M.C.Q (1 Marks) - Maths STD 10 Questions - Vidyadip

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M.C.Q (1 Marks)

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18 questions · timed · auto-graded

MCQ 11 Mark

If the difference of mode and median of a data is $24,$ then the difference of median and mean of the same data is:

A
$8$
✓
$12$
C
$34$
D
$24$

Answer

Correct option: B.

$12$

Given,
$\text { mode }- \text { median }=24$
$\text { median }- \text { mean }=\text { ? }$
$\text { we know that, }$
$\text { mode }=3 \text { median }-2 \text { mean }$
$\text { mode }=\text { median }+2 \text { median }-2 \text { mean }$
$\text { mode }- \text { median }=2 \text { median }-2 \text { mean }$
$24=2 \text { (median }- \text { mean })$
$\text { median }- \text { mean }=\frac{24}{2}=12$

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MCQ 21 Mark

A number is chosen from the numbers $1, 2, 3$ and denoted as $x,$ and a number is chosen from the numbers $1, 4, 9$ and denoted as $y.$ Then $P(xy < 9)$ is:

A
$\frac{7}{9}$
✓
$\frac{5}{9}$
C
$\frac{3}{9}$
D
$\frac{1}{9}$

Answer

Correct option: B.

$\frac{5}{9}$

Numbers $x =1,2,3$ and $y =1,4,9$
Now $x y=\{1,4,9,2,8,18,3,12,27\}=9$
$\therefore n=9$
and $xy <9$ are $1,2,3,4,8$
$\therefore m=5$
$\therefore P(xy<9)=\frac{5}{9}$

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MCQ 31 Mark

A bag contains $100$ cards numbered $1$ to $100.$ A card is drawn at random from the bag. What is the probability that the number on the card is a perfect cube?

A
$\frac{1}{20}$
✓
$\frac{1}{25}$
C
$\frac{7}{100}$
D
$\frac{3}{50}$

Answer

Correct option: B.

$\frac{1}{25}$

$n(S)=100$
$E=\{1,8,27,64\}$
$n(E)=4$
the probability of drawing a number on the card that is a cube is
$P(E)=\frac{4}{100}=\frac{1}{25}$

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MCQ 41 Mark

Pankaj has a motorcycle with wheels of diameter 91 cm. There are 22 spokes in the wheel. Find the length of arc between two adjoining spokes.

✓
13 cm
B
26 cm
C
15 cm
D
18 cm

Answer

Correct option: A.

13 cm

(a) 13 cm
Explanation:Radius of wheel $=\frac{91}{2} cm$
Angle between two adjoining spokes, $\theta=\frac{360^{\circ}}{22}$
$\therefore$ Length of arc $=\frac{\theta}{360^{\circ}} \times 2 \pi r$
$=\frac{360^{\circ}}{360^{\circ} \times 22} \times 2 \times \frac{22}{7} \times \frac{91}{2}=13 cm$

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MCQ 51 Mark

The area of the sector of a circle with radius $6 \ cm$ which subtends an angle of $60^{\circ}$ at the centre of the circle is:

A
$\frac{152}{7} \ cm^2$
✓
$\frac{132}{7} \ cm^2$
C
$\frac{142}{7} \ cm^2$
D
$\frac{122}{7} \ cm^2$

Answer

Correct option: B.

$\frac{132}{7} \ cm^2$

Angle of the sector is $60^{\circ}$
Area of sector $=\left(\frac{\theta}{360^{\circ}}\right) \times \pi r ^2$
$\therefore$ Area of the sector with angle $60^{\circ}=\left(\frac{60^{\circ}}{360^{\circ}}\right) \times \pi r ^2 \ cm^2$
$=\left(\frac{36}{6}\right) \pi \ cm^2$
$=6 \times\left(\frac{22}{7}\right) \ cm^2$
$=\frac{132}{7} \ cm^2$

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MCQ 61 Mark

A ladder $15 m$ long just reaches the top of a vertical wall. If the ladder makes an angle of $60^{\circ}$ with the wall, then the height of the wall is

A
$15 \sqrt{3} m$
B
$15 m$
C
$\frac{15 \sqrt{3}}{2} m$
✓
$\frac{15}{2} m$

Answer

Correct option: D.

