CBSE Class 10 Maths Standard 2024 Question Paper 30/1/2 with Solutions

AD is a median of $△ABC$ with vertices A$(5,-6)$, B$(6,4)$ and C$(0,0)$. Length AD is equal to :

1. (A)$\sqrt{68}$ units
2. (B)$2\sqrt{15}$ units
3. (C)$\sqrt{101}$ units
4. (D)10 units

If $\sec \theta -\tan \theta =m$, then the value of $\sec \theta +\tan \theta$ is :

1. (A)$1-\frac{1}{m}$
2. (B)$m^2-1$
3. (C)$\frac{1}{m}$
4. (D)$-m$

If the distance between the points $(3,-5)$ and $(x,-5)$ is 15 units, then the values of $x$ are :

1. (A)$12,-18$
2. (B)$-12,18$
3. (C)18, 5
4. (D)$-9,-12$

If $\sin A=\frac{2}{3}$, then value of $\cot A$ is :

1. (A)$\frac{\sqrt{5}}{2}$
2. (B)$\frac{3}{2}$
3. (C)$\frac{5}{4}$
4. (D)$\frac{2}{3}$

In the given figure, graphs of two linear equations are shown. The pair of these linear equations is :

1. (A)consistent with unique solution.
2. (B)consistent with infinitely many solutions.
3. (C)inconsistent.
4. (D)inconsistent but can be made consistent by extending these lines.

The centre of a circle is at $(2,0)$. If one end of a diameter is at $(6,0)$, then the other end is at :

1. (A)(0, 0)
2. (B)(4, 0)
3. (C)$(-2,0)$
4. (D)$(-6,0)$

Which of the following is not probability of an event ?

1. (A)0.89
2. (B)52%
3. (C)$\frac{1}{13}\%$
4. (D)$\frac{1}{0.89}$

The zeroes of a polynomial $x^2+px+q$ are twice the zeroes of the polynomial $4x^2-5x-6$. The value of $p$ is :

1. (A)$-\frac{5}{2}$
2. (B)$\frac{5}{2}$
3. (C)$-5$
4. (D)10

The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is :

1. (A)$\frac{4\pi }{3}$ cu cm
2. (B)$\frac{5\pi }{3}$ cu cm
3. (C)$\frac{8\pi }{3}$ cu cm
4. (D)$\frac{2\pi }{3}$ cu cm

The middle most observation of every data arranged in order is called :

1. (A)mode
2. (B)median
3. (C)mean
4. (D)deviation

If the roots of equation $a{x}^2+bx+c=0,a\ne 0$ are real and equal, then which of the following relation is true ?

1. (A)$a=\frac{b^2}{c}$
2. (B)$b^2=ac$
3. (C)$ac=\frac{b^2}{4}$
4. (D)$c=\frac{b^2}{a}$

If the probability of a player winning a game is 0.79, then the probability of his losing the same game is :

1. (A)1.79
2. (B)0.31
3. (C)0.21%
4. (D)0.21

If the sum and the product of zeroes of a quadratic polynomial are $2\sqrt{3}$ and 3 respectively, then a quadratic polynomial is :

1. (A)$x^2+2\sqrt{3}x-3$
2. (B)$(x-\sqrt{3}{)}^2$
3. (C)$x^2-2\sqrt{3}x-3$
4. (D)$x^2+2\sqrt{3}x+3$

For some data $x_1,x_2,......x_n$ with respective frequencies $f_1,f_2,.....f_n$, the value of ${\sum }_1^n{f}_i\left(x_i-\bar{x}\right)$ is equal to :

1. (A)$n\bar{x}$
2. (B)1
3. (C)$\sum f_i$
4. (D)0

A solid sphere is cut into two hemispheres. The ratio of the surface areas of sphere to that of two hemispheres taken together, is :

1. (A)1 : 1
2. (B)1 : 4
3. (C)2 : 3
4. (D)3 : 2

If two positive integers $p$ and $q$ can be expressed as $p=18a^2{b}^4$ and $q=20a^3{b}^2$, where $a$ and $b$ are prime numbers, then LCM $(p,q)$ is :

1. (A)$2a^2{b}^2$
2. (B)$180a^2{b}^2$
3. (C)$12a^2{b}^2$
4. (D)$180a^3{b}^4$

$n$th term of an A.P. is $7n+4$. The common difference is :

1. (A)$7n$
2. (B)4
3. (C)7
4. (D)1

From the data 1, 4, 7, 9, 16, 21, 25, if all the even numbers are removed, then the probability of getting at random a prime number from the remaining is :

1. (A)$\frac{2}{5}$
2. (B)$\frac{1}{5}$
3. (C)$\frac{1}{7}$
4. (D)$\frac{2}{7}$

Assertion (A) : If the graph of a polynomial touches $x$-axis at only one point, then the polynomial cannot be a quadratic polynomial.
Reason (R) : A polynomial of degree $n(n>1)$ can have at most $n$ zeroes.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

Assertion (A) : The tangents drawn at the end points of a diameter of a circle, are parallel.
Reason (R) : Diameter of a circle is the longest chord.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

Diagonals AC and BD of a trapezium ABCD intersect at O, where AB $\parallel$ DC. If $\frac{DO}{OB}=\frac{1}{2}$, then show that AB = 2CD

Prove that $5-2\sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.

