CBSE Class 10 Maths Standard 2024 Question Paper 30/2/3 with Solutions

The distance between the points $(a\cos \theta ,-a\sin \theta )$ and $(a\sin \theta ,a\cos \theta )$ is

1. (A)a
2. (B)$a\sqrt{2}$
3. (C)0
4. (D)2a

In the given figure, tangents PA and PB to the circle centred at O, from point P are perpendicular to each other. If PA = 5 cm, then length of AB is equal to

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

Which term of the A.P. $-29,-26,-23,\dots ,61$ is 16 ?

1. (A)$11^{th}$
2. (B)$16^{th}$
3. (C)$10^{th}$
4. (D)$31^{st}$

A box contains cards numbered 6 to 55. A card is drawn at random from the box. The probability that the drawn card has a number which is a perfect square, is

1. (A)$\frac{7}{50}$
2. (B)$\frac{7}{55}$
3. (C)$\frac{1}{10}$
4. (D)$\frac{5}{49}$

AB and CD are two chords of a circle intersecting at P. Choose the correct statement from the following :

1. (A)$△ADP\sim △CBA$
2. (B)$△ADP\sim △BPC$
3. (C)$△ADP\sim △BCP$
4. (D)$△ADP\sim △CBP$

Two dice are rolled together. The probability of getting the sum of the two numbers to be more than 10, is

1. (A)$\frac{1}{9}$
2. (B)$\frac{1}{6}$
3. (C)$\frac{7}{12}$
4. (D)$\frac{1}{12}$

After an examination, a teacher wants to know the marks obtained by maximum number of the students in her class. She requires to calculate _____ of marks.

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

The roots of the quadratic equation $4x^2-5x+4=0$ are

1. (A)irrational
2. (B)rational and distinct
3. (C)not real
4. (D)rational and equal

The common difference of an A.P. in which $a_{20}-a_{15}=20$, is

1. (A)4
2. (B)5
3. (C)4d
4. (D)5d

If value of each observation in a data is increased by 2, then median of the new data

1. (A)increases by 2
2. (B)increases by 2n
3. (C)remains same
4. (D)decreases by 2

The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. PQ/AB is equal to

1. (A)$\frac{7}{8}$
2. (B)$\frac{6}{7}$
3. (C)$\frac{7}{6}$
4. (D)$\frac{8}{7}$

If $\alpha$ and $\beta$ $(\alpha >\beta )$ are the zeroes of the polynomial $-x^2+8x+9$, then $(\alpha -\beta )$ is equal to

1. (A)$-10$
2. (B)10
3. (C)$\pm 10$
4. (D)8

The value of k for which the system of equations $3x-y+8=0$ and $6x-ky+16=0$ has infinitely many solutions, is

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

If $\sin \theta =\cos \theta$, $(0^{\circ }<\theta <90^{\circ })$, then value of $(\sec \theta \cdot \sin \theta )$ is :

1. (A)$\frac{1}{\sqrt{2}}$
2. (B)$\sqrt{2}$
3. (C)1
4. (D)0

Point P divides the line segment joining the points A(4, $-5$) and B(1, 2) in the ratio 5:2. Co-ordinates of point P are

1. (A)$\left(\frac{5}{2},\frac{-3}{2}\right)$
2. (B)$\left(\frac{11}{7},0\right)$
3. (C)$\left(\frac{13}{7},0\right)$
4. (D)$\left(0,\frac{13}{7}\right)$

In the given figure $△ABC$ is shown. DE is parallel to BC. If AD = 5 cm, DB = 2.5 cm and BC = 12 cm, then DE is equal to

1. (A)10 cm
2. (B)6 cm
3. (C)8 cm
4. (D)7.5 cm

If the HCF (2520, 6600) = 40 and LCM (2520, 6600) = $252\times k$, then the value of k is

1. (A)1650
2. (B)1600
3. (C)165
4. (D)1625

In the given figure, AT is tangent to a circle centred at O. If $\angle CAT=40^{\circ }$, then $\angle CBA$ is equal to

1. (A)$70^{\circ }$
2. (B)$50^{\circ }$
3. (C)$65^{\circ }$
4. (D)$40^{\circ }$

Assertion (A) : If $\sin A=\frac{1}{3}$ $(0^{\circ }<A<90^{\circ })$, then the value of $\cos A$ is $\frac{2\sqrt{2}}{3}$
Reason (R) : For every angle $\theta$, ${\sin }^2\theta +{\cos }^2\theta =1$.

