If $a_n$, $S_n$ and $S_{n-1}$ represent $n^{th}$ term, sum of first $n$ terms and sum of first $(n-1)$ terms of an arithmetic progression respectively, then the correct relation among the following is
a.
$S_n + a_n = S_{n-1}$
b.
$a_n = S_n - S_{n-1}$
Correct Answer
c.
$a_n = S_n - S_{n-1}$
d.
$S_{n-1} = a_n - S_n$
Step-by-Step AI Explanation
Verified Method
By definition, the sum of the first $n$ terms is $S_n = a_1 + a_2 + ... + a_{n-1} + a_n$. Since $S_{n-1} = a_1 + a_2 + ... + a_{n-1}$, subtracting $S_{n-1}$ from $S_n$ leaves only $a_n$. Thus, $a_n = S_n - S_{n-1}$. Options A and D are algebraic rearrangements that do not hold true, and C is identical to B (though B is the standard representation).
Q2mcq1 mark
The angle between the tangent at a point of a circle and radius through the point of contact is
a.
30°
b.
60°
c.
90°
Correct Answer
d.
180°
Step-by-Step AI Explanation
Verified Method
According to the theorem on tangents to a circle, the tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, the angle between them is always 90°.
Q3mcq1 mark
The lines represented by a pair of linear equations in two variables having no solution are
a.
Parallel
Correct Answer
b.
Coincident
c.
Intersecting
d.
Perpendicular
Step-by-Step AI Explanation
Verified Method
For a system of linear equations in two variables, having no solution corresponds to the lines being parallel, meaning they never intersect at any point. Coincident lines have infinitely many solutions, and intersecting lines have a unique solution.
Q4mcq1 mark
Volume of a cuboid whose length is ‘l’ unit, breadth is ‘b’ unit and height is ‘h’ unit, is
a.
$l + b + h$ cubic units
b.
$2(b + h)l$ cubic units
c.
$2(lb + bh + hl)$ cubic units
d.
$lbh$ cubic units
Correct Answer
Step-by-Step AI Explanation
Verified Method
The volume of a cuboid is calculated as the product of its length, breadth, and height. Thus, Volume = $l \times b \times h = lbh$ cubic units. Other options correspond to perimeter or surface area formulas.
Q5mcq1 mark
Which of the following can be the probability of an event?
a.
$1/5$
Correct Answer
b.
$1.5$
c.
$5/2$
d.
$-5$
Step-by-Step AI Explanation
Verified Method
The probability $P(E)$ of an event must satisfy $0 \leq P(E) \leq 1$. Option A ($1/5 = 0.2$) is within this range. Option B ($1.5$) and C ($2.5$) are greater than 1, and option D ($-5$) is less than 0, so none of those can be probabilities.
Q6mcq1 mark
The value of $\sec^2 \theta - \tan^2 \theta$ is
a.
2
b.
-1
c.
0
d.
1
Correct Answer
Step-by-Step AI Explanation
Verified Method
From the fundamental trigonometric identity $1 + \tan^2 \theta = \sec^2 \theta$, it follows that $\sec^2 \theta - \tan^2 \theta = 1$.
Q7mcq1 mark
In the figure, $DE \parallel BC$. If $AD = 2 \text{ cm}$, $AE = 1 \text{ cm}$ and $BD = 6 \text{ cm}$ then the measure of $EC$ is
a.
2 cm
b.
3 cm
Correct Answer
c.
4 cm
d.
5 cm
Step-by-Step AI Explanation
Verified Method
By Thales Theorem (Basic Proportionality Theorem), since $DE \parallel BC$, then $\frac{AD}{BD} = \frac{AE}{EC}$. Given $AD = 2$, $BD = 6$, and $AE = 1$, we have $\frac{2}{6} = \frac{1}{EC}$. Simplifying $\frac{2}{6}$ gives $\frac{1}{3} = \frac{1}{EC}$, therefore $EC = 3 \text{ cm}$.
Q8mcq1 mark
The mean and mode of a data are 7.5 and 6 respectively. The median for this data is
a.
3
b.
5
Correct Answer
c.
7
d.
9
Step-by-Step AI Explanation
Verified Method
The empirical relationship between mean, median, and mode is given by the formula: $3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean}$. Substituting the given values: $3 \times \text{Median} = 6 + 2(7.5) = 6 + 15 = 21$. Therefore, $\text{Median} = 21 / 3 = 7$. Wait, recalculating: $21/3=7$. Actually, the calculation $6 + 15 = 21$, $21/3 = 7$. Let's re-verify inputs. $3(M) = 6 + 2(7.5) = 21$, $M=7$. Checking options: A is 3, B is 5, C is 7, D is 9. Choice C is correct.