Show that the number $5\times 11\times 17+3\times 11$ is a composite number.

Solve the following system of linear equations :
$2p+3q=13$ and $5p-4q=-2$

Evaluate : $2\sqrt{2}\cos 45^{\circ }\sin 30^{\circ }+2\sqrt{3}\cos 30^{\circ }$

If $A=60^{\circ }$ and $B=30^{\circ }$, verify that :
$\sin (A+B)=\sin A\cos B+\cos A\sin B$

In a pack of 52 playing cards one card is lost. From the remaining cards, a card is drawn at random. Find the probability that the drawn card is queen of heart, if the lost card is a black card.

Prove that : $\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \mathrm{cosec}\theta$

In a chemistry lab, there is some quantity of 50% acid solution and some quantity of 25% acid solution. How much of each should be mixed to make 10 litres of 40% acid solution ?

Find the ratio in which the point $\left(\frac{8}{5},y\right)$ divides the line segment joining the points $(1,2)$ and $(2,3)$. Also, find the value of $y$.

ABCD is a rectangle formed by the points A$(-1,-1)$, B$(-1,6)$, C$(3,6)$ and D$(3,-1)$. P, Q, R and S are mid–points of sides AB, BC, CD and DA respectively. Show that diagonals of the quadrilateral PQRS bisect each other.

A wooden toy is made by scooping out a hemisphere of same radius as of cylinder, from each end of a wooden solid cylinder. If the height of the cylinder is 20 cm and its base is of radius 7 cm, find the total surface area of the toy.

In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.

In the given figure, AB is a diameter of the circle with centre O. AQ, BP and PQ are tangents to the circle. Prove that $\angle POQ=90^{\circ }$.

A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If AD is perpendicular to DC, BC = 30 cm and BS = 24 cm, then find the length DC.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

In the given figure PA, QB and RC are each perpendicular to AC. If AP = $x$, BQ = $y$ and CR = $z$, then prove that $\frac{1}{x}+\frac{1}{z}=\frac{1}{y}$

From the top of a building 60 m high, the angles of depression of the top and bottom of the vertical lamp post are observed to be $30^{\circ }$ and $60^{\circ }$ respectively.
(i) Find the horizontal distance between the building and the lamp post.
(ii) Find the distance between the tops of the building and the lamp post.

The sum of first and eighth terms of an A.P. is 32 and their product is 60. Find the first term and common difference of the A.P. Hence, also find the sum of its first 20 terms.

In an A.P. of 40 terms, the sum of first 9 terms is 153 and the sum of last 6 terms is 687. Determine the first term and common difference of A.P. Also, find the sum of all the terms of the A.P.

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area and the volume of the vessel.

BINGO is game of chance. The host has 75 balls numbered 1 through 75. Each player has a BINGO card with some numbers written on it. The participant cancels the number on the card when called out a number written on the ball selected at random. Whosoever cancels all the numbers on his/her card, says BINGO and wins the game.
The table given below, shows the data of one such game where 48 balls were used before Tara said 'BINGO'.
Numbers announced: 0-15, 15-30, 30-45, 45-60, 60-75
Number of times: 8, 9, 10, 12, 9
Based on the above information, answer the following :
(i) Write the median class. (1)
(ii) When first ball was picked up, what was the probability of calling out an even number ? (1)
(iii) (a) Find median of the given data. (2)
OR (b) Find mode of the given data. (2)

A backyard is in the shape of a triangle ABC with right angle at B. AB = 7 m and BC = 15 m. A circular pit was dug inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that AP = $x$ m.
Based on the above information, answer the following questions :
(i) Find the length of AR in terms of $x$. (1)
(ii) Write the type of quadrilateral BQOR. (1)
(iii) (a) Find the length PC in terms of $x$ and hence find the value of $x$. (2)
OR (b) Find $x$ and hence find the radius $r$ of circle. (2)

A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.
(i) Assuming the original length of each side of a tile be $x$ units, make a quadratic equation from the above information. (1)
(ii) Write the corresponding quadratic equation in standard form. (1)
(iii) (a) Find the value of $x$, the length of side of a tile by factorisation. (2)
OR (b) Solve the quadratic equation for $x$, using quadratic formula. (2)