Real Numbers
Que1 ✔ Prove that √5 is an irrational number.
Prove that 2 + 5√3 is an irrational number, if it is given that 3 is an irrational number.
The ratio of HCF to LCM of the least composite number and the least prime number is?
Que2 ✔ Explain why 7x11x13+13 is a composite number.
If 1080 = 2^x × 3^y × 5, then (x – y) is equal to
The sum of exponents of prime factors in the prime-factorisation of 196 is:
On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. Find the minimum distance each should walk so that each can cover the same distance in complete steps.
Que3 ✔ Can the number (15)n , n being a natural number, end with the digit 0? Give reasons.
Show that 6² cannot end with digit 0 for any natural number 'n'.
Que4 ✔ Two positive integers m and n are expressed as m = p5 q2 and n = p3 q 4 , where p and q are prime numbers. The LCM of m and n is?
Que5 ✔ Find the HCF and the LCM of the following by prime factorization.
a) 360 , 756
b) Find the HCF and LCM of 72 and 120.
c) 2x 4 y 3 z , 32x3 y4 p2
Que6 ✔ State the Fundamental theorem of Arithmetic.
Que7 ✔ The HCF of 2 numbers is 75 and their LCM is 1500. If one of the numbers is 300, find the other.
Que8 ✔ Find the smallest number which when divided by 30, 40 and 60 leaves the remainder 7 in each case.
Que9 ✔ The dimensions of a room are 6 m 75 cm, 4 m 50 cm and 2 m 25 cm. Find the length of the largest measuring rod which can measure the dimensions in exact number of times.
Que10 ✔ If a and b are two prime numbers, write their HCF and LCM.
Que11 ✔ If p and q are two coprime numbers, write their HCF and LCM.
Que12 ✔ Can 72 and 20 respectively be the LCM and HCF of two numbers. Write down the reason.
Que13 ✔ Without actual division, state whether the decimal form of is terminating OR recurring.
Que14 ✔ Find the HCF and LCM of 350 and 400 and verify that HCFxLCM=Product of the numbers.
Que15 ✔ Find the largest number which divides 245 and 1205 leaving the remainder 5 in each case.
Que16 ✔ Find the largest number which divides 303, 455 and 757 leaving the remainder 3, 5 and 7 respectively.
Polynomials
If α and β are roots of the quadratic equation x² – 7x + 10 = 0, find the quadratic equation whose roots are α² and β².
Find the value of 'p' for which the quadratic equation px(x – 2) + 6 = 0 has two equal real root
The number of quadratic polynomials having zeroes – 5 and – 3 is?
If a, b are zeroes of the polynomial x2 – 1, then value of (a + b) is:
Find the zeroes of the quadratic polynomials p(t) = 5t2 + 12t + 7 and verify the relationship between the zeroes and the coefficients.
Quadratic Equation
If the quadratic equation ax2 + bx + c = 0 has two real and equal roots, then 'c' is equal to
Linear equations in two variable
Que1 ✔ Using graphical method, solve the following system of equations: 3x + y + 4 = 0 and 3x – y + 2 = 0
The pair of linear equations 2x = 5y + 6 and 15y = 6x – 18 represents two lines which are?
A motorboat whose speed in still water is 9 km/h, goes 15km downstream and comes back to the same spot, in a total time of 3 hours 45 minutes. Find the speed of the stream.
A takes 6 days less than the time taken by B to finish a piece of work. If both A And B together can finish it in 4 days, find the time taken by B to finish the work.
Coordinate Geometry
Que1 ✔ Find the type of triangle ABC formed whose vertices are A(1, 0), B(–5, 0) and C(–2, 5).
The distance of the point (–6, 8) from x-axis is?
A line intersects y-axis and x-axis at point P and Q, respectively. If R(2, 5) is the mid-point of line segment PQ, then find the coordinates of P and Q.
Find the ratio in which the line segment joining the points A(6, 3) and B(–2, –5) is divided by xaxis
Triangle
Que1 ✔ (BPT)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.
Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.
If ∆PQR ~ ∆ABC; PQ = 6 cm, AB = 8 cm and the perimeter of ∆ABC is 36 cm, then the perimeter of ∆PQR is
Area Related to Circle
Que1 ✔ The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector.
In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find the area of the sector formed by the arc. Also, find the length of the arc.
Circle
Two tangents TP and TQ are drawn to a circle with centre O from an external point T (see below left figure). Prove that ∠PTQ = 2 ∠OPQ.
What is the area of a semi-circle of diameter 'd' ?
A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use Ï€ = 3.14 and √3 = 1.73)
Que1 ✔ If a hexagon PQRSTU circumscribes a circle, prove that, PQ + RS + TU = QR + ST + UP
Que2 ✔ In the given figure, two concentric circles have radii 3 cm and 5 cm. Two tangents TR and TP are drawn to the circles from an external point T such that TR touches the inner circle at R and TP touches the outer circle at P. If TR = 4√10 cm, then find the length of TP.
Height and Distance
Que1 ✔ From the top of a 45 m high light house, the angles of depression of two ships, on the opposite side of it, are observed to be 30° and 60°. If the line joining the ships passes through the foot of the light house, find the distance between the ships. (Use √3 = 1.73)
Find the length of the shadow on the ground of a pole of height 18 m when angle of elevation θ of the sun is such that tanθ = 6/7.
A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After 30 seconds, the angle of elevation reduces to 30° (see the below figure).
A.P.
The ratio of the 11th term to the 18th term of an A.P. is 2 : 3. Find the ratio of the 5th term to the 21st term. Also, find the ratio of the sum of first 5 terms to the sum of first 21 terms.
If the sum of first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of first 10 terms.
The next term of the A.P.: √6 , √24, √54 is?
trigonometry
Prove that sec A (1 – sin A) (sec A + tan A) = 1.
