CBSE Class 10 Maths (Standard) 2026 — Paper 30/1/1

The HCF of $960$ and $432$ is :

The natural number $2$ is :

For any natural number $n$, $6^n$ ends with the digit :

The graph of $y=f(x)$ is given. The number of zeroes of $f(x)$ is :

If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has :

The common difference of the AP : $\sqrt{2},2\sqrt{2},3\sqrt{2},4\sqrt{2},\dots$ is :

If $△ABC$ and $△DEF$ are similar such that $2AB=DE$ and $BC=8$ cm, then $EF$ is equal to :

The mid-point of the line segment joining the points $(5,-4)$ and $(6,4)$ lies on :

Given that $\sin \theta =\frac{a}{b}$, then $\cos \theta$ is equal to :

If $\cos A=\frac{1}{2}$, then the value of ${\sin }^{2}A+2{\cos }^{2}A$ is :

A car is moving away from the base of a $30$ m high tower. The angle of elevation of the top of the tower from the car at an instant, when the car is $10\sqrt{3}$ m away from the base of the tower, is :

If $TP$ and $TQ$ are two tangents to a circle with centre $O$ from an external point $T$ so that $\angle POQ=120^{\circ }$, then $\angle PTQ$ is equal to :

In the given figure, $PA$ is a tangent from an external point $P$ to a circle with centre $O$. If $\angle POB=125^{\circ }$, then $\angle APO$ is equal to :

The length of the arc of the sector of a circle with radius $21$ cm and of central angle $60^{\circ }$, is :

The hour hand of a clock is $7$ cm long. The angle swept by it between $7:00$ a.m. and $8:10$ a.m. is :

The total surface area of a solid hemisphere of diameter '$2d$' is :

If the mean and mode of a data are $12$ and $21$ respectively, then its median is :

A die is thrown once. Probability of getting a number other than $3$ is :

In the question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option :

If $\alpha$, $\beta$ are the zeroes of the polynomial $p(x)=x^2-3x-1$, then find the value of $\frac{1}{\alpha }+\frac{1}{\beta }$.

In $△ABC$, $DE\parallel BC$. If $AD=x$, $DB=x-2$, $AE=x+2$ and $EC=x-1$, then find the value of $x$.

In the figure given above, $△ABC\sim △XYZ$, then find the values of $x$ and $y$.

The coordinates of the centre of a circle are $(x-7,2x)$. Find the value(s) of '$x$', if the circle passes through the point $(-9,11)$ and has radius $5\sqrt{2}$ units.

If $\tan \theta =\frac{24}{7}$, then find the value of $\sin \theta +\cos \theta$.

If $\cot \theta =\frac{7}{8}$, then find the value of $\frac{(1+\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(1-\cos \theta )}$.

Two concentric circles are of radii $5$ cm and $4$ cm. Find the length of the chord of the larger circle which touches the smaller circle.

Find the ratio in which the $x$-axis divides the line segment joining the points $(-6,5)$ and $(-4,-1)$. Also, find the point of intersection.

If $x=h+a\cos \theta$, $y=k+b\sin \theta$, then prove that : ${\left(\frac{x-h}{a}\right)}^2+{\left(\frac{y-k}{b}\right)}^2=1$

Prove that : $\frac{\tan A}{1+\sec A}-\frac{\tan A}{1-\sec A}=2\mathrm{cosec}A$

In the given figure, $△ABC$ is a right triangle in which $\angle B=90^{\circ }$, $AB=4$ cm and $BC=3$ cm. Find the radius of the circle inscribed in the triangle $ABC$.

In the given figure, if a circle touches the side $QR$ of $△PQR$ at $S$ and extended sides $PQ$ and $PR$ at $M$ and $N$ respectively, then prove that : $PM=\frac{1}{2}(PQ+QR+PR)$

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is $20$ cm and the diameter of the cylinder is $7$ cm. Find the total volume of the solid. $\left(\text{Use }\pi =\frac{22}{7}\right)$

Determine graphically, the coordinates of vertices of a triangle whose equations are $2x-3y+6=0$; $2x+3y-18=0$ and $x=0$. Also, find the area of this triangle.

A faster train takes one hour less than a slower train for a journey of $200$ km. If the speed of the slower train is $10$ km/hr less than that of the faster train, find the speeds of the two trains.

The sum of the areas of two squares is $640{\text{m}}^2$. If the difference in their perimeters is $64$ m, find the sides of the two squares.

State and prove Basic Proportionality Theorem.

The mean of the following frequency distribution is $35$. Find the values of $x$ and $y$, if the sum of frequencies is $25$ :

In a potato race, a bucket is placed at the starting point, which is $5$ m from the first potato. The other potatoes are arranged $3$ m apart in a straight line, with a total of $10$ potatoes, as shown in the figure : A competitor starts from the bucket, picks up the nearest potato, runs back to the bucket to drop it in, then returns to pick up the next potato. This process continues until all the potatoes are in the bucket. Based on the above information, answer the following questions :

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure. On a similar concept, a radio station tower was built in two sections 'A' and 'B'. Tower is supported by wires from a point 'O' (as shown in figure). Distance between the base of the tower and point 'O' is $6$ m. From point 'O', the angle of elevation of the top of the section 'B' is $30^{\circ }$ and the angle of elevation of the top of section 'A' is $60^{\circ }$. Based on the above information, answer the following questions :

A brooch is crafted from silver wire in the shape of a circle with a diameter of $35$ cm. The wire is also used to create $5$ diameters, dividing the circle into $10$ equal sectors as shown in figure. Based on the above information, answer the following questions :