$ then
(a) $x=\frac{1}{2}, y=\frac{3}{2}$
(b) $x=\frac{5}{2}, y=\frac{1}{2}$
(c) $x=\frac{3}{2}, y=\frac{1}{2}$
(d) $x=\frac{1}{2}, y=\frac{5}{2}$
Answer: (b) $x=\frac{5}{2}, y=\frac{1}{2}$ Solution: The given system of equations are $\begin{array}{l} \frac{3}{x+y}+\frac{2}{x-y}=2\dots \dots(i) \\ \frac{9}{x+y}-\frac{4}{x-y}=1\dots \dots(ii)...
If $\frac{2 x+y+2}{5}=\frac{3 x-y+1}{3}=\frac{3 x+2 y+1}{6}$ then
(a) x=1, y=1
(b) x=-1, y=-1
(c) x=1, y=2
(d) x=2, y=1
Answer: $($ a) $x=1, y=1$ Solution: Considering $\frac{2 x+y+2}{5}=\frac{3 x-y+1}{3}$ and $\frac{3 x-y+1}{3}=\frac{3 x+2 y+1}{3}$. Now, on simplifying these equations, we obtain $\begin{array}{l}...
If $\frac{1}{x}+\frac{2}{y}=4$ and $\frac{3}{y}-\frac{1}{x}=11$ then
(a) $x=2, y=3$
(b) x=-2, y=3
(c) $x=\frac{-1}{2}, y=3$
(d) $x=\frac{-1}{2}, y=\frac{1}{3}$
Answer: (d) $x=\frac{-1}{2}, y=\frac{1}{3}$ Solution: The given system is $\begin{array}{l} \frac{1}{x}+\frac{2}{y}=4\dots \dots(i) \\ \frac{3}{y}-\frac{1}{x}=11\dots \dots(ii) \end{array}$ Adding...
If $\frac{2 x}{3}-\frac{y}{2}+\frac{1}{6}=0$ and $\frac{x}{2}+\frac{2 y}{3}=3$ then
(a) x=2, y=3
(b) x=-2, y=3
(c) x=2, y=-3
(d) x=-2, y=-3
Answer: $($ a) $x=2, y=3$ Solution: The given system is $\begin{array}{l} \frac{2 x}{3}-\frac{y}{2}=-\frac{1}{6}\dots \dots(i) \\ \frac{x}{2}+\frac{2 y}{3}=3\dots \dots(ii) \end{array}$ Multiplying...
If $x-y=2$ and $\frac{2}{x+y}=\frac{1}{5}$ then
(a) x=4, y=2
(b) x=5, y=3
(c) x=6, y=4
(d) x=7, y=5
Answer: (c) $x=6, y=4$ Solution: The given system is $\begin{array}{l} x-y=2\dots (i) \\ x+y=10\dots (ii) \end{array}$ Adding equation(i) and equation(ii), we get $2 x=12 \Rightarrow x=6$ Now,...
If $2 x+3 y=12$ and $3 x-2 y=5$ then
(a) x=2, y=3
(b) x=2, y=-3
(c) x=3, y=2
(d) x=3, y=-2
Answer: (c) $x=3, y=2$ Solution: The given system is $\begin{array}{l} 2 x+3 y=12\dots \dots(i) \\ 3 x-2 y=5\dots \dots(ii) \end{array}$ Multiplying equation(i) by 2 and equation(ii) by 3 and then...
Which of the following rational numbers is expressible as a non-terminating decimal? $(a)\frac{1351}{1250}(b)\frac{2017}{250}(c)\frac{3219}{1800}$
Correct Answer: Option (c) Explanation: 2, 3 and 5 are not the factors of 3219. So, the given rational is in its simplest form. ∴ (23 × 52 × 32) ≠ (2m × 5n) for some integers m, n. This rational...
If a = (22 × 33 × 54) and b = (23 × 32 × 5), then HCF (a, b) = ? (a) 90 (b) 180 (c) 360 (d) 540
Correct Answer: (b) 180 Explanation: Given, a = (22 × 33 × 54) b = (23 × 32 × 5) HCF (a,b) = 22× 32 × 5 HCF (a,b) = 180
HCF of (23 × 32 × 5), (22 × 33 × 52) and (24 × 3 × 53 × 7) is (a) 30 (b) 48 (c) 60 (d) 105
Correct Answer: (c) 60 Explanation: HCF = 22 × 3 × 5 HCF = 60
LCM of (23 × 3 × 5) and (24 × 5 × 7) is (a) 40 (b) 560 (c) 1120 (d) 1680
Correct Answer: (c) 1680 Explanation: LCM = 24 × 3 × 5 × 7 LCM = 16 × 3 × 5 × 7 LCM = 1680
The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number? (a) 36 (b) 45 (c) 9 (d) 81
Correct Answer: (d) 81 Explanation: Let the two numbers be x and y. Given, x = 54 HCF = 27 LCM = 162 x × y = HCF × LCM 54 × y = 27 × 162 54 y = 4374 y = 81
The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is (a) 8000 (b) 1600 (c) 320 (d) 1605
Correct Answer: (c) 320 Explanation: Let the two numbers be x and y. Given, x × y = 1600 HCF = 5 HCF × LCM = x × y 5 × LCM = 1600 LCM = 320
What is the largest number that divided each one of the 1152 and 1664 exactly? (a) 32 (b) 64 (c) 128 (d) 256
Correct Answer: (c) 128 Explanation: Largest number that divides each one of 1152 and 1664 = HCF (1152, 1664) HCF = 27 HCF = 128
What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively? (a) 13 (b) 9 (c) 3 (d) 585
Correct Answer: (a) 13 Explanation: The number divides 65 (70 – 5) and 117 (125 – 8) is HCF (65, 117) 65 = 13 × 5 117 = 13 × 3 × 3 ∴ HCF = 13
What is the largest number that divides 245 and 1029, leaving remainder 5 in each case? (a) 15 (b) 16 (c) 9 (d) 5
Correct Answer: (b) 16 Explanation: The number divides 240 (245 – 5) and 1024 (1029 – 5) is HCF (240, 1024) 240 = 2 × 2 × 2 × 2 × 3 × 5 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 ∴ HCF = 2 × 2 × 2...
