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Q9
mcq
1 mark
The slope of the straight line perpendicular to x-axis is :
Correct Answer: C —
Solution
The line perpendicular to the x-axis is a vertical line. A vertical line has undefined slope, often written as $\infty$ in such school-level MCQs. So the correct choice is (c) $\infty$. The other options are not correct because slope $0$ is for a line parallel to the x-axis, and $1$ or $-1$ are slopes of slanting lines.
Q13
mcq
If the sum and mean of a data are 407 and 11 respectively, then the number of observations in the data are:
-
A.
37
-
B.
4477
-
C.
396
-
D.
418
Correct Answer: A —
Solution
Mean = $\dfrac{\text{sum}}{\text{number of observations}}$.
Given sum = 407 and mean = 11,
$$11 = \frac{407}{n}$$
So,
$$n = \frac{407}{11} = 37$$
Hence, the number of observations is 37. Therefore, the correct option is (a).
Q14
mcq
If a letter is chosen at random from the English alphabets {a, b, ..., z}, then the probability that the letter chosen precedes x:
-
A.
\(\frac{12}{13}\)
-
B.
\(\frac{1}{13}\)
-
C.
\(\frac{23}{26}\)
-
D.
\(\frac{3}{26}\)
Correct Answer: B —
Solution
There are 26 English letters in total: $a, b, c, \dots, z$.
Letters that precede $x$ are $a$ through $w$, which is 23 letters.
So the probability is
$$\frac{23}{26}$$
Wait: the question asks for the probability that the chosen letter precedes $x$, which means any letter before $x$ in the alphabet. That is 23 out of 26.
Therefore the correct answer is (c) $\frac{23}{26}$.
Q15
short answer
If $B \times A = \{(-2, 3), (-2, 4), (0, 3), (0, 4), (3, 3), (3, 4)\}$, find $A$ and $B$.
Solution
Given
$$B \times A = \{(-2,3),(-2,4),(0,3),(0,4),(3,3),(3,4)\}$$
In a Cartesian product $B \times A$, the first components come from $B$ and the second components come from $A$.
So,
$$B = \{-2,0,3\}$$
and
$$A = \{3,4\}$$
Hence, $A = \{3,4\}$ and $B = \{-2,0,3\}$.
Q16
short answer
Find $k$ if $f(k)=5$ where $f(k)=2k-1$.
Solution
Given $f(k)=2k-1$ and $f(k)=5$.
So,
$$2k-1=5$$
$$2k=6$$
$$k=3$$
Therefore, $k=3$.
Q17
short answer
Find $x$ so that $x^{16}$, $x^{112}$ and $x^{115}$ are consecutive terms of a Geometric Progression.
Q18
short answer
Simplify:
$$\frac{2x^2+2x-6}{4x^2-12y}$$
Q19
short answer
Determine the nature of roots for the following quadratic equation:
$$2x^2-x-1=0$$
Q20
short answer
In the figure AD is the bisector of $\angle BAC$, if $AB=10$ cm, $AC=14$ cm and $BC=6$ cm. Find $BD$ and $DC$.
Q21
short answer
A cat is located at the point $(-6,-4)$ in $xy$ plane. A bottle of milk is kept at $(5,11)$. The cat wishes to consume the milk travelling through shortest possible distance. Find the equation of the path it needs to take the milk.
Q22
short answer
If the straight lines $12y=-(P13)x112$, $12x-7y=16$ are perpendicular then find $P$.
Q23
short answer
Prove that
$$\frac{\sec \theta - \sin \theta}{\cot \theta} = \frac{\sin \theta}{\sin \theta \cos \theta}$$
Q24
short answer
The radius of a conical tent is 7 m and height is 24 m. Calculate the length of the canvas used to make the tent if the width of the rectangular canvas is 4 m.
Q25
short answer
If the ratio of radii of two spheres is 4 : 7, find the ratio of their volumes.
Q26
short answer
Find the range and co-efficient of range of the following data:
63, 89, 98, 125, 79, 108, 117, 68
Q27
short answer
A and B are two candidates seeking admission to IIT. The probability that A getting selected is 0.5 and the probability that both A and B getting selected is 0.3. Prove that probability of B being selected is at the most 0.8.
Q28
short answer
If $p^2 \times q^1 \times r^4 \times s^3 = 3,15,000$ then find $p, q, r$ and $s$.
Q29
long answer
Let $f : A \to B$ be a function defined by $f(x)=\frac{1}{2}x$. Where $A=\{2,4,6,10,12\}$, $B=\{0,1,2,4,5,9\}$. Represent $f$ by:
(i) set of ordered pairs
(ii) a table
(iii) an arrow diagram
(iv) a graph
Q30
short answer
The houses of a street are numbered from 1 to 49. Senthil’s house is numbered such that the sum of numbers of the houses prior to Senthil’s house is equal to the sum of the numbers of the houses following Senthil’s house. Find Senthil’s house number.
Q31
short answer
Find the sum to $n$ terms of the series 515515551.....
Q41
short answer
Two unbiased dice are rolled once. Find the probability of getting: (i) a doublet (equal numbers on both dice) (ii) the product as a prime number (iii) the sum as a prime number (iv) the sum as 1
Q42
short answer
Let A={x∈W/x < 3}, B={x∈N/1 < x ≤ 5} and C={3, 5, 7} verify that A×(B∪C)=(A×B)∪(A×C).