Q9short answer1 mark
Write the LCM of 5 and 7.
Step-by-Step AI Explanation
Verified Method
Since 5 and 7 are prime numbers, their least common multiple (LCM) is their product: $5 \times 7 = 35$.
Q10short answer1 mark
Write the roots of the quadratic equation $ax^2 + bx + c = 0, a \neq 0$.
Step-by-Step AI Explanation
Verified Method
The roots of the quadratic equation $ax^2 + bx + c = 0$ are given by the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
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The length of the tangent drawn from a point ‘P’ to the circle with centre ‘O’ is 8 cm. If the distance of point ‘P’ from the centre is 10 cm, then find the radius of the circle.
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Find the roots of the quadratic equation $x^2 + 5x + 6 = 0$ by factorisation method.
OR
Find the nature of the roots of the quadratic equation $x^2 + 5x + 4 = 0$.
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The length and breadth of a rectangular field are 147 m and 56 m respectively. Calculate the maximum length of a rod that can be used to measure both length and breadth exactly. How many times this rod has to be used to measure the length of the field?
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A box contains 6 blue and some red marbles. The probability of picking a blue marble from the box is $\frac{3}{8}$. Find the number of red marbles in the box and also find the probability of picking a red marble from the box. OR The six faces of a die show the numbers as given below: 1, 2, 3, 4, 1, 5. This die is rolled once. Find the probability of getting a perfect square number.
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Find the mean for the following data:
Class-interval: 0-10, 10-20, 20-30, 30-40, 40-50
Frequency: 2, 3, 3, 1, 1
OR
Find the mode for the following data:
Class-interval: 0-5, 5-10, 10-15, 15-20, 20-25
Frequency: 4, 10, 6, 4, 5
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$\Delta ABC$ is an equilateral triangle. Arcs are drawn in $\Delta ABC$ with $A, B$ and $C$ as the centres of the circles of radius 3.5 cm as shown in the figure. If the area of the $\Delta ABC$ is $49\sqrt{3} \text{ cm}^2$, then find the perimeter of the shape $PQRSTUP$.
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Find the coordinates of the point which divides the line segment joining the points $(4, -3)$ and $(8, 5)$ in the ratio 3 : 1 internally. OR Show that the three points $A(3, 1), B(6, 4)$ and $C(8, 6)$ are collinear using ‘distance formula’.
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In a fraction, denominator is one more than twice its numerator. Sum of this fraction and its reciprocal is $\frac{29}{10}$. Then find the fraction. OR Some students planned to visit a museum. The total cost of food for them was Rs. 300. But 4 of them were unable to visit the museum and therefore the cost of food for each student was increased by Rs. 20. Find the number of students who visited the museum.
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In $\Delta ABC$, a line is drawn parallel to $BC$. Let this line intersect $AB$ at $P$ and $AC$ at $Q$. Draw the figure and prove that $\Delta APQ \sim \Delta ABC$.
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In an Arithmetic progression 7th term is five times its 2nd term and the 5th term is 1 less than twice the 3rd term. Write the Arithmetic progression. Find the 15th term of this progression using formula.
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A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. If a right circular cylinder circumscribes the toy as shown in the figure, find the difference of the volumes of the cylinder and the toy. (Take $\pi = 3.14$) OR From a solid cylinder whose height is 2.4 m and diameter is 1.4 m, a conical cavity of the same height and same diameter as that of the cylinder is hollowed out. Find the total surface area of the remaining solid.
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A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. If a right circular cylinder circumscribes the toy as shown in the figure, find the difference of the volumes of the cylinder and the toy. ( Take $\pi = 3.14$ )
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Two towers AB and CD are standing vertically on a level ground as shown in the given figure. The angles of elevation of the top of the towers AB and CD from a point ‘M’ on the line BD joining the foot of the towers are found to be $30^{\circ}$ and $60^{\circ}$ respectively. The distance of the foot of the tower AB from the point ‘M’ is $90\text{ m}$. The height of the tower CD is double the height of the tower AB. Calculate the height of the two towers, distance between the towers and also distance from the top of the towers to the point ‘M’ on the ground. ( Take $\sqrt{3} = 1.7$ )
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From a solid cylinder whose height is $2.4\text{ m}$ and diameter is $1.4\text{ m}$, a conical cavity of the same height and same diameter as that of the cylinder is hollowed out. Find the total surface area of the remaining solid.
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