The product of HCF and LCM of two numbers 15 and 20 is
a.
15
b.
20
c.
300
Correct Answer
d.
35
Step-by-Step AI Explanation
Verified Method
For any two positive integers a and b, the relationship is HCF(a, b) * LCM(a, b) = a * b. Here, 15 * 20 = 300.
Q2mcq1 mark
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = ax^2 + bx + c$, then $\alpha\beta$ is
a.
$\frac{b}{a}$
b.
$-\frac{b}{a}$
c.
$-\frac{c}{a}$
d.
$\frac{c}{a}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
According to the relationship between zeroes and coefficients of a quadratic polynomial ax^2 + bx + c, the product of the zeroes is given by the constant term divided by the coefficient of x^2, which is c/a.
Q3mcq1 mark
If $\sin \theta = \frac{4}{5}$, then the value of $\sqrt{1 - \cos^2 \theta}$ is
a.
$\frac{16}{25}$
b.
$\frac{4}{5}$
Correct Answer
c.
$\frac{5}{4}$
d.
$\frac{9}{25}$
Step-by-Step AI Explanation
Verified Method
Using the identity sin^2 ΞΈ + cos^2 ΞΈ = 1, we know that sqrt(1 - cos^2 ΞΈ) = sqrt(sin^2 ΞΈ) = sin ΞΈ. Since sin ΞΈ = 4/5, the value is 4/5.
Q4mcq1 mark
The probability of a sure event is
a.
1
Correct Answer
b.
0
c.
$-1$
d.
1.5
Step-by-Step AI Explanation
Verified Method
By the definition of probability, an event that is certain to occur (a sure event) has a probability of 1.
Q5mcq1 mark
The secant of the circle in the figure, is
a.
MN
Correct Answer
b.
OE
c.
CD
d.
AB
Step-by-Step AI Explanation
Verified Method
A secant of a circle is a line that intersects the circle at two distinct points. Looking at the figure, line MN passes through the circle at two points, making it a secant line.
Q6mcq1 mark
The volume of the frustum of a cone whose base radii are $r_1$ and $r_2$ and height 'h', is
$\frac{1}{3} \pi h (r_1^2 + r_2^2 - r_1 \cdot r_2)$
c.
$\frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 \cdot r_2)$
d.
$\frac{1}{3} \pi h (r_1^2 - r_2^2 - r_1 \cdot r_2)$
Step-by-Step AI Explanation
Verified Method
The standard formula for the volume of a frustum of a cone with radii r1 and r2 and vertical height h is V = (1/3)Οh(r1^2 + r2^2 + r1*r2).
Q7mcq1 mark
If 2, x, 26 are in Arithmetic progression, then the value of x is
a.
12
b.
14
Correct Answer
c.
28
d.
24
Step-by-Step AI Explanation
Verified Method
In an Arithmetic progression (AP), if a, x, b are in AP, then x is the arithmetic mean: x = (a + b) / 2. Here, x = (2 + 26) / 2 = 28 / 2 = 14.
Q8mcq1 mark
If $\tan(90^{\circ} - \theta) = \sqrt{3}$, then the value of $\cot \theta$ is
a.
$\frac{1}{\sqrt{3}}$
b.
1
c.
0
d.
$\sqrt{3}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
We know that tan(90Β° - ΞΈ) = cot ΞΈ. Since we are given tan(90Β° - ΞΈ) = sqrt(3), it follows directly that cot ΞΈ = sqrt(3).
Q9short answer1 mark
In the figure, $\Delta ADE \sim \Delta ABC$ and $DE : BC = 2 : 3$. Find $\frac{\text{Area of } \Delta ADE}{\text{Area of } \Delta ABC}$.
Step-by-Step AI Explanation
Verified Method
According to the theorem on the areas of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Since $\Delta ADE \sim \Delta ABC$ and $\frac{DE}{BC} = \frac{2}{3}$, we have: $$\frac{\text{Area of } \Delta ADE}{\text{Area of } \Delta ABC} = \left(\frac{DE}{BC}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$$
Q10short answer1 mark
The radii of the base and the height of a cylinder and a cone are same. If the volume of the cylinder is 27 cubic units, then find the volume of the cone.