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

Assertion (A) : Two cubes each of edge length 10 cm are joined together. The total surface area of newly formed cuboid is 1200 cm${}^2$.
Reason (R) : Area of each surface of a cube of side 10 cm is 100 cm${}^2$.

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

In what ratio is the line segment joining the points $(3,-5)$ and $(-1,6)$ divided by the line $y=x$ ?

A(3, 0), B(6, 4) and C($-1$, 3) are vertices of a triangle ABC. Find length of its median BE.

In the given figure, AB and CD are tangents to a circle centred at O. Is $\angle BAC=\angle DCA$ ? Justify your answer.

In the given figure, a circle centred at origin O has radius 7 cm, OC is median of $△OAB$. Find the length of median OC.

Evaluate : $2{\sin }^{2}30^{\circ }\sec 60^{\circ }+{\tan }^{2}60^{\circ }$.

If $2\sin (A+B)=\sqrt{3}$ and $\cos (A-B)=1$, then find the measures of angles A and B. $0\le A,B,(A+B)\le 90^{\circ }$.

Can the number $8^n$, n being a natural number, end with the digit 0 ? Give reasons.

Prove that $\frac{{\mathrm{cosec}}^2\theta -{\sec }^2\theta }{{\mathrm{cosec}}^2\theta +{\sec }^2\theta }=\frac{3}{4}$, if $\tan \theta =\frac{1}{\sqrt{7}}$

If the sum of first m terms of an A.P. is same as sum of its first n terms $(m\ne n)$, then show that the sum of its first $(m+n)$ terms is zero.

In an A.P., the sum of three consecutive terms is 24 and the sum of their squares is 194. Find the numbers.

Prove that $\sqrt{5}$ is an irrational number.

In the given figure, PQ is tangent to a circle centred at O and $\angle BAQ=30^{\circ }$; show that BP = BQ.

In the given figure, AB, BC, CD and DA are tangents to the circle with centre O forming a quadrilateral ABCD.
Show that $\angle AOB+\angle COD=180^{\circ }$

A dealer sells an article for ₹ 75 and gains as much percent as the cost price of the article. Find the cost price of the article.

In a test, the marks obtained by 100 students (out of 50) are given below :
Marks obtained : 0 – 10, 10 – 20, 20 – 30, 30 – 40, 40 – 50
Number of students : 12, 23, 34, 25, 6
Find the mean marks of the students.

A person standing on the bank of a river observes that the angle of elevation of the top of a tower on the opposite bank is $60^{\circ }$. When he moves 30 m away from the bank, he finds the angle of elevation to be $30^{\circ }$. Find the height of the tower and width of the river. (Take $\sqrt{3}=1.732$)

The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector.

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.

Sides AB and AC and median AD to $△ABC$ are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that $△ABC\sim △PQR$.

Using graphical method, solve the following system of equations :
$3x-2y=10$ and $5x+3y=4$

If three times the greater of two numbers is divided by the smaller one, we get 4 as the quotient and 3 as the remainder. Also, if seven times the smaller number is divided by greater one, we get 5 as the quotient and 1 as the remainder. Find the numbers.

In a survey on holidays, 120 people were asked to state which type of transport they used on their last holiday. The following pie chart shows the results of the survey.
Observe the pie chart and answer the following questions :
(i) If one person is selected at random, find the probability that he/she travelled by bus or ship. (1)
(ii) Which is most favourite mode of transport and how many people used it ? (1)
(iii) (a) A person is selected at random. If the probability that he did not use train is 4/5, find the number of people who used train. (2)
OR
(b) The probability that randomly selected person used aeroplane is 7/60. Find the revenue collected by air company at the rate of ₹ 5,000 per person. (2)

The word ‘circus’ has the same root as ‘circle’. In a closed circular area, various entertainment acts including human skill and animal training are presented before the crowd.
A circus tent is cylindrical upto a height of 8 m and conical above it. The diameter of the base is 28 m and total height of tent is 18.5 m.
Based on the above, answer the following questions :
(i) Find slant height of the conical part. (1)
(ii) Determine the floor area of the tent. (1)
(iii) (a) Find area of the cloth used for making tent. (2)
OR
(b) Find total volume of air inside an empty tent. (2)

A ball is thrown in the air so that t seconds after it is thrown, its height h metre above its starting point is given by the polynomial $h=25t-5t^2$.
Observe the graph of the polynomial and answer the following questions :
(i) Write zeroes of the given polynomial. (1)
(ii) Find the maximum height achieved by ball. (1)
(iii) (a) After throwing upward, how much time did the ball take to reach to the height of 30 m ? (2)
OR
(b) Find the two different values of t when the height of the ball was 20 m. (2)