$\frac{15}{2} m$

Let $AB$ be the ladder and $BC$ be the wall Then, $\angle \text{ABC} =60^{\circ}$
$\Rightarrow \angle \text{CAB} =\left(90^{\circ}-60^{\circ}\right)=30^{\circ}$
Let $BC = h m$. then,
$\frac{B C}{A B}=\sin 30^{\circ}$
$\Rightarrow \frac{h}{15}=\frac{1}{2}$
$\Rightarrow h=\frac{15}{2}$

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MCQ 71 Mark

$\left(3 \sin ^2 30^{\circ}-4 \cos ^2 60^{\circ}\right)$ is equal to:

✓
$-\frac{1}{4}$
B
$\frac{5}{4}$
C
$-\frac{3}{4}$
D
$-\frac{9}{4}$

Answer

Correct option: A.

$-\frac{1}{4}$

$\left(3 \sin ^2 30^{\circ}-4 \cos ^2 60^{\circ}\right)$
$\Rightarrow 3 \times\left(\frac{1}{2}\right)^2-4 \times\left(\frac{1}{2}\right)^2$
$\Rightarrow-\frac{1}{4}$

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MCQ 81 Mark

The value of $\frac{\sin 90^{\circ}+\cos 60^{\circ}}{\sec 45^{\circ}+\tan 45^{\circ}}$ is:

A
$\frac{3}{2}(\sqrt{2}-1)$
B
$\frac{1+\sqrt{3}}{\sqrt{2}+1}$
✓
$\frac{3}{2}(\sqrt{2}+1)$
D
$1$

Answer

Correct option: C.

$\frac{3}{2}(\sqrt{2}+1)$

$=\frac{\sin 90^{\circ}+\cos 60^{\circ}}{\sec 45^{\circ}+\tan 45^{\circ}}$
$=\frac{1+\frac{1}{2}}{\sqrt{2}+1}$
$=\frac{\frac{3}{2}}{\sqrt{2}+1}$
upon rationalization
$=\frac{3}{2} \times \frac{1}{\sqrt{2}+1} \times \frac{\sqrt{2}-1}{\sqrt{2}-1}$
$=\frac{3}{2} \times \frac{\sqrt{2}-1}{\sqrt{2}-1}$
$=\frac{3}{2}(\sqrt{2}-1)$

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MCQ 91 Mark

From an external point $Q,$ the length of tangent to a circle is $12 \ cm$ and the distance of $Q$ from the centre of circle is $13 \ cm.$ The radius of circle $($ in $cm)$ is

A
$10$
✓
$5$
C
$7$
D
$12$

Answer

Correct option: B.

$5$

$\because$ We know that $\ce{PQ \perp OP}$
$\therefore \triangle \text{QPO}$ is right angled $\triangle$
$\therefore$ By python theory.
$\ce{QO^2=QP^2 + OP^2}$
$(13)^2=(12)^2 + OP^2$
$r^2=169-144$
$r^2=25$
$r=5 \ cm$

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MCQ 101 Mark

In the given figure $ ,PA$ and $PB$ are tangents from external point $P$ to a circle with centre $C$ and $Q$ is any point on the circle. Then the measure of $\angle AQB$ is

A
$90^{\circ}$
B
$125^{\circ}$
✓
$62 \frac{1}{2}^{\circ}$
D
$55^{\circ}$

Answer

Correct option: C.

$62 \frac{1}{2}^{\circ}$

Explanation:
Given, $\angle APB =55^{\circ}$
$\therefore \angle ACB =180^{\circ}-55^{\circ}=125^{\circ} \ \ (\because \angle APB$ and $\angle ACB$ are supplementary angles $)$
Now, as we know that
Angle subtended by an arc at the centre $=2 \ \times$ angle subtended by arc at any point on the remaining part of the circle
$\therefore 125^{\circ}=2 \times \angle AQB$
$\Rightarrow \angle AQB=\frac{125}{2}$
$=62.5^{\circ} $ or $62 \frac{1}{2}^{\circ}$

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MCQ 111 Mark

In the given figure, $\ce{AB \| PQ.}$ If $AB = 6 \ cm, PQ = 2 \ cm$ and $OB = 3 \ cm,$ then the length of $OP$ is:

✓
$1 \ cm$
B
$4 \ cm$
C
$9 \ cm$
D
$3 \ cm$

Answer

Correct option: A.