Step-by-Step AI Explanation
Verified Method
The volume of a cylinder is $V_{cyl} = \pi r^2 h = 27$. The volume of a cone is $V_{cone} = \frac{1}{3} \pi r^2 h$. Substituting $V_{cyl}$ into the cone formula, we get $V_{cone} = \frac{1}{3} (27) = 9$ cubic units.
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In an Arithmetic progression, sum of the first six terms and sum of the first five terms are 78 and 55 respectively. Then find the sixth term of the progression.
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Find the coordinates of the point P and Q in the given graph and hence find the length of PQ using distance formula.
OR
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A basket contains 36 mangoes. $\frac{1}{4}$th of them are rotten and others are good. If one mango is drawn at random from the basket, then find the probability of getting a good mango.
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Divide $p(x) = x^3 + 3x^2 + 4x + 5$ by $g(x) = x^2 - x + 1$ and find the quotient $[q(x)]$ and remainder $[r(x)]$.
OR
When the polynomial $p(x) = x^3 + 4x^2 + 5x - 2$ is divided by the polynomial $g(x)$, the quotient $[q(x)]$ and remainder $[r(x)]$ are $x^2 - x + 2$ and 4 respectively. Find $g(x)$.
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Find the mean for the following data:
Class-interval: 2-6, 7-11, 12-16, 17-21, 22-26
Frequency: 2, 4, 5, 3, 1
OR
Find the mode for the following data:
Class-interval: 1-5, 5-9, 9-13, 13-17, 17-21
Frequency: 1, 3, 7, 10, 9
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'D' is a point on the side BC of a $\Delta ABC$ such that $\angle BAC = \angle ADC$. Then prove that $AC^2 = BC \cdot CD$.
OR
In the figure, $\Delta ABC$ and $\Delta AMP$ are right angled triangles, right angled at $B$ and $M$ respectively. Then prove that $\frac{CA}{PA} = \frac{BC}{MP}$.
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In the figure area of sector AOBPA of radius 'r' is $231 \text{ cm}^2$ and the length of the arc APB is 22 cm. Find the radius of the sector and angle $\theta$.
OR
In the figure a rectangle ROQP is inscribed in the quadrant of a circle. If the length and breadth of the rectangle are 16 cm and 12 cm respectively, find the area of the shaded region.
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Age of mother is twice the square of age of her son. After 8 years motherβs age becomes 4 years more than the thrice of age of her son. Find their present ages.
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In the figure, ABC is a triangle whose vertices are A $(x, 10)$, B $(2, 2)$ and C $(12, 2)$. If Q $(9, 6)$ is the mid-point of AC and area of $\Delta APQ$ is $12 \text{ cm}^2$, then find the area of quadrilateral PBCQ.
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The ages of 100 patients admitted in a hospital are as follows. Draw a βless than type ogiveβ for the given data :
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Construct a triangle with sides 6 cm, 8 cm and 9 cm and then construct another triangle whose sides are $\frac{2}{3}$ of the corresponding sides of the first triangle.
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In an Arithmetic progression the sum of first $n$ terms is 210 and the sum of first $(n - 1)$ terms is 171. If the first term of the Arithmetic progression is 3, then find the Arithmetic progression and find its 20th term.
OR
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In an Arithmetic progression the sum of first $n$ terms is 210 and the sum of first $(n - 1)$ terms is 171. If the first term of the Arithmetic progression is 3, then find the Arithmetic progression and find its 20$^{th}$ term.
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OR
The sum of interior angles of a polygon of βnβ sides is $(n - 2) 180^{\circ}$. If the interior angles of a pentagon are in Arithmetic progression and its least angle is 72$^{\circ}$, then find all the interior angles of the pentagon.
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In the figure the poles AB and CD of different heights are standing vertically on a level ground. From a point P on the line joining the foots of the poles on the level ground, the angles of elevation to the tops of the poles are found to be complementary. The height of CD and the distance PD are $20\sqrt{3}$ m and 20 m respectively. If BP is 10 m, then find the length of the pole AB and the distance AC between the tops of the poles.
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An insect control device made of a cylinder, a frustum of a cone and a hemisphere attached to each other is as shown in the figure. Sticky liquid is completely filled in the hemispherical part. If the radii of hemisphere and cylinder are 21 cm and 14 cm respectively and total height of the device is 60 cm and height of the cylinder is 15 cm, then calculate the curved surface area of the device and also find the quantity of the sticky liquid in the hemisphere.
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