The range of the relation $R=\{(x, x^2) \mid x$ is a prime number less than $13\}$ is :
a.
$\{2, 3, 5, 7\}$
b.
$\{2, 3, 5, 7, 11\}$
c.
$\{4, 9, 25, 49, 121\}$
Correct Answer
d.
$\{1, 4, 9, 25, 49, 121\}$
Step-by-Step AI Explanation
Verified Method
The prime numbers less than 13 are 2, 3, 5, 7, and 11. For the relation R = {(x, x^2)}, the range is the set of second elements, เคฏเคพเคจเฅ {x^2}. So the range is {4, 9, 25, 49, 121}. Hence option (c) is correct. Options (a), (b), and (d) list the first elements or include 1, which is not a prime number.
Q2mcq1 mark
Let $n(A)=m$ and $n(B)=n$, then the total number of functions that exist from $B$ to $A$ is :
a.
$m^n$
Correct Answer
b.
$n^m$
c.
$2^{mn-1}$
d.
$2^{mn}$
Step-by-Step AI Explanation
Verified Method
If set B has n elements and set A has m elements, then each element of B can be mapped to any one of the m elements of A. So for each of the n elements, there are m choices. Therefore, the total number of functions from B to A is $m^n$. Hence option (a) is correct.
Q3mcq1 mark
Euclid's division lemma states that for positive integers $a$ and $b$, there exist unique positive integers $q$ and $r$ such that $a=bq+r$, where $r$ must satisfy.
a.
$1<r<b$
b.
$0<r<b$
c.
$0\le r<b$
Correct Answer
d.
$0<r\le b$
Step-by-Step AI Explanation
Verified Method
By Euclid's division lemma, for positive integers a and b, there exist unique integers q and r such that $a = bq + r$, where the remainder r satisfies $0 \le r < b$. Hence option (c) is correct. The other options are incorrect because the remainder can be 0 and must always be less than b.
Q4mcq1 mark
If 6 times of $6^{\text{th}}$ term of an A.P. is equal to 7 times the $7^{\text{th}}$ term, then the $13^{\text{th}}$ term of the A.P. is :
a.
0
Correct Answer
b.
6
c.
7
d.
13
Step-by-Step AI Explanation
Verified Method
Let the nth term of the A.P. be $T_n = a + (n-1)d$. Then
$T_6 = a + 5d$ and $T_7 = a + 6d$.
Given that 6 times the 6th term is equal to 7 times the 7th term:
$6(a+5d) = 7(a+6d)$
$6a + 30d = 7a + 42d$
$a + 12d = 0$.
Now the 13th term is
$T_{13} = a + 12d = 0$.
So the correct answer is (a) 0.
Q5mcq1 mark
If $(x-6)$ is the H.C.F. of $x^2-2x-24$ and $x^2-kx-6$, then the value of $k$ is :
a.
3
b.
5
c.
6
d.
8
Correct Answer
Step-by-Step AI Explanation
Verified Method
Since $(x-6)$ is the H.C.F. of the two polynomials, it must be a common factor of both.
First polynomial:
$x^2 - 2x - 24 = (x-6)(x+4)$
For the second polynomial $x^2 - kx - 6$ to have factor $(x-6)$, substitute $x=6$:
$6^2 - 6k - 6 = 0$
$36 - 6k - 6 = 0$
$30 - 6k = 0$
$k = 5$
So the correct answer should be 5. However, among the given options, this corresponds to option (b), not (d).
Q6mcq1 mark
Transpose of a column matrix is :
a.
unit matrix
b.
diagonal matrix
c.
column matrix
d.
row matrix
Correct Answer
Step-by-Step AI Explanation
Verified Method
A column matrix has only one column. Its transpose changes rows into columns and columns into rows, so the transpose of a column matrix is a row matrix. Hence option (d) is correct.
Q7mcq1 mark
A tangent is perpendicular to the radius of a circle at the :
a.
Centre
b.
Point of Contact
Correct Answer
c.
Infinity
d.
Chord
Step-by-Step AI Explanation
Verified Method
A tangent to a circle is perpendicular to the radius drawn to the point where it touches the circle. This point is called the point of contact. Hence option (b) is correct.
Q8mcq1 mark
The equation of a line passing through the origin and parallel to the line $7x-3y+14=0$ is :
a.
$7x-3y+14=0$
b.
$3x-7y+14=0$
c.
$3x+17y=0$
d.
$7x-3y=0$
Correct Answer
Step-by-Step AI Explanation
Verified Method
The given line is $7x - 3y + 14 = 0$. A line parallel to it must have the same coefficients of x and y, so its general form is $7x - 3y + c = 0$. Since it passes through the origin, substitute $(0,0)$:
$7(0) - 3(0) + c = 0 \Rightarrow c = 0$.
Therefore, the equation is $7x - 3y = 0$.
So option (d) is correct.
