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Match with the correct value of $\tan C\times sinA$ in each case.

If (a, 0) is a point on the line represented by the equation $2x-3y=4,$, find the value of a.

Solve each of the following given systems of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:

$x-2y+2=0,2x+y-6=0.$

Prove the following identities:

$\frac{sinA}{secA+tanA-1}+\frac{cosA}{cosecA+cotA-1}=1$

In the given figure, LMN is tangent to the circle with centre O. If $\angle$ PMN = ${60}^{\circ }$, find $\angle$MOP.

The distribution below gives the weight of 30 students in a class. Find the median weight of students:

Weight (in kg): 40-45 45-50 50-55 55-60 60-65 65-70 70-75

No. of students: 2 3 8 6 6 3 2

Construct triangle BCP, where CB = 5 cm, BP = 4 cm, $\angle$PBC = 45${}^{\circ }$. Complete the rectangle ABCD such that

i.P is equidistant from AB and BC; and

ii.P is equidistant from C and D. Measure the length of AB.

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are ${30}^{\circ }$ and ${45}^{\circ }$, respectively. If the width of the river is 12 m , find the height of the bridge.

Let $a_n$ be the $n^{th}$ term of the G.P. of positive numbers. Let ${\sum }_{n=1}^{100}{a}_{2n}=\alpha$ and ${\sum }_{n=1}^{100}{a}_{2n-1}=\beta$, such that such that $\alpha \ne \beta$. Prove that the common ratio of the G.P. is $\frac{\alpha }{\beta }$.

If from a positive number, twice its reciprocal is subtracted, we get 1. Then the number is ___.

If only 2 workers were employed on this job on Day 4, what could be the minimum amount that Mr. Shyam pays the workers for this job for 6 days?

12 is the mean proportional of _____________.

In fig., two circles with centres A and B touch each other externally at k. The length of PQ (in cm) is

A = {: 11 – 5 > 7 + 3, ε R} & B = {: 18 – 9 > 12 + 15, ε R}. Find the range of set A ∩ B and represent it on a number line.

Given that ∆ABC~∆DEF, ∠A = 50^o, ∠C = 35^o, ∠E = ?

D, E and F are the points on sides BC, CA and AB respectively of $\Delta ABC$ such that All bisects $\angle A$, BE bisects $\angle VB$and CF bisects $\angle C$. If AB = 5 cm, BC = 8 cm and CA = 4cm determine AF. CE and BD.

A plane left 40 minutes late due to bad weather and in order to reach its destination, 1600km away in time, it had to increase its speed by 400km/hr from its usual speed. Find the usual speed of the plane.

Using factor theorem show that ( – 3) is a factor of ³ – 7² + 15 – 9. Hence factorise the expression completely.

In the picture O is the centre of the circle and A, B, C are points on it. Then $\angle OAC+\angle ABC$ equals ___

Competent Ltd. issued a prospectus inviting applications for 50,000 Equity Shares of ₹ 10 each, payable ₹ 5 as per application (including ₹ 2 as premium), ₹ 4 as per allotment and the balance towards first and final call.

Applications were received for 65,000 shares. Application money received on 5,000 shares was refunded with letter of regret and allotments were made on pro rata basis to the applicants of 60,000 shares. Money overpaid on applications including premium was adjusted on account of sums due on allotment.

All the forfeited shares were subsequently sold to Mr. Jain credited as fully paid-up for ₹ 9 per share.

If $S_n$ is the sum of the first n even natural numbers, then which of the following leave no remainder on dividing $S_n$ ?

Which one of the following has point symmetry about the origin?

If $p\left(x\right)=x^2-2\sqrt{2}x+1$, then $p\left(2\sqrt{2}\right)$ is equal to

A) $0$
B) $1$
C) $4\sqrt{2}$
D) $8\sqrt{2}+1$

The following table gives the number of children of 150 families in a village
No. of children (x) : 0 1 2 3 4 5
No. of families (f) : 10 21 55 42 15 7
Find the average number of children per family.

Question 111

Living on his own Sanjay is looking for one bedroom apartment for rent. At Neelgiri apartment, rent for the first two months is 20% off. The one bedroom rate at Neelgiri is Rs.6000 per month. At Savana apartment, the first month is 50% off. The one bedroom rate at Savana apartment is 7000 per month. Which apartment will be cheaper for the first two months? By how much?

Question 4 (ix)
Which of the following are APs? If they form an A.P. Find the common difference d and write three more terms.
(ix) 1, 3, 9, 27, …

Two circles touch each other externally at a point C and P is a point on the common tangent at C. If PA and PB are tangents to the two circles, prove that PA = PB.