$1 \ cm$

In $\triangle \text{ABO}$ and $\triangle \text{QPO}$
$\angle \text{BAO}=\angle \text{PQO} ($ by alt. angle $)$
$\angle \text{AOB}=\angle \text{QOP} ($ vert. oppo. angle $)$
$\therefore \triangle \text{ABO} \sim \triangle \text{QPO} ($ by Similarity $)$
$\therefore \frac{AB}{QP}=\frac{OB}{OP}$
$\frac{6}{2}=\frac{3}{OP}$
$OP=1 \ cm$

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MCQ 121 Mark

The mid$-$point of the line$-$segment $AB$ is $P(0, 4).$ If the coordinates of $B$ are $(-2, 3)$ then the co$-$ordinates of $A$ are

A
$(2, 9)$
B
$(-2, -5)$
✓
$(2, 5)$
D
$(-2, 11)$

Answer

Correct option: C.

$(2, 5)$

Let cordinate of $A ( x , y )$
Then cordinate of mid point are $\left[\frac{(x-2)}{2}, \frac{(y+3)}{2}\right]$
On comparing the cordinates of mid points
$\frac{(x-2)}{2}=0$
$x=2$
$\frac{(y+3)}{2}=4$
$y = 5$
Cordinates of $A$ are $(2, 5).$

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MCQ 131 Mark

The diameter of a circle is of length $6 \ cm.$ If one end of the diameter is $(-4, 0),$ the other end on $x-$axis is at:

A
$(6, 0)$
B
$(0, 2)$
C
$(4, 0)$
✓
$(2, 0)$

Answer

Correct option: D.

$(2, 0)$

Let the other point on $x$ axis is $( x , 0)$
By distance formula
$\sqrt{(x+4)^2+(0-0)^2}=(6)$
$x+4=6$
$x=2$
Hence the point is $(2,0)$

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MCQ 141 Mark

How many terms are there in the $A.P.$ given below? $14,19,24,29,........ 119$

✓
$22$
B
$21$
C
$18$
D
$14$

Answer

Correct option: A.

$22$

Given, $a _1=14, t _{ n }=119$
$d=a_2-a_1=19-14=5$
$t_n=a_1+(n-1) d$
$119=14+(n-1) 5$
$119-14=5 n-5$
$105+5=5 n$
$110=5 n$
$n=22$

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MCQ 151 Mark

The value of $\lambda$ for which $\left(x^2+4 x+\lambda\right)$ is a perfect square, is

A
$1$
B
$16$
✓
$4$
D
$9$

Answer

Correct option: C.

$4$

For a quadratic equation to be a perfect square
$D=0$
$b^2-4 a c=0$
$(4)^2-4(1)(\lambda)=0$
$16-4 \lambda=0$
$16=4 \lambda$
$\lambda=4$

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MCQ 161 Mark

A system of linear equations is said to be inconsistent if it has

A
one solution
B
at least one solution
C
two solutions
✓
no solution

Answer

Correct option: D.

no solution

(d) no solution
Explanation: A system of linear equations is said to be inconsistent if it has no solution means two lines are running parallel
and not cutting each other at any point.

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MCQ 171 Mark

In the figure, the graph of the polynomial p(x) is given. The number of zeroes of the polynomial is:

✓
$2$
B
$1$
C
$0$
D
$3$

Answer

Correct option: A.

$2$

(a) 2
Explanation: 2
The number of zeroes is 2 as the graph does cut the x-axis 2 times.

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MCQ 181 Mark

The total number of factors of a prime number is:

✓
$2$
B
$1$
C
$3$
D
$0$

Answer

Correct option: A.

$2$

(a) 2
Explanation: The total number of factors of a prime number $=2$ i.e. 1 and itself

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M.C.Q (1 Marks) - Maths STD 10 Questions - Vidyadip