Q9mcq1 mark
The slope of the line which is perpendicular to a line joining the points $(0, 0)$ and $(-8, 8)$ is :
a.
-1
b.
1
Correct Answer
c.
$\frac{1}{3}$
d.
-8
Step-by-Step AI Explanation
Verified Method
The line joining $(0,0)$ and $(-8,8)$ has slope
$$m=\frac{8-0}{-8-0}=-1.$$
A line perpendicular to it has slope equal to the negative reciprocal, so
$$m_\perp=1.$$
Hence the correct choice is **(b) 1**. The other options are not the negative reciprocal of $-1$.
Q10mcq1 mark
If the ratio of the height of a tower and the length of its shadow is $3:1$, then the angle of elevation of the sun has measure :
a.
45$^{\circ}$
b.
30$^{\circ}$
c.
90$^{\circ}$
d.
60$^{\circ}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
If the height of the tower is $3$ and the shadow length is $1$, then
$$\tan\theta=\frac{\text{height}}{\text{shadow}}=\frac{3}{1}=3.$$
The angle whose tangent is $3$ is approximately $71.6^\circ$, which is not among the given options. However, in such board MCQs, the intended ratio for standard angles is often interpreted from common values; among the options, $60^\circ$ corresponds to \(\tan 60^\circ=\sqrt{3}\), so none matches exactly. Since the question asks for a single correct choice from the listed options, the expected answer is **(d) $60^\circ$** only if the ratio were meant to be $\sqrt{3}:1$. As written, the data does not match any option exactly.
🎉 10 Free Sample Solutions Completed
Unlock All 38 Questions & Solutions
Sign up free in seconds to unlock step-by-step AI answers for the remaining 28 questions.
A purse contains 10 notes of $2000, 15 notes of $500 and 25 notes of $200. One note is drawn at random. What is the probability that the note is either a $500 note or $200 note ?
a.
$\frac{1}{5}$
b.
$\frac{3}{10}$
c.
$\frac{2}{3}$
d.
$\frac{4}{5}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
From the top of a rock $50\sqrt{3}$ m high, the angle of depression of a car on the ground is observed to be $30^\circ$. Find the distance of the car from the rock.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Find the range of the following distribution:
Age (in years): 16 - 18, 18 - 20, 20 - 22, 22 - 24, 24 - 26, 26 - 28
Number of Students: 0, 4, 6, 8, 2, 2
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A flower is selected at random from a basket containing 80 yellow, 70 red and 50 white flowers. Find the probability of selecting a yellow coloured flower.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Let $A$ be the set of all natural numbers less than 8, $B$ the set of all prime numbers less than 8 and $C$ the set of even prime numbers. Verify that $(A\cap B)\times C=(A\times C)\cap(B\times C)$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Let $f : A \to B$ be a function defined by $f(x)=\frac{1}{2}x-1$. Here, $A=\{2,4,6,10,12\}$, $B=\{0,1,2,4,5,9\}$. Represent $f$ by (i) set of ordered pairs (ii) a table (iii) an arrow diagram (iv) a graph.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
If $p_1^{31} \times p_2^2 \times p_3^4 \times p_4^{113400} = x_1 \times x_2 \times x_3 \times x_4$, where $p_1, p_2, p_3, p_4$ are primes in ascending order and $x_1, x_2, x_3, x_4$ are integers, find the value of $p_1, p_2, p_3, p_4$ and $x_1, x_2, x_3, x_4$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Find the value of $k$, if the area of a quadrilateral is 28 sq. units, whose vertices are taken in the order $(-4,-2)$, $(-3,k)$, $(3,-2)$ and $(2,3)$.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Nathan, an engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of the model that Nathan made.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
The total marks scored by two students, Sathya and Vidhya in 5 subjects are 460 and 480 with standard deviation 4.6 and 2.4 respectively. Who is more consistent in performance?
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A bag contains 5 red balls, 6 white balls, 7 green balls, 8 black balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is
a.
(i) white
b.
(ii) black or red
c.
(iii) not white
d.
(iv) neither white nor black
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
(A) If $BC=8$ cm, $\angle A=60^\circ$ and the perpendicular bisector of $\angle A$ meets $BC$ at $D$ such that $BD=6$ cm, construct triangle $ABC$. Or (B) Draw a circle of radius 4 cm and mark a point at a distance of 11 cm from its center. Draw the two tangents from that point to the circle.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
(a)Draw a triangle ABC of base BC=8 cm, โ A=608 and the bisector of โ A meets BC at D such that BD=6 cm. OR (b)Take a point which is 11 cm away from the centre of a circle of radius 4 cm and draw the two tangents to the circle from that point.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
(a)Graph the quadratic equation $x^2-9x120=0$ and state the nature of solution. OR (b)A bus is travelling at a uniform speed of 50 km/hr. Draw the distance - time graph and hence find. (i)the constant of variation. (ii)how far will it travel in 90 minutes ? (iii)the time required to cover a distance of 300 km.
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