The simplest form of $\frac{1095}{1168}(a)\frac{17}{26}(b)\frac{25}{26}(c)\frac{13}{16}(d)\frac{15}{16}$
Correct Answer: Option (d) Explanation: HCF of 1095 and 1168 = 73.
Euclid’s division lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy (a) 1 ˂ r ˂ v (b) 0 ˂ r ≤ b (c) 0 ≤ r ˂ b (d) 0 ˂ r ˂ b
Correct Answer: (c) 0 ≤ r ˂ b Explanation: Euclid’s division lemma, states that for any positive integers a and b, there exist unique integers q and r, such that a = bq + r where r must satisfy 0 ≤...
A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13? (a) 0 (b) 1 (c) 3 (d) 5
Correct Answer: (d) 5 Explanation: Dividend = Divisor × Quotient + Remainder. Given, Divisor = 143 Remainder = 13 The number is in the form of 143x + 31, where x is the quotient. ∴ 143x + 31 = 13...
Which of the following is an irrational number? $(a)\frac{22}{7}(b)3.1416(c)3.\overline{1416}(d)3.141141114….$
Correct Answer: (d) 3.141141114….. Explanation: 3.141141114 is an irrational number because it is a non-repeating and non-terminating decimal.
π is (a) an integer (b) a rational number (c) an irrational number (d) none of these
Correct Answer: (c) π is an irrational number Explanation: π is an irrational number because it is a non-repeating and non-terminating decimal.
$2.\overline{35}$ is (a) an integer (b) a rational number (c) an irrational number (d) none of these
Correct Answer: (b) $2.\overline{35}$ is a rational number Explanation: $2.\overline{35}$ is a rational number because it is a repeating decimal.
2.13113111311113…… is (a) an integer (b) a rational number (c) an irrational number (d) none of these
Correct Answer: (c) an irrational number Explanation: It is an irrational number because it is a non-terminating and non-repeating decimal.
$1.23\overline{48}$ is (a) an integer (b) a rational number (c) an irrational number (d) none of these
Correct Answer: (b) a rational number Explanation: $1.23\overline{48}$ is a rational number because it is a repeating decimal.
Which of the following rational numbers is expressible as a terminating decimal? $(a)\frac{124}{165}(b)\frac{131}{30}(c)\frac{2027}{625}(d)\frac{1625}{462}$
Correct Answer: Option (c) Explanation: \frac{2027}{625} = 3.2432 is a terminating decimal.
The decimal expansion of the rational number $\frac{37}{{{2}^{5}}\times 5}$ will terminate after (a) one decimal place (b) two decimal places (c) three decimal places (d) four decimal places
Correct Answer: (b) two decimal places Explanation: 1.85 is the decimal expansion of the rational number terminates after two decimal places.
The decimal expansion of the number $\frac{14753}{1250}$ will terminate after (a) one decimal place (b) two decimal places (c) three decimal places (d) four decimal places
Correct Answer: (d) four decimal places Explanation: 11.8024 is the decimal expansion of the number will terminate after four decimal places.
The number 1.732 is (a) an integer (b) a rational number (c) an irrational number (d) none of these
Correct Answer: (b) a rational number Explanation: 1.732 is a terminating decimal.
If a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a + b) is (a) 2 (b) 3 (c) 5 (d) 8
Correct Answer: (a) 2 Explanation: 5 + 3 = 8, the least prime factor of a + b has to be 2, unless a + b is a prime number greater than 2.
√2 is (a) an integer (b) an irrational number (c) a rational number (d) none of these
Correct Answer: (b) an irrational number Explanation: √2 is an irrational number.
$\frac{1}{\sqrt{2}}$ is (a) a fraction (b) a rational number (c) an irrational number (d) none of these
Correct Answer: (c) an irrational number Explanation: $\frac{1}{\sqrt{2}}$ is an irrational number.
(2 + √2) is (a) an integer (b) a rational number (c) an irrational number (d) none of these
Correct Answer: (c) an irrational number Explanation: 2 + √2 is an irrational number. if it is rational, then the difference of two rational is rational. (2 + √2) – 2 = √2 is...
What is the least number that is divisible by all the natural numbers from 1 to 10 (both inclusive)?(a)10 (b)100 (c)504 (d)2520
Correct Answer: (c) 2520 Explanation: The least number that is divisible by all numbers from 1 to 10. LCM (1 to 10) = 23 × 32 × 5 × 7 LCM = 2520 Hence, 2520 is the least number that is divisible by...